Exact Minimisation of Treatment Time for the Delivery of ...iii Abstract This thesis investigates the exact minimisation of treatment delivery time for Intensity Modulated Radiation

Exact Minimisation of Treatment Time

for the Delivery of

Intensity Modulated Radiation Therapy

Giulia M.G.H. Wake

B.Sc. (Hons) in Mathematics and Physics

The University of Western AustraliaSchool of Mathematics and Statistics

February 2009

This thesis is presented for the degree of Doctor of Philosophy ofThe University of Western Australia

iii

Abstract

This thesis investigates the exact minimisation of treatment delivery time for Intensity ModulatedRadiation Therapy (IMRT) for the treatment of cancer using Multileaf Collimators (MLC). Al-though patients are required to remain stationary during the delivery of IMRT, inevitably somepatient movement will occur, particularly if treatment times are longer than necessary. Thereforeminimising the treatment delivery time of IMRT may result in less patient movement, less inaccu-racy in the dosage received and a potentially improved outcome for the patient. When IMRT isdelivered using multileaf collimators in ‘step and shoot’ mode, it consists of a sequence of multileafcollimator configurations, or shape matrices; for each, time is needed to set up the configuration,and in addition the patient is exposed to radiation for a specified time, or beam-on time. The‘step and shoot leaf sequencing’ problems for minimising treatment time considered in this thesisare the constant set-up time Total Treatment Time (TTT) problem and the Beam-on Time Con-strained Minimum Cardinality (BTCMC) problem. The TTT problem minimises a weighted sumof total beam-on time and total number of shape matrices used, whereas the BTCMC problemlexicographically minimises the total beam-on time then the number of shape matrices used in asolution. The vast majority of approaches to these strongly NP-hard problems are heuristics; of thefew exact approaches, the formulations either have excessive computation times or their solutionmethods do not easily incorporate multileaf collimator mechanical constraints (which are presentin most currently used MLC systems).

In this thesis, new exact mixed integer and integer programming formulations for solving the TTTand BTCMC problems are developed. The models and solution methods considered can be appliedto the unconstrained and constrained versions of the problems, where ‘constrained’ refers to themodelling of additional MLC mechanical constraints. Within the context of integer programmingformulations, new and existing methods for improving the computational efficiency of the modelspresented are investigated. Numerical results for all variations considered are provided.

This thesis demonstrates that significant computational improvement can be achieved for the exactmixed integer and integer programming models investigated, via solution approaches based on anidea of systematically ‘stepping-up’ through the number of shape matrices used in a formulation,via additional constraints (particularly symmetry breaking constraints) and via the applicationof improved bounds on variables. This thesis also makes a contribution to the wider field ofinteger programming through the examination of an interesting substructure of an exact integerprogramming model.

In summary, this thesis presents a thorough analysis of possible integer programming models for thestrongly NP-hard ‘step and shoot’ leaf sequencing problems and investigates and applies methodsfor improving the computational efficiency of such formulations. In this way, this thesis contributesto the field of leaf sequencing for the application of Intensity Modulated Radiation Therapy usingMultileaf Collimators.

iv Abstract

v

Contents

Abstract iii

List of Tables xi

List of Figures xv

List of Algorithms xvii

Glossary xix

Acknowledgements xxv

Preface xxvii

1 Introduction 1

2 Mixed Integer Programming Approaches to Exact Minimisation of Total Treat-ment Time in Cancer Radiotherapy Using Multileaf Collimators 17

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 A New MIP Formulation for the Total Treatment Time Problem . . . . . . . . . . 19

2.3 The Step-up Algorithm for the Total Treatment Time Problem . . . . . . . . . . . 23

2.3.1 The Cardinality Constrained Minimum Beam-on Time Subproblem. . . . . 26

2.3.2 The Step-up Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 The Case of Integer Beam-On Times . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Exact Integer Programming Models for the Beam-on Time Constrained Mini-mum Cardinality Problem in Cancer Radiotherapy Using Multileaf Collimators 35

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

vi CONTENTS

3.2 The Johnston and Sadinlija (JS) Model . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.1 Variable s- and b- based Symmetry Breaking Constraints. . . . . . . . . . . 37

3.2.2 Variable s- and x- based Symmetry Breaking Constraints. . . . . . . . . . . 39

3.3 The Johnston and Sadinlija Leaf Explicit (JS-LE) Model . . . . . . . . . . . . . . . 41

3.4 The Johnston and Sadinlija Leaf Explicit Asymmetric (JS-LEA) Model . . . . . . 43

3.5 The Johnston and Sadinlija Leaf Explicit Pairs (JS-LEP) Model . . . . . . . . . . 45

3.6 The Unit Radiation Pattern (URP) Model . . . . . . . . . . . . . . . . . . . . . . . 46

3.6.1 Variable p-based Symmetry Breaking Constraints of Type 1. . . . . . . . . 49

3.6.2 Variable p-based Symmetry Breaking Constraints of Type 2. . . . . . . . . 50

3.6.3 Variable x-based Symmetry Breaking Constraints. . . . . . . . . . . . . . . 50

3.6.4 Variable p- and x- based Symmetry Breaking Constraints. . . . . . . . . . . 50

3.7 The Binary Expansion (BE) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.7.1 Variable e- based Symmetry Breaking Constraints. . . . . . . . . . . . . . . 54

3.7.2 Variable d- based Symmetry Breaking Constraints. . . . . . . . . . . . . . . 54

3.7.3 Additional Constraints Tested on the BE Model. . . . . . . . . . . . . . . . 55

3.8 The Counter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.8.1 The Cumulative Counter (CC) Model. . . . . . . . . . . . . . . . . . . . . . 59

3.9 Comparison of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4 Polyhedral Analysis of the Equality Switch Polytope 71

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 The Equality Switch Polytope (ESP ) . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3 An Example of Points Satisfying ESP and the Corresponding Facets of ESP . . . 75

4.3.1 Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

CONTENTS vii

4.4 Constraint ‘2a-any-x’ is a Facet of ESP . . . . . . . . . . . . . . . . . . . . . . . . 82

4.5 Constraint ‘na-any-x’ is a Facet of ESP . . . . . . . . . . . . . . . . . . . . . . . . 84

4.6 Facets of ESP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.7 ESP with the Strict Consecutive-1-Constraint (C1) . . . . . . . . . . . . . . . . . 86

4.8 Numerical Results for the Application of the Facets of ESP and ESP -C1 to ESPand ESP -C1 Models Respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.9 Numerical Results for the Application of the Facets of ESP and ESP -C1 to the JSModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.9.1 Application of Facets ‘2a-any-x’, ‘1a-big-x-small-x’, ‘3a-any-x’ and ‘3a-big-x-any-x’ of ESP to the JS Model. . . . . . . . . . . . . . . . . . . . . . . . 92

4.9.2 Application of Facets ‘2a-any-x-eq’, ‘2a-1diff-x-eq’, ‘3a-coeff2-x-eq’ and ‘4a-any-x-eq’ of ESP -C1 to the JS Model. . . . . . . . . . . . . . . . . . . . . . 94

4.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5 Novel Bounds on the Beam-on Time Related Variables in Exact Integer Pro-gramming Models for the Modulation of Intensity Beams in Cancer Radiother-apy Using Multileaf Collimators 103

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.2 The Sum Constrained Sorted Multiset System . . . . . . . . . . . . . . . . . . . . . 105

5.3 Bounds Conversion Between Models . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.4 Bounds Arising from Properties of the Intensity Matrix . . . . . . . . . . . . . . . 113

5.4.1 Initialisation of Bounds on Beam-on Time Variables. . . . . . . . . . . . . . 113

5.4.2 Extension of the Work of Baatar et al. [1, 2]: Properties of Intensity Matricesand Decompositions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.5 Improving Bounds Using the Integrality of Variables . . . . . . . . . . . . . . . . . 124

5.6 Flow Models for Improving Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.7 Bounds On Radiation Delivered To Cell (i, j) . . . . . . . . . . . . . . . . . . . . . 133

5.7.1 Returning to Possible Initialisations for Upper Bounds on Beam-on TimeVariables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

viii CONTENTS

5.8 Application of Improved Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.8.1 Bounds Application within the JS Model. . . . . . . . . . . . . . . . . . . . 137

5.8.2 Additional Constraints Tested on the JS Model. . . . . . . . . . . . . . . . 138

5.8.3 Case A: Application of Initialise Multiset Bounds and Cell Based Bounds Initialiseto the JS Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.8.4 Cases B, C and D: Application of Leaf Pair Bounds Algorithms, UB Consistency,LB Consistency, Bounds Propagate and Cell Based Bounds Initialise to theJS Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

5.8.5 Case E: Application of IP Bounds Initialise, LP1 Bounds Initialise orLP2 Bounds Initialise, UB Consistency, LB Consistency, Bounds Propagate,Leaf Pair Bounds Improve and Cell Based Bounds Initialise to the JS Model. 147

5.8.6 Bounds Application within the CC Model. . . . . . . . . . . . . . . . . . . . 153

5.8.7 Case A: Application of Initialise Multiset Bounds, Consistent b N Bounds,UB Consistency and LB Consistency to the CC Model. . . . . . . . . . . . 155

5.8.8 Cases B, C and D: Application of Leaf Pair Bounds Algorithms, Bounds Propagate,Consistent b N Bounds, UB Consistency and LB Consistency to the CCModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.8.9 Case E: Application of IP Bounds Initialise, LP1 Bounds Initialise orLP2 Bounds Initialise, Bounds Propagate, Leaf Pair Bounds Improve,Consistent b N Bounds, UB Consistency and LB Consistency to the CCModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

5.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6 Conclusion 171

A Modified Langer et al. [3] Model 177

B Additional Facets, Examples and Numerical Results for Polytopes ESP andESP -C1 181

B.1 Facets of ESP of Small Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

B.1.1 Constraint ‘1a-big-x-small-x’ is a Facet of ESP . . . . . . . . . . . . . . . . 181

B.1.2 Constraint ‘3a-any-x’ is a Facet of ESP . . . . . . . . . . . . . . . . . . . . . 182

CONTENTS ix

B.1.3 Constraint ‘3a-big-x-any-x’ is a Facet of ESP . . . . . . . . . . . . . . . . . 185

B.1.4 Constraint ‘3a-middle-x-small-x’ is a Facet of ESP . . . . . . . . . . . . . . 187

B.1.5 Constraint ‘3a-middle-x-big-x’ is a Facet of ESP . . . . . . . . . . . . . . . 189

B.1.6 Constraint ‘3a-big-x-small-x’ is a Facet of ESP . . . . . . . . . . . . . . . . 192

B.1.7 Constraint ‘3a-big-x-middle-x’ is a Facet of ESP . . . . . . . . . . . . . . . 196

B.1.8 Proof that Constraint ‘na-any-x’ of Section 4.5 is a Facet of ESP . . . . . . 200

B.2 Examples of Points Satisfying ESP -C1 and Facets of ESP -C1 of Small Support forthe Equal I’s Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

B.2.1 Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

B.2.2 Example 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

B.2.3 Constraint ‘2a-any-x-eq’ is a Facet of ESP -C1. . . . . . . . . . . . . . . . . 214

B.2.4 Constraint ‘2a-1diff-x-eq’ is a Facet of ESP -C1. . . . . . . . . . . . . . . . 215

B.2.5 Constraint ‘3a-coeff2-x-eq’ is a Facet of ESP -C1. . . . . . . . . . . . . . . . 217

B.2.6 Constraint ‘4a-any-x-eq’ is a Facet of ESP -C1. . . . . . . . . . . . . . . . . 219

B.3 Tables of Results for the Application of Facets of ESP to the JS Model . . . . . . 221

Bibliography 227

x CONTENTS

xi

List of Tables

2.2.1 Numerical results for the TTT model. . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.1 Time taken to solve the TTT model versus the time taken to run the Step-upalgorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.1 Computation times for the Real TTT model, the Integer TTT model, the RealStep-up algorithm and the Integer Step-up algorithm. . . . . . . . . . . . . . . . . 32

2.4.2 Numerical results for the Integer TTT model compared with the Modified Langeret al. [3] model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.1 Numerical results for the JS model solving the BTCMC problem: sk and bk sym-metry constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.2 Numerical results for the JS model solving the BTCMC problem: xijk symmetryconstraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.5.1 Numerical results for the JS model versus the JS-LE model, the JS-LEA model andthe JS-LEP model solving the BTCMC problem. . . . . . . . . . . . . . . . . . . . 46

3.5.2 Numerical results for the JS model versus the JS-LE model with and without xijkvariables solving the BTCMC problem. . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.6.1 Numerical results for the URP-pSCT1 model, the URP-pSCT2 model, the URP-xSCmodel and the URP-pxSC model solving the BTCMC problem. . . . . . . . . . . . 51

3.7.1 Numerical results for the BE model with variable e-based symmetry constraintsand the BE model with variable d-based symmetry constraints solving the BTCMCproblem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.7.2 Numerical results for the BE model with variable d- based symmetry constraintsand various combinations of constraints (3.7.11) and (3.7.12) solving the BTCMCproblem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.8.1 Numerical results for the Counter model and the CC model, with and without simplebounds on variables solving the BTCMC problem. . . . . . . . . . . . . . . . . . . 61

3.9.1 Numerical results for the JS model, the URP-xSC model, the BE model and theCounter model with simple bounds solving the BTCMC problem. . . . . . . . . . . 65

3.9.2 Summary of numerical results for the JS model, BE model, URP-xSC model and CCmodel with simple bounds compared with the Counter model with simple bounds,over batches of problems where all models were solved as given in Table 3.9.1. . . . 66

xii List of Tables

3.9.3 Numerical results for the Counter model with simple bounds using medical data setssolving the BTCMC problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3.1 Example 1: points satisfying ESP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.6.1 Facets of ESP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.7.1 Facets of ESP -C1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.8.1 Results for the application of Facets of ESP to the ESP model and Equal I Facetsof ESP -C1 to the ESP -C1 model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.9.1 Results for the application of the ‘2a-any-x’ Facet of ESP to the JS model. . . . 93

4.9.2 Results for the application of the ‘2a-any-x-eq’ Facet of ESP -C1 to the JS model. 96

4.9.3 Results for the application of the ‘2a-1diff-x-eq’ Facet of ESP -C1 to the JS model. 97

4.9.4 Results for the application of the ‘3a-coeff2-x-eq’ Facet of ESP -C1 to the JS model. 98

4.9.5 Results for the application of the ‘4a-any-x-eq’ Facet of ESP -C1 to the JS model. 99

4.9.6 Results for the simultaneous application of the ‘2a-any-x-eq’, ‘2a-1diff-x-eq’, ‘3a-coeff2-x-eq’ and ‘4a-any-x-eq’ Facets of ESP -C1 to the JS model. . . . . . . . . . 100

5.5.1 Application of Proposition 5.5.1 to upper bounds on the beam-on time variables ofExample 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.6.1 Application of Z Bounds Initialise, with Z=IP, Z=LP1 and Z=LP2 respectively, toExample 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.8.1 The specific initialisation, and/or bounds propagation and/or bounds improvementsteps trialled on the JS and CC models with Case Y, where Y=A, B, C, D or E. . 141

5.8.2 Numerical results for Algorithm JS 5.8.2 Case A applied to the JS model solvingthe BTCMC problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.8.3 Numerical results for Algorithm JS 5.8.2 Cases A, B, C and D applied to the JSmodel solving the BTCMC problem. . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.8.4 Summary of numerical results for Algorithm JS 5.8.2 Cases A, B, C and D appliedto the JS model solving the BTCMC problem. . . . . . . . . . . . . . . . . . . . . 146

List of Tables xiii

5.8.5 Computation time and upper bounds returned by IP Bounds Initialise, LP1 Bounds Initialiseand LP2 Bounds Initialise of Section 5.6. Algorithm JS 5.8.2 Case E with each ofIP Bounds Initialise, LP1 Bounds Initialise and LP2 Bounds Initialise respectively,applied to the JS model, solving the BTCMC problem. . . . . . . . . . . . . . . . . 148

5.8.6 Numerical results for Algorithm JS 5.8.2 Cases D and E applied to the JS modelsolving the BTCMC problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.8.7 Summary of numerical results for Algorithm JS 5.8.2 Cases D and E with andwithout the Step-up Method applied solving the BTCMC problem. . . . . . . . . . 152

5.8.8 Numerical results for Algorithm CC 5.8.3 Case A applied to the CC model solvingthe BTCMC problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

5.8.9 Numerical results for Algorithm CC 5.8.3 Cases A, B, C and D applied to the CCmodel solving the BTCMC problem. . . . . . . . . . . . . . . . . . . . . . . . . . . 158

5.8.10 Summary of numerical results for Algorithm CC 5.8.3 Cases A, B, C and D appliedto the CC model solving the BTCMC problem. . . . . . . . . . . . . . . . . . . . . 160

5.8.11 Numerical results for Algorithm CC 5.8.3 Cases D and E applied to the CC modelsolving the BTCMC problem. Algorithm CC 5.8.3 Case E trials IP Bounds Initialise,LP1 Bounds Initialise and LP2 Bounds Initialise respectively in the appropriatestep of the algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.8.12 Summary of numerical results for Algorithm CC 5.8.3 Cases D and E applied tothe CC model solving the BTCMC problem. . . . . . . . . . . . . . . . . . . . . . . 164

5.8.13 Numerical results for Algorithm CC 5.8.3 Case E applied to the CC model and theCounter model of Baatar et al. [4], with simple bounds, as described in Chapter 3,Section 3.8, solving the BTCMC problem. . . . . . . . . . . . . . . . . . . . . . . . 166

B.2.1 Example 2: points satisfying ESP -C1 . . . . . . . . . . . . . . . . . . . . . . . . 203

B.2.2 Example 3: points satisfying ESP -C1 . . . . . . . . . . . . . . . . . . . . . . . . 211

B.3.1 Results for the application of the ‘1a-big-x-small-x’ Facet of ESP to the JS model. 222

B.3.2 Results for the application of the ‘3a-any-x’ Facet of ESP to the JS model. . . . 223

B.3.3 Results for the application of the ‘3a-big-x-any-x’ Facet of ESP to the JS model. 224

B.3.4 Results for the simultaneous application of the ‘2a-any-x’, ‘1a-big-x-small-x’, ‘3a-any-x’ and ‘3a-big-x-any-x’ Facets of ESP to the JS model. . . . . . . . . . . . . . 225

xiv List of Tables

xv

List of Figures

1.0.1 A Medical Linear Accelerator [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.0.2 A Multileaf Collimator [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.0.3 An Example of the Leaf Representation of Shape Matrices . . . . . . . . . . . . . . 3

2.3.1 A Typical Set of Feasible Points and the Feasible Frontier (?) . . . . . . . . . . . . 25

3.3.1 An Example Representation of Row i in Shape Matrix k . . . . . . . . . . . . . . . 42

5.6.1 An Example Bipartite Graph, for a particular Row i in an Intensity Matrix withcolumn dimension 3, which exhibits the characteristics of the Flow Model (5.6.1) . 128

xvi List of Figures

xvii

List of Algorithms

2.3.2 Step-up Algorithm 28

5.2.1 t :=UB Consistency(D,t) 106

5.2.2 t :=LB Consistency(D,t) 106

5.2.3 (t, t) :=Bounds Propagate(D,B,B, t, t) 111

5.3.1(bmax, N , N

):=Consistent b N Bounds(K,Klb, b, b, b

max, N , N) 112

5.3.2 (K, b, b) :=Consistent N b Bounds(K,Klb, b, b, b

max, N , N)

112

5.4.1 (b, b) :=Initialise Multiset Bounds(K,Klb, B,B, b1, I(1), I(2)) 115

5.4.2 (b, b, {U ipq}) :=Leaf Pair Bounds Initialise(K,Klb, Beammin, {Bi}, I, I(1), I(2), L, R,P) 123

5.5.1 b :=Leaf Pair Bounds Improve(K, b, {U ipq}) 127

5.6.1 (b, b) :=Z Bounds Initialise(K,Klb, Beammin, {Bi}, I, I(1), I(2), L, R) 132

5.7.1 Bu(B) :=Cell Based Bounds Initialise(B, {Bi}, I, L, R) 135

5.8.1 (Klb, {Bi}, I(1), I(2), I, L, R,P) :=Initialise Parameters Algorithms(I) 137

5.8.2 Algorithm JS 5.8.2 Cases A to E inclusive 140

5.8.3 Algorithm CC 5.8.3 Cases A to E inclusive 154

xviii List of Algorithms

xix

Glossary

Aci(j−1)h the number of shape matrices using leaf position (j − 1, h) in row i that can be given

radiation level c or more, for c = 1, . . . , b1, i = 1, . . . ,m, j = 1, . . . , n+ 1, h = j, . . . , n+ 1

aijk the intensities corresponding to each cell of shape matrix k, for i = 1, . . . ,m, j = 1, . . . , n, k =1, . . . ,K

B the total beam-on time to be used in a decomposition

B an upper bound on the total beam-on time of a decomposition

B a lower bound on the total beam-on time of a decomposition

Bi the minimum total beam-on time for the delivery of row i of an intensity matrix, for i = 1, . . . ,m

Buij(B) an upper bound on the amount of radiation that can be applied to shape matrices exposing

cell (i, j) when total beam-on time for the decomposition is B, for i = 1, . . . ,m, j = 1, . . . , n

bk the beam-on time applied to shape matrix k in a decomposition, where k = 1, . . . ,K

bk an upper bound on the beam-on time that can be applied to shape matrix k, for k = 1, . . . ,K

bk a lower bound on the beam-on time that can be applied to shape matrix k, for k = 1, . . . ,K

bmax an upper bound on the most radiation that can be applied to a shape matrix

Beammin the minimum total beam-on time for which a problem instance is feasible

Beamupper the total beam-on time used in a decomposition when the number of shape matricesused is Kmin

BB the total number of branch and bound nodes searched

BE model Binary Expansion model which solves the BTCMC problem. Later, the best of versiontested of the BE - type models (with variable d-based symmetry constraints)

BTCMC Beam-on Time Constrained Minimum Cardinality problem

C1 the Strict Consecutive-1-Constraint

CCMBT Cardinality Constrained Minimum Beam-on Time problem

CC model Cumulative Counter model which solves the BTCMC problem. Later, the CumulativeCounter model with simple bounds

Counter model Counter model of Baatar et al. [4] which solves the BTCMC problem

dijhk the binary variables indicating if cell (i, j) of shape matrix k receives h units of radiation,for i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K, h = 1, . . . ,Hk

ehk the binary variables indicating if shape matrix k receives h units of radiation, for k =1, . . . ,K, h = 1, . . . ,Hk

xx Glossary

ep the pth unit vector ∈ Rn or ∈ Zn, for p = 1, . . . , n

ESP Equality Switch Polytope

ESP -C1 Equality Switch Polytope with the addition of the Strict Consecutive-1-Constraint (C1)

F a facet of ESP

Fi(j−1)h the beam-on time applied to shape matrices with left leaf in position j− 1 and right leafin position h in row i, for i = 1, . . . ,m, j = 1, . . . , n+ 1, h = j, . . . , n+ 1

FK the set of all possible delivery plans (b,X) using K shape matrices

fi01k the binary variables indicating if the leaf position in row i of shape matrix k is (0, 1), fori = 1, . . . ,m, k = 1, . . . ,K

filrk the binary variables indicating if the leaf position in row i of shape matrix k is (l, r), fori = 1, . . . ,m, l = 0, . . . , n− 1, r = l + 2, . . . , n+ 1, k = 1, . . . ,K

Gijk an upper bound on bk, for i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K

GHA the unconstrained Greedy Heuristic Algorithm of Baatar et al. [1] which solves for a smallnumber of shape matrices with total beam-on time equal to Beammin

Hk an upper bound on bk, for k = 1, . . . ,K

I an intensity matrix of dimension m× n

I a matrix of differences in adjacent intensity values in an intensity matrix

I(1) the smallest non-zero intensity value in an intensity matrix

I(2) the second smallest, different, non-zero intensity value in an intensity matrix

i∗ a row, ∈ {1, . . . ,m}, in an intensity matrix yielding Beammin

IMRT Intensity Modulated Radiation Therapy

IP(c, i) integer program to find an upper bound on the number of shape matrices that can receiveradiation level c or more, for c = 1, . . . , b1, i = 1, . . . ,m

IP OBJ(c, i) the objective function for integer program IP(c, i), for c = 1, . . . , b1, i = 1, . . . ,m

ITS the total number of simplex iterations

JS model Johnston and Sadinlija model: the BTCMC version of the integer TTT model, basedon Johnston and Sadinlija [6]

JS-LE model Johnston and Sadinlija Leaf Explicit model: integer program based on the JSmodel using ‘leaf’ variables first described by Langer et al. [3]

JS-LEA model Johnston and Sadinlija Leaf Explicit Asymmetric model: integer program basedon the JS model using ‘leaf’ variables first described by Queyranne [7]

JS-LEP model Johnston and Sadinlija Leaf Explicit Pairs model: integer program based on theJS model using ‘leaf’ pair variables

Glossary xxi

K the maximum number of shape matrices to be used in a decomposition

Kbm the number of shape matrices found when solving BTCMC with the unconstrained GreedyHeuristic Algorithm (GHA) of Baatar et al. [1]

Kbm the minimum number of shape matrices that can be used when the total beam-on time isrestricted to Beammin

Klb a known lower bound on the minimum number of shape matrices that can be used in a solution

Kmin the smallest number of shape matrices for which a solution exists

K∗T the number of shape matrices used in an optimal solution to the TTT mixed integer program-

ming model with set-up time equal to T

L a matrix of differences in adjacent intensity values in an intensity matrix which are greater thanor equal to zero

Lij the total beam-on time applied to shape matrices with left leaf in position j − 1, in row i, fori = 1, . . . ,m, j = 1, . . . , n+ 1

lijk the binary variables equaling zero if column j is covered by the left leaf in row i of shapematrix k and one otherwise, for i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K

lijk the binary variables equaling one if column j is covered by the left leaf in row i of shapematrix k and zero otherwise, for i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K

leftik(x) the left leaf position in row i of shape matrix k, for i = 1, . . . ,m, k = 1, . . . ,K

LP1(c) the same linear program as LP1(c, i) other than the objective which maximises a sumover all rows i = 1, . . . ,m

LP1(c, i) the linear program obtained when we remove the integrality conditions from the vari-ables of integer program IP(c, i), for c = 1, . . . , b1, i = 1, . . . ,m

LP1 OBJ(c) the objective function for linear program LP1(c), for c = 1, . . . , b1

LP1 OBJ(c, i) the objective function for linear program LP1(c, i), for c = 1, . . . , b1, i = 1, . . . ,m

LP2(c) the same linear program as LP2(c, i) other than the objective which maximises a sumover all rows i = 1, . . . ,m

LP2(c, i) another relaxation of integer program IP(c, i), for c = 1, . . . , b1, i = 1, . . . ,m

LP2 OBJ(c) the objective function for linear program LP2(c), for c = 1, . . . , b1

LP2 OBJ(c, i) the objective function for linear program LP2(c, i), for c = 1, . . . , b1, i = 1, . . . ,m

Mijk an upper bound on the most radiation that can be delivered by a shape matrix k exposingcell (i, j) and therefore an upper bound on aijk, for i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K

m the number of rows in an intensity matrix

MLC Multileaf Collimator

xxii Glossary

Nb the number of shape matrices given radiation level b, for b = 1, . . . , bmax

Nb the number of shape matrices given radiation level b or more, for b = 1, . . . , bmax

Nb an upper bound on the number of shape matrices given radiation level b or more, for b =1, . . . , bmax

Nb a lower bound on the number of shape matrices given radiation level b or more, for b =1, . . . , bmax

N i the maximum of, the number of positive I values and the number of negative I values, in rowi, for i = 1, . . . ,m

n the number of columns in an intensity matrix

P the set of left and right leaf ‘row intervals’ possible for a shape matrix with n columns

pt the binary variables indicating if shape matrix t is the last shape matrix of a run and t + 1 isthe new shape matrix, or if shape matrix t equals shape matrix t+ 1, for t = 1, . . . , B − 1

πi a one-to-one function such that {1, . . . , |P|} → P which sorts the multiset U i such that U iπi(h) ≥U iπi(h′) whenever h ≥ h′

Qijb the number of shape matrices exposing cell (i, j) with radiation level b, for i = 1, . . . ,m, j =1, . . . , n, b = 1, . . . , bmax

R = L− I

Rij the total beam-on time applied to shape matrices with right leaf in position j in row i, fori = 1, . . . ,m, j = 1, . . . , n+ 1

rijk the binary variables equaling one if column j is covered by the right leaf in row i of shapematrix k and zero otherwise, for i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K

rijk equals rijk, for i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K

rightik(x) the right leaf position in row i of shape matrix k, for i = 1, . . . ,m, k = 1, . . . ,K

RNLB root node lower bound

Sijb the number of shape matrices exposing cell (i, j) with radiation level b, in excess of the numberthat expose cell (i, j− 1) with radiation level b, for i = 1, . . . ,m, j = 1, . . . , n, b = 1, . . . , bmax

sk the binary variables indicating whether shape matrix k is non-zero in a solution, for k =1, . . . ,K

σ(i, j) a sorting of the intensity matrix, for i = 1, . . . ,m, j = 1, . . . , n

SCSMS Sum Constrained Sorted Multiset System

Step-up Method method for minimising the cardinality of the solution with or without theadded restriction of minimal total beam-on time, by successively increasing the value of Kuntil a feasible solution is found

T set-up time

Glossary xxiii

td a ‘beam-on time related’ variable for d = 1, . . . , D

TTT Total Treatment Time problem

TTTBest the current best value for the TTT problem

TTT model Total Treatment Time model

U i a multiset of upper bounds for possible beam-on times that can be applied to any shape matrixwith left and right leaf ‘row interval’ (p, q) in row i, for i = 1, . . . ,m, (p, q) ∈ P

U ipq an upper bound on the beam-on time that can be applied to any shape matrix with left and

right leaf ‘row interval’ (p, q) in row i, for i = 1, . . . ,m, (p, q) ∈ P

URP model Unit Radiation Pattern model which solves the BTCMC problem

URP-pSCT1 model Unit Radiation Pattern model with variable p-based Symmetry Constraintsof Type 1

URP-pSCT2 model Unit Radiation Pattern model with variable p-based Symmetry Constraintsof Type 2

URP-pxSC model Unit Radiation Pattern model with variable p- and x-based Symmetry Con-straints

URP-xSC model Unit Radiation Pattern model with variable x-based Symmetry Constraints

Wij the ‘excess’ beam-on time applied to shape matrices with left leaf in position j − 1 in row i,for i = 1, . . . ,m, j = 1, . . . , n+ 1

X the set of all valid, non-zero, shape matrices

Xk the kth shape matrix in a decomposition, where k = 1, . . . ,K

xijk the binary variables indicating if cell (i, j) is exposed in shape matrix k, for i = 1, . . . ,m, j =1, . . . , n, k = 1, . . . ,K

xijt the binary variables indicating if cell (i, j) is exposed in the shape matrix corresponding tothe tth unit of radiation, for i = 1, . . . ,m, j = 1, . . . , n, t = 1, . . . , B

Case Y a particular bounds algorithm where Y=A, B, C, D or E, the details of which are givenin Table 5.8.1

Z a parameter indicating which of IP(c, i), LP1(c) or LP2(c), for c = 1, . . . , b1, i = 1, . . . ,m, shouldbe tested within algorithm Z Bounds Initialise. Z equals IP, LP1 or LP2

1a-big-x-small-x Facet of ESP a facet of ESP containing 1 a variable, 1 x variable corre-sponding to the largest of the ordered I parameters and the complement of the x variablecorresponding to the smallest of the ordered I parameters

2a-any-x Facet of ESP a facet of ESP containing 2 a variables and the complement of anyallowed x variable

2a-any-x-eq Facet of ESP -C1 a facet of ESP -C1 containing 2 a variables and the comple-ment of any allowed x variable when I parameters are equal

xxiv Glossary

2a-1diff-x-eq Facet of ESP -C1 a facet of ESP -C1 containing 2 a variables and the comple-ment of an x variable which has a subscript different from the a variables, and equal Iparameters

3a-any-x Facet of ESP a facet of ESP containing 3 a variables and the complement of anyallowed x variable

3a-big-x-any-x Facet of ESP a facet of ESP containing 3 a variables, 1 x variable corre-sponding to the largest of the ordered I parameters and the complement of any remainingallowed x variable

3a-middle-x-small-x Facet of ESP a facet of ESP containing 3 a variables, 1 x variablecorresponding to the middle of 3 ordered I parameters and the complement of the x variablecorresponding to the smallest of the ordered I parameters

3a-middle-x-big-x Facet of ESP a facet of ESP containing 3 a variables, 1 x variable cor-responding to the middle of 3 ordered I parameters and the complement of the x variablecorresponding to the largest of the ordered I parameters

3a-big-x-small-x Facet of ESP a facet of ESP containing 3 a variables, 1 x variable corre-sponding to the largest of the ordered I parameters and the complement of the x variablecorresponding to the smallest of the ordered I parameters

3a-big-x-middle-x Facet of ESP a facet of ESP containing 3 a variables, 1 x variable corre-sponding to the largest of the ordered I parameters and the complement of the x variablecorresponding to the middle of 3 ordered I parameters

3a-coeff2-x-eq Facet of ESP -C1 a facet of ESP -C1 containing 3 a variables and the com-plement of 1 x variable which has a subscript equal to the subscript for the a variable withcoefficient 2, and equal I parameters

4a-any-x-eq Facet of ESP -C1 a facet of ESP -C1 containing 4 a variables and the comple-ment of any allowed x variable when I parameters are equal

na-any-x Facet of ESP a facet of ESP containing n a variables and the complement of anyallowed x variable

xxv

Acknowledgements

First and foremost I wish to sincerely thank my supervisors, Associate Professor Les Jennings andProfessor Natashia Boland. Les and Natashia have provided me with invaluable guidance andencouragement throughout my PhD. Les has always given up his time to see me. I thank himfor the many hours of discussion, his patience and understanding, and mostly for inspiring mewith his knowledge. Although Natashia was at The University of Melbourne and has now movedto The University of Newcastle, she too has always made time for me and again we have spentnumerous hours on the telephone discussing my thesis. I would like to thank Natashia for herinspirational ideas, dedication and for passing on some small part of her extensive knowledge ofOperations Research. The time and effort of my supervisors, Natashia and Les, has meant thatmy PhD experience has been truly rewarding and enjoyable.

Thanks must also go to Dr Davaatseren Baatar, recently of The University of Melbourne, for thetime he spent, early in my PhD, helping me with some of the more difficult literature and forkindly providing his code for use within the models of this thesis.

I would also like to thank Professor Robert Johnston (Honorary Professorial Fellow in the Depart-ment of Mathematics and Statistics at The University of Melbourne) for helpful discussions andfor the ideas he contributed to Chapter 5 of this thesis.

Throughout the duration of my PhD I was supported financially by an Australian PostgraduateAward and a Completion Scholarship. My supervisor Associate Professor Les Jennings also kindlyprovided me with additional financial support.

Finally I would like to thank my family for their understanding, and my husband Geoff for allhis love and encouragement, particularly during the difficult time towards the end of my research.Now our life begins again!

xxvi Acknowledgements

xxvii

Preface

Chapter 2 (with minor amendments for consistency with this thesis), and parts of the Introduc-tion to this thesis (Chapter 1), have been published in the journal of Computers and OperationsResearch. The published work, [8], is co-authored and we detail the contributions of each authorbelow.

Basing our Total Treatment Time formulation on the cutting stock model of Johnston and Sadinlija[6] was the idea of my supervisor Professor Natashia Boland.

The idea for the Step-up Algorithm as a method for solving the Total Treatment Time model wasthat of Giulia Wake. Refinements to the algorithm were discussed with my supervisors, ProfessorNatashia Boland and Associate Professor Les Jennings, and implemented by Giulia Wake.

The idea for additional constraints applied within the model and algorithm was that of GiuliaWake and my supervisors.

Finally, all modelling, coding and experimental work was conducted by Giulia Wake, as was thewriting of the paper other than minor amendments.

The remaining work in this thesis, is a synthesis of my own ideas and those of my supervisors, unlessotherwise stated. All the ideas were explored and implemented by Giulia Wake. Professor RobertJohnston (an Honorary Professorial Fellow in the Department of Mathematics and Statistics atThe University of Melbourne) also contributed to the ideas for Chapter 5 of this thesis.

1CHAPTER 1

Introduction

The Cancer Council Australia reports that around 106,000 new cases of cancer are diagnosed inAustralia annually and that of these more than 39,000 people are expected to die each year [9].Due to the significant impact this clearly has on society, it is important that we investigate noveland improved methods for treating cancer patients. A relatively new method of external beamradiation treatment, known as Intensity Modulated Radiation Therapy (IMRT), is becoming morewidespread in Australia and around the world for the treatment of certain types of tumours. Inparticular, IMRT is used to treat cancers in areas of the body where the treatment area can befixed in position, for example the head and neck. IMRT allows the intensity of the radiationbeam to vary across the patient surface and thus it applies the prescribed radiation dose to thetumour shape whilst minimising exposure to the surrounding ‘normal’ organs. As new planningand delivery technologies are developed, IMRT is being viewed as a viable alternative to othercurrently used conformal radiotherapies in Australia [10].

Treatment design for Intensity Modulated Radiation Therapy is complicated. The process beginswith the tumour being imaged. It is essential that the patient moves as little as possible duringimaging as this is the basis from which a treatment plan is constructed. Using multiple images, athree dimensional representation of the tumour and surrounding region is obtained which allowsidentification of critical organs to be irradiated or avoided. The absorbed dose of radiation (theenergy actually deposited in the body) is determined by a physician, then a dosimetrist or medicalphysicist algorithmically calculates the amount of radiation and the direction/s from which theradiation should be applied such that the absorbed dose is achieved within the patient. Theradiation to be delivered from each chosen angle is determined as a two dimensional intensity matrixor profile comprising individual beamlets of differing intensities. We describe intensity matrices inmore detail to follow. Finally an efficient delivery sequence, for the radiation applied from eachdirection, must also be calculated prior to application, to ensure minimal patient exposure time.This is to limit the possibility of inaccuracies in dosage received due to patient movement resultingfrom longer than necessary treatment times, and to reduce wear and tear of delivery equipment.The calculation of radiation beam angles and the intensity matrix to be delivered from each angleis known as treatment planning in the IMRT literature. General approaches to planning IMRTtreatment can be found in the work of Hamacher and Kufer [11], Lee et al. [12], and Olafssonand Wright [13], to give three examples. See also the very recent survey by Ehrgott et al. [14].Determination of a delivery sequence for the radiation to be applied from each angle, to minimisetreatment time, is known as leaf sequencing. We focus on leaf sequencing in this thesis and discussthe problems we solve and the literature to follow. In practice, the process for devising IMRTtreatment for a patient combines trial and error and the experience of the physician/planner,with optimisation software. Given the rate at which new technology is becoming available toclinics it is imperative that new optimisation techniques are developed and utilised to fully realisethe advantages of the available technological advancements and in turn pass these benefits on topatients.

The treatment of cancerous tumours using Intensity Modulated Radiation Therapy requires amedical linear accelerator, a gantry to focus the beam of accelerated particles and a mechanism

2 Chapter 1. Introduction

by which the radiation beam can be modulated such that the beam conforms to the cancer shape.The present study focuses on the multileaf collimator (MLC) as the shaping mechanism for theradiation beam. (Other shaping devices such as compensators and jaws are discussed in, forexample, [15] and [16] respectively). Figures 1.0.1 and 1.0.2 demonstrate the configuration of theIMRT equipment.

Figure 1.0.1: A Medical Linear Accelerator [5]

Figure 1.0.2: A Multileaf Collimator [5]

The gantry of the medical linear accelerator can rotate around a patient and is therefore the meansby which radiation can be delivered from any number of angles. In clinical practice the number ofangles used is generally between 3 and 7 [17]. As mentioned, the treatment plan, for each stop of the

Chapter 1. Introduction 3

gantry, can be represented as a two dimensional matrix, known as an intensity matrix. The valuesin the matrix arise from discretisation of the desired intensity profile, and are typically mappedto integer values, usually in quite a small range, e.g. 0–10, although other approaches have beenconsidered (see, for example, [18]), that indicate a finer level of discretisation may be beneficial.For each row of the intensity matrix, the multileaf collimator has an associated pair of left andright metal ‘leaves’ which move to control the amount and shape of the radiation beam passingthrough the collimator. At a single stop of the gantry, with the radiation beam on, the multileafcollimator can be either fixed in position (static) or moving (dynamic). Static multileaf collimationachieves modulation of the intensity beam by moving the leaves with the beam switched off andirradiating different patterns of uniform intensity with the beam turned on and the leaves fixedin position. Dynamic multileaf collimation modulates the beam by varying the speed of the leafmotion with the radiation beam continuously on. This work considers a single stop of the gantryand static multileaf collimation, which is also known as ‘step and shoot’ within the leaf sequencingliterature of IMRT. For a more detailed description of the technology and delivery mode, see, forexample, Galvin et al. [15], Bortfeld et al. [19] or Siochi [20].

When a multileaf collimator is in static mode, each set of left/right leaf positions of the MLC forall rows can be interpreted as a binary matrix with zeros representing cells of the matrix coveredby a leaf, and ones representing cells that are exposed. A zero row corresponds to any left/rightleaf positions in which the leaves are closed and do not overlap. The binary matrices formed bythe MLC have a special configuration known as the strict consecutive-1-property which does notallow patterns of the form 1,0,. . . ,0,1 in any row of a binary matrix. These matrices, called shapematrices, when irradiated with a uniform intensity beam, accumulate to produce the intensityprofile required by the dosimetrist. An example decomposition of an intensity matrix into shapematrices and the corresponding multileaf collimator representation is given below: 5 5 7

1 3 9

8 6 1

= 3

0 0 1

0 0 1

1 0 0

+ 3

1 1 1

0 0 1

1 1 0

+ 2

1 1 0

0 1 1

1 1 0

+ 1

0 0 1

1 1 1

0 1 1

= 3 + 3 + 2 + 1

Figure 1.0.3: An Example of the Leaf Representation of Shape Matrices

The ‘step and shoot’ leaf sequencing problem is to determine a minimal treatment time decom-position of a given intensity matrix into shape matrices. In this thesis we develop and compareexact mixed integer and integer programming models for the ‘step and shoot’ leaf sequencing prob-lem and apply techniques to improve the computational efficiency of the programs. We give anoverview of the models and techniques we use later in the Introduction. We now describe what ismeant by a minimal treatment time decomposition of an intensity matrix. When receiving IMRT,patients must remain motionless, so minimising the time needed for treatment may improve the


radiation delivery to the cancer and, hence, enhance the likelihood of a successful outcome for thepatient. The total treatment time for ‘step and shoot’ IMRT delivery is a combination of the totalbeam-on time of the gantry and the total set-up/verification and record time of the MLC, at asingle stop of the gantry. Set-up time is the amount of time it takes for an MLC to change from theconfiguration of one shape matrix to the next. Verification and record time is the amount of timeit takes to check that the actual leaf positions of the MLC match the planned leaf positions, and torecord the actual leaf positions. See Siochi [20] for further details. Set-up/verification and recordtime can be considered to be variable, depending on leaf movement, or can be approximated by aconstant value per set-up. This work investigates constant set-up/verification and record time perset-up, which we call the set-up time and denote by T > 0. We now mathematically formulate theminimal treatment time problems for leaf sequencing. We first define:

• a set of K shape matrices X = {X1, . . . , XK}, where Xk ∈ X for all k = 1, . . . ,K and Xdenotes the set of all valid, non-zero, shape matrices

• a set of beam-on time values b = {b1, . . . , bK}

• a corresponding delivery plan (b,X)

• an intensity matrix I, and

• the set of all possible delivery plans using K shape matrices,

FK = {(b,X) ∈ ZK+ × XK :K∑k=1

bkXk = I}, where the vector b can also be in RK+ . We

investigate the effect of non-integer beam-on time in Chapter 2 of this thesis, where we alsogive our motivation for the investigation.

The Minimum Beam-on Time (MBT) problem:

The Minimum Beam-on Time problem minimises the total beam-on time of a decomposition ofthe intensity matrix regardless of the number of shape matrices used in a solution. We have:

Beammin = minK,b,X

{K∑k=1

bk : (b,X) ∈ FK},

where Beammin is the minimum total beam-on time for which a problem instance is feasible.

The Minimum Cardinality (MC) problem:

The Minimum Cardinality problem minimises the total number of shape matrices (cardinality)used in a solution regardless of the total beam-on time of the decomposition. We have:

Kmin = minK,b,X

{K : (b,X) ∈ FK},

where Kmin is the smallest number of shape matrices for which a solution exists. The minimumtotal beam-on time for a decomposition using Kmin shape matrices we denote by Beamupper.

The Total Treatment Time (TTT) problem:


Finally the Total Treatment Time problem minimises a sum of the two time components for the‘step and shoot’ leaf sequencing problem, the total beam-on time and the total set-up time, wheretotal set-up time can reasonably be approximated by the number of shape matrices multiplied bya constant overhead time per shape matrix. We have:

TTT ∗ = minK,b,X

{K∑k=1

bk + TK : (b,X) ∈ FK

},

where TTT ∗ is the optimal total treatment time for a problem instance. We also define

The Beam-on Time Constrained Minimum Cardinality (BTCMC) problem:

The Beam-on Time Constrained Minimum Cardinality problem lexicographically minimises totalbeam-on time then the number of shape matrices used in a solution. We have:

Kbm = minK,b,X

{K : (b,X) ∈ FK},

subject toK∑k=1

bk = Beammin,

where Kbm is the minimum number of shape matrices that can be used when the total beam-ontime is restricted to Beammin.

It is important to note that the number of shape matrices, K, needed to minimise treatment timeis unknown, and must be found as part of solving the treatment time problems.

The MLC may have a mechanical restriction which prohibits the left and right leaves in adjacentrows of a shape matrix to overlap. This is known as the ‘interleaf motion constraint’. Mathematicalformulations of the constraint can be applied to a model of the leaf sequencing problem to enforcethis restriction. The first shape matrix in Figure 1.0.3 demonstrates a violation of the interleafmotion constraint. Furthermore, the leaves of the MLC may have a tongue-and-groove designto minimise radiation leakage, however this mechanism may also result in underdosage along theborder between leaves in adjacent rows. To avoid this underdosage a tongue-and-groove constraintcan be applied to a model. We do not consider interleaf motion or tongue-and-groove effectsin this study, or any other mechanical restrictions of the MLC, though if necessary the relevantconstraints can be incorporated into all the models we present. Whilst the constraints to apply maynot be simple to formulate, they nevertheless can be formulated within our models, and nothingin our solution method precludes their inclusion. Indeed, a reason we chose to focus our work oninteger programming models is the ease with which additional technical requirements can generallybe included in such models. The treatment time problems without MLC mechanical restrictionsare referred to as unconstrained problems in the leaf sequencing literature, whereas with MLCmechanical restrictions, the problems are referred to as constrained. For an example of an integerprogramming formulation of interleaf and tongue-and-groove constraints see Langer et al. [3].

There is already quite a large body of literature on methods of finding delivery plans for IMRTwith an MLC in ‘step and shoot’ mode [1, 2, 15, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31].Many describe heuristics that seek a value K and a delivery plan (b,X) ∈ FK so that either or both


the total beam-on time,K∑k=1

bk, and the total number of shape matrices, K, are small. A concise

and exhaustive review of the leaf sequencing literature is provided by Ehrgott et al. [14, 32]. Theinterested reader is referred to these articles for a comprehensive overview of the subject. However,for completeness, in this chapter, we summarise the main works, for each of the minimum treatmenttime leaf sequencing problems, analysed in the Ehrgott et al. [32] review article. We then describehow the work of this thesis contributes to the overall field of leaf sequencing.

The MBT problem can be solved in polynomial time with and without the added restriction ofMLC mechanical constraints. We focus first on the case of no MLC mechanical constraints, theunconstrained case and present some of the main results. Engel [27], Ahuja and Hamacher [33]and Baatar et al. [1] independently determine the minimum beam-on time of a problem instancefor the unconstrained case, utilising the fact that, when solving the unconstrained MBT problem,rows of an intensity matrix can be considered individually: there are no constraints linking theallowed multileaf collimator leaf positions in each row. Engel [27] provides the following closedform solution to the problem:

Beammin = maxi=1...,m

n+1∑j=1

max(0, Iij),

where i and j are the row and column indices respectively for the intensity matrix Iij , i = 1, . . . ,m,j = 1, . . . , n and

Iij =

Iij − Ii,j−1, 1 < j ≤ nIi1, j = 1−Iin, j = n+ 1

,

where we have used notation consistent with this thesis. Engel [27] also refers to the expression forBeammin as the total number of monitor units (TNMU) complexity of the intensity matrix, andpresents an algorithm, the TNMU algorithm, to calculate Beammin and heuristically determinea corresponding ‘small’ number of shape matrices. The TNMU algorithm subtracts an admissiblesegmentation pair (comprising a positive integer beam-on time value and a shape matrix) from theintensity matrix, at each step, so that the entries of the residual intensity matrix are greater thanor equal to zero and the complexity of the intensity matrix is reduced by the value of the beam-ontime in the admissible segmentation pair. When the residual intensity matrix becomes the zeromatrix the algorithm terminates. Each step of the TNMU algorithm has time complexity O(mn)and the TNMU algorithm itself has time complexity O(m2n2).

Ahuja and Hamacher [33] formulate a linear program for solving the unconstrained MBT problemfor each row of an intensity matrix. They then transpose the row vectors in the linear program tocolumn vectors (comprising zeros and consecutive ones) and demonstrate that the resulting linearprogram can be transformed into a minimum cost flow model in a directed network, using simplerow operations. Their algorithm utilises the structure of the network (namely that it is acyclic,complete and that each arc has a cost of 1 and infinite capacity) to determine the solution to theMBT problem for each row in O(n) time and hence Beammin in O(mn) time. The algorithm ofAhuja and Hamacher [33] also proves the optimality of the sweep algorithm of Bortfeld et al. [19]which was the first algorithm to determine Beammin for the unconstrained MBT problem butwhich had not previously been proved optimal. It should be noted that Kamath et al. [34] and


Baatar et al. [1] also provide algorithms for the unconstrained MBT problem and also prove theoptimality of the sweep algorithm of Bortfeld et al. [19]. We discuss some of the ideas of Baataret al. [1] in detail in Chapter 5 of this thesis.

We now turn our attention to the constrained MBT problem. Boland et al. [17] develop a networkflow model with side constraints to formulate the constrained MBT problem, where paths from asource to sink describe feasible shape matrices. They define a layered digraph of m layers, and asource and sink node, where each layer of the digraph corresponds to a row of a shape matrix. Eachnode in each layer is described by the row number and the possible left and right leaf positions ofthe multileaf collimator, taking into account any mechanical restrictions of the MLC that occurwithin a row. Furthermore, arcs are possible between layers where the appropriate MLC interleafconstraints are not violated. If interleaf constraints are unnecessary the arcs connect all nodes inone layer to all nodes in the next. There is also a return arc from sink to source, and arcs betweenthe source and the nodes of the first layer, and between the nodes of the last layer and the sink.The size of the network is O(mn2). The formulation of Boland et al. [17], which minimises flowfrom source to sink subject to the particular circulation satisfying the given intensity matrix, solvesthe constrained MBT problem in polynomial time.

Kalinowski [30] formulates the dual of a linear program for the constrained MBT problem andshows that the longest path length in a layered digraph (from source to sink) corresponds to thesolution of the dual problem. He demonstrates that the formula of Engel [27] for the unconstrainedproblem can be interpreted as the longest path length in a layered digraph and then develops asimilar digraph to incorporate interleaf constraints. Kalinowski [30] uses a duality argument toprove the optimality of his approach and demonstrates that the maximal weight of a path in thedigraph is a lower bound for the total beam-on time of a decomposition and further that thereexists a decomposition with total beam-on time equal to this bound.

Baatar and Hamacher [35] also consider a network approach to solving the constrained MBTproblem. The network utilises 2m layers comprising separate sets of vertices for the left and rightleaf positions of the multileaf collimator respectively, and a source and sink node. The allowed leafpositions and any MLC constraints that are necessary within a row are now achieved with arcsbetween layers of ‘left’ and ‘right’ nodes corresponding to the same row and interleaf constraintsare achieved with arcs between ‘left’ and ‘right’ nodes corresponding to adjacent rows. Arcs alsoconnect the source with the layer of ‘left’ nodes of the first row, and the layer of ‘right’ nodes in thefinal row with the sink. The number of variables used in the Baatar and Hamacher [35] network isan order of magnitude smaller than the Boland et al. [17] network, by a factor of n, and furthermorethe coefficient matrix for the corresponding mixed integer programming formulation of the networkis totally unimodular. Hence it is sufficient to solve the linear relaxation of the mixed integerprogram and computational improvement over the Boland et al. [17] formulation is achieved.Finally, Baatar et al. [1] extend this work and instead solve the integer programs recursively usinga sequence of multiobjective integer programs. The multiobjective integer programs are solvedvia a combinatorial algorithm in O(m2) time and the entire program for the constrained MBTproblem is in turn solved in polynomial time, with time complexity O(m2n).

It is clear from the above discussion of the MBT literature that since the Minimum Beam-on Time


problem is solvable in polynomial time many exact algorithms have been developed for its solution.On the other hand, the problems of interest in this thesis, the Beam-on Time Constrained Mini-mum Cardinality problem and the Total Treatment Time problem are strongly NP-hard problemswhether MLC mechanical restrictions are imposed or otherwise. This result follows from the NP-hardness of the Minimum Cardinality problem (unconstrained and constrained) which has beenshown to be strongly NP-hard even if the intensity matrix has only a single row [1], or only asingle column [36]. (In the case of the TTT problem, if set-up time T is sufficiently large, theproblem is equivalent to minimising cardinality alone). Only if the intensity matrix is a posi-tive integer multiple of a binary matrix can the (unconstrained and constrained) MC problem besolved in polynomial time, Baatar et al. [1]. Therefore, most of the literature dealing with the MC,BTCMC and TTT problems are heuristics. We now briefly mention some of the heuristics to befound. We first discuss the unconstrained and constrained MC problem, then the unconstrainedand constrained BTCMC problem and finally the unconstrained and constrained TTT problem.We discuss exact approaches to these problems subsequently.

Xia and Verhey [21] utilise an heuristic method, for the unconstrained and constrained MC prob-lem, which delivers shape matrices with beam-on times which are powers of 2, dependent on themaximum intensity level in an intensity matrix. The beam-on times are factored out of the intensitymatrix first, then the decomposition of the resultant binary matrices can be solved in polynomialtime by virtue of the property of Baatar et al. [1] stated above. The Xia and Verhey [21] algorithmreturns smaller numbers of shape matrices when compared with the earlier methods of Bortfeld etal. [19] and Galvin et al. [15]. Crooks et al. [24] improve on the unconstrained results of Xia andVerhey [21] determining an absolute magnitude norm-minimisation algorithm based on reducingthe sum of the individual intensities in the intensity matrix by the maximum possible amount ateach step of the decomposition into shape matrices. The improvement by Crooks et al. [24], whencompared with Xia and Verhey [21], is at least a 30% reduction in the number of shape matricesand a similar total beam-on time. More recently, Luan et al. [37] have developed heuristics for theunconstrained MC problem: a (blog hc+ 1)-approximation algorithm where h is the largest valuein a given intensity matrix, with time complexity O(mn log h), and a 2(blog Dc+1)-approximationalgorithm, with time complexity O(mn log D), where D is the maximum element of a set whichconsists of the first and final entries of each row of a given intensity matrix and the absolute valueof the difference between consecutive values in each row of an intensity matrix, over all rows. (Thelogarithms are base 2). The heuristics of Luan et al. [37] describe the decomposition of matri-ces with entries based on the binary representation of the original intensity matrix. Finally, wealso mention Chen et al. [38] who consider geometric approaches to the constrained MC problem,modelling shape matrices as ‘x-axis-monotone rectilinear polygons’ and intensity matrices as threedimensional ‘mountains’. Their algorithm determines a trade-off between the number of shapematrices used in a decomposition and the tongue-and-groove error.

With regard to heuristics for the BTCMC problem, Engel [27], Kalinowski [30] and Baatar et al.[1] present greedy algorithms. Engel [27] considers the unconstrained case, Baatar et al. [1] theunconstrained and constrained cases, and Kalinowski [30] extends the algorithm of Engel [27] tothe constrained case. Each of the algorithms of Engel [27], Baatar et al. [1] and Kalinowski [30]utilise an extraction procedure to extract a shape matrix with the maximum possible beam-on timecoefficient whilst ensuring that the decomposition maintains minimum total beam-on time. The


greedy approaches are based on the idea that if the minimum beam-on time for a given probleminstance is small and the beam-on time coefficients are on average large, then the cardinality ofthe solution should be small. The unconstrained version of the Baatar et al. [1] greedy heuristicalgorithm is utilised throughout this thesis as an efficient way to calculate an upper bound onthe number of shape matrices that can be used in any decomposition of an intensity matrix. Wehereafter refer to the unconstrained greedy heuristic algorithm of Baatar et al. [1] as GHA. Finally,Baatar et al. [39] also consider the constrained BTCMC problem, using integer programming withinanother sequential extraction heuristic. In this case however, rather than a single shape matrix,the maximum number of shape matrices that can receive the maximum possible ‘allowed’ beam-ontime value are extracted at each step of the procedure. The algorithm seeks to extract a large ‘totalquantity’ from an intensity matrix at each step without using an ‘unnecessarily’ large number ofshape matrices in the final decomposition. As part of their algorithm, Baatar et al. [39] solve adifficult optimisation problem efficiently using ideas from Baatar et al. [1] and significantly reducecomputation time by introducing bounds on some of the variables in their formulation. (We alsoextend the ideas presented in Baatar et al. [1], in our case, to bounds on variables in exact models.See Chapter 5 of this thesis. We discuss any similar ideas, simultaneously determined for thisthesis, to those presented in Baatar et al. [39], in Chapter 5). Baatar et al. [39] present a ‘best of’algorithm which combines their algorithm, the ‘sequential maximising extraction heuristic’, withthe complementary algorithm of Kalinowski [30], which performs well when the Baatar et al. [39]algorithm does not and vice versa. Baatar et al. [39] demonstrate that the ‘best of’ heuristicalgorithm reduces the optimality gap of the Kalinowski [30] algorithm by more than half.

The algorithm of Siochi [20] minimises total beam-on time and is an heuristic for the constrainedTTT problem [32]. Siochi [20] utilises a geometric approach which considers an intensity matrixas a three dimensional array of rods with heights corresponding to the values in the intensitymatrix. Siochi [20] implements two techniques: extraction and rod-pushing. The rod-pushing partof the algorithm is a reformulation of the algorithm of Bortfeld et al. [19], though amended forthe constrained case. The result of rod-pushing is that a ‘slice’ through each unit height of the 3Darray of intensity rods represents a valid shape matrix. In the extraction process, each extractedmatrix is converted to shape matrices satisfying the constrained case and the shape matrix ofthose determined with the largest number of exposed cells then becomes the extracted matrix.The solution sequences considered in the Siochi [20] algorithm are tested to determine which yieldsthe least value for a measure of total treatment time incorporating leaf travel time.

Ehrgott et al. [32] compare the ‘state-of-the-art’ unconstrained and constrained MC, BTCMC andTTT heuristics numerically. They also consider the case of variable set-up time and apply a post-processor to sequence delivery of the shape matrices in a given decomposition so as to minimise thetotal set-up time. They find that algorithms that perform best on one objective (e.g. minimising thenumber of shape matrices), are not necessarily best for the variable set-up TTT problem. Ehrgottet al. [32] report that for the unconstrained case, the Engel [27] algorithm outperforms the Baataret al. [1] algorithm when solving the Minimum Cardinality problem, though the opposite is truefor the variable set-up time TTT solution. Both algorithms have similar run times. With regard tothe constrained algorithms, the Kalinowski [30] algorithm is superior in terms of the variable set-uptime TTT problem and the MC problem, when compared with the Baatar et al. [1], Siochi [20] andXia and Verhey [21] constrained algorithms. The Baatar et al. [1] and Siochi [20] algorithms yield


similar results for every problem variation considered. Whilst the Xia and Verhey [21] algorithm isthe fastest of the unconstrained and constrained algorithms tested it does not compare in terms ofthe quality of solution for any of the problem types considered. Ehrgott et al. [32] also report thatthe traveling salesman algorithm they apply to minimise variable set-up time results in significantimprovement over the set-up time values otherwise obtained for the decompositions. Finally asmentioned, Baatar et al. [39] have since compared their constrained heuristic with the algorithmof Kalinowski [30] and have determined that a ‘best of’ algorithm reduces the optimality gap ofthe Kalinowski [30] algorithm by more than half.

As discussed, the models we consider in this thesis are exact mixed integer and integer programmingformulations for the strongly NP-hard problems BTCMC and TTT. The only exact approaches weare aware of for leaf sequencing with static MLC which solve the BTCMC, MC and TTT problemsare: for the constrained BTCMC problem, the integer programming models of Langer et al. [3] andBaatar [2]; for the unconstrained BTCMC problem, the integer programming model of Baatar etal. [4], the constraint programming approach also in [4] and the specialised enumerative algorithmof Kalinowski [28, 40]; for the unconstrained MC problem, the generalisation of the Kalinowski[28, 40] algorithm by Nußbaum [31] and the ‘tailored branch and bound type search methods’ ofErnst et al. [41] and Mak [42]; and for the unconstrained TTT problem, the very recent, as yetunpublished, integer programming/combinatorial search approach of Taskin et al. [43]. We nowdescribe these works in more detail.

The integer program of Langer et al. [3] solves the constrained BTCMC problem and utilisesbinary beam-on time variables such that each shape matrix in a solution is irradiated with unitbeam-on time. All variables in the model are indexed on individual ‘monitor units of radiation’.In particular, Langer et al. [3] use left and right multileaf collimator leaf binary variables to modelvalid shape matrices. These variables equal one if the left (resp. right) leaf position covers aparticular column in a particular row during the delivery of a particular monitor unit of radiation,and are zero otherwise. The integer program presented first solves the MBT problem (with afixed upper bound on the total beam-on time that can be used in a solution), then the minimumbeam-on time returned is applied as a constraint in the model and the objective is altered tominimise cardinality. The new objective minimises cardinality using further binary variables toindicate the number of times adjacent shape matrices in a solution are different. The Langer et al.[3] model easily incorporates interleaf and tongue-and-groove constraints by virtue of its flexiblestructure, however the integer program uses a large number of variables generally and does notconsider symmetry breaking constraints. Furthermore, no solution computation times for probleminstances are reported. The Langer et al. [3] model is shown to be computationally inefficient,in Chapter 2 (where it is converted to solve the TTT problem) (see also Wake et al. [8]) and inBaatar et al. [4], when compared with the models presented in the respective works. The details ofthe converted model, which we name the Modified Langer et al. [3] model, are given in AppendixA.

The ‘leaf implicit’ integer program of Baatar [2] also solves the constrained BTCMC problem. (Theunconstrained version of the Baatar [2] model is presented in Baatar et al. [4]). Like Langer etal. [3], Baatar [2] utilises left and right leaf variables though in this case the left leaf variable hasa single one in the position of the left most exposed cell in a particular row of a particular shape


matrix and the right leaf variable has a single one in the position ‘one to the right’ of the right mostexposed cell in a particular row of a particular shape matrix. Furthermore, rather than monitorunits of radiation as in Langer et al. [3], the ‘leaf implicit’ model indexes its variables on shapematrices k for k = 1, . . . ,K, where K is an upper bound on the number of shape matrices thatcan be used in a solution. Baatar [2] also defines integer variables to represent the beam-on timefor shape matrix k if the left leaf in a particular row covers a particular set of columns and similarinteger variables for the right leaf. Results from Baatar [2]/Baatar et al. [1] on characterisations ofintensity matrix decompositions are then used to express the intensity cover constraint in terms ofthese variables, a constraint is applied to restrict the total beam-on time to Beammin and finally,whether a shape matrix is used in a solution is indicated with further binary variables, the sum overwhich (for all k = 1, . . . ,K) is minimised in the objective. Again, the ‘leaf implicit’ integer programof Baatar [2] can easily incorporate constraints for the constrained case using their left and rightbinary leaf variables. Furthermore, Baatar et al. [4] apply symmetry breaking constraints to theunconstrained version of the ‘leaf implicit’ model. The symmetry breaking constraints enforce onlyone closed leaf position and a non-increasing ordering on the beam-on time values in the resultingdecomposition. (We consider similar symmetry breaking constraints in Chapters 2 to 5 of thisthesis). Even with the application of a lower bound on the number of shape matrices to be usedin a solution, developed by Baatar [2]/Baatar et al. [1] (see Chapter 2 Section 2.3.1 of this thesisfor a definition), and the application of other techniques for reducing computation time, Baatar [2]reports that the ‘leaf implicit’ model is computationally inefficient, only able to solve small problemsizes. Baatar et al. [4] confirm this result reporting that the unconstrained model does not performwell. The results of Baatar et al. [4] demonstrate that the ‘leaf implicit’ model performs similarlyto the Langer et al. [3] model and that the application of symmetry breaking constraints to eachmodel improves overall computation time. Finally, Baatar et al. [4] present a second model, theCounter model, to solve the unconstrained BTCMC problem, which they formulate as an integerprogram and using a constraint programming method. The integer programming version of theCounter model is discussed in detail in Chapter 3, Section 3.8 of this thesis. We simply mentionhere that the model utilises new integer variables which enumerate shape matrices based on thelevel of beam-on time they receive and that therefore the model indexes its variables on radiationlevel and requires an upper bound on the most beam-on time that can be applied to any shapematrix. We omit the detail for the constraint programming method applied to the Counter model,as constraint programming is beyond the scope of this thesis. We do note that the constraintprogramming search strategy is based on solving independent row problems after fixing values forthe number of shape matrices given particular levels of radiation. A row-wise decomposition ispossible since interleaf constraints are not considered in the problem formulation and hence thereare no constraints linking allowable leaf positions in adjacent rows. Baatar et al. [4] report thatthe Counter model solved with the constraint programming method performs substantially betterthan the integer programming formulation of the Counter model (since the constraint programmingapproach solves individual row problems) which in turn significantly outperforms the unconstrainedversion of the Langer et al. [3] model with symmetry constraints developed by Baatar et al. [4]and the ‘leaf implicit’ model also in [4].

Kalinowski [28, 40] develops an intricate specialised enumerative algorithm for the unconstrainedBTCMC problem and demonstrates that the problem is solvable in pseudo-polynomial time(O(mn2L+2)) if the values within the given intensity matrix are bounded by a constant, L. The


idea of checking the feasibility of partitions of the minimal beam-on time against individual rowsof the intensity matrix (since the unconstrained case is considered) is also utilised by Kalinowski[28, 40] and thus the approach appears to be similar to the constraint programming methodimplemented in [4]. Inspired by ideas from multicriteria optimisation, Nußbaum [31] considersthe total beam-on time of a decomposition to be a parameter and applies the Kalinowski [28, 40]algorithm iteratively to solve the unconstrained MC problem. At each step in the Nußbaum[31] algorithm the parameter for the total beam-on time is increased by one unit (beginning atBeammin) and the Kalinowski [28, 40] algorithm applied. The Kalinowski [28, 40] algorithmensures the (unconstrained) minimum cardinality of a solution for a given total beam-on time,hence the Nußbaum [31] algorithm returns the solution to the unconstrained MC problem. TheNußbaum [31] algorithm requires an upper bound on the total beam-on time parameter. In theworst case, the bound is set to mnL. Correspondingly, the time complexity of the Nußbaum[31] algorithm for solving the unconstrained MC problem is O((mn)2L+3). The efficiency of theKalinowski [28, 40] and Nußbaum [31] algorithms are therefore both highly dependent on the sizeof the constant L for the given intensity matrix.

Recently, there has been development in the area of ‘tailored branch and bound type searchmethods’, where, for the unconstrained MC problem in particular, specialised partitioning searchschemes have been used to determine exact solutions. The Ernst et al. [41] algorithm is suchan exact method for solving the unconstrained MC problem. Since Ernst et al. [41] consider theunconstrained MC problem, the feasibility of beam-on time sequences is evaluated for individualrows of an intensity matrix, and feasible decompositions into ‘shape rows’ and finally shape ma-trices are determined, in this instance utilising a ‘constrained tree search’. The ‘constrained treesearch’ begins with an initial feasible solution determined with the heuristic of Engel [27] and thendecreases the number of shape matrices sequentially or according to a binary search, allowing thetotal beam-on time to increase. More specifically, a ‘MU’ (monitor unit) tree search is used to finda feasible beam-on time sequence for a decomposition using less shape matrices than that returnedby the Engel [27] algorithm, where the number of levels in the MU tree corresponds to the numberof shape matrices being considered. If a feasible sequence cannot be found, the solution returnedby the Engel [27] algorithm is optimal for the unconstrained MC problem, otherwise a feasiblebeam-on time sequence is found and a ‘CP’ tree is searched to determine whether a feasible rowdecomposition exists using the current beam-on time sequence. Each CP tree considered in theErnst et al. [41] algorithm corresponds to a row of the given intensity matrix. Again, the numberof levels in each CP tree is equal to the number of shape matrices being considered and eachnode in a CP tree represents a possible left and right leaf position for a row decomposition. If abeam-on time sequence is infeasible for any row of the intensity matrix then a different beam-ontime sequence is considered, otherwise, if a sequence can be shown to be feasible for each CP treeand therefore for each row, a feasible decomposition of the intensity matrix has been found withthe current total beam-on time and number of shape matrices. If this is the case, the number ofshape matrices is decreased by one unit (or the new number of shape matrices to trial is foundusing a binary search) and the algorithm returns to the MU tree search step. If all beam-on timesequences corresponding to a particular number of shape matrices are infeasible then the solutionusing the previous number of shape matrices trialled is optimal for the unconstrained MC problem.Finally, Ernst et al. [41] also investigate various techniques for reducing the search space of theirMU and CP trees. For example, utilising beam-on time sequences with non-increasing beam-on


times. (As mentioned, we too consider this symmetry breaking constraint in this thesis). Mak [42]also develops a ‘tailored branch and bound type search method’ to exactly solve the unconstrainedMC problem. Like Baatar et al. [4], Mak [42] utilises variables with indexing based on radiationlevel. Mak [42] first formulates a master integer program for the unconstrained MC problem, thendefines a row and column aggregated integer program and demonstrates that the linear program-ming relaxation of the aggregated integer program has smaller dimension and is at least as strongas the master integer program. To determine optimal solutions to the master integer program,Mak [42] proposes an iterative integer aggregation disaggregation algorithm which, partitions thesearch space for the master program into disjoint sets creating subproblems, and solves the ag-gregated integer program for each subproblem. The ‘most promising’ subproblem is then solvedvia a disaggregated integer program and the iterative algorithm results in a solution to the masterinteger program. Ernst et al. [41] report that the ‘constrained tree search’ generally outperformsthe Nußbaum [31] algorithm and the Mak [42] iterative integer aggregation disaggregation algo-rithm. Furthermore, when altered to solve the BTCMC problem, the ‘constrained tree search’ alsooutperforms the Kalinowski [28, 40] enumerative algorithm. As an example, the Ernst et al. [41]exact algorithm for the unconstrained MC problem can solve 50 random intensity matrices of size8× 8 with intensities ranging from 0 to 20 in approximately 8 hours.

In general it appears that approaches using ‘tailored branch and bound type search methods’are very promising and yield significant computational savings over other exact algorithms in theliterature for the unconstrained case. We note however that these approaches, and the specialisedenumerative algorithms, depend on the fact that no MLC mechanical constraints are considered.The algorithms of [28, 31, 40, 41] all solve individual row problems to determine optimal solutionsto intensity matrix decompositions. (Again, row problems can be solved when considering theunconstrained problem since there are no constraints linking possible leaf positions in adjacentrows). Furthermore, it is not clear how extensions could be made to these algorithms to handlethe constrained case. In this thesis we limit our focus to the study of exact mixed integer andinteger programming models for the TTT and BTCMC problems, and within that framework weexamine the possibilities for computational improvement. As mentioned, whilst the models ofthis thesis do not incorporate MLC mechanical constraints they can be applied due to the flexiblestructure of the integer programs we investigate; the solving techniques we use do not preclude theirinclusion. Since the majority of the most common multileaf collimator systems currently in use, forexample the Elekta and Siemens systems, prohibit interleaf motion it is clear that models for theleaf sequencing problem and solution techniques must be able to enforce these MLC mechanicalconstraints. Furthermore, most current MLC systems also have a tongue-and-groove leaf design.Hence to eliminate tongue-and-groove effects, models must be able to handle tongue-and-grooveconstraints. See Chen et al. [38] for further details of MLC machinery constraints in current MLCsystems. Finally, the incorporation of MLC mechanical constraints into the models of this thesisis solely limited by how difficult the constraints are to formulate with the variables of the integerprogram under consideration. We note that with the incorporation of MLC mechanical constraints,the solutions to the integer programs of this thesis will in general differ from the numerical resultspresented for the unconstrained problems we solve. We now discuss one final exact model whichsolves the unconstrained TTT problem.

Very recently, in as yet unpublished work, Taskin et al. [43] present an exact algorithm for the


unconstrained TTT problem which again uses a row decomposition solution approach reliant onthe fact that MLC mechanical constraints are not considered in the model formulation. TheTaskin et al. [43] approach combines integer programming and combinatorial search techniques.The method finds an ‘allowable intensity multiset’ which minimises total treatment time and thentests whether a subset of the multiset is feasible for all rows of an intensity matrix, in whichcase it is a ‘feasible intensity multiset’. Taskin et al. [43] first define a master problem in termsof integer variables for the number of times a particular radiation level occurs in an allowableintensity multiset (which are equivalent to variables in the Counter model of Baatar et al. [4] andsimilar to those in Mak [42]) and variables for a binary representation of the integer variables.The objective for the master integer program minimises a weighted sum of total beam-on timeand total numbers of shape matrices used in a solution (with constant set-up time) to obtain anallowable intensity multiset. A combinatorial search algorithm is then used to solve a subproblemfor each row of the intensity matrix to determine if a feasible solution corresponding to the allowableintensity multiset can be found. (Specifically, decompositions of cells of intensity matrices are usedto construct decompositions of rows and finally components of row decompositions with the samebeam-on time applied are combined to form shape matrices). Taskin et al. [43] prove that thesubproblem, which finds a feasible decomposition for a row given an allowable intensity multiset, isstrongly NP-hard. If a feasible solution is determined, the unconstrained TTT problem is solved,otherwise, a constraint is applied to the master integer program to cut off the infeasible solutionand a new allowable intensity multiset is found via the solution of the modified program. Taskinet al. [43] also heuristically minimise tongue-and-groove effects using the tongue-and-groove indexderived by Que et al. [44]. Whilst the Taskin et al. [43] algorithm has an exponential timecomplexity, by virtue of the strongly NP-hard subproblem, in practice computation times for thealgorithm are relatively efficient for clinical as well as random intensity matrices. For example, theTaskin et al. [43] algorithm for the unconstrained TTT problem can solve a single random intensitymatrix of size 20×20 with intensities ranging from 0 to 10 in approximately 16 minutes on average,where 100 problems were tested. Taskin et al. [43] do not compare their computational resultswith those of Ernst et al. [41] and a direct comparison is not possible here since the same sizerandom intensity matrices have not been computed. Finally, this very recent, unpublished, workof Taskin et al. [43], which is an exact algorithm for the unconstrained TTT problem, appearsto perform very well, and warrants further investigation, though as mentioned, the algorithm isunable to solve the more general case of the constrained TTT problem. In future work we willconsider combining some of the ideas of the Taskin et al. [43] algorithm for the unconstrainedproblem with the ideas of this thesis (see the Conclusion of this thesis for detail on future work).

We now provide an overview of the exact models and techniques we investigate in this thesis. Wealso reiterate our motivation for studying exact models and for the particular problems we solve.

The aim of this thesis is to increase the body of knowledge of exact models, in particular mixedinteger and integer programming formulations, for solving the Total Treatment Time and Beam-on Time Constrained Minimum Cardinality problems. This study is significant since it has beenshown that there are few exact models in the literature for solving the TTT and BTCMC problems.Approaches have centered primarily on heuristics for the NP-hard problems and current exactmodels either solve only small problem instances to optimality or are unable to incorporate MLCmechanical constraints. Clearly further investigation is required into exact approaches and there


are many possibilities for exact models using integer programming formulations, since the TTTand BTCMC problems are naturally expressed using such formulations. Exact models whichcan incorporate MLC mechanical constraints are also necessary to validate corresponding heuristicmethods. Our objective is therefore to develop new mixed integer and integer programming modelsfor the TTT and BTCMC problems, using solution methods which do not preclude the inclusionof MLC mechanical constraints, and to develop techniques for reducing the computation time ofthe mixed integer and integer programming models for the determination of exact solutions.

This thesis is set out as follows. In Chapter 2 we present a new mixed integer formulation forthe Total Treatment Time problem with constant set-up time. The formulation is based on amodel for solving cutting stock problems [6] and does not require beam-on times to be integer.We investigate whether non-integral beam-on time reduces overall computation time for the TTTmodel, since the beam-on time variables will no longer require branching in the branch and boundprocess. We consider direct solution of the mixed integer program for TTT and then exploit thebi-criteria structure of the objective to derive an algorithm that ‘steps up’ through the number ofshape matrices used, leading to substantial computational savings. As we have discussed earlier inthe Introduction both the total beam-on time and total set-up time contribute to overall treatmenttime for intensity modulated radiation therapy: therefore it is important to investigate the TotalTreatment Time problem. By varying the value of the constant set-up time T we determine thetrade-off between total beam-on time and the number of shape matrices used. Chapter 2 (andparts of the Introduction to this thesis) have been published in the Journal of Computers andOperations Research, see [8]. (Compared with the paper, minor amendments have been made toSections 2.1 and 2.5 of Chapter 2 for consistency with this thesis, otherwise there are no otherchanges. We note that since Chapter 2 is published it does not consider the very recent as yetunpublished work of Taskin et al. [43]. At the time of acceptance for publication no other workdirectly considered modelling the exact solution of the TTT problem with constant set-up time). InChapter 3 we turn our attention to the Beam-on Time Constrained Minimum Cardinality problem.We present a comparison of four exact integer programming models for the BTCMC problem, threeof which are new, and include the integer version of the model introduced in Chapter 2 (whichwe call the JS model) and an improved formulation of the Langer et al. [3] model. The threenew models are tested against the Counter model of Baatar et al. [4]. In general, the modelswe consider either index their variables on radiation level, similarly to the Counter model, andhence are pseudo-polynomial in size, or index their variables on the number of shape matrices tobe used or individual ‘monitor units’ of radiation, and hence are polynomial in size. We considerdifferent variables for modelling valid shape matrices, numerous symmetry breaking constraintsand we also formulate a variation of the Counter model which utilises cumulative variables todescribe the number of shape matrices given a particular radiation level or more. Finally weapply simple bounds to the Counter model of Baatar et al. [4] and to our cumulative version ofthe Counter model. We test the appropriate form of the ‘step-up’ idea developed in Chapter 2on the different formulations and provide numerical results for all the models and variations weconsider. As discussed earlier in the Introduction, the study of the Beam-on Time ConstrainedMinimum Cardinality problem is important: in fact it is generally viewed by clinicians as thepreferable solution for the delivery of intensity modulated radiation therapy [4]. In Chapters 4and 5 we consider techniques for further improving the computation time of the better performingpolynomial and pseudo-polynomial sized models of Chapter 3. In Chapter 4 we apply polyhedral


analysis to a significant polyhedral structure within the JS model. Facets of the polytope aredetermined and applied to a formulation of the polyhedron and to the JS model itself. We theninclude a further constraint within the polyhedron which models the strict consecutive-1-propertyand determine facets of the new polytope under special conditions. The facets are applied to aformulation of the new polytope and to the JS model. Numerical results for all model variationsare provided. As far as we are aware the particular polytopes we analyse are yet to be investigatedin the literature, though polytopes describing the strict consecutive-1-property (where the strictconsecutive-1-property may have a more general definition than the definition we use in this thesis)have been widely studied, see for example [45, 46], and Chapter 4 for examples of other applicationsfor the strict consecutive-1-property. Finally in Chapter 5 we develop novel bounds, for integervariables related to beam-on time, for the JS and Cumulative Counter models of Chapter 3.We again consider the Beam-on Time Constrained Minimum Cardinality problem for numericalexperiments however the majority of results derived in Chapter 5 apply to any upper bound ontotal beam-on time and therefore any objective for the leaf sequencing problem as described earlierin the Introduction. Given an allowable intensity multiset, in this case, a set of bounds for thebeam-on time related variables, the problem of determining a feasible decomposition of even a rowof an intensity matrix is strongly NP-hard, [43]. However, improving bounds on variables reducesthe search space and in Chapter 3 we demonstrate that even the application of simple boundsreduces computation time. In Chapter 5, to determine improved bounds, we consider a structurecommon to both the JS and Cumulative Counter models and demonstrate that properties of thestructure can be used to tighten bounds. We also extend ideas on properties of intensity matricesand decompositions developed by Baatar et al. [1], utilise the integrality of variables and deriveinteger and linear programs to improve bounds. We again apply the appropriate form of the ‘step-up’ idea of Chapter 2 where relevant. Finally, we show that bounds on the variables of the JSmodel can be easily converted to bounds on the variables of the Cumulative Counter model andvice versa and we systematically test all new bounds and techniques to determine computationalimprovement for each model. Significant improvement in computation time for both models isachieved with the application of bounds.

Therefore, in summary, this thesis presents a thorough analysis of integer programming formu-lations for the TTT and BTCMC problems: new models and algorithms are developed and newtechniques for reducing the computation time of these formulations are investigated and applied.All the bounds results, models and techniques we present in this thesis are applicable to the un-constrained and constrained cases (with minor amendments) of the treatment time problems weconsider. Within the framework of integer programming formulations, this thesis increases thebody of knowledge of exact models for the leaf sequencing problem for the application of IntensityModulated Radiation Therapy.

17CHAPTER 2

Mixed Integer Programming Approaches to Exact

Minimisation of Total Treatment Time in Cancer

Radiotherapy Using Multileaf Collimators

2.1 Introduction

In this chapter we are interested in solving the problem of delivering a given intensity matrix,I, using a multileaf collimator in step-and-shoot mode, so as to minimise the number of shapematrices, weighted by constant set-up time T , plus the sum of the beam-on times associated witheach shape matrix. As discussed in the Introduction to this thesis (Chapter 1) we call this the TotalTreatment Time (TTT) problem. We re-define the TTT problem here. A solution to the TTTproblem is characterised mathematically by a set of K beam-on time values, b = {b1, . . . , bK}, anda corresponding set of K shape matrices, X = {X1, . . . , XK}, where Xk ∈ X for all k = 1, . . . ,Kand X denotes the set of all valid, non-zero, shape matrices. These define a delivery plan, (b,X).The TTT problem is thus formulated mathematically as

TTT ∗ = minK,b,X

{K∑k=1

bk + TK : (b,X) ∈ FK

},

where

FK =

{(b,X) ∈ RK+ ×XK :

K∑k=1

bkXk = I

}is the set of all possible delivery plans using K shape matrices. The number of shape matrices,K, in the Total Treatment Time definition is unknown and must be determined as part of solvingthe TTT problem. We discuss the implications of unknown parameter K for our TTT modelformulation and the motivation for using real beam-on time variables in our definition of the TTTproblem later in this chapter. As mentioned in the Introduction (Chapter 1) we consider theunconstrained TTT problem in this chapter, though all the models and algorithms we present herecan be amended to incorporate MLC mechanical constraints.

The TTT problem is strongly NP-hard and at the time of publication of Wake et al. [8], no workin the literature considered modelling the exact solution of the TTT problem with constant set-uptime directly. (See the Introduction to this thesis, Chapter 1, for a discussion of the TTT heuristicof Siochi [20] and the post-processor applied by Ehrgott et al. [32] for minimising the variableset-up time of current state-of-the-art heuristics). We note that the method given in Chapter 5 ofthe thesis by Nußbaum [31] for minimising K can also be used to solve the TTT problem, althoughthis is not explicitly done in the thesis. We discuss this approach further below.

No methods we are aware of explicitly optimise the trade-off between K and total beam-on time:Fu et al. [25] demonstrate that there is such a trade-off, but no work we are aware of makes use ofa value for T to find the best trade-off. To date, experiments with clinical data (see, for example,the work of Langer et al. [47]) suggest that, in practice, sacrificing beam-on time can decreaseK by at most one. Our own experiments and those of others, (see, for example, [32] and [31]),suggest that this is also true for randomly generated problems. However Nußbaum [31] (see Table

18Chapter 2. Mixed Integer Programming Approaches to Exact Minimisation of Total Treatment Time in

Cancer Radiotherapy Using Multileaf Collimators

5.4 in [31]) reports one instance in which two shape matrices can be saved. He also finds that insome cases three or four beam-on time units are required to save one shape matrix. Furthermore,optimal solutions are not yet known for larger problems, so we argue that it is not yet clear howmuch can be gained by making the trade-off. In any case, we believe the TTT model to be ofvalue: it is a flexible model that naturally captures the key features of the problem; it is clear thatprogress in solving it, whether directly or indirectly, will be useful.

In this work, we consider exact solution methods for the TTT problem. The only exact approacheswe are aware of for IMRT with static MLC are those that solve the problem of minimising totalbeam-on time alone, for example, [1, 17, 27, 33], and those that address the problem of minimisingK alone, in some cases subject to the requirement that the total beam-on time is lexicographicallyminimised first. The former are in no way adaptable to the TTT problem; minimising beam-ontime alone is polynomially solvable [17, 27], and approaches to the problem do not capture K.The latter category consists of the integer programming models of Langer et al. [3] and Baatar etal. [4], the constraint programming approach also in [4], the specialised enumerative algorithm ofKalinowski [28] and the generalisation of this algorithm by Nußbaum [31]. All of these approachesassume that the beam-on times must be integer, and, under that assumption, only those of Langeret al. [3], Baatar et al. [4], and Nußbaum [31] appear to be readily adaptable to address theTTT problem. As a point of comparison for our work, we have chosen to adapt the Langer et al.[3] integer programming model for the TTT problem, for reasons discussed further below. Sinceinteger programming models are the focus of our work, we did not compare against specialisedenumerative approaches such as those of Kalinowski [28] and Nußbaum [31], although results inthe latter suggest these may be highly effective computationally.

Our approach was motivated by noticing a similarity between the TTT problem, and problemsthat arise in cutting stock applications, where Johnston and Sadinlija [6] had success with theirMIP model. We noticed that the model was readily adaptable to solve the TTT problem, andfurthermore did not require integer beam-on times. The model explicitly indexes on the shapematrices, for example, using variables bk for k = 1, . . . , K, thus directly controlling the number ofshape matrices that can be used with the parameter K. We noticed that if we solved this modelfor values of K which were infeasible, i.e. if we chose K so that FK = ∅, then the model solvedvery quickly, much more so than if K were set to some known upper bound on the number of shapematrices in an optimal solution for the TTT problem, as would be required for a valid model of theTTT problem. This lead us to formulate our “step-up” algorithm, in which we solve a feasibilityproblem for successively increasing values of K. We note that Langer et al. [3] used a similarapproach in order to find the minimum value of K. Our approach is somewhat more sophisticated:we go beyond this value to prove TTT optimality, and we show how information gained on the waycan be used to further constrain the feasibility problems solved. We also note that Nußbaum [31]uses a step-up idea, but instead of stepping up through the number of shape matrices, he steps upthrough the (integer) beam-on time, applying Kalinowski’s algorithm [28] at each step.

In our “step-up” algorithm, the feasibility problem is modelled as a variation on the Johnstonand Sadinlija MIP [6]. The only other model in the literature that would have been a candidatefor this approach would have been the “leaf implicit” model in Baatar et al. [4], since this toohad parameter K explicit, but this model is shown in [4] to have very similar computational

2.2. A New MIP Formulation for the Total Treatment Time Problem 19

performance to that of Langer et al. [3]; here our results show that our MIP model based on thatin [6] performs far better in this context than that of [3].

In what follows, we first give our MIP formulation of the TTT problem, and give the results ofnumerical experiments on test problems. We then, in Section 2.3, describe our “step-up” algo-rithm, using the efficient frontier for the trade-off between total beam-on time and K to motivateour approach. We also give numerical results comparing the performance of the algorithm withour MIP formulation. Finally, in Section 2.4, we consider the case in which beam-on time valuesare constrained to be integer. We find that our MIP model generally solves faster with the integerrestriction (but not always) and that in all cases it gives the optimal solution for the (real) TTTproblem. For T = 0 it is known that there is no integrality gap between real and integer formu-lations, (in this case the formula of Engel [27] applies), but for large T , a question remains. Asalready mentioned, in all our experiments, the solution to the MIP with continuous beam-on timevariables gave precisely the same value as the all-integer version, in every instance. Whether ornot this is true in general remains an open question. We also note that when solving the TTTproblem with real beam-on time variables the large majority of solutions also yield integer valuesfor the individual beam-on time variables bk.

In the case that beam-on times are restricted to be integer, we are able to compare our MIPformulation with that of Langer et al. [3] adapted for the TTT problem, which demonstratescomputational improvements of around three orders of magnitude. We also give results showingthe “step-up” algorithm with the integer beam-on time restriction; this appears to yield the bestperformance overall.

Throughout this chapter, our numerical results are based on problem instances with n×n intensitymatrices, where entries in the matrices are integer values randomly generated between 0 and 15inclusive. We generated 20 instances with n = 4 and 20 with n = 5; most of our results are basedon these. In some cases (for example when comparing with the formulation of Langer et al. [3],which was very slow to solve), we use some instances with n = 3. In most cases, we have solved allinstances with T = 0.5, T = 1, T = 2 and T = 10; in some cases, for brevity of the presentation,and where all values lead to similar conclusions, we show only the results for T = 2. (The unitsfor set-up time, T , are seconds).

2.2 A New MIP Formulation for the Total Treatment Time Problem

As discussed above, our MIP formulation is a modification of a formulation recently proposedfor solving cutting stock problems. The compact mixed integer programming model developed byJohnston and Sadinlija [6] demonstrates a method for resolving the inherent nonlinearity associatedwith pattern variables and pattern usage levels, in the context of the one-dimensional cutting stockproblem. In cancer radiation therapy, shape matrices and beam-on times interact in a similar wayto the patterns and run lengths of the cutting stock problem, respectively. Thus, the techniquesdeveloped by Johnston and Sadinlija [6], which also do not require integrality of the run lengthvariables, would appear to be applicable in the cancer treatment case. (We note that the modelsof Langer et al. [3] and the “counter model” of Baatar et al. [4] not only assume beam-on times



are integer, but index their variables by these integer values.)

Adopting a similar structure to that proposed by Johnston and Sadinlija [6], we first choose aparameter K to be the upper bound on the number of shape matrices that can be used in thesolution. For the resulting model to be a valid model of the TTT problem, parameter K must bechosen large enough that FK contains an optimal solution to the problem instance. Since parameterK dictates the total number of variables required in our model, and since solution computationtimes increase as K increases, we would like the value of K to be as small as possible. However itis difficult to know a “reasonable” value for K in advance; methods to overcome this difficulty willbe discussed later.

In terms of parameter K, the variables of our TTT formulation are as follows. We define variables

xijk =

{1, if cell (i, j) is exposed in shape matrix k0, otherwise

, (2.2.1)

for all i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K, to denote the matrix elements of Xk and thereforedescribe shape matrix k at a cell level. We also define (real) beam-on time variables

bk ≥ 0, for all k, (2.2.2)

to record the length of time the medical linear accelerator is on when the MLC is set in the positioncorresponding to shape matrix k, i.e. to record the beam-on time applied to shape matrix k, andvariables

sk =

{1, if shape matrix k is non-zero0, otherwise

, for all k, (2.2.3)

to indicate whether shape matrix k is used in the solution (is non-zero) or not, for each k = 1, . . . ,K.Finally, we define variables

aijk =

{bk, if cell (i, j) is exposed in shape matrix k0, otherwise

, for all i, j, k, (2.2.4)

to represent the radiation intensity delivered from each cell of shape matrix k, i.e. to model bkxijkand so remove the nonlinearity of the constraint requiring the delivery plan deliver the givenintensity matrix I.

The constraints of our TTT formulation are:

sk ≥ xijk, for all i, j, k, (2.2.5)

0 ≤ bk ≤ Hksk, for all k, (2.2.6)

0 ≤ aijk ≤Mijkxijk, for all i, j, k, (2.2.7)

0 ≤ aijk ≤ bk, for all i, j, k, (2.2.8)

aijk +Gijk(1− xijk) ≥ bk, for all i, j, k, (2.2.9)

K∑k=1

aijk = Iij , for all i, j, and (2.2.10)

2.2. A New MIP Formulation for the Total Treatment Time Problem 21

xij1k − xij2k + xij3k ≤ 1, (2.2.11)

for all i, k, j1 = 1, ..., n− 2, j3 = j1 + 2, ..., n, j2 = j1 + 1, ..., j3− 1, where Hk, Mijk and Gijk denotesufficiently large constants such that the relevant constraints are valid. We discuss our choices foreach of these constants later in this section.

Constraint (2.2.5) is required to partially enforce the definition of our sk variables. It ensures that,for a particular shape matrix k, sk will equal one if any xijk equals one and otherwise that sk isunrestricted.

Constraint (2.2.6) restricts the beam-on time corresponding to shape matrix k to be zero if shapematrix k is identically zero in a solution and to be essentially unrestricted if shape matrix k isnon-zero in a solution.

The first constraint on our aijk variables, (2.2.7), restricts aijk to be zero if cell (i, j) is not exposedin shape matrix k, and to be bounded by Mijk ≥ Iij if cell (i, j) is exposed. Constraint (2.2.8)further limits the upper bound on aijk to be the value of the corresponding beam-on time bk.Constraint (2.2.9) applies a lower bound of bk to aijk, when cell (i, j) is exposed, and when cell(i, j) is not exposed, the constraint needs to be inactive. Constraints (2.2.7), (2.2.8) and (2.2.9)enforce a value of bk for aijk when cell (i, j) is exposed so that these three constraints, consideredsimultaneously, ensure that equation (2.2.4) holds.

The intensity requirement (2.2.10) is now expressed linearly in terms of our variables aijk, whereparameters Iij describe intensity matrix I at a cell level.

We introduce a further restriction (2.2.11) to ensure that our xijk variables induce valid shapematrices. Specifically, we apply a constraint to enforce the strict consecutive-1-property. Constraint(2.2.11) restricts the xijk variable values in a row of a shape matrix so that a zero valued xijk

variable never occurs between xijk variables taking the value 1. In other words, patterns of theform 1, . . . , 0, . . . , 0, . . . , 1 cannot occur anywhere within a row of a shape matrix. Correspondinglyallowed patterns of xijk variables in a row of a shape matrix are those where if ones occur, theyoccur consecutively in a single block [1]. There are various ways to model valid shape matricesdepending on the variables chosen to depict the leaf positionings of the MLC. The study of suchmodels is of interest and is a focal point of future research for the author. This work however,concentrates on other aspects of the TTT model structure.

The objective function for the Total Treatment Time problem can now be expressed in terms ofthe variables of our specific model. We have:

minK,b,s

(K∑k=1

bk + T

K∑k=1

sk

)(2.2.12)

where T > 0 is a constant. Furthermore, in keeping with our definition for variable sk, minimisingthe sk variables in our objective function ensures that sk will be zero if all xijk variables in shapematrix k are zero.

The above is the basic form of the MIP model. We make a few very simple improvements. Toaddress the symmetry caused by the indexing on k, we ask that non-zero shape matrices appear



first in the index order, and that the shape matrices be ordered by decreasing beam-on time, i.e.we ask that

sk ≥ sk+1, for all k = 1, ...,K − 1 (2.2.13)

andbk ≥ bk+1, for all k = 1, ...,K − 1. (2.2.14)

We also add one redundant constraint. Recall that minimising beam-on time alone is polynomiallysolvable, indeed, in the case without interleaf constraints, there is a closed-form expression for it[27]. We use Beammin to denote the minimum beam-on time, i.e.

Beammin = minK,b,X

K∑k=1

bk : (b,X) ∈ FK

.

We found that adding the constraint

K∑k=1

bk ≥ Beammin, (2.2.15)

to the MIP formulation significantly improved computation time. Averaging over set-up time values0.5, 1, 2 and 10, approximately 30% more problem instances of size 5×5 solve to optimality withina 10 hour time limit. Furthermore, again averaging over set-up time values 0.5, 1, 2 and 10, andonly considering problem instances which solve under 10 hours both with and without constraint(2.2.15), the total amount of time saved using the TTT model with additional constraint (2.2.15)is approximately 11 hours, over the problem instances of size 4× 4 and 5× 5.

Finally, the choice of constants Hk, Mijk and Gijk influences the MIP efficacy, and is an additionalfocus of future study for the author. For the purposes of the current research, the values for Hk,Mijk and Gijk implemented in our model are I, Iij and I, respectively, where I = max

i,jIij .

In summary, the TTT model is as follows:

(2.2.12)

s.t. (2.2.5), (2.2.6), (2.2.7), (2.2.8), (2.2.9), (2.2.10), (2.2.11), (2.2.13),

(2.2.14), (2.2.15),

x ∈ {0, 1}m×n×K and s ∈ {0, 1}K .

As noted in the previous section, the Total Treatment Time problem is known to be difficult tosolve. Table 2.2.1 shows the computation times for our TTT model, for different values of set-up time T . In all cases, we take K to be the number of shape matrices found by the greedyheuristic algorithm of Baatar et al. [1], which finds a delivery plan achieving minimal beam-ontime, Beammin, while using a very small number of shape matrices. We call this algorithm GHAand use the notation Kbm to denote the number of shape matrices in the solution it finds; theseare shown in the fourth column of the table. It is not hard to see why Kbm is a valid choice forK: there must be an optimal solution to the TTT problem having total beam-on time β and usingnumber of shape matrices K with β + TK ≤ Beammin + TKbm, since the GHA delivery plan is a

2.3. The Step-up Algorithm for the Total Treatment Time Problem 23

feasible solution for the TTT problem. But for any delivery plan, it must be that β ≥ Beammin.It follows that K ≤ Kbm as required.

In the first column of Table 2.2.1, 4- implies a problem instance of size 4 × 4 and 5- a probleminstance of size 5× 5. The minimum beam-on time value for each problem instance, calculated byGHA, is shown in the last column of the table. For interest, we have also given for each problemthe value of Kmin, the smallest number of shape matrices for which a solution exists, and Kbm,the smallest number of shape matrices for which a solution having beam-on time Beammin exists.(These were found using a variation of the step-up method, which we discuss in Section 2.3.1.Blank entries in columns 2 and 3 indicate these values were not found within a 10-hour CPU timelimit.) Instances in which Kmin and Kbm differ are those that offer a trade-off between set-up andbeam-on times; results for these instances have been highlighted in bold in Table 2.2.1.

It is evident from Table 2.2.1 that the Kbm values determined by GHA differ from the Kbm valuesby at most 2 cardinality units for all problem instances, and are equal in 61% of the probleminstances trialled where Kbm was computed. Clearly, the Kbm values calculated by GHA areextremely good values to use for parameter K in our TTT model.

For each value of T tested, the columns labelled K∗T and∑

bk show the number of shape matricesand total beam-on time respectively in the optimal solution found by solving the TTT MIP model,for each of the problem instances. The column labelled “Time” reports the CPU time in seconds,to two decimal places. A time limit of 10 hours was set; blank rows indicate the time required wasmore than 10 hours.

Table 2.2.1 shows that computation times are typically longer for larger problems. It also showsthat for problems of these sizes, there are relatively limited opportunities for trade-off betweenbeam-on time and set-up time. Nevertheless, any general algorithm for solving the TTT problemmust be able to detect and solve cases where no trade-off is possible, thus we continue to useall problem instances in our later tests. Clearly the run times for many of these problems areunacceptably long, thus we seek new approaches that can reduce run times.

2.3 The Step-up Algorithm for the Total Treatment Time Problem

The algorithm we describe in this section is motivated in part from our observation that the MIPmodel in the previous section solved very quickly when the parameter K was set so small thatFK was empty, i.e. the problem was infeasible, and in part by a view of the TTT problem as abicriteria problem.

Clearly, the TTT problem combines two criteria, the total beam-on time, and the number of shapematrices, K, in a single objective, with the latter term weighted by T . Similar visualisation ideasto those standard in multicriteria optimisation can be helpful in our context: we thus consider the“feasible frontier” of possible solutions to the TTT problem, shown in Figure 2.3.1.



Table 2.2.1: Numerical results for the TTT model using CPLEX version 8.1 and AMPL version8.1 on a 2GHz AMD 64 3000+: time in seconds, 10-hour time limit.

T = 0.5 T = 1 T = 2 T = 10

Kmin Kbm Kbm K∗0.5∑bk Time K∗1

∑bk Time K∗2

∑bk Time K∗10

∑bk Time Beammin

4a 5 5 5 5 17 2.09 5 17 4.15 5 17 0.68 5 17 1.38 17

4b 6 6 6 6 26 2.21 6 26 2.01 6 26 1.25 6 26 2.31 26

4c 5 5 5 5 17 0.30 5 17 0.19 5 17 0.27 5 17 0.42 17

4d 5 5 5 5 21 0.15 5 21 0.16 5 21 0.18 5 21 0.36 21

4e 5 6 6 6 16 4.87 6 16 5.18 5 17 15.15 5 17 1.15 16

4f 5 5 7 5 23 1.01 5 23 3.54 5 23 1.82 5 23 4.42 23

4g 5 5 5 5 26 0.05 5 26 0.07 5 26 0.04 5 26 0.03 26

4h 5 6 6 6 28 50.35 6 28 102.48 6 28 12.99 5 30 96.38 28

4i 5 5 6 5 24 2.76 5 24 6.94 5 24 8.61 5 24 3.16 24

4j 5 5 5 5 14 0.57 5 14 0.81 5 14 3.71 5 14 0.96 14

4k 5 5 6 5 25 8.02 5 25 5.04 5 25 10.51 5 25 0.20 25

4l 6 6 6 6 23 0.65 6 23 0.76 6 23 0.77 6 23 0.77 23

4m 5 5 6 5 17 13.74 5 17 38.28 5 17 2.32 5 17 41.81 17

4n 5 5 5 5 19 0.21 5 19 0.22 5 19 0.21 5 19 0.24 19

4o 5 5 5 5 18 0.23 5 18 0.18 5 18 0.53 5 18 0.28 18

4p 5 5 6 5 15 1.49 5 15 1.71 5 15 0.41 5 15 2.06 15

4q 5 5 6 5 23 1.30 5 23 7.08 5 23 3.80 5 23 7.38 23

4r 5 5 6 5 17 0.27 5 17 0.42 5 17 0.57 5 17 0.25 17

4s 5 5 5 5 21 1.73 5 21 8.85 5 21 3.38 5 21 3.02 21

4t 5 5 6 5 21 3.32 5 21 1.46 5 21 0.39 5 21 0.31 21

5a 6 7 6 21 6417.51 21

5b 6 6 7 6 26 7812.17 6 26 7898.52 6 26 1062.03 26

5c 6 6 6 6 22 66.57 6 22 64.20 6 22 48.29 6 22 88.48 22

5d 6 6 6 6 22 21.88 6 22 14.71 6 22 99.50 6 22 113.85 22

5e 6 6 7 6 19 10.88 6 19 84.37 6 19 57.73 6 19 653.79 19

5f 6 6 6 6 23 5.42 6 23 11.42 6 23 10.82 6 23 5.10 23

5g 6 6 6 6 27 489.24 6 27 477.79 6 27 120.86 6 27 46.78 27

5h 6 7 6 24 5980.10 6 24 2334.17 6 24 1676.75 24

5i 6 6 6 6 26 597.27 6 26 56.45 6 26 207.67 6 26 48.44 26

5j 6 6 6 6 18 43.67 6 18 32.70 6 18 88.69 6 18 40.35 18

5k 6 6 8 6 25 6112.00 6 25 4741.00 25

5l 6 6 7 6 22 225.08 6 22 913.92 6 22 128.31 6 22 296.20 22

5m 7 7 24

5n 6 7 7 7 22 2216.38 6 24 13746.20 22

5o 7 7 19

5p 7 7 8 7 28 1084.43 7 28 40.71 7 28 175.04 7 28 314.54 28

5q 6 8 6 25 144.59 6 25 682.93 6 25 153.29 6 25 16337.10 25

5r 6 6 7 6 23 32.43 6 23 110.78 6 23 112.74 6 23 143.81 23

5s 7 7 7 30 16662.50 7 30 7531.75 7 30 1362.20 30

5t 6 7 7 7 27 34524.60 7 27 24362.20 6 28 391.73 27


Figure 2.3.1: A Typical Set of Feasible Points and the Feasible Frontier (?)

Beamupper

Beammin

KbmKmin K

K∑k=1

bk

B

A

The graph in Figure 2.3.1 illustrates the trade-off between minimising the number of shape matricesused and minimising the total beam-on time of a solution to the Total Treatment Time problem.In particular, point B on the graph represents the solution to the problem of lexicographically

minimising K and then total beam-on time, β =K∑k=1

bk, which we write as lexmin(K,β). We

use the notation (Kmin, Beamupper) to denote the optimal values of lexmin(K,β). Point A onthe graph corresponds to the solution to the lexmin(β,K), which has optimal value denoted by(Beammin,Kbm), where Beammin and Kbm are as defined earlier. Points A and B are also optimalsolutions to the TTT problem for different values of set-up time T . We define the “feasible frontier”to be the set of points (K,β) such that FK 6= ∅ and β is the minimum beam-on time achievablein any decomposition using K or fewer shape matrices.

The feasible frontier between A and B shown in Figure 2.3.1 is indicative only; it might look quitedifferent in different instances. We can, however, say that if any point is a solution to a givenproblem instance, then all (integer) points to the “right” of the solution are also feasible points.We can also say that since T > 0, the optimal solution to TTT must lie on the boundary of theconvex hull between A and (heading “west” and “north”) B of the finite set of feasible points, andthat therefore the optimal solution to TTT must have a number of shape matrices at least Kmin.The boundary of the convex hull for this particular feasible frontier is depicted by the dashed linein Figure 2.3.1.

The algorithm we propose steps up through possible K values. Starting with K set to any lowerbound on Kmin, we seek to minimise the total beam-on time subject to the number of shapematrices not exceeding K, which may well be infeasible. We can also ask that the TTT objectivevalue of any solution found be at least as good as that of the best known value found so far, andcan stop when we can prove that no higher value of K can yield a better value. In what follows,



we define the subproblem we solve for each value of K. We then go on to formally define thealgorithm, and to verify that it does indeed terminate with the optimal TTT solution. Finally, wegive the results of numerical experiments, comparing its performance with that of the MIP modelgiven in Section 2.2.

2.3.1 The Cardinality Constrained Minimum Beam-on Time Subproblem. As ex-plained above, at each step of the algorithm we propose, we seek to minimise the total beam-ontime subject to the number of shape matrices not exceeding a given value K. We call this theCardinality Constrained Minimum Beam-on Time (CCMBT) problem, where “cardinality” refersto the number of shape matrices in the solution. We formulate the CCMBT problem similarly tothe MIP model in Section 2.2:

CCMBT (K) =

min

K∑k=1

bk

s.t. (2.2.7), (2.2.8), (2.2.9), (2.2.10), (2.2.11), (2.2.14),x ∈ {0, 1}m×n×K and b ≥ 0.

We use the convention that if the problem is infeasible, CCMBT (K) =∞.

Apart from the objective, the main point of difference between the Cardinality Constrained Min-imum Beam-on Time model and the Total Treatment Time model is that the former does notutilise the sk variables and related constraints.

Before proceeding to our algorithm, we first note that the CCMBT model can be used to calculatepoints A and B in Figure 2.3.1, using the idea of stepping up through values of K. We use this asan initial test of the step-up idea.

To begin, we need a lower bound on Kmin. This can be calculated using a formula developed byBaatar et al. [1] as follows: set

Klb = maxi=1,...,m

Ni (2.3.1)

whereN i = max{|{j ∈ {1, . . . , n+ 1} : Iij > 0}|, |{j ∈ {1, . . . , n+ 1} : −Iij > 0}|},

for all i = 1, . . . ,m and

Iij =

Iij − Ii,j−1, 1 < j ≤ nIi1, j = 1−Iin, j = n+ 1

,

for all i = 1, . . . ,m, j = 1, . . . , n+ 1.

Now starting with K = Klb, we solve the CCMBT model to get CCMBT (K), increment K by 1,and repeat until the first time that CCMBT (K) is finite. At this point, we have found point B,and can set Kmin = K, and Beamupper = CCMBT (K).

We can find point A similarly by adding the constraint

K∑k=1

bk = Beammin (2.3.2)


to the CCMBT model; let CCMBT−(K) denote the value of the resulting problem. Again,starting with K = Kmin if this value is known, or K = Klb otherwise, we solve the CCMBT modelwith constraint (2.3.2) to get CCMBT−(K), increment K by 1, and repeat until the first timethat CCMBT−(K) is finite. At this point, we have found point A, and can set Kbm = K.

In further experiments conducted on the CCMBT model and the TTT model, we observed thatcomputation times increased when K was set greater than Kmin in the case of the CCMBT modeland when K was set greater than K∗T in the case of the TTT model. This observation furtherreinforces the importance of knowing a small upper bound on the optimal number of shape matricesfor a given problem instance, and further motivates the algorithm we now develop, which solvesthe TTT problem by systematically stepping up K.

2.3.2 The Step-up Algorithm. The Step-up algorithm extends the ideas discussed abovefor finding points A and B on Figure 2.3.1 to finding the best possible solution for the TTTproblem, on the convex hull of the feasible frontier between A and B. The method starts by settingK = Klb, as defined in the previous section, solving the CCMBT model, incrementing K, andrepeating. Since redundant constraint (2.2.15) improves solution time in the MIP formulation, (insome cases dramatically), we also add it to the CCMBT model. Furthermore, we have a feasiblesolution readily available at the outset – we run the GHA algorithm that finds a feasible solutionwith minimum beam-on time, and use Kbm to denote the cardinality of the solution – so we caninitialise the value of the best solution found so far, TTTBest = Beammin + TKbm, and add theconstraint

K∑k=1

bk ≤ TTTBest − TK, (2.3.3)

which simply says that the value of any solution found

(K∑k=1

bk + TK

)should be at least as good

as that of one we already know. We use the notation CCMBT+(K) to denote the solution of theCCMBT model with both constraints (2.2.15) and (2.3.3) added.



We formally define the Step-up algorithm as follows.

Calculate Klb using (2.3.1)Calculate Kbm and Beammin using GHASet TTTBest := Beammin + TKbm, θ := Kbm and K := Klb

while K < θ doSolve the CCMBT model, with additional constraints (2.2.15) and (2.3.3)while CCMBT+(K) is not finite and K + 1 < θ do

K := K + 1Solve the CCMBT model, with additional constraints (2.2.15) and (2.3.3)

enddoif K < θ and CCMBT+(K) is finite then

Set TTTBest := CCMBT+(K) + TK

Set θ :=TTTBest −Beammin

TendifK := K + 1

enddo

It is not hard to see that the Step-up algorithm does, in fact, return the optimal solution to theTTT problem.

In the first place, it is easy to justify incrementing K when CCMBT+(K) is not finite, since thedefinition of the corresponding model implies that there is no feasible solution with TTT value atleast as good as TTTBest using K shape matrices. By induction on K, and since K starts at thesmallest possible value for which any feasible solution could exist, whenever CCMBT+(K) is notfinite it must be impossible to find a better value feasible solution using K or fewer shape matrices.

Secondly, the solution to the CCMBT model with constraints (2.2.15) and (2.3.3) must use Kshape matrices, as we will now show. Let K ′ be the number of matrices used in the solution,and note that the beam-on time of the solution is given by CCMBT+(K). Now either this isthe very first time CCMBT+(K) is finite, in which case it obviously must be that K ′ = K, orTTTBest was the value of the best solution using fewer than K shape matrices. Thus if K ′ < K

it must be that CCMBT+(K) + TK ′ ≥ TTTBest. But by constraint (2.3.3) it must be thatCCMBT+(K) ≤ TTTBest−TK. Thus TTTBest−TK ≥ CCMBT+(K) ≥ TTTBest−TK ′, andit follows that K ′ ≥ K, which is a contradiction.

Thus the expression CCMBT+(K) + TK is indeed the TTT value of the solution found whenCCMBT+(K) is finite, and by (2.3.3) it must be at least as good as any found so far, so we arejustified in updating TTTBest with this expression.

Finally, we prove that at any stage of the algorithm, any solution with value better than TTTBest

must satisfy K < θ. First, observe that at both the initialisation, and all later stages of the

algorithm, θ =TTTBest −Beammin

T. Now suppose there is a feasible solution to the TTT problem

with beam-on time B′ using K ′ shape matrices, with TTT value better than TTTBest. Then


B′+TK ′ < TTTBest, and so Beammin +TK ′ < TTTBest, since no feasible solution has beam-on

time less than Beammin, hence K ′ <TTTBest −Beammin

T= θ as required. Hence we are justified

in stopping when K < θ is no longer satisfied.

Table 2.3.1 shows the results of solving the TTT model versus the results of the Step-up algorithm,with set-up time T = 2. The column labelled “Number of Loops” gives the number of CCMBT+

model solves in executing the Step-up algorithm. The column labelled “Speed-up Factor” givesthe time taken to solve the TTT model divided by the time taken for the Step-up algorithm. Ofthe 5 × 5 matrices trialled, 20% did not solve within the 10 hour time limit with either the TTTmodel or the Step-up algorithm when set-up time was 2. The blank spaces in Table 2.3.1 indicatewhere the TTT model or the Step-up algorithm could not be solved within a 10 hour time limitfor the given problem instance.

The results in Table 2.3.1 show that although in a small number of cases, the TTT model is faster,generally speaking, the Step-up algorithm decreases the computation time, and across the boardgives a substantial decrease.



Table 2.3.1: Time taken to solve the TTT model versus the time taken to run the Step-up algorithmusing CPLEX version 8.1 and AMPL version 8.1 on a 2GHz AMD 64 3000+: time in seconds,T = 2, 10-hour time limit. An asterisk under the Speed-up Factor column indicates where theSpeed-up Factor is undefined. This occurs when the Step-up algorithm has zero computation time.

TTT model Step-up algorithm

Total Time Total Time Number of Loops Speed-up Factor

4a 0.68 0.02 2 34.0

4b 1.25 1.18 3 1.1

4c 0.27 0.04 2 6.8

4d 0.18 0.02 2 9.0

4e 15.15 4.48 3 3.4

4f 1.82 0.21 3 8.7

4g 0.04 0.00 2 *

4h 12.99 2.48 3 5.2

4i 8.61 1.34 3 6.4

4j 3.71 0.09 2 41.2

4k 10.51 2.93 3 3.6

4l 0.77 0.55 2 1.4

4m 2.32 0.44 2 5.3

4n 0.21 0.00 2 *

4o 0.53 0.00 1 *

4p 0.41 0.49 2 0.8

4q 3.80 30.11 3 0.1

4r 0.57 0.07 2 8.1

4s 3.38 0.02 2 169.0

4t 0.39 0.11 3 3.5

5a 6417.51 4341.72 3 1.5

5b 3

5c 48.29 0.38 2 127.1

5d 99.50 0.46 2 216.3

5e 57.73 3.87 3 14.9

5f 10.82 0.15 2 72.1

5g 120.86 43.72 2 2.8

5h 412.23 4 >87.0

5i 207.67 0.07 2 2966.7

5j 88.69 7.06 2 12.6

5k 23.57 3 >1527.0

5l 128.31 2395.62 3 0.1

5m 3

5n 3

5o 4

5p 175.04 318.68 4 0.5

5q 153.29 7953.63 2 0.0

5r 112.74 569.47 3 0.2

5s 7531.75 137.34 4 54.8

5t 391.73 110.78 3 3.5

2.4. The Case of Integer Beam-On Times 31

2.4 The Case of Integer Beam-On Times

All previous approaches to IMRT delivery of the type we consider here require the beam-on timesto be integer. We are interested in whether or not this requirement makes a difference. We arealso interested in integer beam-on time versions of the TTT model and Step-up algorithm, sincewe can get a basis for comparison with previous approaches.

In what follows, we will refer to the TTT model presented earlier as the Real TTT model, and definethe Integer TTT model to be the same model with the additional requirement that the beam-ontime variables, i.e. the b and a variables, be integer. As for the Real version of our TTT model,computation times decreased when the additional constraint (2.2.15) was applied to the IntegerTTT model. With the addition of this constraint, and averaging over set-up time values 0.5, 1,2 and 10, approximately 20% more problem instances of size 5 × 5 solved within a 10 hour timelimit. Furthermore, again averaging over set-up time values 0.5, 1, 2 and 10, and only consideringproblem instances which solve under 10 hours under both variations, the average amount of timesaved over all 4 × 4 and 5 × 5 instances using the Integer TTT model with additional constraint(2.2.15) is approximately 6 hours.

The second and third columns of Table 2.4.1 display computation times for the Real TTT modelversus the Integer TTT model for problem instances of size 5× 5, with set-up time set to 2, usinga time limit of 10 hours. It is clear from Table 2.4.1 that computation times for the Integer TTTmodel are generally much faster than for the real version. In fact, averaged over set-up time equalto 0.5, 1, 2 and 10, 73% of problem instances of size 5 × 5 which solved under at least one of theReal or Integer versions had shorter run times under the Integer TTT model. In all instances,the Integer TTT model had the same objective value as the Real TTT model, showing that theadditional requirement of integer beam-on times is not restrictive, in practice. Thus, and given thelarge improvement in computation time, the Integer TTT model may be considered a reasonablyefficient heuristic for the Total Treatment Time problem.

We also compare the performance of the Integer TTT model to that of the model of Langer et al.[3], modified to mirror the Integer TTT model as closely as possible. The objective of the Langeret al. [3] model was altered to a total treatment time objective rather than a minimum cardinalityobjective, the upper bound on the total number of binary beam-on time variables was set toBeammin + T (Kbm −Klb) and the appropriate minimum beam-on time constraint correspondingto constraint (2.2.15) was applied. We also strengthened the model with the use of symmetry-breaking constraints adapted from [4]; these significantly reduce computation time. Full details ofthe model we use are given in Appendix A.

Table 2.4.2 displays the numerical results of the Integer TTT model and the Modified Langer etal. [3] model, when set-up time equals 2, with a time limit of 10 hours. The data sets labelled 3-are 3×3 subsets of randomly generated 10×10 integer matrices, with intensity values in the range0-15 inclusive.

Clearly, the Integer TTT model significantly outperforms the Modified Langer et al. [3] model,taking, on average, 2.2% of the run time when set-up time is 2, considering problem instances



Table 2.4.1: Computation times for the Real TTT model, the Integer TTT model, the Real Step-upalgorithm and the Integer Step-up algorithm using CPLEX version 8.1 and AMPL version 8.1 ona 2GHz AMD 64 3000+: time in seconds, T = 2, 10-hour time limit.

Real TTT model Integer TTT model Real Step-up algorithm Integer Step-up algorithm

Total Time Total Time Total Time Total Time

5a 6417.51 4341.72 1841.11

5b 3181.70 19.90

5c 48.29 4.66 0.38 0.52

5d 99.50 108.84 0.46 0.41

5e 57.73 16.15 3.87 400.24

5f 10.82 2.76 0.15 0.22

5g 120.86 94.80 43.72 0.77

5h 1960.87 412.23 254.14

5i 207.67 8.88 0.07 0.05

5j 88.69 116.41 7.06 6.00

5k 13642.80 23.57 13.76

5l 128.31 173.20 2395.62 1.06

5m

5n 22766.00 504.84

5o 28054.73

5p 175.04 30.60 318.68 38.58

5q 153.29 5148.07 7953.63 6773.77

5r 112.74 32.19 569.47 1.09

5s 7531.75 37.80 137.34 5.86

5t 391.73 541.23 110.78 20735.54

which solved under both models.

We also test an integer version of the Step-up algorithm, in which variables b and a in the CCMBT+

model are restricted to take integer values.

The fourth and fifth columns of Table 2.4.1 show the comparison of run times between the Realversion of the Step-up algorithm and the Integer version of the Step-up algorithm. The fastest runtime for each problem instance is highlighted in bold. It is clear that the Integer Step-up algorithmis the best approach, solving more problems within the time limit, (indeed always solving withinthe time limit if one of the other methods did), and often substantially faster. The Integer versionof the Step-up algorithm outperforms the Integer TTT model in at least 75% of instances, andalso outperforms the Real Step-up algorithm, again faster in at least 75% of instances.

2.5. Conclusion 33

Table 2.4.2: Numerical results for the Integer TTT model compared with the Modified Langer etal. [3] model using CPLEX version 8.1 and AMPL version 8.1 on a 2GHz AMD 64 3000+: timein seconds, T = 2, BB = number of branch and bound nodes, ITS = number of simplex iterations,10-hour time limit.

Integer TTT model Modified Langer et al. [3] modelBB ITS Time BB ITS Time K∗

2 Obj.

Value

3a 6 86 0.02 320 7274 1.66 4 17

3b 57 447 0.04 478 12765 2.56 4 17

3c 161 1278 0.12 42798 1231593 229.35 4 24

3d 0 0 0.01 4 256 0.05 3 11

3e 0 0 0.00 1371 29274 7.54 4 22

3f 216 2218 0.16 5878 197488 36.31 4 24

3g 189 1177 0.13 26629 965490 180.84 5 27

3h 240 2122 0.11 3547 141042 25.25 4 26

3i 0 25 0.01 87 1169 0.25 3 11

3j 0 49 0.01 217 4602 1.17 4 15

4a 540 7892 0.58 199067 8438103 2267.77 5 27

4b 1065 16571 1.20 6 38

4c 413 4657 0.33 90179 5088800 1402.02 5 27

4d 221 2568 0.26 46416 1714569 440.23 5 31

2.5 Conclusion

In this chapter we have presented an exact mixed integer programming formulation for the modu-lation of intensity beams in cancer radiotherapy which minimises the total treatment delivery timeof the “step and shoot” optimisation problem. Our model is based on a formulation proposed byJohnston and Sadinlija [6] for solving cutting stock problems, and does not require beam-on timesto be integer. Considering the “frontier” of possible solutions to the combined objective for theTotal Treatment Time problem, we determine that the TTT problem reduces to a choice betweena very small number of possible solutions. The Step-up algorithm is developed to systematicallyfind these solutions, and hence solve the TTT problem. Numerical results show that the Step-upalgorithm typically decreases the computation time for the Total Treatment Time problem, inmany cases by a substantial margin.

We also consider variations in which beam-on times are required to be integer. For T > 0 thequestion of whether or not this always yields the optimal value of the TTT problem with continuousbeam-on times remains open. However, we verify that, in practice, assuming beam-on times tobe integer has no impact on solution quality. We found that the integer version of our modelsignificantly outperforms the corresponding version of the pure integer model of Langer et al. [3],taking, on average, 2.2% of the run time. It is also faster than the real version of our model. Theinteger version of our Step-up algorithm outperforms all other approaches.



Although this work represents some progress in exact solution techniques for the Total TreatmentTime problem, clearly our best results still show excessive run times for a few problems of size5 × 5, and are unlikely to be practical for larger problems. Whilst particular applications, suchas a nasopharyngeal tumor, utilise intensity matrices that range between 5 × 5 and 15 × 15 cells(Olafsson and Wright [13]), the “effective area” of typical intensity matrices, more generally, isreported to be closer to 20×20 cells, (Lee et al. [12]). Thus in the remaining chapters of this thesis,we concentrate on further decreasing the computation times for larger problem instances. Thisinvolves alternative formulations, for example, with different leaf movement/shape matrix modelcomponents, and strengthening formulations with the use of cutting planes and preprocessingtechniques. Even if exact approaches are unlikely to provide practical solution techniques forproblems of realistic size, by solving bigger instances exactly, we hope to find better benchmarksto measure heuristic methods against.

35CHAPTER 3

Exact Integer Programming Models for the Beam-on Time

Constrained Minimum Cardinality Problem in Cancer

Radiotherapy Using Multileaf Collimators

3.1 Introduction

In this chapter we extend the ideas of Chapter 2 and develop new exact formulations for the leafsequencing problem with the aim of further reducing the overall computation time for the deter-mination of exact solutions. We now turn our attention to solving the Beam-on Time ConstrainedMinimum Cardinality problem, in fact we do so for the remainder of this thesis. Like the TotalTreatment Time problem, exact solutions to the BTCMC problem are required, in particular sinceclinicians prefer solutions with guaranteed minimum total beam-on time and a small number ofshape matrices, [4]. For ease of referral we re-define the BTCMC problem here:

Kbm = minK,b,X

{K : (b,X) ∈ FK},

subject toK∑k=1

bk = Beammin,

where Kbm is the minimum number of shape matrices that can be used when the total beam-on

time is restricted to Beammin and FK = {(b,X) ∈ ZK+ × XK :K∑k=1

bkXk = I}. In this way

total beam-on time is lexicographically minimised before the number of shape matrices used in asolution. In this chapter, and in the remainder of this thesis, we focus on integer beam-on timevariables in model formulations since the integer TTT model outperforms the real TTT model inChapter 2 with no loss of solution quality in practice. Utilising integer beam-on time variables alsoallows us to easily compare against exact formulations currently in the literature.

As discussed in the Introduction to this thesis (Chapter 1) the strongly NP-hard BTCMC problemis well studied from an heuristic view point however exact solutions are few. Within the frameworkof integer programming, the exact models in the literature comprise the models of Langer et al. [3],Baatar [2] and Baatar et al. [4]. All these models can incorporate MLC mechanical constraints.In this chapter we reformulate the integer TTT model of Chapter 2 to solve the BTCMC problem,we improve the formulation of Langer et al. [3], (the Baatar [2] and Langer et al. [3] models havepreviously been shown to have similar computational performance, [4]) we develop a new exactformulation and we compare against the Counter model of Baatar et al. [4]. The formulationswe consider either index their variables on shape matrices or individual monitor units of radiationand hence are polynomial in size, or, index their variables on radiation level and hence are pseudo-polynomial in size. We wish to investigate the benefits and limitations of both types of model. Dueto the relative success of the Counter model of Baatar et al. [4] amongst the integer programmingmodels for the BTCMC problem in the literature, we expect better performance from similar typesof models. In addition to the four core models we present, we therefore develop a variation onthe Counter model which is also pseudo-polynomial in size and which utilises cumulative variables

36Chapter 3. Exact Integer Programming Models for the Beam-on Time Constrained Minimum Cardinality

Problem in Cancer Radiotherapy Using Multileaf Collimators

to model the number of shape matrices that can be given a particular radiation level or morein a solution. Finally, we consider the unconstrained BTCMC problem, though, again, MLCmechanical constraints can be included if necessary into all the models we present. The MLCmechanical constraints may not be simple to formulate; nevertheless the models and solutionapproaches we use in this chapter (and thesis) allow their incorporation. We note again that ingeneral, the solutions and computation times for the constrained versions of the integer programswe consider will be different from the unconstrained results presented in this work.

In this chapter we explore different methods for reducing the computation time of our exact for-mulations. We consider alternative ways to formulate valid shape matrices, numerous constraintsfor breaking symmetry and the effect of applying simple bounds to variables. To model valid shapematrices we must model the strict consecutive-1-property. The strict consecutive-1-property arisesin applications other than the delivery of intensity modulated radiation therapy (see Chapter 4for examples in the literature) and therefore the different variables and constraints we consider inthis chapter for modelling the strict consecutive-1-property may have broader application. Sym-metry breaking ideas are well documented in the integer programming literature generally, see forexample [48]. Symmetry breaking constraints are useful to prevent a branch and bound solverfrom searching and testing alternative symmetric solutions: therefore we explore many symmetrybreaking constraints in this chapter based on the different variables of our different exact formula-tions. Finally, the application of bounds to variables reduces the feasible region to be searched fora model formulation. We investigate some simple ideas in this chapter and expand this approachto improving computational efficiency in Chapter 5. We also apply the version of the Step-up Al-gorithm (which we call the Step-up Method) discussed in Chapter 2, Section 2.3.1 for determiningpoint A, the solution to the BTCMC problem, to all models of this chapter. We apply the Step-upMethod to determine its effect on computational efficiency of the models given the relative successof the Step-up Algorithm in solving the TTT problem with the formulation given in Chapter 2.

The exact integer programming models we investigate in the following sections are numericallytested using batches of 100 randomly generated intensity matrices of varying size and varyingmaximum intensity level and we enforce a 2 hour time limit on all individual problem computations.For the majority of our comparisons we use 100 examples of size 4×4 with intensities ranging from 0to 15 inclusive, 5×5 with intensities ranging from 0 to 5 inclusive, 5×5 with intensities ranging from0 to 10 inclusive and 6× 6 with intensities ranging from 0 to 5 inclusive. The best models are thencompared using randomly generated data sets, each containing 100 problems, with sizes rangingfrom 4× 4 to 18× 18 and intensity values ranging from 0 to 5 inclusive, 0 to 10 inclusive and 0 to15 inclusive. If the total computation time for any batch of 100 problems exceeded approximately100,000 seconds we no longer tested problems of increased size with this same intensity range. Inall tables of comparative results presented in the following sections we indicate in bold face themodel variation which achieves the lowest value for total time for each batch of problems over allvariations considered. As a final experiment, we test the best model of this chapter using medicaldata sets, of varying sizes and maximum intensity levels, obtained from Baatar [49]. We removeany zero rows and columns from the medical data prior to solving. Consequently, the smallestmedical data we utilise, when considering row dimension, is a problem of size 9×9 with intensitiesranging from 0 to 10 inclusive and the smallest medical data when considering column dimensionis a problem of size 11× 7 with intensities ranging from 0 to 21 inclusive. The smallest maximum

3.2. The Johnston and Sadinlija (JS) Model 37

intensity level considered in any of the medical data is 9 and the largest is 35. The notationwe use to describe the medical data is data number row dimension column dimension minimumintensity maximum intensity.

In summary, in Sections 3.2 to 3.8 to follow we present four exact integer programming models forsolving the BTCMC problem (3 of which are new), with numerous variations, and the numericalresults for each. The best model variations are then numerically compared in Section 3.9 wherewe also summarise our main results.

3.2 The Johnston and Sadinlija (JS) Model

3.2.1 Variable s- and b- based Symmetry Breaking Constraints. The first modelwe consider, the Johnston and Sadinlija (JS) model, is the BTCMC version of the integer TotalTreatment Time model discussed in Chapter 2. To solve the BTCMC problem, we amend theobjective function of the integer TTT formulation to minimise cardinality and apply a constraintto restrict the total beam-on time used in the decomposition to Beammin. In terms of the variablesof the formulation, the objective function becomes

minK∑k=1

sk (3.2.1)

and the additional beam-on time constraint is given by (2.3.2). Again, by minimising the sk

variables in the objective we ensure that zero shape matrices will not be used in a solution, whichis consistent with our definition for variable sk.

In Chapter 2, the mixed integer and integer Total Treatment Time models had an additionalredundant constraint applied to the formulations which was shown, numerically, to decrease com-

putation time. We appliedK∑k=1

bk ≥ Beammin. In this chapter, since we are now considering the

BTCMC problem, we replace the additional redundant constraint of the formulations of Chapter2 with the necessary equality constraint, (2.3.2).

As discussed in the Introduction to Chapter 3, we are interested in investigating the effect thatdifferent symmetry breaking constraints have on the different models considered in this chapter.For the JS model, we first apply symmetry breaking constraints equivalent to those applied to theTTT formulation of Chapter 2, (2.2.13) and (2.2.14), which ensure than non-zero shape matricesoccur first in the index order and that beam-on time values are non-increasing in a solution.We trial both ‘non-decreasing’ and ‘non-increasing’ versions of the constraints, though it shouldbe noted that both constraints must be either non-decreasing, or non-increasing, when appliedtogether, for consistency with constraint (2.2.6). (The ‘non-decreasing’ form of each constraintsimply contains a less than or equal to sign rather than a greater than or equal to sign). We applythe non-decreasing and non-increasing versions of constraints (2.2.13) and (2.2.14) to determinethe effect on computational efficiency for the JS model.



To summarise, the formulation for the JS model for solving the BTCMC problem is as follows:

(3.2.1)

s.t. (2.2.5), (2.2.6), (2.2.7), (2.2.8), (2.2.9), (2.2.10), (2.2.11), (2.2.13),

(2.2.14), (2.3.2),

x ∈ {0, 1}m×n×K , s ∈ {0, 1}K , b ∈ ZK+ and a ∈ Zm×n×K+ .

As for the TTT formulations of Chapter 2, we again use K = Kbm in the JS model, whereKbm is the number of shape matrices determined using the Greedy Heuristic Algorithm (GHA)of Baatar et al. [1]. Beammin and Kbm are calculated in relatively negligible time, however allresults given in this chapter incorporate this time component where necessary. Furthermore, all JSmodel variations considered in Sections 3.2 to 3.5 of this chapter use K = Kbm in their respectiveformulations.

As part of our investigation, we apply the Step-up Method of Chapter 2, Section 2.3.1 to the JSmodel. The formulation for the application of the Step-up Method to the JS model differs fromthe formulation of the JS model itself in that the sk variables and related constraints are no longerrequired. The formulation becomes:

minK∑k=1

bk (3.2.2)

s.t. (2.2.7), (2.2.8), (2.2.9), (2.2.10), (2.2.11), (2.2.14), (2.3.2),

x ∈ {0, 1}m×n×K , b ∈ ZK+ and a ∈ Zm×n×K+ .

K in this case is set to Klb initially and then stepped up by one until a feasible solution is attained,with K always less than Kbm, since Kbm is already a solution to BTCMC. (Klb is a known lowerbound on the minimum number of shape matrices that can be used in a solution. See Chapter 2,equation (2.3.1), for the mathematical definition of Klb). Whenever we apply the Step-up Methodto the JS model or its variations (in Sections 3.2 to 3.5 of this chapter) parameter K is set in thisway.

Results of experiments on the JS model, and for the Step-up Method applied to the JS model,are shown in Table 3.2.1. The experiments indicate that non-increasing bk and sk symmetryconstraints generally produce slightly faster results than non-decreasing symmetry constraints.As there is no major difference between the non-decreasing and non-increasing versions of theseconstraints, we choose to utilise non-increasing bk and sk symmetry constraints, where appropriate,in the remaining models of this chapter.

Furthermore, the integer programming solver we utilise, CPLEX version 8.1, follows differentsolution paths depending on the ordering of constraints within model files. There are too manydifferent orderings of constraints to consider within any one model, however one combination wedid trial was to position the symmetry constraints at the beginning of the constraint list ratherthan at the end. The results of these experiments are also given in Table 3.2.1. The results showthat computation times are generally faster when the symmetry constraints are positioned firstin the constraint listing. Considering this result and for consistency, we position all symmetry

3.2. The Johnston and Sadinlija (JS) Model 39

Table 3.2.1: Numerical results for the JS model solving the BTCMC problem using CPLEX version8.1 and AMPL version 8.1 on a 2GHz AMD 64 3000+: time in seconds, 2-hour time limit onindividual problem instances, sk and bk symmetry constraints non-decreasing versus non-increasing,sk and bk symmetry constraints positioned at the end versus at the start of the constraint ordering.

Batches

of 100

problems

BTCMC JS

model: sk

and bk non-

decreasing,

end

BTCMC JS

model: sk

and bk non-

increasing,

end

BTCMC JS

model: sk

and bk non-

increasing,

start

BTCMC

JS model

Step-up:

bk non-

decreasing,

end

BTCMC

JS model

Step-up:

bk non-

increasing,

end

BTCMC

JS model

Step-up:

bk non-

increasing,

start

4 4 0 15 764.84 526.31 329.33 182.94 133.77 107.21

5 5 0 5 799.05 191.89 205.35 83.68 24.12 18.73

5 5 0 10 77814.24 58371.94 43832.30 10196.69 8899.84 1788.22

6 6 0 5 98858.10 97690.50 96619.01 20012.53 38789.30 17023.71

constraints at the start of the constraint listing in the remaining models we investigate in thischapter.

Finally, just as the Step-up Algorithm applied to the (real and integer) TTT formulations ofChapter 2 improves overall computation time for these models, so the Step-up Method applied tothe JS model significantly improves the computational efficiency of the JS model over all problemsizes and maximum intensity levels tested in Table 3.2.1. We also note that in general computationtime for the JS model increases with problem size and maximum intensity level. We investigatethis characteristic further in Section 3.9 where we solve an increased number of problems with thebest version of the JS model that we determine in this chapter.

In the following section we investigate whether alternative symmetry breaking constraints for theJS model have a positive effect on computational efficiency.

3.2.2 Variable s- and x- based Symmetry Breaking Constraints. To address symme-try occurring within shape matrices, we now investigate applying symmetry breaking constraints tothe JS model which are based on variables xijk. We replace constraint (2.2.14) with the following:

xijk +∑

(i,j) s.t. σ(i,j)<σ(i,j)

xijk ≥ xij(k+1),

∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K − 1

(3.2.3)

and

xijk + σ(i, j)− 1 ≥ xij(k+1) +∑


xij(k+1),

∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K − 1,

(3.2.4)



where σ(i, j) is a sorting of the intensity matrix.

Constraints (3.2.3) and (3.2.4) enforce the exposition of cells which occur earlier in a particularordering, in the first shape matrix possible, before the exposition of cells which occur later in theordering. In other words, these constraints satisfy the first intensity value in a particular orderingfirst followed by the others, in order. In particular, constraint (3.2.3) requires that, if all cellscorresponding to a lower ordering value than that for cell (i, j) are closed in shape matrix k, thencell (i, j) must be open in shape matrix k before cell (i, j) can be open in shape matrix (k+1). (Wehave used the term open to mean an exposed cell. The corresponding xijk value for an open cellis one. Conversely when we refer to a closed cell the corresponding xijk value is zero). Similarly,constraint (3.2.4) requires that, if all cells with lower ordering values than that for cell (i, j) areopen in shape matrix (k + 1), then again, cell (i, j) must be open in shape matrix k before it canbe open in shape matrix (k+ 1). The σ(i, j)− 1 term on the left hand side of constraint (3.2.4) isequal to the maximum value of the summation term on the right hand side of the expression and isnecessary to ensure a valid constraint. We trial σ(i, j) sortings having increasing values as intensityvalues increase and decreasing values as intensity values increase to determine any difference incomputational efficiency for the JS model. For example, if an intensity matrix is given by

I =

10 4 14 214 13 5 313 8 13 147 14 4 8

then sorting the intensity matrix with ‘increasing values as intensity values increase’ yields a σ

matrix given by

σ =

9 3 13 114 10 5 211 7 12 156 16 4 8

.

Constraint (2.2.14) can not be used with constraints (3.2.3) and (3.2.4) as the non-increasingvalues of individual beam-on time variables may be contradictory to the beam-on time values thatoccur when shape matrices with greater exposure of cells are preferentially chosen. Therefore,the formulations to incorporate the variable x- based symmetry constraints, for the JS modeland the Step-up Method applied to the JS model, are equivalent to those given in Section 3.2.1other than we now include constraints (3.2.3) and (3.2.4) and remove constraint (2.2.14) from eachformulation.

Table 3.2.2 presents the results for the JS model with x-based symmetry constraints. Experimentstrialling whether the intensity matrix should be sorted into increasing or decreasing values ofintensity within the xijk variable symmetry constraints indicate that sorting with decreasing valuesof intensity is much slower computationally. Furthermore, in the case of decreasing intensity values,computation time increases when the Step-up Method is applied to the JS model. On the otherhand, for an increasing sorting of intensity values within the xijk symmetry constraints, the Step-upMethod significantly reduces computation time for the JS model. Therefore, we apply an orderingof increasing intensity to all symmetry breaking constraints in the remainder of this chapter whichutilise a σ(i, j) sorting of intensity values.

3.3. The Johnston and Sadinlija Leaf Explicit (JS-LE) Model 41

Table 3.2.2: Numerical results for the JS model solving the BTCMC problem using CPLEX version8.1 and AMPL version 8.1 on a 2GHz AMD 64 3000+: time in seconds, 2-hour time limit onindividual problem instances, xijk symmetry constraints with the intensity matrix sorted in orderof decreasing intensity versus increasing intensity.

Batches

of 100

problems

BTCMC

JS model:

xijk sym-

metry

constraints

with sigma

decreasing

BTCMC

JS model:

xijk sym-

metry

constraints

with sigma

increasing

BTCMC

JS model

Step-up

version:

xijk sym-

metry

constraints

with sigma

decreasing

BTCMC

JS model

Step-up ver-

sion: xijk

symmetry

constraints

with sigma

increasing

4 4 0 15 34316.48 2728.83 96744.23 285.01

5 5 0 5 2802.97 3144.18 24985.42 320.94

5 5 0 10 253212.72 212940.65 566500.56 88169.45

6 6 0 5 214110.98 161104.66 341898.90 101956.86

Furthermore, we can conclude that the JS model using variable sk and bk symmetry constraintsperforms more efficiently than the JS model using variable sk and xijk symmetry constraints. Sincethis is the case, when we refer to the JS model in the remainder of this chapter we refer to the JSmodel with non-increasing bk and sk symmetry constraints as described in Section 3.2.1. Of coursewhenever we consider the JS model in the remainder of this chapter we also apply the Step-upMethod to further reduce total computation time.

In the following three sections we investigate whether changing the variables used to model thestrict consecutive-1-property, and hence the cell exposure/ coverage within a shape matrix, effectsthe efficiency of the JS model.

3.3 The Johnston and Sadinlija Leaf Explicit (JS-LE) Model

In this section we consider modelling cell exposure/coverage within a shape matrix using variableswhich describe the left and right ‘leaf’ positions of the multileaf collimator. The specific variableswe use are the equivalent of the leaf variables of the integer programming formulation of Langer etal. [3]. We investigate whether an alternative formulation of the JS model based on these variableswill improve computational efficiency. In our new formulation, the more compact xijk variables ofthe JS model are replaced with the following left and right ‘leaf’ variables:

lijk =

{1, if column j is covered by the left leaf in row i of shape matrix k0, otherwise

,

∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K

(3.3.1)



and

rijk =

{1, if column j is covered by the right leaf in row i of shape matrix k0, otherwise

,

∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K,

(3.3.2)

wherexijk = 1− (lijk + rijk), ∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K. (3.3.3)

If the following diagram represents row i in shape matrix k, where closed cells are shaded and opencells are unshaded,

Figure 3.3.1: An Example Representation of Row i in Shape Matrix k

then the corresponding values for lijk, rijk and xijk are:

lijk = (1 1 1 0 0 0),

rijk = (0 0 0 0 1 1)

andxijk = (0 0 0 1 0 0).

The bk, sk and aijk variables are as for the JS model.

Given the new variables, lijk and rijk, we reformulate the JS model accordingly. The formulationbecomes:

(3.2.1)

subject to constraints

sk ≥ 1− (lijk + rijk), ∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K, (3.3.4)

0 ≤ aijk ≤Mijk(1− (lijk + rijk)), ∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K, (3.3.5)

aijk +Gijk(lijk + rijk) ≥ bk, ∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K, (3.3.6)

lijk ≥ li(j+1)k, ∀ i = 1, . . . ,m, k = 1, . . . ,K, j = 1, . . . , n− 1, (3.3.7)

rijk ≤ ri(j+1)k, ∀ i = 1, . . . ,m, k = 1, . . . ,K, j = 1, . . . , n− 1, (3.3.8)

li(j−1)k + rijk ≤ 1, ∀ i = 1, . . . ,m, j = 2, . . . , n, k = 1, . . . ,K, (3.3.9)

ri1k = 0, ∀ i = 1, . . . ,m, k = 1, . . . ,K, (3.3.10)

(2.2.6), (2.2.8), (2.2.10), (2.2.13), (2.2.14), (2.3.2), l ∈ {0, 1}m×n×K , r ∈ {0, 1}m×n×K , s ∈ {0, 1}K ,b ∈ ZK+ and a ∈ Zm×n×K+ . We name this new model the Johnston and Sadinlija Leaf Explicit(JS-LE) model.

3.4. The Johnston and Sadinlija Leaf Explicit Asymmetric (JS-LEA) Model 43

Constraint (3.3.4) of the JS-LE model is equivalent to constraint (2.2.5) of the JS model, constraint(3.3.5) is equivalent to constraint (2.2.7), constraint (3.3.6) is equivalent to constraint (2.2.9) andconstraints (3.3.7) and (3.3.8) replace constraint (2.2.11). We have also defined only one closedleaf position to reduce symmetry via the constraints (3.3.9) and (3.3.10). The closed leaf positionis arbitrarily chosen to be the left leaf covering the entire row and the right leaf fully retracted. Wenote that if we wished to solve the constrained BTCMC problem we would not apply constraints(3.3.9) and (3.3.10), since many different closed leaf positions may be necessary to satisfy interleafconstraints. This observation is true whenever we apply a single closed leaf position set of symmetrybreaking constraints in the remaining JS ‘leaf based’ models of this chapter.

When we apply the Step-up Method the JS-LE model becomes:

(3.2.2)

s.t. (2.2.8), (2.2.10), (2.2.14), (2.3.2), (3.3.5), (3.3.6), (3.3.7), (3.3.8),

(3.3.9), (3.3.10),

l ∈ {0, 1}m×n×K , r ∈ {0, 1}m×n×K , b ∈ ZK+ and a ∈ Zm×n×K+ .

Numerical results for the JS-LE model are given at the end of Section 3.5, where we also discussand compare two other ‘leaf’ variable formulations of the JS model. The two other ‘leaf’ variableformulations are described in detail in the following two sections.

3.4 The Johnston and Sadinlija Leaf Explicit Asymmetric (JS-LEA) Model

In this section we again consider left and right leaf variables to model cell exposure/coverage withinshape matrices. The variables we investigate here are based on those used in a similar formulationdeveloped by Queyranne [7] to model the strict consecutive-1-property.

The left and right leaf variables of this section are asymmetrically defined: with the binary valuesof one variable taking on the opposing meaning of the other variable. These new left and right leafvariables replace the xijk variables of the JS model and are defined as follows:

lijk =

{0, if column j is covered by the left leaf in row i of shape matrix k1, otherwise

,

∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K

(3.4.1)

and

rijk =

{1, if column j is covered by the right leaf in row i of shape matrix k0, otherwise

,

∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K,

(3.4.2)

where

lijk = 1− lijk, ∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K, (3.4.3)

rijk = rijk, ∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K, (3.4.4)



andxijk = lijk − rijk, ∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K. (3.4.5)

Again considering Figure 3.3.1, the corresponding values for lijk and rijk are:

lijk = (0 0 0 1 1 1)

andrijk = (0 0 0 0 1 1).

Variables bk, sk and aijk are as for the JS model. We note that the left and right leaf variablesof the ‘leaf implicit’ model of Baatar [2] are very similar to the leaf variables defined here, thoughinstead of multiple ones, the Baatar [2] variables utilise only a single one to indicate the position ofthe leaves. Considering Figure 3.3.1, the corresponding left and right leaf variables for the Baatar[2] ‘leaf implicit’ model are (0 0 0 1 0 0) and (0 0 0 0 1 0) respectively.

Utilising the new variables, lijk and rijk, our new formulation for the JS model is as follows:

(3.2.1)


sk ≥ lijk − rijk, ∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K, (3.4.6)

0 ≤ aijk ≤Mijk(lijk − rijk), ∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K, (3.4.7)

aijk +Gijk(1− lijk + rijk) ≥ bk, ∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K, (3.4.8)

lijk ≤ li(j+1)k, ∀ i = 1, . . . ,m, k = 1, . . . ,K, j = 1, . . . , n− 1, (3.4.9)

rijk ≤ ri(j+1)k, ∀ i = 1, . . . ,m, k = 1, . . . ,K, j = 1, . . . , n− 1, (3.4.10)

rijk ≤ li(j−1)k, ∀ i = 1, . . . ,m, j = 2, . . . , n, k = 1, . . . ,K, (3.4.11)

ri1k = 0, ∀ i = 1, . . . ,m, k = 1, . . . ,K (3.4.12)

and (2.2.6), (2.2.8), (2.2.10), (2.2.13), (2.2.14), (2.3.2), l ∈ {0, 1}m×n×K , r ∈ {0, 1}m×n×K , s ∈{0, 1}K , b ∈ ZK+ and a ∈ Zm×n×K+ . We name this model the Johnston and Sadinlija Leaf ExplicitAsymmetric (JS-LEA) model.

Constraint (3.4.6) is equivalent to (2.2.5), constraint (3.4.7) is equivalent to constraint (2.2.7),constraint (3.4.8) is equivalent to constraint (2.2.9) and constraints (3.4.9) and (3.4.10) replaceconstraint (2.2.11). We have again defined only one closed leaf position to reduce symmetry viaconstraints (3.4.11) and (3.4.12).

If we apply the Step-up Method the JS-LEA model becomes:

(3.2.2)

s.t. (2.2.8), (2.2.10), (2.2.14), (2.3.2), (3.4.7), (3.4.8), (3.4.9), (3.4.10),

(3.4.11), (3.4.12),

l ∈ {0, 1}m×n×K , r ∈ {0, 1}m×n×K , b ∈ ZK+ and a ∈ Zm×n×K+ .

We present numerical results for the JS-LEA formulation at the end of Section 3.5.

3.5. The Johnston and Sadinlija Leaf Explicit Pairs (JS-LEP) Model 45

3.5 The Johnston and Sadinlija Leaf Explicit Pairs (JS-LEP) Model

We now consider one final variation of the JS model which again describes the strict consecutive-1-property using left and right leaf variables. In this case each possible left and right leaf positionis considered in a pair and we utilise a single binary variable to indicate whether a particular(left,right) leaf position is used in row i of shape matrix k. We define variables filrk to replace thexijk variables of the JS model as follows:

filrk =

{1, if leaf position is (l, r) in row i of shape matrix k0, otherwise

,

∀ i = 1, . . . ,m, k = 1, . . . ,K, l = 0, . . . , n− 1, r = l + 2, . . . , n+ 1,

(3.5.1)

and

fi01k =

{1, if leaf position is (0, 1) in row i of shape matrix k0, otherwise

,

∀ i = 1, . . . ,m, k = 1, . . . ,K,

(3.5.2)

where

xijk =j−1∑l=0

n+1∑r=j+1

filrk, ∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K (3.5.3)

and where we have again defined only one closed leaf position to reduce symmetry. Variables bk,sk and aijk are as for the JS model.

The formulation for our final JS model variation is as follows:

(3.2.1)


sk ≥j−1∑g=0

n+1∑h=j+1

fighk, ∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K, (3.5.4)

0 ≤ aijk ≤Mijk

j−1∑g=0

n+1∑h=j+1

fighk

, ∀ i = 1, . . . ,m, j = 1, . . . ,m, k = 1, . . . ,K, (3.5.5)

aijk +Gijk

1−

j−1∑g=0

n+1∑h=j+1

fighk

≥ bk,∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K,

(3.5.6)

fi01k +n−1∑l=0

n+1∑r=l+2

filrk = 1, ∀ i = 1, . . . ,m, k = 1, . . . ,K, (3.5.7)

(2.2.6), (2.2.8), (2.2.10), (2.2.13), (2.2.14), (2.3.2), f ∈ {0, 1}m× 12 (n+1)n+1×K , s ∈ {0, 1}K , b ∈ ZK+

and a ∈ Zm×n×K+ . We name this model the Johnston and Sadinlija Leaf Explicit Pairs (JS-LEP)model.



Constraint (3.5.4) is equivalent to (2.2.5), constraint (3.5.5) is equivalent to constraint (2.2.7), con-straint (3.5.6) is equivalent to constraint (2.2.9) and constraint (3.5.7) replaces constraint (2.2.11).

Applying the Step-up Method the formulation becomes:

(3.2.2)

s.t. (2.2.8), (2.2.10), (2.2.14), (2.3.2), (3.5.5), (3.5.6), (3.5.7),

f ∈ {0, 1}m× 12 (n+1)n+1×K , b ∈ ZK+ and a ∈ Zm×n×K+ .

Table 3.5.1 compares each of our leaf models, JS-LE, JS-LEA and JS-LEP with the JS model andindicates that the JS-LE model is the best of the leaf models trialled but that it does not comparecomputationally to the JS model.

Table 3.5.1: Numerical results for the JS model versus the JS-LE model, the JS-LEA model andthe JS-LEP model solving the BTCMC problem using CPLEX version 8.1 and AMPL version 8.1on a 2GHz AMD 64 3000+: time in seconds, 2-hour time limit on individual problem instances.

Batches

of 100

problems

BTCMC

JS

model

BTCMC

JS

model

Step-up

version

BTCMC

JS-LE

model

BTCMC

JS-LE

model

Step-up

version

BTCMC

JS-LEA

model

BTCMC

JS-LEA

model

Step-up

version

BTCMC

JS-LEP

model

BTCMC

JS-LEP

model

Step-up

version

4 4 0 15 329.33 107.21 10416.48 1024.19 19438.87 913.01 76969.87 21310.37

5 5 0 5 205.35 18.73 716.42 194.13 3068.13 143.42 22011.83 8511.22

5 5 0 10 43832.30 1788.22 86901.96 17084.17 115243.82 29312.36 294952.88 230117.71

6 6 0 5 96619.01 17023.71 150289.85 49720.79 169074.71 77552.72 221368.03 201054.29

As a final test, we trial whether including xijk variables into the best of our leaf models, JS-LE,improves computation time. To incorporate the xijk variables into the JS-LE formulation, we mustalso include the equation relating the xijk variables to the lijk and rijk variables, (3.3.3). Table3.5.2 compares the JS model to the JS-LE model with and without the additional xijk variables.The results demonstrate that the additional xijk variables do not improve computation time for theJS-LE model, at least for larger problem sizes and larger maximum intensity levels, and thereforethat the JS model itself is the best performing model we have tested using this formulation asa basis. As mentioned previously whenever we consider the JS model in the remainder of thischapter we also apply the Step-up Method to further reduce total computation time.

3.6 The Unit Radiation Pattern (URP) Model

The previous sections of this chapter have focused on the JS model and numerous variations ofthe formulation with regard to the variables used to describe the cells within shape matrices andthe symmetry breaking constraints applied. We now turn our attention to a new exact model forthe Beam-on Time Constrained Minimum Cardinality problem which is an improved formulationof the type of model presented by Langer et al. [3]. We demonstrated in Chapter 2 that the

3.6. The Unit Radiation Pattern (URP) Model 47

Table 3.5.2: Numerical results for the JS model versus the JS-LE model with and without xijkvariables solving the BTCMC problem using CPLEX version 8.1 and AMPL version 8.1 on a2GHz AMD 64 3000+: time in seconds, 2-hour time limit on individual problem instances.

Batches

of 100

problems

BTCMC

JS model

BTCMC

JS model

Step-up

version

BTCMC

JS-LE

model

without

xijk

variables

BTCMC

JS-LE

model

Step-up

version

without

xijk

variables

BTCMC

JS-LE

model

with

xijk

variables

BTCMC

JS-LE

model

Step-up

version

with

xijk

variables

4 4 0 15 329.33 107.21 10416.48 1024.19 7298.17 951.03

5 5 0 5 205.35 18.73 716.42 194.13 2486.46 169.00

5 5 0 10 43832.30 1788.22 86901.96 17084.17 174922.73 54989.12

6 6 0 5 96619.01 17023.71 150289.85 49720.79 160203.69 76811.67

Total Treatment Time model of Langer et al. [3] did not perform well when compared with theTotal Treatment Time version of the JS model. (For a complete description of the Langer et al.[3] model, which solves the Total Treatment Time problem, see Appendix A). In this section weimprove the Langer-type model by decreasing the overall number of variables and including a rangeof symmetry breaking constraints. The new model utilises similar variables to the xijk variables ofthe JS model to describe cell exposure/coverage, however now each shape matrix is given only asingle unit of radiation. We name this model the Unit Radiation Pattern (URP) model and we testa number of different symmetry breaking constraints on the URP model in this section. In Section3.9 we compare the numerical results for the best version of the URP model that we determine inthis section with those of the JS model and other models of this chapter.

As mentioned, the URP model irradiates each shape matrix with a beam-on time of 1 unit, andtherefore the same shape matrix can be repeatedly used. The model assumes fixed total beam-ontime (since we are solving the BTCMC problem, total beam-on time equals Beammin) and deter-mines particular shape matrices corresponding to the delivery of each individual unit of radiation.Where in the JS model we indexed on shape matrices, k, we now index on individual monitorunits of radiation, t, since shape matrices can be repeated. Correspondingly we do not need tocalculate an upper bound on the number of shape matrices to be used, Kbm, for this model, sincein the URP model the number of shape matrices to be used (with unit beam-on time) will equalBeammin.

To distinguish between different ‘runs’ of shape matrices, where a ‘run’ contains just one type ofshape matrix, we define variable pt as follows:

pt =

1, if shape matrix t is the last shape matrix of the run

(t+1 is the new shape matrix)0, if shape matrix t = shape matrix (t+1)

,

∀ t = 1, . . . , B − 1.

(3.6.1)



B for the URP model equals Beammin. In fact, B = Beammin for all variations of the URP modelpresented in Section 3.6.

We also define a variable xijt in a similar way to the xijk variable of the JS model.

xijt =

1, if cell (i, j) is exposed in shape matrix t

(corresponding to the tth unit of radiation)0, otherwise

,

∀ i = 1, . . . ,m, j = 1, . . . , n and t = 1, . . . , B.

(3.6.2)

In the URP model the intensity requirement is written directly in terms of variables xijt, againsince each shape matrix has a beam-on time of 1 unit. We have:

B∑t=1

xijt = Ii,j , ∀ i = 1, . . . ,m, j = 1, . . . , n (3.6.3)

and we use the corresponding version of constraint (2.2.11) from the JS model to apply the strictconsecutive-1-property. This means that we replace parameter k with parameter t in this con-straint.

To mathematically connect variable pt to xijt the following constraints are required:

pt ≥ xijt − xij(t+1), ∀ i = 1, . . . ,m, j = 1, . . . , n, t = 1, . . . , B − 1 (3.6.4)

and

pt ≥ xij(t+1) − xijt, ∀ i = 1, . . . ,m, j = 1, . . . , n, t = 1, . . . , B − 1. (3.6.5)

Constraints (3.6.4) and (3.6.5) state that variable pt is greater than or equal to zero if there is nodifference between all cell exposures/coverages in shape matrix t and shape matrix t+ 1, and thatpt is greater than or equal to 1 if there is any difference between cell exposures/coverages in shapematrix t and shape matrix t+ 1.

The objective function for the Unit Radiation Pattern model, for the solution to the Beam-onTime Constrained Minimum Cardinality problem, is as follows:

min

1 +B−1∑t=1

pt

. (3.6.6)

Minimising the sum over the pt values ensures that no two identical shape matrices will be non-adjacent in an optimal solution and we add one to this sum to correctly count the number ofdifferent shape matrices used rather than the number of changes in shape matrix type.

In summary, the Unit Radiation Pattern model is as follows:

(3.6.6)

s.t. (2.2.11) (with parameter k changed to parameter t), (3.6.3), (3.6.4), (3.6.5),

p ∈ {0, 1}B−1 and x ∈ {0, 1}m×n×B .


In applying the Step-up Method we introduce parameter K into the URP formulation and hencethere is a small additional time component associated with this version of the model to calculatethe upper bound on K, Kbm, with GHA. We have:

min B

s.t. (2.2.11) (with parameter k changed to parameter t), (3.6.3), (3.6.4), (3.6.5),

1 +B−1∑t=1

pt = K, (3.6.7)

p ∈ {0, 1}B−1 and x ∈ {0, 1}m×n×B .

The treatment of parameter K in this formulation (and whenever we apply the Step-up Methodto the URP model variations in the remainder of Section 3.6) follows the Step-up Method rules(see the discussion of the Step-up Method in Section 3.2).

As discussed, a focus of this chapter is to determine the effect of different symmetry breakingconstraints on the different models we investigate. Langer et al. [3] do not apply symmetrybreaking constraints in their formulation of a unit radiation model, however Baatar et al. [4]do apply symmetry breaking to the Langer et al. [3] model, finding that with symmetry breakingconstraints the Langer et al. [3] model performs significantly better. We also apply the appropriateform of the symmetry breaking constraints developed by Baatar et al. [4] to the Total TreatmentTime version of the Langer et al. [3] model in Chapter 2. In the following sections, we describea number of symmetry breaking constraints for the URP model, (including equivalent constraintsto those used by Baatar et al. [4], see Section 3.6.2) and we compare the different formulationsnumerically at the end of Section 3.6. Each time we consider a new set of symmetry breakingconstraints in the following sections, we simply amend the URP model (given in this section) toincorporate the new constraints. No other changes are required to the URP formulation. The sameis true for the formulation of the Step-up Method applied to the URP model: the only change isthe inclusion of the additional symmetry breaking constraints.

3.6.1 Variable p-based Symmetry Breaking Constraints of Type 1. The first sym-metry breaking constraints we trial on the URP model are in keeping with the idea of non-increasingbeam-on time values in a decomposition, which we have considered in previous symmetry breakingconstraints of this chapter. In this case, the constraints involve variables pt. We have:

(B − t)t∑t=1

pt ≤ t

B−1∑t=t+1

pt

+ 1

, ∀ t = 1, . . . , B − 2. (3.6.8)

Constraint (3.6.8) orders the use of shape matrices so that those with longer ‘runs’ are used beforethose with shorter ‘runs’ which also means that shape matrices will occur with non-increasingbeam-on time. The constraint considers any number of pt variables with t ≤ t, where t is as givenin (3.6.8), and requires that these pt variables contain more zero values than the pt variables witht > t.

When we incorporate constraint (3.6.8) into the URP formulation of Section 3.6, we refer to



the URP model as the Unit Radiation Pattern model with variable p-based Symmetry BreakingConstraints of Type 1. We use the acronym, the URP-pSCT1 model.

3.6.2 Variable p-based Symmetry Breaking Constraints of Type 2. The constraintswe refer to as ‘variable p-based symmetry constraints of type 2’ were formulated by Baatar et al.[4] and applied to the Langer et al. [3] model for solving the Beam-on Time Constrained MinimumCardinality problem. Since both the URP and the Langer et al. [3] models are unit radiationpattern models we can utilise these symmetry breaking constraints in this context.

The ‘variable p-based symmetry constraints of type 2’ are as follows:

y∑t=1

pt ≤2y∑

t=y+1

pt, ∀ y = 1, . . . ,⌊B − 1

2

⌋(3.6.9)

and (r+y∑t=r

pt

)− 1 ≤

r+2y∑t=r+y+1

pt, ∀ r = 1, . . . , B − 3, y = 1, . . . ,⌊B − r − 1

2

⌋. (3.6.10)

These constraints specify that earlier ‘runs’ of length y should contain more zero values for pt thansubsequent ‘runs’ of length y in a decomposition. This again leads to longer ‘runs’ of the sameshape matrices occurring earlier in a solution and corresponding beam-on time values occurring innon-increasing order.

We name this version of the URP model, which incorporates constraints (3.6.9) and (3.6.10), theUnit Radiation Pattern model with variable p-based Symmetry Breaking Constraints of Type 2and use the acronym URP-pSCT2.

3.6.3 Variable x-based Symmetry Breaking Constraints. We now consider breakingsymmetry that may be occurring within shape matrices, as we did in Section 3.2.2 for the JS model.In this case we utilise the corresponding versions of constraints (3.2.3) and (3.2.4), which meansthat we replace parameter k with parameter t in each of these constraints. The σ(i, j) sortingmatrix within these constraints sorts intensity values in order of increasing radiation since this wasthe more efficient sorting, at least when applied to the JS model, when compared with that ofdecreasing radiation (see Section 3.2.2).

When we incorporate constraints (3.2.3) and (3.2.4) (with parameter k changed to parameter tin each constraint) into the URP model, we rename the URP model the Unit Radiation Patternmodel with variable x-based Symmetry Breaking Constraints or the URP-xSC model.

3.6.4 Variable p- and x- based Symmetry Breaking Constraints. As a final test ofsymmetry breaking constraints for the URP model, we now combine the ordering of variables ptand xijt so that the xijt variables are only ordered when a new shape matrix is used.

In this case the corresponding versions of symmetry breaking constraints (3.2.3) and (3.2.4) (with


parameter k changed to parameter t) become:

1− pt + xijt +∑


xijt ≥ xij(t+1),

∀ i = 1, . . . ,m, j = 1, . . . , n, t = 1, . . . , B − 1

(3.6.11)

and1− pt + xijt + σ(i, j)− 1−

∑(i,j) s.t. σ(i,j)<σ(i,j)

xij(t+1) ≥ xij(t+1),

∀ i = 1, . . . ,m, j = 1, . . . , n, t = 1, . . . , B − 1.

(3.6.12)

Here we again use the sorting matrix, σ(i, j), to sort intensity values in order of increasing radiation.

We name this version of the URP model, with constraints (3.6.11) and (3.6.12), the Unit RadiationPattern model with variable p- and x-based Symmetry Breaking Constraints or the URP-pxSCmodel.

Table 3.6.1: Numerical results for the URP-pSCT1 model, the URP-pSCT2 model, the URP-xSCmodel and the URP-pxSC model solving the BTCMC problem using CPLEX version 8.1 and AMPLversion 8.1 on a 2GHz AMD 64 3000+: time in seconds, 2-hour time limit on individual probleminstances.

Batches

of 100

problems

BTCMC

URP-

pSCT1

model

BTCMC

URP-

pSCT2

model

BTCMC

URP-xSC

model

BTCMC

URP-xSC

model

Step-up

version

BTCMC

URP-

pxSC

model

4 4 0 15 116720.00 10893.20 14454.30 188032.53 95453.00

5 5 0 5 2043.75 223.26 572.33 8173.95 1155.30

5 5 0 10 139005.00 34350.10 25065.50 483711.63 115745.00

6 6 0 5 31275.00 17004.70 4537.95 257398.91 17938.90

7 7 0 5 230870.00 148811.00 181456.00 177861.00

The results of experiments on the URP model, using the various symmetry constraint combinations,are given in Table 3.6.1. The results indicate that using variable x- based symmetry constraintsor using variable p- based symmetry constraints of type 2 yields similar total computation timeover the problem sizes tested and that these model versions are generally more efficient thanthe versions with variable p- based symmetry constraints of type 1 and variable p- and x- basedsymmetry constraints. We applied the Step-up Method to the URP-xSC model for smaller sizedproblems and found that total computation time increased dramatically for all problems tested.We therefore concluded that in general the application of the Step-up Method to the URP modelswould not improve computation time. It should also be noted that like the JS model, the URPmodel increases in computation time with the size of the intensity matrix and the size of themaximum intensity level in the matrix. We investigate this property further in Table 3.9.1 ofSection 3.9. The experiments with extra data sets in Section 3.9 demonstrate that the URP-xSC



model is the better performing of the URP models and therefore it is this version that we includein Table 3.9.1. We postpone our discussion of the comparative results of the URP-xSC, JS and allother models investigated in this chapter till Section 3.9.

The JS and URP models, studied in the previous sections, are polynomial sized models and aswe have seen, increase in total computation time with problem size and maximum intensity level.In contrast, the Counter model of Baatar et al. [4], to be discussed later in this chapter, is apseudo-polynomial sized model, indexing its variables on radiation level. Indexation on radiationlevel means that the Counter model has the possibility of becoming too large to solve as maximumintensity level increases, however the experiments of Baatar et al. [4] demonstrate that in practicethe Counter model is relatively efficient and outperforms the Langer et al. [3] model and the ‘leafimplicit’ model of Baatar [2]. Given the relative success of the Counter model, we now consider anew exact formulation for the BTCMC problem, of pseudo-polynomial size, which also indexes onradiation level.

3.7 The Binary Expansion (BE) Model

The pseudo-polynomial sized Binary Expansion (BE) model utilises variables which describe theradiation level cell (i, j) in shape matrix k receives, and the radiation level shape matrix k itselfreceives, and therefore indexes on radiation level and shape matrices. We have:

dijhk =

{1, if cell (i, j) of shape matrix k receives h units of radiation0, if cell (i, j) of shape matrix k receives no radiation

,

∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K, h = 1, . . . ,Hk

(3.7.1)

and

ehk =

{1, if shape matrix k receives h units of radiation0, if shape matrix k receives no radiation

,

∀ k = 1, . . . ,K, h = 1, . . . ,Hk,

(3.7.2)

where Hk is as before in the JS model, an upper bound on the beam-on time that can be appliedto shape matrix k. We would like the value for Hk to be as small as possible since the size of Hk

influences the size of the feasible region to be searched when solving our integer program. It is thefocus of Chapter 5 to develop improved bounds on beam-on time variables, and hence any boundson the beam-on time that can be applied to shape matrix k determined in Chapter 5, can be usedas values for Hk in the BE model. In this chapter, we set Hk to a constant for all k = 1, . . . ,Kand hence replace Hk with H = max

i=1,...,m,j=1,...,nIij .

We also utilise variables xijk from the JS model.

To ensure that only one value of beam-on time is applied to each shape matrix k we apply:

H∑h=1

ehk ≤ 1, ∀ k = 1, . . . ,K. (3.7.3)

3.7. The Binary Expansion (BE) Model 53

Then to link variables dijhk and ehk we apply:

dijhk ≤ ehk, ∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K, h = 1, . . . ,H (3.7.4)

which ensures that if dijhk is one (cell (i, j) of shape matrix k receives h units of radiation) thenehk is greater than or equal to 1 (shape matrix k will receive h units of radiation). If dijhk is zero(cell (i, j) of shape matrix k receives no radiation) constraint (3.7.4) implies that ehk is greaterthan or equal to 0 (shape matrix k may or may not receive radiation).

To link variables dijhk and xijk we require the following constraint:

H∑h=1

dijhk = xijk, ∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K. (3.7.5)

Equation (3.7.5) implies that when cell (i, j) is exposed, only one value of beam-on time is appliedto cell (i, j), and when cell (i, j) is not exposed, no value of beam-on time is applied to cell (i, j).

The intensity constraint for the Binary Expansion model takes the form:

K∑k=1

H∑h=1

hdijhk = Iij , ∀ i = 1, . . . ,m, j = 1, . . . , n. (3.7.6)

The constraint states that the intensity received by cell (i, j) is equal to the sum over all shapematrices of the radiation value h applied to open cell (i, j) in each shape matrix k.

We also apply constraint (2.2.11) from the JS model to describe the patterns of allowable shapematrices.

Finally the equation to constrain the total beam-on time of a decomposition to Beammin is:

K∑k=1

H∑h=1

hehk = Beammin (3.7.7)

and the objective function for the BTCMC problem in terms of the variables of the BE model is:

minK∑k=1

H∑h=1

ehk. (3.7.8)

The objective minimises the total number of shape matrices used, as required.

Therefore the BE model is as follows:

(3.7.8)

s.t. (2.2.11), (3.7.3), (3.7.4), (3.7.5), (3.7.6), (3.7.7),

d ∈ {0, 1}m×n×H×K , e ∈ {0, 1}H×K and x ∈ {0, 1}m×n×K ,

where the value of K used in the model is Kbm, calculated using GHA. K equals Kbm in all BEmodel variations in this section.



Applying the Step-up Method to the BE model yields an equivalent formulation other than theobjective. We have:

minK∑k=1

H∑h=1

hehk

s.t. (2.2.11), (3.7.3), (3.7.4), (3.7.5), (3.7.6), (3.7.7),

d ∈ {0, 1}m×n×H×K , e ∈ {0, 1}H×K and x ∈ {0, 1}m×n×K .

The treatment of parameter K in this and all variations of the BE formulation when the Step-upMethod is applied follows the Step-up Method rules (see Section 3.2).

We trial two types of symmetry breaking constraints within the BE model: variable e-basedsymmetry constraints and variable d-based symmetry constraints. When each set of symmetrybreaking constraints is considered in the following sections, we simply amend the BE formulationsgiven in this section (with and without the application of the Step-up Method) to incorporate theappropriate symmetry breaking constraints. No other changes are required to the BE formulations.

3.7.1 Variable e- based Symmetry Breaking Constraints. The constraints we referto as variable e-based symmetry breaking constraints for the BE model are beam-on time basedsymmetry breaking constraints similar to those we have considered previously in this chapter. Wehave:

H∑h=1

hehk ≥H∑h=1

heh(k+1), ∀ k = 1, . . . ,K − 1. (3.7.9)

As before, constraint (3.7.9) ensures that the beam-on time values corresponding to shape matricesin a solution are returned in non-increasing order.

Numerical results for the BE model with variable e-based symmetry breaking constraints are givenin Table 3.7.1 at the end of Section 3.7.

3.7.2 Variable d- based Symmetry Breaking Constraints. The second symmetry con-straint we trial is based on the dijhk variables and again considers an ordering, σ(i, j), of the cellsof an intensity matrix, in particular that the cells are numbered in order of increasing radiation.We have:

H∑h=1

hdijhk +Aij

H∑h=1

∑(i,j) s.t. σ(i,j)<σ(i,j)

dijhk ≥H∑h=1

hdijh(k+1),

∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K − 1,

(3.7.10)

where Aij = Iij is sufficiently large for the constraint to be valid, given that h ≤ Iij when weconsider cell (i, j). Again reducing the size of a parameter such as Aij is the focus of the work weconsider in Chapter 5.

Constraint (3.7.10) is similar to constraint (3.2.3) (trialled with the JS model) in that, constraint(3.7.10) fulfills the intensity requirement of cells occurring earlier in an ordering in earlier shapematrices of a decomposition.

3.7. The Binary Expansion (BE) Model 55

Constraint (3.7.10) cannot be used with constraint (3.7.9) since a non-increasing beam-on timevalue ordering may not coincide with the beam-on time values that result when satisfying lowerintensity values before higher intensity values.

Again numerical results for the BE model with variable d-based symmetry breaking constraintsare given in Table 3.7.1.

3.7.3 Additional Constraints Tested on the BE Model. As a final test, we investigatethe computational benefit, if any, of the following constraints applied to the BE model.

K∑k=1

dH2 −1e∑h=0

dij(2h+1)k ≥ 1, ∀ i = 1, . . . ,m, j = 1, . . . , n with Iij odd (3.7.11)

which implies that if Iij is odd then we must have at least one odd numbered beam-on time valuein a solution, as implemented by at least one non-zero dijhk variable value with h odd, and

K∑k′=1,k′ 6=k

dH2 −1e∑h=0

dij(2h+1)k′ ≥dH2 −1e∑h=1

dij(2h+1)k,

∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K with Iij even,

(3.7.12)

which implies that if Iij is even then if a shape matrix has an odd numbered associated beam-ontime value then there must be at least one other shape matrix with an odd numbered beam-ontime value to satisfy the even intensity cell (i, j).

We also investigate,

H∑h=1

dijhk ≤ 1, ∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K, (3.7.13)

which if considered in conjunction with our intensity constraint, (3.7.6), is a special case of theknapsack problem with partitioning side constraints. Whilst (3.7.13) is a redundant constraintwith respect to the linear relaxation of the BE model, in that constraints (3.7.3) and (3.7.4) imply(3.7.13), CPLEX may recognise this structure and utilise it to speed up computation time.

We apply the additional constraints of this section separately to the BE model formulation (andto the Step-up Method applied to the BE model formulation) to determine any effect they mayhave on computational efficiency.

The numerical results for all variations of the BE model are given in Tables 3.7.1 and 3.7.2. Theresults demonstrate that the variable e-based symmetry constraints are not as efficient as variable d-based symmetry constraints. Furthermore, the additional constraints, (3.7.11) and (3.7.12), appliedto the BE model with variable d-based symmetry constraints, since this is the more efficient versionof the BE model, do not improve computation time significantly, if at all. Finally, the addition ofthe redundant constraint, (3.7.13), did not alter the run time, the number of branch-and boundnodes nor the number of iterations when applied to the BE model. (These experiments are thereforenot included in Table 3.7.2). Hence, in the remainder of this chapter, when we refer to the BE



model, we refer to the BE model with variable d-based symmetry constraints and no ‘additional’constraints. Furthermore, the Step-up Method applied to all versions of the BE model significantlyimproves computation time, therefore whenever we consider the BE model in the remainder of thischapter, we also apply the Step-up Method to further reduce total computation time. It shouldalso be noted that the BE model increases in computation time more rapidly as the size of themaximum value in the intensity matrix increases and more slowly with increased problem size.(The BE model exhibits this characteristic due to its variables being indexed on radiation level).This is in contrast to the JS and URP models which increase in computation time both withincreasing problem size and maximum intensity level. We compare the BE model with the bestversions of the other models considered in this chapter in Section 3.9.

Table 3.7.1: Numerical results for the BE model with variable e-based symmetry constraints and theBE model with variable d-based symmetry constraints solving the BTCMC problem using CPLEXversion 8.1 and AMPL version 8.1 on a 2GHz AMD 64 3000+: time in seconds, 2-hour time limiton individual problem instances.

Batches

of 100

problems

BTCMC

BE model

with e-based

symmetry

w/o (3.7.11),

(3.7.12) and

(3.7.13)

BTCMC BE

model Step-

up version

with e-based

symmetry

w/o (3.7.11),

(3.7.12) and

(3.7.13)

BTCMC

BE model

with d-based

symmetry

w/o (3.7.11),

(3.7.12) and

(3.7.13)

BTCMC BE

model Step-

up version

with d-based

symmetry

w/o (3.7.11),

(3.7.12) and

(3.7.13)

4 4 0 15 33622.08 9624.98 10520.17 4667.09

5 5 0 5 176.69 58.03 138.10 57.15

5 5 0 10 34816.08 5576.22 10976.32 4220.96

6 6 0 5 1702.10 457.18 1211.26 632.73

Table 3.7.2: Numerical results for the BE model with variable d- based symmetry constraintsand various combinations of constraints (3.7.11) and (3.7.12) solving the BTCMC problem us-ing CPLEX version 8.1 and AMPL version 8.1 on a 2GHz AMD 64 3000+: time in seconds,2-hour time limit on individual problem instances.

Batches

of 100

problems

BTCMC

BE model

with

d-based

symmetry

w/o

(3.7.11),

(3.7.12)

and

(3.7.13)

BTCMC

BE model

Step-up

with

d-based

symmetry

w/o

(3.7.11),

(3.7.12)

and

(3.7.13)

BTCMC

BE model

with

d-based

symmetry

with

(3.7.11)

BTCMC

BE model

Step-up

with

d-based

symmetry

with

(3.7.11)

BTCMC

BE model

with

d-based

symmetry

with

(3.7.12)

BTCMC

BE model

Step-up

with

d-based

symmetry

with

(3.7.12)

BTCMC

BE model

with

d-based

symmetry

with

(3.7.11)

and

(3.7.12)

BTCMC

BE model

Step-up

with

d-based

symmetry

with

(3.7.11)

and

(3.7.12)

4 4 0 15 10520.17 4667.09 20038.00 4498.17 22585.03 5488.64 16888.36 4083.43

5 5 0 5 138.10 57.15 133.36 61.69 126.97 59.54 130.32 57.08

5 5 0 10 10976.32 4220.96 8322.61 3766.20 9829.69 4775.38 9015.34 11043.90

6 6 0 5 1211.26 632.73 1034.39 457.72 1185.89 1494.67 1035.31 514.99

7 7 0 5 29056.05 24460.84 36195.64 21363.66 32785.04 27310.62 35720.06 20708.44

3.8. The Counter Model 57

Prior to making our final comparisons, we first investigate the Counter model of Baatar et al. [4],another pseudo-polynomial sized model, and a variation of the Counter model which considerscumulative variables, in the following two sections.

3.8 The Counter Model

In the Counter model, Baatar et al. [4] define the following variables:

Nb ≥ 0 integer, ∀ b = 1, . . . , bmax, (3.8.1)

which indicate the number of shape matrices given radiation level b,

Qijb ≥ 0 integer, ∀ i = 1, . . . ,m, j = 1, . . . , n, b = 1, . . . , bmax, (3.8.2)

which indicate the number of shape matrices exposing cell (i, j) with radiation level b, and

Sijb ≥ 0 integer, ∀ i = 1, . . . ,m, j = 1, . . . , n, b = 1, . . . , bmax, (3.8.3)

which indicate the number of shape matrices exposing cell (i, j) with radiation level b, in excessof the number that expose cell (i, j − 1) with radiation level b, where bmax is an upper boundon the most radiation that can be applied to any shape matrix. In the current work we setbmax = max

i=1,...,m,j=1,...,nIij . In Chapter 5 we develop improved values for this parameter.

The intensity requirement for the Counter model utilises the Qijb variables directly. We have:

bmax∑b=1

bQijb = Iij , ∀ i = 1, . . . ,m, j = 1, . . . , n. (3.8.4)

To model the strict consecutive-1-property with the Qijb variables, on the other hand, is not soobvious. We know that the number of shape matrices exposing cell (i, j) and given radiation levelb, Qijb, can not exceed the number of shape matrices given radiation level b, Nb. However, thisrelationship is not sufficient to correctly count shape matrices. Instead we require that for anyradiation level b, and for each row i, the sum of the ‘step ups’ of the Qijb variables, captured bythe Sijb variables, does not exceed Nb. (We need only consider ‘step ups’ in Qijb variables sincethe sum of the ‘step ups’ across any row must be equal to the sum of the ‘step downs’). Therefore,to ensure that shape matrices satisfy the strict consecutive-1-property Baatar et al. [4] ask that

n∑j=1

Sijb ≤ Nb, ∀ i = 1, . . . ,m, b = 1, . . . , bmax. (3.8.5)

The ≤ sign is necessary since different rows may have different sums of ‘step ups’ and we need toapply the maximum of these. Furthermore, the Nb variables are bounded above when minimisedin our objective function (to follow). In essence, the underlying argument that Baatar et al. [4]use to justify that this is a valid model of the strict consecutive-1-property is that matrix Q canbe decomposed into a sum of Nb matrices having the strict consecutive-1-property each with unitmultiplicity. Then just as a single-row intensity matrix can be decomposed into consecutive onesmatrices with coefficients summing to the sum of the positive differences in intensity across therow, so Nb must be the smallest Nb satisfying (3.8.5). For a complete discussion of the Countermodel, see Baatar et al. [4].



The relationship required between the Sijb and Qijb variables is as follows:

Sijb =

{Qijb, j = 1max(Qijb −Qi(j−1)b, 0), j ≥ 2

∀ i = 1, . . . ,m, j = 1, . . . , n, b = 1, . . . , bmax.

(3.8.6)

Since (3.8.5) bounds the sum of Sijb variables from above, it is sufficient to ask that

Sijb ≥

{Qijb, j = 1max(Qijb −Qi(j−1)b, 0), j ≥ 2

∀ i = 1, . . . ,m, j = 1, . . . , n, b = 1, . . . , bmax

(3.8.7)

and this is easily modelled with linear constraints as follows:

Si1b = Qi1b, ∀ i = 1, . . . ,m, b = 1, . . . , bmax (3.8.8)

andSijb ≥ Qijb −Qi(j−1)b, ∀ i = 1, . . . ,m, j = 2, . . . , n, b = 1, . . . , bmax (3.8.9)

combined with the non-negativity of the Sijb variables.

Finally, to restrict the total beam-on time of the solution to equal Beammin the following constraintis applied:

bmax∑b=1

bNb = Beammin, (3.8.10)

and we minimise the number of shape matrices used in a solution via the objective

bmax∑b=1

Nb. (3.8.11)

Therefore the formulation for the Counter model, for solving the BTCMC problem, is:

min (3.8.11)

s.t. (3.8.4), (3.8.5), (3.8.8), (3.8.9), (3.8.10),

N ∈ Zbmax

+ , Q ∈ Zm×n×bmax

+ and S ∈ Zm×n×bmax

+ ,

or we introduce parameter K into the model and apply the Step-up Method to

minbmax∑b=1

bNb

s.t. (3.8.4), (3.8.5), (3.8.8), (3.8.9), (3.8.10),

bmax∑b=1

Nb = K, (3.8.12)

N ∈ Zbmax+ , Q ∈ Zm×n×bmax


+ ,

where the treatment of parameter K follows the Step-up Method rules (see Section 3.2).


We extend the formulation given by Baatar et al. [4] to investigate the effect of simple bounds onthe variables of the Counter model. We apply:

Nb ≤⌊Beammin

b

⌋, ∀ b = 1, . . . , bmax,

Qijb ≤⌊Iijb

⌋, ∀ i = 1, . . . ,m, j = 1, . . . , n, b = 1, . . . , bmax,

and

Sijb ≤⌊Iijb

⌋, ∀ i = 1, . . . ,m, j = 1, . . . , n, b = 1, . . . , bmax.

The first two bounds are straight forward, making use of the integrality of the variables, and thebound on the Sijb variables comes about by considering equation (3.8.6). We see that in the ‘worstcase scenario’ Sijb must be equal to a positive integer valued Qijb. Hence the bound on Qijb canbe used as a bound on Sijb.

The numerical results for the Counter model with and without the addition of the simple boundsabove are given in Table 3.8.1 at the end of Section 3.8.

3.8.1 The Cumulative Counter (CC) Model. In Chapter 5 we develop improved boundson beam-on time and beam-on time related variables. The cumulative version of the Counter model,which considers cumulative Nb variables, allows a direct application of the bounds we determine inChapter 5, whereas the Counter model requires additional constraints to be applied to the modelto incorporate the new bounds. Since bounds are handled more efficiently than constraints in theSimplex solving method, we investigate the cumulative version of the Counter model.

The Cumulative Counter (CC) model is as follows. We have variables

Nb ≥ 0 integer, ∀ b = 1, . . . , bmax,

which indicate the number of shape matrices given radiation level b or higher, where bmax =max

i=1,...,m,j=1,...,nIij in our current work, and the relationship between Nb and Nb is as follows:

Nb = Nb − Nb+1, ∀ b = 1, . . . , bmax − 1 and Nbmax = Nbmax .

Variables Qijb and Sijb, and constraints (3.8.4), (3.8.8) and (3.8.9) are as before in the Countermodel. Constraint (3.8.5) on the other hand is modified to

n∑j=1

Sijb ≤ Nb − Nb+1, ∀ i = 1, . . . ,m, b = 1, . . . , bmax, (3.8.13)

with Nbmax+1 = 0. Since variables Sijb are non-negative,n∑j=1

Sijb ≥ 0 and constraint (3.8.13)



enforce symmetry breaking on the Nb variables by implicitly creating the constraints Nb ≥ Nb+1

for all b = 1, . . . , bmax with Nbmax+1 = 0.

The total beam-on time constraint becomes:

bmax∑b=1

Nb = Beammin (3.8.14)

and the new objective function takes the form:

min N1. (3.8.15)

So the CC model is:

(3.8.15)

s.t. (3.8.4), (3.8.8), (3.8.9), (3.8.13), (3.8.14),

N ∈ Zbmax

+ , Q ∈ Zm×n×bmax


+ ,

or we introduce parameter K into the model and apply the Step-up Method to

minbmax∑b=1

Nb

s.t. (3.8.4), (3.8.8), (3.8.9), (3.8.13), (3.8.14),

N1 = K, (3.8.16)

N ∈ Zbmax+ , Q ∈ Zm×n×bmax


+ ,

where K is set following the Step-up Method rules.

We again investigate the effect of simple bounds on the variables of the CC model. We have:

Nb ≤⌊Beammin

b

⌋, ∀ b = 1, . . . , bmax, (3.8.17)

Qijb ≤⌊Iijb

⌋, ∀ i = 1, . . . ,m, j = 1, . . . , n, b = 1, . . . , bmax (3.8.18)

and

Sijb ≤⌊Iijb

⌋, ∀ i = 1, . . . ,m, j = 1, . . . , n, b = 1, . . . , bmax. (3.8.19)

The bounds on variables Qijb and Sijb follow directly from the Counter model and the bound onvariable Nb follows from constraint (3.8.14).

The numerical results for the CC model with and without the addition of the simple bounds aregiven in Table 3.8.1.

Table 3.8.1 indicates primarily that the Counter model and the CC model are extremely efficientand far superior to the other models we have tested in this chapter. This being the case, we nowtest the Counter and CC models to their limits. In particular, if the total run time for a batch of


Table 3.8.1: Numerical results for the Counter model and the CC model, with and without simplebounds on variables solving the BTCMC problem using CPLEX version 8.1 and AMPL version 8.1on a 2GHz AMD 64 3000+: time in seconds, 2-hour time limit on individual problem instances.

Batches

of 100

problems

BTCMC

Counter

model

w/o

simple

bounds

BTCMC

Counter

model

Step-up

version

w/o

simple

bounds

BTCMC

Counter

model

with

simple

bounds

BTCMC

Counter

model

Step-up

version

with

simple

bounds

BTCMC

CC

model

w/o

simple

bounds

BTCMC

CC

model

Step-up

version

w/o

simple

bounds

BTCMC

CC

model

with

simple

bounds

BTCMC

CC

model

Step-up

version

with

simple

bounds

4 4 0 5 0.98 1.77 1.42 1.83 1.28 1.99 1.51 1.99

4 4 0 10 5.38 7.16 5.44 7.38 7.77 8.09 7.35 8.42

4 4 0 15 21.99 39.02 24.87 46.44 40.29 42.11 35.91 51.49

5 5 0 5 1.92 2.95 2.01 3.15 2.33 3.09 2.20 3.39

5 5 0 10 14.81 26.60 14.92 27.18 22.51 27.81 25.00 26.55

5 5 0 15 112.13 400.64 137.73 2681.01 507.20 972.51 355.84 1034.34

6 6 0 5 3.51 5.04 3.43 8.07 3.92 5.86 3.46 5.64

6 6 0 10 53.81 166.75 61.41 527.70 334.67 376.96 151.41 351.86

6 6 0 15 2019.03 29853.84 2560.16 64673.63 36526.30 63437.60 15483.70 49942.73

7 7 0 5 6.32 8.48 5.41 9.40 7.98 9.76 6.23 10.07

7 7 0 10 7960.63 54025.91 5140.46 54923.37 16589.30 76652.67 17745.40 56240.59

7 7 0 15 61834.40 380590.05 55119.10 481764.03 188568.00 413982.22 177213.00 428292.52

8 8 0 5 11.98 23.47 8.07 32.27 16.58 120.24 13.27 60.22

8 8 0 10 22006.60 220195.08 24041.30 293719.83 61665.20 274225.86 66811.50 242451.26

8 8 0 15 377181.00 328547.00

9 9 0 5 14.62 27.29 12.39 66.09 23.63 635.01 14.58 43.25

9 9 0 10 150764.00 128140.00 304153.00 297533.00

9 9 0 15

10 10 0 5 86.08 9365.02 21.60 12543.78 2199.48 29136.31 87.25 23047.67

10 10 0 10

10 10 0 15

11 11 0 5 7267.89 61162.81 221.55 54772.68 18100.10 100154.20 1620.69 56360.39

12 12 0 5 8785.24 133111.30 59.62 122444.23 36804.30 20846.00 144991.17

13 13 0 5 15171.10 7384.16 117493.00 63777.80

14 14 0 5 26438.10 17090.40 83977.00

15 15 0 5 48435.70 33738.30 138135.00

16 16 0 5 75396.80 59622.90

17 17 0 5 101532.00 30842.70

18 18 0 5 100525.00

4 8 0 15 4351.14 79763.96 2610.16 177346.29 19746.80 101828.31 21639.90 110065.96

8 4 0 15 84.39 690.36 109.19 417.56 190.13 437.53 539.60 995.45



100 problems exceeds at least 100,000 seconds, no further batches with higher dimension and thesame maximum intensity level are run for the model under consideration.

It is clear from Table 3.8.1 that computation times increase more rapidly with increasing maximumintensity level than with increasing problem size (which is in keeping with the fact that both theCounter and CC models are pseudo-polynomial in size, indexing their variables on radiation level)and that the Step-up versions of the models do not outperform the standard versions. The Step-upversion of the Counter model and the CC model includes the computation time needed to run theGreedy Heuristic Algorithm of Baatar et al. [1] to determine the Kbm stopping condition for theStep-up Method. For small problem sizes, this time component is significant when considering thesmall numbers returned by the Counter and CC models themselves. The standard Counter modeland CC model, without the application of the Step-up Method, do not require a parameter, large‘K’, and hence this computation time is saved for the standard versions. For larger problems, itis the Step-up Method application itself that is the major reason for the increase in computationtime in the Counter model and the CC model, due to the increase in the number of sub-problemsto be solved.

Table 3.8.1 also demonstrates that the inclusion of simple bounds on the variables of the Countermodel and the CC model significantly improves computation time for the standard versions ofthe models, that is without the Step-up Method applied. For the standard Counter model, forsmall sized problems, the simple bounds tend to increase computation time, however overall, andin particular for larger matrices with small maximum intensity level, the simple bounds decreasecomputation time greatly. For the standard CC model, computation time generally decreasesacross all problems when simple bounds are applied and again in some cases by a large margin.Total computation time over the batches of problems given in Table 3.8.1, where both standardmodels were solved with and without simple bounds, decreases by approximately 59 hours for thestandard Counter model and by approximately 33 hours for the standard CC model, when simplebounds are applied.

When the Step-up Method is applied with simple bounds to the Counter and CC models, thetotal computation time increases for the Counter model by approximately 82 hours and for theCC model decreases by approximately 26 hours when compared with the Counter model appliedwith the Step-up Method and the CC model applied with the Step-up Method respectively. Whilstcomputation time improves for the application of simple bounds to the Step-up Method version ofthe CC model, the total computation time still does not compare with the total computation timefor the standard CC model with simple bounds.

We also test batches of intensity matrices of size 4 × 8 and 8 × 4 each with intensities rangingfrom 0 to 15 with the Counter and CC models. The first number in the problem dimension is thenumber of rows and the second is the number of columns. Although we only test two batches of 100problems, it would appear that it is more difficult to solve the BTCMC problem when the intensitymatrix has a smaller number of rows and larger number of columns than when the intensity matrixhas a larger number of rows and a smaller number of columns.

Finally, again considering batches of problems given in Table 3.8.1 where both the Counter model

3.9. Comparison of Models 63

and the CC model with simple bounds are solved, we see that the Counter model with simplebounds takes approximately 175 hours less total computation time than the CC model with simplebounds.

In Section 3.9, to follow, we present a final comparison of the best models investigated in thischapter.

3.9 Comparison of Models

Table 3.9.1 shows the best versions of the JS model, the URP model, the BE model and theCounter model over batches of 100 problems. We now test each of these models to their limits:again, when the total run time for a batch exceeds at least 100,000 seconds, we no longer testfurther batches with the same maximum intensity level for the model under consideration. In thisway, the table clearly shows where one model improves on another in terms of problem size andmaximum intensity level.

Table 3.9.1 demonstrates that depending on the model chosen to solve the BTCMC problem thereis great variability in the computation times that may result when solving a batch of problems.For example, a batch of size 7× 7 with intensities ranging from 0 to 5 takes over 300,000 secondsto run with the JS model but only 5 seconds to run using the Counter model with simple bounds.We again see in this table that the JS and URP-xSC models increase in computation time withproblem size and maximum intensity level, whereas the BE model and Counter model with simplebounds are more affected by increases in maximum intensity level than in problem dimension.With models no longer being tested once computation time for a batch exceeds 100,000 seconds foreach maximum intensity level considered, we see that within this ‘limit’, the JS model, the Step-upMethod applied to the JS model and the URP-xSC model can solve problems of dimension 7× 7with intensities ranging from 0 to 5, and when considering maximum intensity level, problems ofdimension 5× 5 with intensities ranging from 0 to 15, 6× 6 with intensities ranging from 0 to 15and 5 × 5 with intensities ranging from 0 to 15 respectively. The Step-up Method applied to theURP-xSC model is particularly inefficient, only able to solve problems of size 6×6 with intensitiesranging from 0 to 5 and problems of size 4 × 4 with intensities ranging from 0 to 15 within the100,000 second ‘limit’. The BE model performs slightly better than the JS and URP-xSC modelsand is able to solve problems of size 8 × 8 with intensities ranging from 0 to 5 and problems ofsize 5× 5 with intensities ranging from 0 to 15, with both the standard BE model and the Step-upMethod applied to the BE model. Therefore the polynomial sized models (JS and URP-xSC)are outperformed by the pseudo-polynomial sized models (BE and Counter with simple bounds)considered in this chapter. We also see that in general the JS model and the BE model improvewith the application of the Step-up Method but that the URP-xSC model and the Counter modelwith simple bounds do not.

The Counter model with simple bounds is the best of all models tested in this chapter by asignificant margin. The next best model is the CC model with simple bounds, however in thissection we compare only the best versions of the four core models we have studied. (The CCmodel with simple bounds is a variation of the Counter model with simple bounds). The Counter



model with simple bounds outperforms the BE model (which is the best of the JS model, the URP-xSC model and the BE model) by approximately 230 hours of computation time for the standardmodels and approximately 160 hours of computation time for the application of the Step-up Method(to both models), over batches of problems solved by both models as shown in Table 3.9.1. Whenwe compare the Counter model with simple bounds to the JS model, the Step-up Method appliedto the JS model, the URP-xSC model and the Step-up Method applied to the URP-xSC model,over the batches of problems where both models under consideration were solved, we see that theCounter model with simple bounds achieves time savings of approximately 331 hours, 325 hours,270 hours and 262 hours respectively. For a direct comparison involving each of the best modelsagainst the Counter model with simple bounds, we provide a ‘summary’ table, Table 3.9.2, whichdetails the total time saved in hours when using the Counter model with simple bounds versusthe other best models, where we only consider problem batches where all models were solved.The specific problem sizes we consider are the batches solved with the URP-xSC model when theStep-up method is applied as shown in Table 3.9.1. We also include the CC model with simplebounds in the summary table.

Again, Table 3.9.2 demonstrates that the Counter model with simple bounds is the best performingmodel we have considered in this chapter, followed by the CC model with simple bounds, the Step-up Method applied to the Counter model with simple bounds, the Step-up Method applied to theCC model with simple bounds, the Step-up Method applied to the BE model, the Step-up Methodapplied to the JS model, the BE model, the URP-xSC model, the JS model and the Step-upMethod applied to the URP-xSC model.

Finally, we also test 42 medical data sets obtained from Baatar [49] on the Counter model withsimple bounds. The results are shown in Table 3.9.3. We removed any zero rows and columns fromthe medical data sets before the trial and the sizes of the amended problems are given in Table3.9.3.


Table 3.9.1: Numerical results for the JS model, the URP-xSC model, the BE model and theCounter model with simple bounds solving the BTCMC problem using CPLEX version 8.1 andAMPL version 8.1 on a 2GHz AMD 64 3000+: time in seconds, 2-hour time limit on individualproblem instances.

Batches

of 100

problems

BTCMC

JS

model

BTCMC

JS

model

Step-up

BTCMC

URP-

xSC

model

BTCMC

URP-

xSC

model

Step-up

BTCMC

BE

model

BTCMC

BE

model

Step-up

BTCMC

Counter

model

with

simple

bounds

BTCMC

Counter

model

with

simple

bounds

Step-up

4 4 0 5 8.23 3.09 13.08 7.55 17.93 6.89 1.42 1.83

4 4 0 10 37.72 13.17 306.21 4862.13 536.47 268.32 5.44 7.38

4 4 0 15 329.33 107.21 14454.30 188032.53 10520.17 4667.09 24.87 46.44

5 5 0 5 205.35 18.73 572.33 8173.95 138.10 57.15 2.01 3.15

5 5 0 10 43832.30 1788.22 25065.50 483711.63 10976.32 4220.96 14.92 27.18

5 5 0 15 260609.65 83603.06 333759.00 275807.21 157354.33 137.73 2681.01

6 6 0 5 96619.01 17023.71 4537.95 257398.91 1211.26 632.73 3.43 8.07

6 6 0 10 447870.26 268304.83 412880.00 246147.50 155001.58 61.41 527.70

6 6 0 15 574389.47 2560.16 64673.63

7 7 0 5 342394.16 226664.03 181456.00 29056.05 24460.84 5.41 9.40

7 7 0 10 5140.46 54923.37

7 7 0 15 55119.10 481764.03

8 8 0 5 253242.70 234088.67 8.07 32.27

8 8 0 10 24041.30 293719.83

8 8 0 15 328547.00

9 9 0 5 12.39 66.09

9 9 0 10 128140.00

9 9 0 15

10 10 0 5 21.60 12543.78

10 10 0 10

10 10 0 15

11 11 0 5 221.55 54772.68

12 12 0 5 59.62 122444.23

13 13 0 5 7384.16

14 14 0 5 17090.40

15 15 0 5 33738.30

16 16 0 5 59622.90

17 17 0 5 30842.70

18 18 0 5 100525.00

4 8 0 15 2610.16 177346.29

8 4 0 15 109.19 417.56



Table 3.9.2: Summary of numerical results for the JS model, BE model, URP-xSC model and CCmodel with simple bounds compared with the Counter model with simple bounds, over batches ofproblems where all models were solved as given in Table 3.9.1. We use CPLEX version 8.1 andAMPL version 8.1 on a 2GHz AMD 64 3000+: 2-hour time limit on individual problem instances.

Approximate Total Time Saved in Hours using the Counter model with simple bounds compared with the ...

JS model 39

Step-up Method applied to the JS

model

5

URP-xSC model 12

Step-up Method applied to the URP-

xSC model

262

BE model 6

Step-up Method applied to the BE

model

2

Step-up Method applied to the

Counter model with simple bounds

0.01

CC model with simple bounds 0.01

Step-up Method applied to the CC

model with simple bounds

0.01


Table 3.9.3: Numerical results for the Counter model with simple bounds using medical data sets solving the BTCMC

problem using CPLEX version 8.1 and AMPL version 8.1 on a 2GHz AMD 64 3000+: time in seconds, 2-hour time

limit on individual problem instances, BB = number of branch and bound nodes, ITS = number of simplex iterations.

Problem BTCMC Counter model with simple bounds

Beammin Objective BB ITS Time

1 9 9 0 10 17 8 875 9743 2.20

2 9 9 0 10 20 8 1456 18247 2.54

3 9 10 0 10 19 7 14420 175014 14.79

4 9 10 0 35 35 time limit >764911 >28264638 >7213.38

5 9 10 0 40 59 time limit >1355668 >37517762 >7211.93

6 9 12 0 29 46 time limit >535658 >29752358 >7219.76

7 9 12 0 31 45 time limit >814932 >39579540 >7211.39

8 9 13 0 29 45 time limit >756400 >38210284 >7211.91

9 10 9 0 10 18 7 34358 287591 29.15

10 10 9 0 10 15 7 407 7091 1.18

11 10 9 0 10 16 7 288 4540 0.77

12 10 10 0 10 16 7 6200 98894 7.31

13 10 14 0 26 49 12 3355 75840 7.94

14 11 7 0 21 21 time limit >5044493 >45700485 >7219.72

15 11 9 0 14 23 8 484047 7848937 820.79

16 11 8 0 16 20 7 1610 27346 3.30

17 11 11 0 22 28 time limit >4602416 >44954692 >7215.84

18 11 10 0 16 19 9 3452 42005 8.37

19 11 11 0 19 26 time limit >6058673 >32881453 >7221.52

20 11 11 0 26 43 time limit >2789912 >36751685 >7221.76

21 11 13 0 22 24 10 2625 59511 7.72

22 14 10 0 10 22 time limit >2549013 >55593485 >7216.71

23 14 10 0 10 26 9 495899 5565744 747.18

24 14 10 0 10 22 time limit >8871620 >32121809 >7215.48

25 14 10 0 10 23 9 1757 36485 6.47

26 14 10 0 10 23 9 120805 1497856 184.44

27 15 10 0 10 22 time limit >5602011 >36666952 >7227.34

28 15 28 0 9 12 9 1823 23256 12.18

29 16 27 0 10 12 9 182 5537 5.28

30 16 28 0 10 10 7 106 5345 4.90

31 16 28 0 10 11 time limit >1387090 >23995076 >7220.20

32 16 29 0 10 13 time limit >1876547 >33481347 >7215.61

33 16 30 0 10 12 8 24846 587462 154.26

34 20 23 0 10 11 6 36 737 1.06

35 20 25 0 9 17 11 335 10910 5.44

36 22 14 0 26 33 time limit >969120 >19334580 >7291.43

37 22 18 0 31 41 time limit >1222424 >17964628 >7315.83

38 22 20 0 31 50 time limit >523913 >20631107 >7273.47

39 22 21 0 22 47 time limit >657822 >18136108 >7204.65

40 22 23 0 24 33 time limit >421294 >23698015 >7250.25

41 23 15 0 33 35 time limit >610605 >23943989 >7271.56

42 23 17 0 27 46 time limit >783792 >20200814 >7305.85



Table 3.9.3 shows the value for Beammin, the value of the objective, the number of branch andbound nodes searched, iterations and total time for each problem. The notation used for each dataset gives the data number, the row dimension, column dimension, minimum intensity value andmaximum intensity value respectively. In general the Counter model with simple bounds eithersolves the medical data extremely quickly or the computation times out taking longer than ourtime limit of 2 hours on individual problem instances. Exactly half of the medical data we trialsolves within the time frame.

3.10 Conclusion

In this chapter we have presented four integer programming models (3 of which are new) for solvingthe Beam-on Time Constrained Minimum Cardinality problem. We considered polynomial sizedmodels, the JS model and the URP model, and pseudo-polynomial sized models, the BE modeland Counter model of Baatar et al. [4]. The JS model is the BTCMC version of the integerTTT formulation of Chapter 2, the URP model is an improved formulation of the Langer et al. [3]model and the BE model is based on the idea of Baatar et al. [4] for indexing variables on radiationlevel. All models we have presented can be amended to solve the constrained BTCMC problem(provided the appropriate MLC mechanical constraints can be formulated) since the models andsolution approaches presented in this chapter do not preclude their inclusion. Again, in general,the solutions and computation times for the constrained versions of the integer programs discussedin this chapter will be different from the unconstrained results presented.

We investigated different approaches for reducing the computation time of our exact models:varying the formulation of the strict consecutive-1-property, trialling many and varied symme-try breaking constraints, applying the Step-up Method of Chapter 2, Section 2.3.1 and utilisingsimple bounds on variables within models.

We also developed a new formulation based on the Counter model which utilised cumulative vari-ables for the number of shape matrices given a particular radiation level or more. Simple boundswere applied to the Counter and Cumulative Counter models, with both models demonstratingsignificant improvement in computation time with the application. However the application of theStep-up Method to the Counter and Cumulative Counter models did not decrease computationtime further. Neither did the Step-up Method improve computation time for the URP model. Onthe other hand, significant improvement in computation time was seen for the JS and BE modelswith the application of the Step-up Method.

The best versions of each model type (amongst the variations tested), with regard to the formula-tion of the strict consecutive-1-property, the particular symmetry breaking constraints utilised, theapplication of the Step-up Method and the inclusion of simple variable bounds, were numericallycompared. The best model overall was the Counter model of Baatar et al. [4] with our extensionto incorporate simple bounds on variables. The Counter model with simple bounds took approx-imately 230 hours less computation time than the standard BE model, approximately 331 hoursless computation time than the standard JS model and approximately 270 hours less computationtime than the standard URP-xSC model, over batches of problems where both models were solved.

3.10. Conclusion 69

The CC model with simple bounds was the next best model investigated, followed by the Step-upMethod applied to the Counter model with simple bounds, the Step-up Method applied to the CCmodel with simple bounds, the Step-up Method applied to the BE model, the Step-up Method ap-plied to the JS model, the BE model, the URP-xSC model, the JS model and the Step-up Methodapplied to the URP-xSC model.

We conclude that the pseudo-polynomial sized models, the Counter, CC and BE models, outper-form the polynomial sized models, the JS and URP-xSC models. Both types of model eventuallybecome too large to solve (even the Counter model with simple bounds was unable to solve half theavailable medical data within a 2 hour individual problem time limit) and therefore in Chapters4 and 5 we continue to investigate both polynomial sized models and pseudo-polynomial sizedmodels, considering their different properties, to further reduce computation time for both modeltypes. Furthermore, it is not clear whether a ‘Counter model type’ formulation is directly appli-cable to the cutting stock application, from which the JS model is derived. Hence it is importantthat we continue to investigate the JS model (the Step-up Method applied to the JS model is thebest of the polynomial sized models we have considered in this chapter) and to further improve itscomputational efficiency.

As mentioned, whilst the Counter model with simple bounds has been shown to solve certain ‘realworld’ problems, in general the exact formulations we consider are still unable to solve the entirespectrum of medical data in reasonable time. To address this issue, in Chapter 4 we investigatethe polyhedral structure of particular constraints in the JS model and apply the facets that resultto determine computational improvement. In Chapter 5 we determine improved bounds for thevariables of the JS and CC model formulations in preprocessing routines to reduce the sizes ofthe feasible regions to be solved. The promising results achieved with the application of simplebounds to the Counter and CC models in this chapter motivates the extension of this work inChapter 5. We focus on the CC model with simple bounds rather than the Counter model withsimple bounds in Chapter 5 since the CC model formulation has similar properties to the JS model(which we discuss later) and allows a direct application of the bounds we determine. On the otherhand, the Counter model with simple bounds would require additional constraints to be applied toincorporate the new bounds. This distinction is important as bounds are handled more efficientlythan constraints in the Simplex solving method. (The CC model with simple bounds is also thesecond best performing model considered in this chapter). Finally, we mention that the applicationof improved bounds to the JS and CC models in Chapter 5 results in significant computationalsavings.



71CHAPTER 4

Polyhedral Analysis of the Equality Switch Polytope

4.1 Introduction

In this chapter we investigate the best of the polynomial sized models considered in Chapter 3for solving the Beam-on Time Constrained Minimum Cardinality problem with the aim of furtherreducing the computation time for its exact solution. We now explore the technique of polyhedralanalysis for reducing the search space of the JS model.

The JS model is essentially a weak model in that three of the constraints in the formulationcontain ‘big-M’ parameters. The ‘big-M’ parameters are Hk for all k = 1, . . . ,K, Mijk for alli = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K and Gijk for all i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K,as given in constraints (2.2.6), (2.2.7) and (2.2.9) respectively, in Chapter 2, Section 2.2. Modelscontaining ‘big-M’ parameters are weak in the sense that their linear programming relaxationsare a long way from their integer programs and hence these types of models may also have largesolution computation times.

In this chapter, we focus on the three constraints on variables aijk of the JS model, (2.2.7), (2.2.8)and (2.2.9), which require that all the aijk variable values for the same k are equal to bk or 0,where constraints (2.2.7) and (2.2.9) utilise ‘big-M’ parameters to achieve this. The relationshipwe require can be illustrated with four variables as follows:

{(xijk, aijk, xhlk, ahlk) :0 ≤ aijk ≤ Iij ,0 ≤ ahlk ≤ Ihl,xijk, xhlk ∈ {0, 1},if xijk = 0 then aijk = 0,if xhlk = 0 then ahlk = 0,xijk = xhlk only if aijk = ahlk},

for all i, h = 1, . . . ,m, j, l = 1, . . . , n and k = 1, . . . ,K.

Investigating the linear relaxation of the JS model, we determine that the model ‘cheats’ by settingthe aijk values for the same k and different i’s and j’s to different positive values rather than thesame positive value, bk, or 0. To solve the linear relaxation of the JS model, the xijk and sk

variables are altered to real variables greater than or equal to 0 and less than or equal to 1 andthe bk and aijk variables are altered to real variables greater than or equal to 0. An example ofthe linear relaxation of the JS model ‘cheating’ is given below for data set 4 4 0 5 1 (a randomlygenerated problem of size 4× 4 with intensity ranging from 0 to 5 inclusive and problem number1):

(Iij) =

0 3 2 03 5 0 43 4 2 10 1 1 1

= (aij1) + (aij2) + (aij3) + (aij4) =

72 Chapter 4. Polyhedral Analysis of the Equality Switch Polytope

0 0 1.2 0

1.8 3 0 1.61.2 2.4 1.2 0.50 0.5 0.5 0.5

+

0 1.2 0.8 0

1.2 2 0 2.41.8 1.6 0.8 00 0 0 0.5

+

0 1.8 0 00 0 0 00 0 0 0.50 0.5 0.5 0

+

0 0 0 00 0 0 00 0 0 00 0 0 0

,

b1 = b2 = b3 = 3.

We also give the same example with the integrality conditions restored:

(Iij) =

0 3 2 03 5 0 43 4 2 10 1 1 1

= (aij1) + (aij2) + (aij3) + (aij4) =

0 0 0 00 0 0 40 0 0 00 0 0 0

+

0 3 0 03 3 0 03 3 0 00 0 0 0

+

0 0 1 00 1 0 00 1 1 00 1 1 1

+

0 0 1 00 1 0 00 0 1 10 0 0 0

,

b1 = 4, b2 = 3, b3 = 1, b4 = 1.

We clearly see that in the linear relaxation, the aijk variable values for the same k are not equalas they should be.

This being the case we investigate the Equality Switch Polytope (ESP ) which describes the convexhull of the generalisation of our 4 variable illustration above. (The convex hull of a set of pointsis the minimal convex set containing the set of points). We choose to analyse this simplifiedpolyhedral structure, rather than the polyhedron describing the entire JS model, since facetsshould be more easily determined and for the possibility that the simplified polyhedron may beapplied with broader application. The Equality Switch Polytope is defined in Section 4.2, where,without loss of generality, we drop subscripts i and k from the definition. We then seek facetsof the Equality Switch Polytope to apply to a model of ESP itself, and to the JS model in anattempt to prevent the differing aijk variable values occurring in the same matrix and to reducethe size of the feasible region for the JS model. The complete list of facets we determine for ESPis given in Section 4.6.

Following our investigation of ESP , we add the appropriate form of constraint (2.2.11), whichenforces the strict consecutive-1-property (C1) in the JS model, to the definition of ESP andconsider a new polytope which we name the ESP -C1 polytope. We consider facets of ESP -C1 under special conditions and apply these to a model of ESP -C1 and to the JS model. Weinvestigate ESP -C1 since its feasible region should more closely resemble the feasible region of theJS model itself and therefore the facets determined should be more effective in reducing the searchspace. The definition of ESP -C1 and a list of corresponding facets is given in Section 4.7.

A search of the mathematical programming literature suggests that the ESP and ESP -C1 poly-topes are yet to be investigated, however the C1 polytope (where C1 may have a more general

4.2. The Equality Switch Polytope (ESP ) 73

definition than that considered in this work) has been studied extensively, see for example [45, 46],since the consecutive-1-constraint also exists in other mathematical and physical applications suchas in the context of unit commitment problems [50, 51], block cave mining problems [7] and stopsign design problems [1].

Numerical results for the application of facets to the ESP , ESP -C1 and JS models are given inSections 4.8 and 4.9 respectively. We compare the number of branch and bound nodes, iterations,computation time, root node lower bound value and, in the case of the JS model, the number ofconstraints with and without the addition of facets. Finally, whilst we focus on solving the Beam-on Time Constrained Minimum Cardinality problem with the JS model, as we do in Chapter 3,ESP and ESP -C1 are derived independently of the objective used in the JS model and of theconstraint which minimises total beam-on time. We mention the particular problem we solve withthe JS model, BTCMC, simply so that our numerical results can be replicated.

4.2 The Equality Switch Polytope

We define the Equality Switch Polytope as follows:

ESP = conv(ES),

where

ES = {(x, a) ∈ {0, 1}n × Rn+,0 : ap ≤ Ipxp, for all p = 1, . . . , n andif for some p, q ∈ {1, . . . , n}, p 6= q,

xp = xq = 1 then ap = aq}.

(4.2.1)

I ∈ Rn+ is given and we have used the notation, Rn+,0, to mean the n-dimensional set of positiveReals including zero.

Alternatively we can write:

ES′ =

∑p∈S

ep, α∑p∈S

ep

: S ⊆ {1, . . . , n}; 0 ≤ α ≤ minp∈S

Ip

, (4.2.2)

where ep ∈ Rn is the pth unit vector.

The definition for ES clearly defines the points in ESP by directly stating the conditions/constraintsthat must be satisfied by the variables of the JS model and therefore this is a very natural wayto describe the substructure of the model we are interested in. However, the benefit of the ES′

definition is that with this description we can easily construct points and check that they belongin ES and hence more easily prove that any resulting constraints are valid and facet defining.

In order to prove formally that the two definitions are equivalent, i.e. that ES = ES′, we firstprove the following lemma which is also useful in subsequent polyhedral analysis.

Lemma 4.2.1. If (x, a) ∈ ES, x 6= 0, and S = {p ∈ {1, . . . , n} : xp = 1} then ap ≤ minq∈S

Iq for all

p = 1, . . . , n.


Proof. Assume to the contrary. Suppose there exists an ap > minq∈S

Iq = ar. Then from our

definition of ES, ar > 0 implies that xr = 1 and ap > 0 implies that xp = 1. The definition of ESalso states that if xr = xp = 1 then ar = ap, which is a contradiction. Hence ap ≤ min

q∈SIq for all

p = 1, . . . , n as required and Lemma 4.2.1 holds.

We now give the proposition.

Proposition 4.2.1. The two definitions (4.2.1) and (4.2.2) for ESP are equivalent, i.e. ES = ES′.

Proof. We first show that ES ⊆ ES′. Let (x, a) ∈ ES. We construct S = {p ∈ {1, . . . , n} :xp = 1} and set α = max

p∈{1,...,n}ap. Note that by the definition of ES the values of ap are equal

for all p ∈ S (and ap = 0 is enforced if p 6∈ S), so in fact α = ap for all p ∈ S and α = 0 if S is

empty. We claim that (x, a) =

∑p∈S

ep, α∑p∈S

ep

and hence (x, a) ∈ ES′ as required. This is not

difficult to show. First, x =∑p∈S

ep is obvious from the definition of S. Now for each p = 1, . . . , n,

either xp = 0 or xp = 1, since x ∈ {0, 1}n by the definition of ES. If xp = 0 then p /∈ S, and, since0 ≤ ap ≤ Ipxp from the definition of ES, it must be that ap = 0. On the other hand, if xp = 1then p ∈ S, and ap = α by the definition of α and the concomitant discussion. Thus ap = α ifp ∈ S and ap = 0 if p 6∈ S, i.e. a = α

∑p∈S

ep as required. Finally by Lemma 4.2.1 we conclude

ES ⊆ ES′.

Now to show ES′ ⊆ ES let S ⊆ {1, . . . , n} and α ∈ [0,minp∈S

Ip] and set x =∑p∈S

ep and a = α∑p∈S

ep.

By this construction, it is obvious that x ∈ {0, 1}n and that a ≥ 0. Furthermore if xp = 0 thenp 6∈ S so ap = 0 and ap ≤ Ipxp is satisfied. On the other hand, if xp = 1 then ap = α ≤ Ip by thedefinition of α and since p ∈ S. Thus again we have ap ≤ Ipxp satisfied. Finally, if p, q ∈ {1, . . . , n},p 6= q and xp = xq = 1, then p, q ∈ S and so ap = α and aq = α by the definition of a and soap = aq. Hence (x, a) ∈ ES as required. We conclude ES′ ⊆ ES.

The above proposition confirms that ESP = conv(ES) = conv(ES′) and hence the two definitions(4.2.1) and (4.2.2) for ESP are equivalent.

For polyhedral analysis, it is convenient to assume that Ip > 0 for all p = 1, . . . , n. (Clearly anyvariable for which this is not the case can be fixed to zero and removed from the polytope.)

Proposition 4.2.2. ESP is a full-dimensional polytope.

Proof. We must demonstrate that, since ESP is defined over 2n variables, there are 2n+1 affinelyindependent vectors in ESP .

Clearly {(ep, 0) : p = 1, . . . , n} ∪ {(ep, Ipep) : p = 1, . . . , n} ∪ {(0, 0)} ⊆ ESP are 2n + 1 affinelyindependent vectors. (If we delete {(0, 0)} from the set of vectors, the remaining vectors are linearlyindependent.)

4.3. An Example of Points Satisfying ESP and the Corresponding Facets of ESP 75

4.3 An Example of Points Satisfying ESP and the Corresponding Facets of ESP

4.3.1 Example 1. Consider the following example of dimension 4 with intensity valuesI1 = 7, I2 = 4, I3 = 12 and I4 = 21. The corresponding complete set of points satisfying ESP aregiven in Table 4.3.1. Note that the differences, and sums of pairs, of Ip values for p = 1, . . . , 4 are3, 5, 8, 9, 11, 14, 16, 17, 19, 25, 28 and 33 which are different to the Ip values themselves. This isnecessary to recognise patterns in the equations for facets.

Utilising the web-based tool Polymake [52], the facets of the Equality Switch Polytope resultingfrom the points in Example 1 are:

4.3.1.1 Facets containing just one a variable

a1 ≥ 0, (4.3.1)

a2 ≥ 0, (4.3.2)

a3 ≥ 0, (4.3.3)

a4 ≥ 0, (4.3.4)

4.3.1.2 Facets containing just one x variable

1− x1 ≥ 0, (4.3.5)

1− x2 ≥ 0, (4.3.6)

1− x3 ≥ 0, (4.3.7)

1− x4 ≥ 0, (4.3.8)

4.3.1.3 Facets containing one x variable and one a variable

7x1 − a1 ≥ 0, (4.3.9)

4x2 − a2 ≥ 0, (4.3.10)

12x3 − a3 ≥ 0, (4.3.11)

21x4 − a4 ≥ 0, (4.3.12)

4.3.1.4 Facets containing one x variable and two a variables: ‘2a-any-x’

4− 4x1 + a1 − a2 ≥ 0, (4.3.13)

12− 12x1 + a1 − a3 ≥ 0, (4.3.14)

21− 21x1 + a1 − a4 ≥ 0, (4.3.15)

7− 7x2 − a1 + a2 ≥ 0, (4.3.16)

12− 12x2 + a2 − a3 ≥ 0, (4.3.17)


Table 4.3.1: Example 1: points satisfying ESP .

x1 x2 x3 x4 a1 a2 a3 a4

0 0 0 0 0 0 0 01 0 0 0 0 0 0 01 0 0 0 7 0 0 00 1 0 0 0 0 0 00 1 0 0 0 4 0 00 0 1 0 0 0 0 00 0 1 0 0 0 12 00 0 0 1 0 0 0 00 0 0 1 0 0 0 211 1 0 0 0 0 0 01 1 0 0 4 4 0 01 0 1 0 0 0 0 01 0 1 0 7 0 7 01 0 0 1 0 0 0 01 0 0 1 7 0 0 70 1 1 0 0 0 0 00 1 1 0 0 4 4 00 1 0 1 0 0 0 00 1 0 1 0 4 0 40 0 1 1 0 0 0 00 0 1 1 0 0 12 121 1 1 0 0 0 0 01 1 1 0 4 4 4 01 0 1 1 0 0 0 01 0 1 1 7 0 7 71 1 0 1 0 0 0 01 1 0 1 4 4 0 40 1 1 1 0 0 0 00 1 1 1 0 4 4 41 1 1 1 0 0 0 01 1 1 1 4 4 4 4


21− 21x2 + a2 − a4 ≥ 0, (4.3.18)

7− 7x3 − a1 + a3 ≥ 0, (4.3.19)

4− 4x3 − a2 + a3 ≥ 0, (4.3.20)

21− 21x3 + a3 − a4 ≥ 0, (4.3.21)

7− 7x4 − a1 + a4 ≥ 0, (4.3.22)

4− 4x4 − a2 + a4 ≥ 0, (4.3.23)

12− 12x4 − a3 + a4 ≥ 0, (4.3.24)

4.3.1.5 Facets containing one x variable and three a variables: ‘3a-any-x’

12− 12x1 + 3a1 − 2a2 − a3 ≥ 0, (4.3.25)

21− 21x1 +214a1 −

174a2 − a4 ≥ 0, (4.3.26)

28− 28x1 +73a1 − a3 −

43a4 ≥ 0, (4.3.27)

845− 84

5x2 − a1 +

125a2 −

75a3 ≥ 0, (4.3.28)

21− 21x2 − 2a1 + 3a2 − a4 ≥ 0, (4.3.29)

28− 28x2 +73a2 − a3 −

43a4 ≥ 0, (4.3.30)

283− 28

3x3 −

43a1 − a2 +

73a3 ≥ 0, (4.3.31)

21− 21x3 − 2a1 + 3a3 − a4 ≥ 0, (4.3.32)

21− 21x3 −174a2 +

214a3 − a4 ≥ 0, (4.3.33)

283− 28

3x4 −

43a1 − a2 +

73a4 ≥ 0, (4.3.34)

845− 84

5x4 − a1 −

75a3 +

125a4 ≥ 0, (4.3.35)

12− 12x4 − 2a2 − a3 + 3a4 ≥ 0, (4.3.36)

4.3.1.6 Facets containing one x variable and four a variables

28− 28x2 −53a1 + 4a2 − a3 −

43a4 ≥ 0, (4.3.37)

28− 28x1 + 7a1 −143a2 − a3 −

43a4 ≥ 0, (4.3.38)

21− 21x3 − 2a1 −94a2 +

214a3 − a4 ≥ 0, (4.3.39)

845− 84

5x4 − a1 −

95a2 −

75a3 +

215a4 ≥ 0, (4.3.40)


4.3.1.7 Facets containing two x variables and one a variable: ‘1a-big-x-small-x’

3 + 4x1 − 3x2 − a1 ≥ 0, (4.3.41)

5− 5x1 + 7x3 − a3 ≥ 0, (4.3.42)

14− 14x1 + 7x4 − a4 ≥ 0, (4.3.43)

8− 8x2 + 4x3 − a3 ≥ 0, (4.3.44)

17− 17x2 + 4x4 − a4 ≥ 0, (4.3.45)

9− 9x3 + 12x4 − a4 ≥ 0, (4.3.46)

4.3.1.8 Facets containing two x variables and three a variables ‘3a-big-x-any-x’

8− 8x1 + 4x3 + 2a1 − 2a2 − a3 ≥ 0, (4.3.47)

4 +163x1 − 4x3 −

43a1 − a2 + a3 ≥ 0, (4.3.48)

17− 17x1 + 4x4 +174a1 −

174a2 − a4 ≥ 0, (4.3.49)

4 +163x1 − 4x4 −

43a1 − a2 + a4 ≥ 0, (4.3.50)

12− 12x1 + 16x4 + a1 − a3 −43a4 ≥ 0, (4.3.51)

7− 7x2 +495x3 − a1 + a2 −

75a3 ≥ 0, (4.3.52)

14− 14x2 + 7x4 − 2a1 + 2a2 − a4 ≥ 0, (4.3.53)

12− 12x2 + 16x4 + a2 − a3 −43a4 ≥ 0, (4.3.54)

14− 14x3 + 7x4 − 2a1 + 2a3 − a4 ≥ 0, (4.3.55)

7 +495x3 − 7x4 − a1 −

75a3 + a4 ≥ 0, (4.3.56)

8 + 4x3 − 8x4 − 2a2 − a3 + 2a4 ≥ 0, (4.3.57)

17− 17x3 + 4x4 −174a2 +

174a3 − a4 ≥ 0, (4.3.58)

‘3a-middle-x-small-x’

12 +203x1 − 12x2 −

53a1 + a2 − a3 ≥ 0, (4.3.59)

21 +563x1 − 21x2 −

143a1 + a2 − a4 ≥ 0, (4.3.60)

21− 21x1 +635x3 + a1 −

95a3 − a4 ≥ 0, (4.3.61)

21− 21x2 +92x3 + a2 −

98a3 − a4 ≥ 0, (4.3.62)

‘3a-middle-x-big-x’

7 +283x1 − 7x3 −

73a1 − a2 + a3 ≥ 0, (4.3.63)

7 +283x1 − 7x4 −

73a1 − a2 + a4 ≥ 0, (4.3.64)


12 +845x3 − 12x4 − a1 −

125a3 + a4 ≥ 0, (4.3.65)

12 + 6x3 − 12x4 − a2 −32a3 + a4 ≥ 0, (4.3.66)

‘3a-big-x-small-x’563− 56

3x1 +

283x4 + a1 − a3 −

43a4 ≥ 0, (4.3.67)

565− 56

5x2 +

285x3 − a1 + a2 −

75a3 ≥ 0, (4.3.68)

17− 17x2 + 4x4 − 2a1 + 2a2 − a4 ≥ 0, (4.3.69)

683− 68

3x2 +

163x4 + a2 − a3 −

43a4 ≥ 0, (4.3.70)

‘3a-big-x-middle-x’563− 56

3x1 +

283x3 + a1 − a2 −

73a3 ≥ 0, (4.3.71)

1193− 119

3x1 +

283x4 + a1 − a2 −

73a4 ≥ 0, (4.3.72)

1685− 168

5x3 +

845x4 − a1 + a3 −

125a4 ≥ 0, (4.3.73)

512− 51

2x3 + 6x4 − a2 + a3 −

32a4 ≥ 0, (4.3.74)

4.3.1.9 Facets containing two x variables and four a variables

28 +1409x1 − 28x2 −

359a1 +

73a2 − a3 −

43a4 ≥ 0, (4.3.75)

3929− 392

9x1 +

283x3 +

8327a1 − a2 −

73a3 −

5627a4 ≥ 0, (4.3.76)

21 +563x1 − 21x3 −

143a1 − 2a2 + 3a3 − a4 ≥ 0, (4.3.77)

21− 21x1 +92x3 +

214a1 −

174a2 −

98a3 − a4 ≥ 0, (4.3.78)

21 +563x1 − 21x3 −

143a1 −

174a2 +

214a3 − a4 ≥ 0, (4.3.79)

12− 12x1 + 16x4 + 3a1 − 2a2 − a3 −43a4 ≥ 0, (4.3.80)

845

+283x1 −

845x4 −

73a1 − a2 −

75a3 +

125a4 ≥ 0, (4.3.81)

1193− 119

3x1 +

283x4 +

114a1 − a2 −

74a3 −

73a4 ≥ 0, (4.3.82)

683− 68

3x1 +

163x4 +

173a1 −

143a2 − a3 −

43a4 ≥ 0, (4.3.83)

12 +203x1 − 12x4 −

53a1 − 2a2 − a3 + 3a4 ≥ 0, (4.3.84)

21− 21x2 +635x3 − 2a1 + 3a2 −

95a3 − a4 ≥ 0, (4.3.85)

21− 21x2 +92x3 −

8556a1 +

14156

a2 −98a3 − a4 ≥ 0, (4.3.86)

845− 84

5x2 +

1125x4 − a1 +

125a2 −

75a3 −

2815a4 ≥ 0, (4.3.87)


563− 56

3x2 +

283x4 −

53a1 +

83a2 − a3 −

43a4 ≥ 0, (4.3.88)

683− 68

3x2 +

163x4 −

53a1 +

83a2 − a3 −

43a4 ≥ 0, (4.3.89)

283

+19615

x3 −283x4 −

43a1 − a2 −

2815a3 +

73a4 ≥ 0, (4.3.90)

14− 14x3 + 7x4 − 2a1 −32a2 +

72a3 − a4 ≥ 0, (4.3.91)

1685

+845x3 −

1685x4 − a1 −

95a2 −

215a3 +

145a4 ≥ 0, (4.3.92)

563

+283x3 −

563x4 −

53a1 − a2 −

73a3 +

83a4 ≥ 0, (4.3.93)

3575− 357

5x3 +

845x4 − a1 −

95a2 +

145a3 −

215a4 ≥ 0, (4.3.94)

1193− 119

3x3 +

283x4 −

143a1 − a2 +

173a3 −

73a4 ≥ 0, (4.3.95)

17− 17x3 + 4x4 − 2a1 −94a2 +

174a3 − a4 ≥ 0, (4.3.96)

565

+285x3 −

565x4 − a1 −

95a2 −

75a3 +

145a4 ≥ 0, (4.3.97)

4.3.1.10 Facets containing three x variables and two a variables

5 +203x1 − 5x2 + 7x3 −

53a1 − a3 ≥ 0, (4.3.98)

8 +203x1 − 8x2 + 4x3 −

53a1 − a3 ≥ 0, (4.3.99)

14 +563x1 − 14x2 + 7x4 −

143a1 − a4 ≥ 0, (4.3.100)

17 +563x1 − 17x2 + 4x4 −

143a1 − a4 ≥ 0, (4.3.101)

9− 9x1 +635x3 + 12x4 −

95a3 − a4 ≥ 0, (4.3.102)

14− 14x1 +635x3 + 7x4 −

95a3 − a4 ≥ 0, (4.3.103)

9− 9x2 +92x3 + 12x4 −

98a3 − a4 ≥ 0, (4.3.104)

17− 17x2 +92x3 + 4x4 −

98a3 − a4 ≥ 0, (4.3.105)

4.3.1.11 Facets containing three x variables and four a variables

21 +563x1 − 21x2 +

635x3 −

143a1 + a2 −

95a3 − a4 ≥ 0, (4.3.106)

21 +856x1 − 21x2 +

92x3 −

8524a1 + a2 −

98a3 − a4 ≥ 0, (4.3.107)

12 +203x1 − 12x2 + 16x4 −

53a1 + a2 − a3 −

43a4 ≥ 0, (4.3.108)

563

+1409x1 −

563x2 +

283x4 −

359a1 + a2 − a3 −

43a4 ≥ 0, (4.3.109)


683

+1409x1 −

683x2 +

163x4 −

359a1 + a2 − a3 −

43a4 ≥ 0, (4.3.110)

12 +283x1 +

845x3 − 12x4 −

73a1 − a2 −

125a3 + a4 ≥ 0, (4.3.111)

7 +283x1 +

495x3 − 7x4 −

73a1 − a2 −

75a3 + a4 ≥ 0, (4.3.112)

1193− 119

3x1 +

1475x3 +

283x4 + a1 − a2 −

215a3 −

73a4 ≥ 0, (4.3.113)

5 +203x1 + 7x3 − 5x4 −

53a1 −

54a2 − a3 +

54a4 ≥ 0, (4.3.114)

1685

+283x1 −

1685x3 +

845x4 −

73a1 − a2 + a3 −

125a4 ≥ 0, (4.3.115)

14 +563x1 − 14x3 + 7x4 −

143a1 − 2a2 + 2a3 − a4 ≥ 0, (4.3.116)

563− 56

3x1 +

283x3 +

2249x4 + a1 − a2 −

73a3 −

5627a4 ≥ 0, (4.3.117)

78427− 784

27x1 +

283x3 +

39227

x4 + a1 − a2 −73a3 −

5627a4 ≥ 0, (4.3.118)

9− 9x1 +92x3 + 12x4 +

94a1 −

94a2 −

98a3 − a4 ≥ 0, (4.3.119)

14 +563x1 − 14x3 + 7x4 −

143a1 −

72a2 +

72a3 − a4 ≥ 0, (4.3.120)

17− 17x1 +92x3 + 4x4 +

174a1 −

174a2 −

98a3 − a4 ≥ 0, (4.3.121)

8 +203x1 + 4x3 − 8x4 −

53a1 − 2a2 − a3 + 2a4 ≥ 0, (4.3.122)

17 +563x1 − 17x3 + 4x4 −

143a1 −

174a2 +

174a3 − a4 ≥ 0, (4.3.123)

563− 56

3x1 +

8417x3 +

22451

x4 + a1 − a2 −2117a3 −

5651a4 ≥ 0, (4.3.124)

17 +563x1 − 17x3 + 4x4 −

143a1 −

177a2 +

177a3 − a4 ≥ 0, (4.3.125)

1535

+ 4x1 −1535x3 +

365x4 − a1 −

65a2 +

65a3 −

95a4 ≥ 0, (4.3.126)

8 +203x1 + 4x3 − 8x4 −

53a1 −

87a2 − a3 +

87a4 ≥ 0, (4.3.127)

725

+ 4x1 +365x3 −

725x4 − a1 −

65a2 −

95a3 +

65a4 ≥ 0, (4.3.128)

9− 9x2 +635x3 + 12x4 −

97a1 +

97a2 −

95a3 − a4 ≥ 0, (4.3.129)

14− 14x2 +635x3 + 7x4 − 2a1 + 2a2 −

95a3 − a4 ≥ 0, (4.3.130)

17− 17x2 +635x3 + 4x4 − 2a1 + 2a2 −

95a3 − a4 ≥ 0, (4.3.131)

565− 56

5x2 +

285x3 +

22415

x4 − a1 + a2 −75a3 −

5645a4 ≥ 0, (4.3.132)

14− 14x2 +92x3 + 7x4 −

8556a1 +

8556a2 −

98a3 − a4 ≥ 0, (4.3.133)

17− 17x2 +92x3 + 4x4 −

8556a1 +

8556a2 −

98a3 − a4 ≥ 0, (4.3.134)


4.3.1.12 Facets containing four x variables and three a variables

14 +563x1 − 14x2 +

635x3 + 7x4 −

143a1 −

95a3 − a4 ≥ 0, (4.3.135)

9 + 12x1 − 9x2 +635x3 + 12x4 − 3a1 −

95a3 − a4 ≥ 0, (4.3.136)

17 +563x1 − 17x2 +

635x3 + 4x4 −

143a1 −

95a3 − a4 ≥ 0, (4.3.137)

9 +152x1 − 9x2 +

92x3 + 12x4 −

158a1 −

98a3 − a4 ≥ 0, (4.3.138)

14 +856x1 − 14x2 +

92x3 + 7x4 −

8524a1 −

98a3 − a4 ≥ 0, (4.3.139)

17 +856x1 − 17x2 +

92x3 + 4x4 −

8524a1 −

98a3 − a4 ≥ 0. (4.3.140)

The naming convention we adopt for all facets described above and in the following sectionsenumerates the aj variables in the facet and describes the xj variables in terms of whether thexj variable itself and/or its complement, xj = (1 − xj), appears in the facet. We also use thewords ‘big’, ‘small’, ‘middle’, ‘any’, ‘diff’ (to mean different) and/or ‘coeff’ (to mean coefficient)to describe the allowed xj variables in the facets, which is often dependent on the ordering of theallowed Ij parameters. If the Ij parameters are equal in the facet we indicate this with ‘eq’ (tomean equality). For example, the ESP facets of Section 4.3.1.8 all contain two x variables andthree a variables. We further discriminate the facets in this section by the particular x variables inthe equations. The ‘3a-big-x-small-x’ facets contain three a variables, one x variable with an indexequal to the index of the a variable in the equation with the largest corresponding I parameter andthe complement of an x variable with an index equal to the index of the a variable in the equationwith the smallest corresponding I parameter.

We now prove that generalisations of the facets, of small support, of Example 1 above are facets ofESP . We provide the details for two facets of ESP and the proof of the first in the following twosections. The proof of the second facet is given in Appendix B, Section B.1.8, as are the details ofthe remaining facets we investigate for ESP . The method we use for determining the validity andfacet defining nature of each of the constraints we consider is equivalent in each case. We providea list of proven facets for ESP in Section 4.6.

4.4 Constraint ‘2a-any-x’ is a Facet of ESP

Proposition 4.4.1. For any s, t = 1, . . . , n, s 6= t the constraint

It(1− xs)− at + as ≥ 0 (4.4.1)

is facet defining for ESP . We call this constraint the ‘2a-any-x’ facet of ESP . The ‘2a-any-x’facet is a generalisation of the constraints given in Section 4.3.1.4.

Proof. Without loss of generality, we take t = 1 and s = 2. Constraint (4.4.1) becomes:

I1(1− x2)− a1 + a2 ≥ 0. (4.4.2)

4.4. Constraint ‘2a-any-x’ is a Facet of ESP 83

First we must check the validity of constraint (4.4.2). Let (x, a) ∈ ESP and define S = {p ∈{1, . . . , n} : xp = 1}. We show that (x, a) satisfies (4.4.2) by checking all possible cases for valuesof x1 and x2 (and hence for a1 and a2).

Case (i): x1 = 0 and x2 = 0. In this case, a1 = a2 = 0. Since I1 ≥ 0 by assumption, (4.4.2) issatisfied.

Case (ii): x1 = 0 and x2 = 1. In this case, a1 = 0. Since a2 ≥ 0 by definition, (4.4.2) is satisfied.

Case (iii): x1 = 1 and x2 = 0. In this case, a2 = 0. Since 1 ∈ S and Lemma 4.2.1 gives a1 ≤minp∈S

Ip ≤ I1, then (4.4.2) is satisfied.

Case (iv): x1 = 1 and x2 = 1. In this case, from the definition of ES it must be that a1 = a2 andso (4.4.2) is satisfied.

We now show that (4.4.2) defines a facet of ESP . Let F = {(x, a) ∈ ESP : I1(1−x2)−a1+a2 = 0}.We will show that F is a facet of ESP using Theorem 3.6 in Nemhauser and Wolsey [53]. Thuswe note that constraint (4.4.2) can be equivalently expressed as(

e2,1I1e1 −

1I1e2

)(x, a) ≤ 1,

sinceI1 − I1x2 − a1 + a2 ≥ 0

⇔ I1x2 + a1 − a2 ≤ I1⇔ x2 +

1I1a1 −

1I1a2 ≤ 1

⇔(e2,

1I1e1 −

1I1e2

)(x, a) ≤ 1.

We begin by supposing that for some µ0 ∈ R and µ, λ ∈ Rn, (µ, λ)(x, a) = µ0 for all (x, a) ∈ F . Inwhat follows, we deduce that

(µ, λ) = µ0

(e2,

1I1e1 −

1I1e2

)and so are able to conclude that F is a facet. We do this by considering five classes of points in Fand observing what each implies about the values of µ, λ and µ0. We note that in each case thepoints take the form of those in ES′ and so are easily seen to be in ESP . In each case we assertthat they are also in F ; this is readily checked by observation.

1. (e2, 0) ∈ F . Now by our supposition it must be that (µ, λ)(e2, 0) = µ0, implying that µ2 = µ0.

2. (e2 + ep, 0) ∈ F for p = 1, 3, . . . , n. Now for each p ∈ {1, 3, . . . , n} it must be that (µ, λ)(e2 +ep, 0) = µ0 and so µ2 + µp = µ0, implying that µp = 0.

3. (e1, I1e1) ∈ F . Now (µ, λ)(e1, I1e1) = µ0, so µ1 + I1λ1 = µ0 and thus λ1 =1I1µ0.

4. (e1 + e2,min(I1, I2)(e1 + e2)) ∈ F . Now (µ, λ)(e1 + e2,min(I1, I2)(e1 + e2)) = µ0, so µ1 +

µ2 + min(I1, I2)(λ1 + λ2) = µ0 and thus λ2 = − 1I1µ0.

5. (e1 + e2 + ep,min(I1, I2, Ip)(e1 + e2 + ep)) ∈ F for p = 3, . . . , n. Now for each p ∈ {3, . . . , n}it must be that (µ, λ)(e1 + e2 + ep,min(I1, I2, Ip)(e1 + e2 + ep)) = µ0 and so µ1 + µ2 + µp +min(I1, I2, Ip)(λ1 + λ2 + λp) = µ0. Hence min(I1, I2, Ip)λp = 0, implying that λp = 0.


Thus (µ, λ) = µ0

(e2,

1I1e1 −

1I1e2

)as required.

Therefore the ‘2a-any-x’ constraint, (4.4.1), is facet defining for ESP .

Note that the enumerated steps above are equivalent to substituting 2n points (x, a) into (µ, λ)(x, a) =µ0 and solving the resulting linear equations.

4.5 Constraint ‘na-any-x’ is a Facet of ESP

Facet ‘2a-any-x’ (of the previous section) and facet ‘3a-any-x’ (discussed in Appendix B, SectionB.1.2), corresponding to constraints (4.4.1) and (B.1.3) respectively, contain one x variable, and,2 and 3 a variables respectively. We now generalise these constraints to consider n a variables.

For j ∈ {1, . . . , n}, we define σj such that {1, . . . , n− 1} → {1, . . . , n} \ {j} to be any 1-1 mappingsatisfying

Iσj(1) ≥ Iσj(2) ≥ · · · ≥ Iσj(n−1),

i.e. σj is a permutation of the indices that sorts I in decreasing order. We use σj in the followinglemma and proposition.

Lemma 4.5.1. For any j ∈ {1, . . . , n}, if for some r ∈ {1, . . . , n− 1} we have that aσj(y) ≤ α for ally = 1, . . . , r, then

1Iσj(1)

aσj(1) +r∑y=2

(1

Iσj(y)− 1Iσj(y−1)

)aσj(y) ≤

1Iσj(r)

α, (4.5.1)

where if r = 1 we take the value of the sum to be zero.

Proposition 4.5.1. For any j ∈ {1, . . . , n} the constraint

xj −1

Iσj(n−1)aj +

1Iσj(1)

aσj(1) +n−1∑y=2

(1


)aσj(y) ≤ 1 (4.5.2)

defines a facet of ESP , where Iσj(1) ≥ Iσj(2) ≥ · · · ≥ Iσj(n−1). We call this constraint the na-any-xfacet of ESP . The na-any-x facet is a generalisation of the constraints given in Sections 4.3.1.4,4.3.1.5 and 4.3.1.6.

4.6 Facets of ESP

In addition to the facets presented in Sections 4.4 and 4.5, as mentioned, we give the detail forproving that other constraints of ESP of small support are facets of ESP in Appendix B, SectionB.1. (The proof for facet ‘na-any-x’ is also given in Appendix B in Section B.1.8). In this section,Table 4.6.1 lists the complete set of facets we have determined for ESP , those determined in thischapter and those determined in Appendix B, Section B.1. In Sections 4.8 and 4.9 we numericallytest facets of ESP on a model of ESP and on the JS model respectively.

4.6. Facets of ESP 85

Table 4.6.1: Facets of ESP .

Facet Corresponding Generalisation ofto constraint

‘2a-any-x’ (4.4.1) the constraints given in Section4.3.1.4

‘1a-big-x-small-x’ (B.1.1) the constraints given in Section4.3.1.7

‘3a-any-x’ (B.1.3) the constraints given in Section4.3.1.5

‘3a-big-x-any-x’ (B.1.5) constraints (4.3.47), (4.3.48),(4.3.49), (4.3.50), (4.3.51), (4.3.52),(4.3.53), (4.3.54), (4.3.55), (4.3.56),(4.3.57) and (4.3.58) given inSection 4.3.1.8

‘3a-middle-x-small-x’ (B.1.7) constraints (4.3.59), (4.3.60),(4.3.61) and (4.3.62) given inSection 4.3.1.8

‘3a-middle-x-big-x’ (B.1.9) constraints (4.3.63), (4.3.64),(4.3.65) and (4.3.66) given inSection 4.3.1.8

‘3a-big-x-small-x’ (B.1.11) constraints (4.3.67), (4.3.68),(4.3.69) and (4.3.70) given inSection 4.3.1.8

‘3a-big-x-middle-x’ (B.1.15) constraints (4.3.71), (4.3.72),(4.3.73) and (4.3.74) given inSection 4.3.1.8

‘na-any-x’ (4.5.2) the constraints given in Sections4.3.1.4, 4.3.1.5 and 4.3.1.6


4.7 ESP with the Strict Consecutive-1-Constraint (C1)

We now apply the strict consecutive-1-constraint to ESP to create a polytope which more accu-rately represents the feasible region of the JS model. The constraint we apply is as follows:

xj1 − xj2 + xj3 ≤ 1, (4.7.1)

for all j1 = 1, . . . , n− 2, j3 = j1 + 2, . . . , n, j2 = j1 + 1, . . . , j3 − 1.

The new polytope we consider is therefore:

ESP -C1 = conv(ES-C1),

where

ES-C1 = {(x, a) ∈ {0, 1}n × Rn+,0 : ap ≤ Ipxp, for all p = 1, . . . , n,xj1 − xj2 + xj3 ≤ 1, for all j1 = 1, . . . , n− 2,j3 = j1 + 2, . . . , n, j2 = j1 + 1, . . . , j3 − 1and, if xp = xq = 1 for p 6= q, then ap = aq}.

(4.7.2)

In what follows we use ES′-C1, which is equivalent to the definition of ES′ other than incorporatingthe strict consecutive-1-property, to enable the easy construction and checking of points belongingto ES-C1. Hence we more easily prove that any resulting constraints are valid and facet defining.The ES′-C1 definition is as follows:

ES′-C1 =

{(r∑p=q

ep, α

r∑p=q

ep

): 1 ≤ q ≤ r ≤ n; 0 ≤ α ≤ min{Ip : q ≤ p ≤ r}

}∪ {(0, 0)}. (4.7.3)

Again, we assume that Ip > 0 for all p.

Proposition 4.7.1. ESP -C1 is a full-dimensional polytope.

Proof. We must demonstrate that, since ESP -C1 is defined over 2n variables, there are 2n + 1affinely independent vectors in ESP -C1.

Clearly {(ep, 0) : p = 1, . . . , n} ∪ {(ep, Ipep) : p = 1, . . . , n} ∪ {(0, 0)} ⊆ ESP -C1 are 2n+ 1 affinelyindependent vectors which have the strict consecutive-1-property. (If we delete {(0, 0)} from theset of vectors, the remaining vectors are linearly independent.)

If we now consider sets of points which satisfy ESP -C1, we see that with the introduction of thestrict consecutive-1-constraint, we now have an ordering on the x variables as well as an orderingon the I parameters. Therefore facets determined for the fixed x variable ordering and a particularI parameter ordering cannot be generalised to other I parameter orderings since particular Iparameters correspond to particular x variables. Even a single ordering of the I parameters, withthe ordered x variables, leads to a large number of facets for ESP -C1. Example 2 in AppendixB, Section B.2.1 illustrates this. We therefore simplify this problem by considering the specialcase of equal I values and apply the resulting facets to each row of each shape matrix, wheneverthe appropriate number of equal I parameters occur. (The different facets we determine require

4.7. ESP with the Strict Consecutive-1-Constraint (C1) 87

Table 4.7.1: Facets of ESP -C1.

Facet Corresponding Generalisation ofto constraint

‘2a-any-x-eq’ (4.4.1) withIt = Is

constraints (B.2.171), (B.2.172),(B.2.173), (B.2.174), (B.2.175),(B.2.176), (B.2.177), (B.2.178),(B.2.179), (B.2.180), (B.2.181) and(B.2.182) given in Section B.2.2.4

‘2a-1diff-x-eq’ (B.2.232) constraints (B.2.183), (B.2.184),(B.2.185), (B.2.186), (B.2.187),(B.2.188), (B.2.189) and (B.2.190)given in Section B.2.2.4

‘3a-coeff2-x-eq’ (B.2.234) the constraints given in SectionB.2.2.5

‘4a-any-x-eq’ (B.2.236) the constraints given in SectionB.2.2.6

certain numbers of I parameters to be equal in a row of an intensity matrix for the facet to apply,depending on the number of different indices on x and a variables within a facet). Furthermore,the equal I parameters do not need to occur consecutively within a row of an intensity matrixfor a facet to apply, since in enforcing the strict consecutive-1-constraint, equal I values are alsoenforced in a solution, (though of course the intensity applied to any resulting shape matrix maybe less than the equal I values occurring in the intensity row, for example if lower intensity valuesoccur between the equal I values). This simplification to ‘equal I’s’ is relevant if we considerapplying bounds to variables bk, and hence to variables aijk, of the JS model, which we investigatein Chapter 5. In this case, there is potential for the same bound to be applied to different aijkvariables, and therefore, the analog of this, in the context of facets for ESP -C1, is to set the Iparameters to be equal. Furthermore, it should be noted that medical data, whilst too large tobe tested here, has the property of much repetition of intensity values within rows of an intensitymatrix. Therefore the application of the facets of ESP -C1 (with equal I values) to the JS model,together with other appropriate techniques for reducing computation time (for example the boundstechniques we develop in Chapter 5), could prove beneficial and is a point for future work.

We provide an example of points satisfying ESP -C1, where we consider equal I values, in AppendixB, Section B.2.2, Example 3. Example 3 demonstrates that this special case has significantly lessfacets than the case with differing I values, Example 2, and Example 2 is just one of manycombinations of I parameter orderings which could be considered. In Appendix B, Sections B.2.3to B.2.6 inclusive, we prove that generalisations of the facets, of small support, of Example 3 arefacets of ESP -C1. The complete list of facets we investigate for ESP -C1, for the case of equal Iparameters, is given in Table 4.7.1. We test these facets numerically in the following two sectionson a model for ESP -C1 and the JS model respectively.


4.8 Numerical Results for the Application of the Facets of ESP and ESP -C1 toESP and ESP -C1 Models Respectively

In this section we first define linear models for ESP and ESP -C1 respectively. We then apply facets‘2a-any-x’, ‘1a-big-x-small-x’, ‘3a-any-x’ and ‘3a-big-x-any-x’ respectively to the ESP model andfacets ‘2a-any-x-eq’, ‘2a-1diff-x-eq’, ‘3a-coeff2-x-eq’ and ‘4a-any-x-eq’ respectively to the ESP -C1model, to determine their effect on the computational efficiency of the problems solved.

The ESP model utilises real variables defined as follows:

0 ≤ xj ≤ 1, for all j = 1, . . . , n,

b ≥ 0, and

aj ≥ 0, for all j = 1, . . . , n,

and the constraints on the ESP model are the equivalent of the three constraints on the aijk

variables of the JS model: (2.2.7), (2.2.8) and (2.2.9). We have:

0 ≤ aj ≤Mjxj , for all j = 1, . . . , n, (4.8.1)

0 ≤ aj ≤ b, for all j = 1, . . . , n (4.8.2)

andaj +Gj(1− xj) ≥ b, for all j = 1, . . . , n, (4.8.3)

where we set ‘big-M’ parameters Mj = Ij for all j = 1, . . . , n and Gj = maxjIj for all j = 1, . . . , n.

Unrestricted variable b is necessary in the ESP model to enforce the ESP requirement that the avariables should be equal to the same value if the corresponding x variables equal 1.

The objective function we apply to the ESP model must capture the property experienced bythe linear relaxation of the JS model, where the aijk variables for the same k take on differentpositive values rather than the same positive value, bk, or zero. We achieve this by weighting thexj variables, with a random positive coefficient, by a small positive number (we use 0.1) and theaj variables, with a random positive coefficient, by a large negative number (we use -1), since theseare conditions which cause the JS model to ‘cheat’. The objective we apply to the ESP model is:

minn∑j=1

(Uniform(1, 2)(0.1xj)−Uniform(1, 2)aj), (4.8.4)

where Uniform(1, 2) returns a random coefficient in the range 1 to 2, not including 2, each timethe function is called. (Small random xj variable values cause constraint (4.8.1) to be active ratherthan constraint (4.8.2) and hence since the objective maximises the aj variable values we end upwith relatively large, different, aj values, which is the scenario we wish to model).

The formulation we utilise for the ESP -C1 model is similar to our formulation for the ESP modelother than the objective function (which is altered to demonstrate the effect of the ESP -C1 equalI facets) and the application of the strict consecutive-1-constraint (4.7.1). Numerical results forthe application of equal I facets to the ESP -C1 model demonstrate no improvement in the real

4.8. Numerical Results for the Application of the Facets of ESP and ESP -C1 to ESP and ESP -C1Models Respectively 89

solution of the ESP -C1 model when objective (4.8.4) is utilised in the formulation. Instead weinvestigate an alternative objective function for the ESP -C1 model which forces xj variable valuesand aj variable values to the extremes of their ranges in different combinations. We implementthe following:

minn∑j=1

(Uniform(−1, 1)(0.1xj)−Uniform(−1, 1)aj), (4.8.5)

where Uniform(−1, 1) returns a random coefficient in the range -1 to 1, not including 1, each timethe function is called.

All numerical results for facets applied to the ESP and ESP -C1 models given in this section (andnumerical results for facets applied to the JS model given in the following section) are computedusing CPLEX version 8.1 and AMPL version 8.1 with all AMPL and CPLEX preprocessing routinesswitched off so that we completely determine the effect of the facets themselves. The numericalresults provided here compare the objective values for real and integer versions of the ESP andESP -C1 models, and the number of branch and bound nodes, iterations and computation time,when each facet (or combination of facets) is separately applied to the integer versions of themodels. (The integer versions of the ESP and ESP -C1 models simply set the x variables tobe binary and the a and b variables to be integers greater than or equal to zero). We considerrandomly generated batches of 100 row problems of size 1 × 5 with intensities ranging from 0 to10 and size 1× 10 with intensities ranging from 0 to 5, 0 to 10 and 0 to 20 inclusive, respectively,for testing on the ESP model. For the ESP -C1 model we utilise single row problems of varyingsizes consisting entirely of 1’s, and four randomly generated batches of 100 row problems, two ofsize 1 × 20 and two of size 1 × 30, each with intensities ranging from 0 to 1 inclusive and 0 to 2inclusive. We increase the dimension of the row problems for testing the ESP -C1 equal I facetssince, for the ‘4a-any-x-eq’ facet in particular, we require 4 values of I to be equal within a rowof a data set for the facet to apply, hence the increased dimension leads to greater opportunity toapply this facet. Furthermore, decreasing the range of random I values in a row from say, 0 to 10,as we used for the ESP model, to 0 to 1 or 2, or 1’s alone, also increases the number of equal Ivalues that occur in a row. Numerical results for facets applied to the ESP and ESP -C1 modelsare given in Table 4.8.1. If a data set within the table is subscripted with a ‘y’ this indicates thata batch of 100 problems is solved, rather than an individual problem.

Finally, when we defined the facets of ESP and ESP -C1 earlier in this chapter we described eachparticular class of facets with a single name, see Tables 4.6.1 and 4.7.1 respectively. For example,we use facet ‘2a-any-x’ to describe the class of facets defined by constraint (4.4.1) which utilisestwo a variables and the complement of one x variable, where the particular a and x variables thatappear in the facets are dependent on the allowed variable indices. Given these definitions, whenwe describe the individual application of facets to models in our numerical results to follow, we aredescribing the application of one entire class of facets defined by the single facet name. For example,when we describe the individual application of facet ‘2a-any-x’ to the ESP model we mean theapplication of the class of facets defined by facet ‘2a-any-x’ to the ESP model. Furthermore, whenwe describe the simultaneous application of facets to models we are describing the simultaneousapplication of multiple classes of facets where the classes of facets are each defined by single facetnames. Again using an example, the simultaneous application of facets ‘2a-any-x’, ‘1a-big-x-small-x’, ‘3a-any-x’ and ‘3a-big-x-any-x’ to the ESP model means the simultaneous application of the


four classes of facets defined by the single facet names ‘2a-any-x’, ‘1a-big-x-small-x’, ‘3a-any-x’and ‘3a-big-x-any-x’ to the ESP model. We now present the numerical results for the addition offacets to the ESP and ESP -C1 models.

Table 4.8.1 demonstrates that the number of branch and bound nodes searched generally decreaseswhen each of the facets of ESP (‘2a-any-x’, ‘1a-big-x-small-x’, ‘3a-any-x’ and ‘3a-big-x-any-x’)is individually applied to the integer version of the ESP model. (Only in the case of the batchof 100 problems of size 1 × 10 with intensities ranging from 0 to 5 does the number of branchand bound nodes searched increase with the application of facet ‘2a-any-x’ to the integer ESPmodel). When we simultaneously apply all four of the facets, ‘2a-any-x’, ‘1a-big-x-small-x’, ‘3a-any-x’ and ‘3a-big-x-any-x’, the number of branch and bound nodes searched is generally smallestwhen compared with the individual applications of the facets. (The only exception is again batch1 × 10 with intensities ranging from 0 to 5 where the application of facet ‘3a-any-x’ yields thesmallest number of branch and bound nodes searched for this batch). Furthermore, Table 4.8.1shows that the average percentage difference between the integer and real objective values forthe ESP model improves when each of the facets ‘2a-any-x’, ‘1a-big-x-small-x’, ‘3a-any-x’ and‘3a-big-x-any-x’ is individually added, and is smallest when all four of the facets we consider areapplied simultaneously. Therefore, we conclude that the ESP facets improve the search space forthe ESP model.

When the equal I facets, ‘2a-1diff-x-eq’, ‘3a-coeff2-x-eq’ and ‘4a-any-x-eq’, are individually appliedto the ESP -C1 model, for all data sets considered in Table 4.8.1, we see that the real solutionimproves in comparison to the corresponding integer solution for the ESP -C1 model. On the otherhand, with the application of facet ‘2a-any-x-eq’, no improvement is seen between real and integersolutions for data sets with maximum intensity level equal to 1 or for data sets containing only asingle intensity value. (We show single row problems comprising only 1’s in Table 4.8.1, howeverthe same result occurs for other row problems where only one intensity level is used). If we increasethe maximum intensity level in the batch data sets to 2, so that the intensity ranges from 0 to 2,improvement in the real objective function value for ESP -C1 is achieved with the application offacet ‘2a-any-x-eq’. It is not clear why facet ‘2a-any-x-eq’ does not improve the real solution whenonly 1 level of intensity is present in a data set or when the intensity range is restricted to 0 to1. We observed that for these data sets the majority of the ‘2a-any-x-eq’ facets were active at thereal solution. We believe that perhaps for such restricted data sets, facet ‘2a-any-x-eq’ intersectswith other constraints already active for the ESP -C1 model at the real solution. Since our focusin this chapter is to determine computational improvement for the JS model however, we do notinvestigate the ‘2a-any-x-eq’ facet application to the ESP -C1 model further. The result that theESP -C1 facets improve the search space for the ESP -C1 model for some data sets is sufficient(for our purposes) to numerically justify the facet defining nature of the ESP -C1 constraints weconsider.

Finally we also mention that the simultaneous application of the ‘2a-any-x-eq’, ‘2a-1diff-x-eq’, ‘3a-coeff2-x-eq’ and ‘4a-any-x-eq’ facets to the ESP -C1 model improves the real solution of ESP -C1for all data sets considered (though of course, further to our discussion above, it is unlikely thatfacet ‘2a-any-x-eq’ contributes to the improvement in search space in this case, for data sets whichcomprise only one intensity level and data sets where the intensity ranges from 0 to 1).

4.8. Numerical Results for the Application of the Facets of ESP and ESP -C1 to ESP and ESP -C1Models Respectively 91

Tab

le4.

8.1:

Resu

lts

for

the

appli

cati

on

of

the

‘2a

-an

y-x

’,‘1a

-big

-x-s

mall

-x’,

‘3a

-an

y-x

’an

d‘3a

-big

-x-a

ny-x

’F

ace

tsofESP

,co

rre

spon

din

gto

Con

stra

ints

(4.4

.1),

(B.1

.1),

(B.1

.3)

an

d(B

.1.5

)re

spec

tively

,to

theESP

model.

Resu

lts

for

the

appli

cati

on

of

the

‘2a

-an

y-x

-eq’,

‘2a

-1diff

-x-e

q’,

‘3a

-coeff

2-x

-eq’

an

d‘4a

-an

y-x

-eq’

Equ

alI

Face

tsofESP

-C1,

corre

spon

din

g

toC

on

stra

int

(4.4

.1)

wit

hIt

=Is,

(B.2

.232),

(B.2

.234)

an

d(B

.2.2

36)

resp

ecti

vely

,to

theESP

-C1

model.

We

use

CP

LE

Xversi

on

8.1

an

dA

MP

Lversi

on

8.1

on

a2G

Hz

AM

D3000+

wit

hpre

pro

cess

ing

opti

on

soff

.A

batc

hof

100

pro

ble

ms

isin

dic

ate

dw

ith

a‘y

’in

the

data

set

desc

rip

tion

.T

ime=

the

tota

lti

me

for

100

pro

ble

ms

inth

eca

seof

aba

tch

an

dth

eti

me

for

asi

ngle

pro

ble

mto

ru

nin

the

case

of

asi

ngle

pro

ble

m,

inse

con

ds,

BB

=th

eto

tal

nu

mbe

rof

bra

nch

an

dbo

un

dn

odes

for

aba

tch

or

sin

gle

pro

ble

m,

ITS

=th

eto

tal

nu

mbe

rof

itera

tion

s

for

aba

tch

or

sin

gle

pro

ble

man

d%

=th

eavera

ge

perc

en

tage

diff

ere

nce

betw

een

the

obje

cti

ve

valu

es

for

the

real

versi

on

of

theESP

(orESP

-C1)

model

an

dth

ein

teger

versi

on

,w

ith

face

tsappli

edas

indic

ate

din

the

table

,w

hen

aba

tch

of

100

pro

ble

ms

isco

nsi

dere

d,

where

as,

ifa

sin

gle

pro

ble

mis

con

sidere

d%

refe

rs

toth

eactu

al

perc

en

tage

diff

ere

nce

betw

een

real

an

din

teger

versi

on

sof

theESP

(orESP

-C1)

model,

again

wit

hfa

cets

appli

edas

indic

ate

din

the

table

.

Facet/s

of

ES

Papplied

to

the

ES

Pm

odel

No

Facets

applied

to‘2a-a

ny-x

’‘1a-b

ig-x

-sm

all-x

’‘3a-a

ny-x

’‘3a-b

ig-x

-any-x

’‘2a-a

ny-x

’,‘1a-b

ig-x

-sm

all-x

’,

ESP

model

‘3a-a

ny-x

’and

‘3a-b

ig-x

-any-x

’

BB

ITS

Tim

e%

BB

ITS

Tim

e%

BB

ITS

Tim

e%

BB

ITS

Tim

e%

BB

ITS

Tim

e%

BB

ITS

Tim

e%

15

010

y189

1605

0.6

110

174

1792

0.6

25

77

1646

0.5

32

31

1537

0.6

01

42

1651

0.6

11

17

1709

0.6

10

110

05

y308

2989

0.7

3337

339

3736

1.2

0167

151

3492

0.8

670

15

2868

1.4

818

73

3894

1.6

130

22

3608

2.2

85

110

010

y1139

4133

0.7

020

723

4957

1.2

411

321

5153

0.9

87

181

3868

1.9

62

242

5642

2.3

44

96

5416

4.2

61

110

020

y1419

4906

0.8

220

963

6410

1.3

211

442

6168

1.2

47

350

4657

2.2

12

337

7020

2.6

54

164

6899

5.3

72

Equal

IFacet/s

of

ES

P-C

1applied

to

the

ES

P-C

1m

odel

No

Facets

applied

to‘2a-a

ny-x

-eq’

‘2a-1

dif

f-x-e

q’

‘3a-c

oeff

2-x

-eq’

‘4a-a

ny-x

-eq’

‘2a-a

ny-x

-eq’,

‘2a-1

dif

f-x-e

q’,

ESP

-C1

model

‘3a-c

oeff

2-x

-eq’

and

‘4a-a

ny-x

-eq’

BB

ITS

Tim

e%

BB

ITS

Tim

e%

BB

ITS

Tim

e%

BB

ITS

Tim

e%

BB

ITS

Tim

e%

BB

ITS

Tim

e%

120

11

270

0.1

325

054

0.0

925

053

0.0

50

069

0.0

40

071

0.7

00

063

0.9

60

125

11

10

199

0.4

478

6186

0.6

778

10

363

2.0

840

6201

0.9

835

6274

32.9

532

4226

42.6

332

130

11

9245

1.2

143

12

350

1.6

743

2175

5.1

812

4147

3.0

68

2166

115.0

07

2173

106.9

87

135

11

15

578

3.6

974

23

636

4.4

274

13

577

15.0

536

19

795

9.9

127

6380

286.2

727

11

596

345.4

927

120

01

y166

4382

5.6

625

172

4559

6.5

425

172

4656

8.1

223

172

5090

6.8

822

162

5522

29.5

922

164

5419

32.6

122

120

02

y1436

14652

13.5

7111

1492

16034

16.9

3108

1081

13882

16.9

382

1013

14074

15.9

973

965

15790

25.8

371

894

15000

31.5

870

130

01

y347

10305

48.2

569

355

11705

54.7

269

359

12618

74.3

760

345

12622

63.9

457

328

14259

491.6

455

326

13922

568.0

355

130

02

y1716

27995

83.0

9131

1813

31091

97.0

5130

1711

30130

110.5

2105

1597

31666

101.0

699

1562

36390

281.3

995

1653

36251

349.4

194


Moreover, we note that in general the number of branch and bound nodes searched tends toincrease when facet ‘2a-any-x-eq’ is applied to the integer version of the ESP -C1 model, thoughwith the application of the remaining facets, individually and simultaneously, generally a decreasein numbers of branch and bound nodes is observed.

In the following section, we provide numerical results for the application of the ESP -C1 and ESPfacets to the JS model.

4.9 Numerical Results for the Application of the Facets of ESP and ESP -C1 to theJS Model

To investigate the effect of the facets of ESP and ESP -C1 on the JS model we test individualproblems of size 4×4 with intensities ranging from 0 to 5 inclusive, and where appropriate, batchesof 100 problems of sizes, 4× 4 with intensities ranging from 0 to 5 inclusive, 4× 4 with intensitiesranging from 0 to 15 inclusive and 5 × 5 with intensities ranging from 0 to 5 inclusive. For theESP -C1 facets we also test problems of size 6 × 6 with intensities ranging from 0 to 5 inclusiveto generate more opportunity for the equal I facets to be applied. Again, in the tables of resultsgiven in this section, a batch of 100 problems is indicated when the notation for a data set containsa subscript ‘y’ rather than a number. Furthermore, each table shows: BB (the number of branchand bound nodes searched), ITS (the number of iterations performed), Time (the total time tosolve the problem, or batch of 100 problems, in seconds), RNLB (the root node lower bound valuefor the problem), bk’s,obj (the individual beam-on time values and objective value returned in anoptimal solution) and the changes in the numbers of constraints of the JS model with and withoutthe addition of the particular facet under consideration. Numbers shown in bold face in the tableshighlight the smallest value for each indicator for each example.

4.9.1 Application of Facets ‘2a-any-x’, ‘1a-big-x-small-x’, ‘3a-any-x’ and ‘3a-big-x-any-x’ of ESP to the JS Model. The individual application of each of the facets of ESP we con-sider, ‘2a-any-x’, ‘1a-big-x-small-x’, ‘3a-any-x’ and ‘3a-big-x-any-x’, corresponding to constraints(4.4.1), (B.1.1), (B.1.3) and (B.1.5) respectively, to the JS model, does not yield improvement inexecution. Instead, for each facet tested, we generally see an increase in branch and bound nodes,iterations and solve time and no change in the root node lower bound value returned by CPLEX.There are a few examples where the number of branch and bound nodes searched and number ofiterations decrease when the facets are individually applied, however computation time increaseseven for these examples. As total number of branch and bound nodes, total iterations and totalcomputation time increases for the three batches of 100 problems we test on the JS model withfacet ‘2a-any-x’, we did not continue with batch runs in testing facets ‘1a-big-x-small-x’, ‘3a-any-x’and ‘3a-big-x-any-x’ respectively on the JS model, as it was clear that computation time wouldnot improve. The results for the application of facet ‘2a-any-x’ of ESP to the JS model are givenin Table 4.9.1. We provide the results for the individual application of the remaining facets ofESP that we consider, to the JS model, in Tables B.3.1, B.3.2 and B.3.3 respectively in AppendixB, Section B.3.

4.9. Numerical Results for the Application of the Facets of ESP and ESP -C1 to the JS Model 93

Tab

le4.

9.1:

Res

ults

for

the

appl

icat

ion

ofth

e‘2a

-any

-x’

Face

tofESP

,C

onst

rain

t(4

.4.1

),to

the

JSm

odel

.W

eus

eC

PL

EX

vers

ion

8.1

and

AM

PL

vers

ion

8.1

ona

2GH

zA

MD

3000

+:

prep

roce

ssin

gop

tion

soff

,ti

me

inse

cond

s,2-

hour

tim

elim

iton

indi

vidu

alpr

oble

min

stan

ces.

Data

JS

mod

elw

ith

ou

t(4

.4.1

)S

olu

tion

Nu

mb

erof

JS

mod

elw

ith

(4.4

.1)

Solu

tion

Nu

mb

erof

Con

stra

ints

Con

stra

ints

wit

hou

t

(4.4

.1)

wit

h

(4.4

.1)

BB

ITS

Tim

eR

NL

Bb k

’s,o

bj

BB

ITS

Tim

eR

NL

Bb k

’s,o

bj

44

05

11070

6280

0.4

01.8

04,3

,1,1

,ob

j=4

347

17

309

0.3

81.8

04,3

,1,1

,ob

j=4

1307

44

05

212

148

0.0

71.0

03,1

,1,o

bj=

3264

18

240

0.2

71.0

02,2

,1,o

bj=

3984

44

05

325

240

0.0

71.2

52,2

,1,o

bj=

3264

21

373

0.3

21.2

52,2

,1,o

bj=

3984

44

05

456

507

0.1

01.4

04,1

,1,1

,ob

j=4

347

87

1671

0.6

21.4

03,2

,1,1

,ob

j=4

1307

44

05

57099

62782

2.8

81.6

03,2

,1,1

,1,o

bj=

5430

4090

57977

8.4

91.6

03,2

,1,1

,1,o

bj=

51630

44

05

6280

2582

0.2

61.8

03,3

,2,1

,ob

j=4

430

1146

21114

3.2

51.8

03,3

,2,1

,ob

j=4

1630

44

05

759

571

0.1

11.5

02,2

,1,1

,ob

j=4

347

97

2040

0.6

31.5

02,2

,1,1

,ob

j=4

1307

44

05

8131

789

0.1

61.0

02,1

,1,1

,ob

j=4

347

123

2338

0.7

91.0

02,1

,1,1

,ob

j=4

1307

44

05

9719

5415

0.3

41.8

04,2

,2,1

,ob

j=4

347

60

947

0.5

81.8

04,2

,2,1

,ob

j=4

1307

44

05

10

72

762

0.1

01.7

53,2

,1,1

,ob

j=4

347

61

841

0.4

41.7

53,2

,1,1

,ob

j=4

1307

44

05

y53418

439493

29.3

556320

745994

130.7

9

44

015

y12173248

197585735

7475.0

318742183

704828838

94339.2

7

55

05

y5592297

52886828

3714.1

97579046

138437252

44123.0

5


Considering that the simultaneous application of facets ‘2a-any-x’, ‘1a-big-x-small-x’, ‘3a-any-x’and ‘3a-big-x-any-x’ to the ESP model itself yields the greatest average percentage improvementin objective values (between integer and real cases) and in general the greatest decrease in thenumber of branch and bound nodes searched for the integer case, we now simultaneously applythese same facets to the JS model. However, given that no improvement is seen when facets‘2a-any-x’, ‘1a-big-x-small-x’, ‘3a-any-x’ and ‘3a-big-x-any-x’ are individually applied to the JSmodel, it is not surprising that the application of the four facets simultaneously also yields noimprovement. These results are shown in Table B.3.4 in Appendix B, Section B.3. Consequently,we do not continue to computationally test the remaining facets of ESP (as given in Table 4.6.1)on the JS model. Furthermore, since the facets of small support do not improve performance forthe JS model there is no reason to expect facets of larger support to improve performance.

4.9.2 Application of Facets ‘2a-any-x-eq’, ‘2a-1diff-x-eq’, ‘3a-coeff2-x-eq’ and ‘4a-any-x-eq’ of ESP -C1 to the JS Model. As mentioned previously, since we consider the specialcase of equal I values for ESP -C1, we apply the facets we have determined for ESP -C1 to eachrow of each shape matrix within the JS model when the appropriate number of I values are equalin that row. This means that there may not always be an opportunity to apply the ESP -C1facets since the appropriate number of equal I values may not occur in any row of our randomlygenerated intensity matrices. This is particularly true since the JS model cannot solve problemswith dimension much larger than 6 × 6 with intensities ranging from 0 to 5 inclusive within areasonable time frame and therefore we cannot test larger problem sizes (with small maximumintensity levels in particular) where repetition of intensity values is more likely to occur. Wherethere is no opportunity for the ESP -C1 facet to be used in the examples we test, we indicate this inthe tables of computational results with the words ‘no facets apply’. Of the 10 individual examples,of size 4× 4 with intensities ranging from 0 to 5, tested with each of the facets ‘2a-1diff-x-eq’ and‘3a-coeff2-x-eq’ respectively, 5 do not have at least 3 equal I values in any row of their intensitymatrix and therefore these facets cannot be applied to these examples. For the application of facet‘4a-any-x-eq’ we require at least 4 equal I values in a row of an intensity matrix. To increase ourchances of achieving this we test individual problem instances of size 6× 6 with intensities rangingfrom 0 to 5 inclusive. Finally, we again highlight in bold face, in each of the tables of computationalresults, the lowest values for the number of branch and bound nodes searched, number of iterationsand computation time for each problem or batch of problems tested when comparing the JS modeland the JS model with particular facets applied.

The application of each of the facets ‘2a-any-x-eq’, ‘2a-1diff-x-eq’, ‘3a-coeff2-x-eq’ and ‘4a-any-x-eq’, individually to the JS model, yields slight improvement when compared with the JS modelwithout facets. For the batch of problems of size 4× 4 with intensities ranging from 0 to 5, facets‘2a-1diff-x-eq’ and ‘3a-coeff2-x-eq’ decrease branch and bound nodes searched, iterations and totalcomputation time though only by small margins, whereas facet ‘2a-any-x-eq’ does not improve anyof the indicator values for this batch. For the batch of size 4× 4 with intensities ranging from 0 to15, all indicators improve for the application of each of the facets ‘2a-any-x-eq’, ‘2a-1diff-x-eq’ and‘3a-coeff2-x-eq’, where there is slightly more improvement with facet ‘2a-any-x-eq’ and marginalimprovement with the other facets. For the batches of size 5 × 5 and 6 × 6, each with intensitiesranging from 0 to 5, the facets ‘2a-any-x-eq’, ‘2a-1diff-x-eq’ and ‘3a-coeff2-x-eq’ do not improve


overall computation time however in the case of the latter batch all three facets improve totalnumbers of branch and bound nodes searched and number of iterations performed even thoughoverall computation time is not simultaneously reduced. Furthermore, in the case of the formerbatch, one particular problem times out, taking longer than our time limit of 2-hours, when eachof the three facets are individually applied to the JS model, and thus it is this one problem whichresults in the excessive run time for this batch for each of the facets under consideration. Withregard to facet ‘4a-any-x-eq’, no indicators improve for the batch of size 5×5 with intensities rangingfrom 0 to 5, however the total number of branch and bound nodes, iterations and computationtime improve for the batch of size 6 × 6 with intensities ranging from 0 to 5. Finally, none ofthe facets we investigate in this section increase the root node lower bound value for any of theindividual problems tested. Therefore, for the batches of smaller dimension that we consider, itappears that there is generally little to no improvement in overall computational efficiency whenfacets ‘2a-any-x-eq’, ‘2a-1diff-x-eq’, ‘3a-coeff2-x-eq’ and ‘4a-any-x-eq’ are applied to the JS model,though for problems of slightly larger dimension (6× 6 with intensities ranging from 0 to 5) facets‘2a-any-x-eq’, ‘2a-1diff-x-eq’ and ‘3a-coeff2-x-eq’ decrease the total number of branch and boundnodes searched and iterations, though not overall computation time, whereas facet ‘4a-any-x-eq’improves each of the indicator values for this batch. As a final trial of the ESP -C1 facets, wealso simultaneously apply facets ‘2a-any-x-eq’, ‘2a-1diff-x-eq’, ‘3a-coeff2-x-eq’ and ‘4a-any-x-eq’ tothe JS model. The results for the simultaneous application are consistent with the results for theindividual application of the facets, and again, the root node lower bound values do not increasefor any example tested when the four facets are all applied to the JS model. The results for theapplication of each of the facets of ESP -C1 investigated in this section, separately applied tothe JS model, are given in Tables 4.9.2, 4.9.3, 4.9.4, and 4.9.5 respectively. The results for thesimultaneous application of the facets are given in Table 4.9.6.

Hence it appears that when we can apply the facets we have determined for ESP -C1, (when theappropriate number of I values are equal in a row of an intensity matrix), in particular when we canapply facet ‘4a-any-x-eq’, computational efficiency of the JS model does improve slightly, at leastin terms of the number of branch and bound nodes searched and number of iterations performed, ifnot also in terms of overall computation time. We conclude that further investigation of ESP -C1facets, with and/or without the added restriction of equal I values, may prove beneficial, thoughonly if additional measures are also applied to reduce computation time for the JS model.


Tab

le4.

9.2:

Res

ults

for

the

appl

icat

ion

ofth

e‘2a

-any

-x-e

q’Fa

cet

ofESP

-C1

(Con

stra

int

(4.4

.1),

wit

hI t

=I s

)to

the

JSm

odel

.W

eus

eC

PL

EX

vers

ion

8.1

and

AM

PL

vers

ion

8.1

ona

2GH

zA

MD

3000

+:

prep

roce

ssin

gop

tion

soff

,ti

me

inse

cond

s,2-

hour

tim

elim

iton

indi

vidu

alpr

oble

min

stan

ces.

Data

JS

mod

elw

ith

ou

tth

eS

olu

tion

Nu

mb

erof

JS

mod

elw

ith

the

Solu

tion

Nu

mb

erof

‘2a-a

ny-x

-eq’

Face

tC

on

stra

ints

‘2a-a

ny-x

-eq’

Face

tC

on

stra

ints

wit

hou

t

Face

t

wit

hF

ace

t

BB

ITS

Tim

eR

NL

Bb k

’s,o

bj

BB

ITS

Tim

eR

NL

Bb k

’s,o

bj

44

05

11070

6280

0.4

01.8

04,3

,1,1

,ob

j=4

347

182

1139

0.1

41.8

04,3

,1,1

,ob

j=4

379

44

05

212

148

0.0

71.0

03,1

,1,o

bj=

3264

17

191

0.0

71.0

03,1

,1,o

bj=

3294

44

05

325

240

0.0

71.2

52,2

,1,o

bj=

3264

9146

0.0

61.2

52,2

,1,o

bj=

3306

44

05

456

507

0.1

01.4

04,1

,1,1

,ob

j=4

347

64

582

0.1

21.4

03,2

,1,1

,ob

j=4

395

44

05

57099

62782

2.8

81.6

03,2

,1,1

,1,o

bj=

5430

16127

139481

6.8

61.6

03,2

,1,1

,1,o

bj=

5460

44

05

6280

2582

0.2

61.8

03,3

,2,1

,ob

j=4

430

216

2227

0.2

81.8

03,3

,2,1

,ob

j=4

470

44

05

759

571

0.1

11.5

02,2

,1,1

,ob

j=4

347

96

804

0.1

31.5

02,2

,1,1

,ob

j=4

403

44

05

8131

789

0.1

61.0

02,1

,1,1

,ob

j=4

347

115

987

0.1

71.0

02,1

,1,1

,ob

j=4

379

44

05

9719

5415

0.3

41.8

04,2

,2,1

,ob

j=4

347

1155

7002

0.5

01.8

04,2

,2,1

,ob

j=4

387

44

05

10

72

762

0.1

01.7

53,2

,1,1

,ob

j=4

347

52

527

0.1

01.7

53,2

,1,1

,ob

j=4

379

44

05

y53418

439493

29.3

5100437

737702

49.1

3

44

015

y12173248

197585735

7475.0

37927374

136122503

5368.6

1

55

05

y5592297

52886828

3714.1

913401055

134808522

10990.3

8

66

05

y188917168

2226706110

276014.4

5165710403

1872404056

302998.6

3


Tab

le4.

9.3:

Res

ults

for

the

appl

icat

ion

ofth

e‘2a

-1di

ff-x

-eq’

Face

tofESP

-C1,

Con

stra

int

(B.2

.232

),to

the

JSm

odel

.W

eus

eC

PL

EX

vers

ion

8.1

and

AM

PL

vers

ion

8.1

ona

2GH

zA

MD

3000

+:

prep

roce

ssin

gop

tion

soff

,ti

me

inse

cond

s,2-

hour

tim

elim

iton

indi

vidu

alpr

oble

min

stan

ces.

Data

JS

mod

elw

ith

ou

t(B

.2.2

32)

Solu

tion

Nu

mb

erof

JS

mod

elw

ith

(B.2

.232)

Solu

tion

Nu

mb

erof

Con

stra

ints

Con

stra

ints

wit

hou

t

(B.2

.232)

wit

h

(B.2

.232)

BB

ITS

Tim

eR

NL

Bb k

’s,o

bj

BB

ITS

Tim

eR

NL

Bb k

’s,o

bj

44

05

11070

6280

0.4

01.8

04,3

,1,1

,ob

j=4

347

123

890

0.1

11.8

04,3

,1,1

,ob

j=4

355

44

05

212

148

0.0

71.0

03,1

,1,o

bj=

3264

no

face

tsap

ply

44

05

325

240

0.0

71.2

52,2

,1,o

bj=

3264

16

196

0.0

71.2

52,2

,1,o

bj=

3270

44

05

456

507

0.1

01.4

04,1

,1,1

,ob

j=4

347

100

751

0.1

21.4

02,2

,2,1

,ob

j=4

355

44

05

57099

62782

2.8

81.6

03,2

,1,1

,1,o

bj=

5430

no

face

tsap

ply

44

05

6280

2582

0.2

61.8

03,3

,2,1

,ob

j=4

430

no

face

tsap

ply

44

05

759

571

0.1

11.5

02,2

,1,1

,ob

j=4

347

112

1148

0.1

71.5

02,2

,1,1

,ob

j=4

363

44

05

8131

789

0.1

61.0

02,1

,1,1

,ob

j=4

347

no

face

tsap

ply

44

05

9719

5415

0.3

41.8

04,2

,2,1

,ob

j=4

347

231

2146

0.1

61.8

04,2

,2,1

,ob

j=4

355

44

05

10

72

762

0.1

01.7

53,2

,1,1

,ob

j=4

347

no

face

tsap

ply

44

05

y53418

439493

29.3

550228

415780

28.2

8

44

015

y12173248

197585735

7475.0

311723455

190553087

7211.3

4

55

05

y5592297

52886828

3714.1

914010308

141649523

10620.2

2

66

05

y188917168

2226706110

276014.4

5181535968

2032868495

300998.8

2


Tab

le4.

9.4:

Res

ults

for

the

appl

icat

ion

ofth

e‘3a

-coe

ff2-x

-eq’

Face

tofESP

-C1,

Con

stra

int

(B.2

.234

),to

the

JSm

odel

.W

eus

eC

PL

EX

vers

ion

8.1

and

AM

PL

vers

ion

8.1

ona

2GH

zA

MD

3000

+:

prep

roce

ssin

gop

tion

soff

,ti

me

inse

cond

s,2-

hour

tim

elim

iton

indi

vidu

alpr

oble

min

stan

ces.

Data

JS

mod

elw

ith

ou

t(B

.2.2

34)

Solu

tion

Nu

mb

erof

JS

mod

elw

ith

(B.2

.234)

Solu

tion

Nu

mb

erof

Con

stra

ints

Con

stra

ints

wit

hou

t

(B.2

.234)

wit

h

(B.2

.234)

BB

ITS

Tim

eR

NL

Bb k

’s,o

bj

BB

ITS

Tim

eR

NL

Bb k

’s,o

bj

44

05

11070

6280

0.4

01.8

04,3

,1,1

,ob

j=4

347

111

738

0.0

81.8

04,3

,1,1

,ob

j=4

351

44

05

212

148

0.0

71.0

03,1

,1,o

bj=

3264

no

face

tsap

ply

44

05

325

240

0.0

71.2

52,2

,1,o

bj=

3264

14

170

0.0

61.2

52,2

,1,o

bj=

3267

44

05

456

507

0.1

01.4

04,1

,1,1

,ob

j=4

347

25

319

0.0

91.4

03,2

,1,1

,ob

j=4

351

44

05

57099

62782

2.8

81.6

03,2

,1,1

,1,o

bj=

5430

no

face

tsap

ply

44

05

6280

2582

0.2

61.8

03,3

,2,1

,ob

j=4

430

no

face

tsap

ply

44

05

759

571

0.1

11.5

02,2

,1,1

,ob

j=4

347

101

804

0.1

41.5

02,2

,1,1

,ob

j=4

355

44

05

8131

789

0.1

61.0

02,1

,1,1

,ob

j=4

347

no

face

tsap

ply

44

05

9719

5415

0.3

41.8

04,2

,2,1

,ob

j=4

347

1211

9898

0.5

91.8

04,2

,2,1

,ob

j=4

351

44

05

10

72

762

0.1

01.7

53,2

,1,1

,ob

j=4

347

no

face

tsap

ply

44

05

y53418

439493

29.3

553015

433619

28.4

7

44

015

y12173248

197585735

7475.0

312020653

196258179

7471.5

5

55

05

y5592297

52886828

3714.1

914629000

130817069

10356.1

8

66

05

y188917168

2226706110

276014.4

5187490706

2140761227

279457.4

8


Tab

le4.

9.5:

Res

ults

for

the

appl

icat

ion

ofth

e‘4a

-any

-x-e

q’Fa

cet

ofESP

-C1,

Con

stra

int

(B.2

.236

),to

the

JSm

odel

.W

eus

eC

PL

EX

vers

ion

8.1

and

AM

PL

vers

ion

8.1

ona

2GH

zA

MD

3000

+:

prep

roce

ssin

gop

tion

soff

,ti

me

inse

cond

s,2-

hour

tim

elim

iton

indi

vidu

alpr

oble

min

stan

ces.

Data

JS

mod

elw

ith

ou

t(B

.2.2

36)

Solu

tion

Nu

mb

erof

JS

mod

elw

ith

(B.2

.236)

Solu

tion

Nu

mb

erof

Con

stra

ints

Con

stra

ints

wit

hou

t

(B.2

.236)

wit

h

(B.2

.236)

BB

ITS

Tim

eR

NL

Bb k

’s,o

bj

BB

ITS

Tim

eR

NL

Bb k

’s,o

bj

66

05

1530536

5210625

693.6

31.8

04,2

,1,1

,1,o

bj=

51370

no

face

tsap

ply

66

05

2>

4971758

>50316572

>7342.3

21.6

0ti

me

lim

it1370

244561

3008423

416.3

61.6

03,2

,1,1

,1,o

bj=

51390

66

05

3>

4279353

>58318807

>7225.1

52.0

0ti

me

lim

it1637

no

face

tsap

ply

66

05

4215070

2200788

334.5

32.0

03,3

,2,1

,1,o

bj=

51637

no

face

tsap

ply

66

05

5>

3833969

>56401722

>7278.4

52.2

0ti

me

lim

it1637

>4250080

>56689956

>7219.2

12.2

0ti

me

lim

it1661

66

05

697307

1252818

193.5

91.8

03,2

,2,1

,1,o

bj=

51637

69942

686327

110.6

11.8

03,2

,2,1

,1,o

bj=

51661

66

05

7>

3456657

>49717792

>7217.4

12.0

0ti

me

lim

it1904

no

face

tsap

ply

66

05

83768980

55255821

6582.5

72.2

03,2

,2,2

,1,1

,ob

j=6

1637

no

face

tsap

ply

66

05

98415

95253

13.7

81.8

04,2

,1,1

,1,o

bj=

51370

135570

1432443

193.5

41.8

04,2

,1,1

,1,o

bj=

51390

66

05

10

4692

49525

7.7

92.2

04,3

,2,1

,1,o

bj=

51370

no

face

tsap

ply

55

05

y5592297

52886828

3714.1

95757717

54620095

3829.9

4

66

05

y188917168

2226706110

276014.4

5181145203

2129281777

269236.7

1

100 Chapter 4. Polyhedral Analysis of the Equality Switch PolytopeT

able

4.9.

6:R

esu

lts

for

the

sim

ult

an

eou

sappli

cati

on

of

the

fou

rfa

cets

dete

rm

ined

forESP

-C1

(‘2a

-an

y-x

-eq’,

‘2a

-1diff

-x-e

q’,

‘3a

-coeff

2-x

-eq’

an

d‘4a

-an

y-x

-eq’)

corre

spon

din

gto

Con

stra

ints

(4.4

.1)

wit

hIt

=Is,

(B.2

.232),

(B.2

.234)

an

d(B

.2.2

36)

resp

ecti

vely

,to

the

JS

model.

We

use

CP

LE

Xversi

on

8.1

an

dA

MP

Lversi

on

8.1

on

a2G

Hz

AM

D3000+

:

pre

pro

cess

ing

opti

on

soff

,ti

me

inse

con

ds,

2-h

ou

rti

me

lim

iton

indiv

idu

al

pro

ble

min

stan

ces.

Data

JS

model

w/o

‘2a-a

ny-x

-eq’,

Solu

tion

JS

model

wit

h‘2a-a

ny-x

-eq’,

Solu

tion

Facets

whic

hdo

not

(B.2

.232),

(B.2

.234)

and

(B.2

.236)

(B.2

.232),

(B.2

.234)

and

(B.2

.236)

apply

BB

ITS

Tim

eR

NL

Bbk’s

,ob

jB

BIT

ST

ime

RN

LB

bk’s

,ob

j

44

05

11070

6280

0.4

01.8

04,3

,1,1

,ob

j=4

20

231

0.0

81.8

04,3

,1,1

,ob

j=4

(B.2

.236)

44

05

212

148

0.0

71.0

03,1

,1,o

bj=

317

191

0.0

71.0

03,1

,1,o

bj=

3(B

.2.2

32),

(B.2

.234),

(B.2

.236)

44

05

325

240

0.0

71.2

52,2

,1,o

bj=

317

181

0.0

71.2

52,2

,1,o

bj=

3(B

.2.2

36)

44

05

456

507

0.1

01.4

04,1

,1,1

,ob

j=4

39

421

0.1

01.4

04,1

,1,1

,ob

j=4

(B.2

.236)

44

05

57099

62782

2.8

81.6

03,2

,1,1

,1,o

bj=

516127

139481

6.6

91.6

03,2

,1,1

,1,o

bj=

5(B

.2.2

32),

(B.2

.234),

(B.2

.236)

44

05

6280

2582

0.2

61.8

03,3

,2,1

,ob

j=4

216

2227

0.3

01.8

03,3

,2,1

,ob

j=4

(B.2

.232),

(B.2

.234),

(B.2

.236)

44

05

759

571

0.1

11.5

02,2

,1,1

,ob

j=4

76

949

0.1

71.5

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j=4

(B.2

.236)

44

05

8131

789

0.1

61.0

02,1

,1,1

,ob

j=4

115

987

0.1

71.0

02,1

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j=4

(B.2

.232),

(B.2

.234),

(B.2

.236)

44

05

9719

5415

0.3

41.8

04,2

,2,1

,ob

j=4

163

1575

0.1

61.8

04,2

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j=4

(B.2

.236)

44

05

10

72

762

0.1

01.7

53,2

,1,1

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j=4

52

527

0.1

01.7

53,2

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j=4

(B.2

.232),

(B.2

.234),

(B.2

.236)

66

05

1530536

5210625

693.6

31.8

04,2

,1,1

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bj=

55

799513

120.4

41.8

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,1,o

bj=

5(B

.2.2

36)

66

05

2>

4971758

>50316572

>7342.3

21.6

0ti

me

lim

it248742

3672703

475.4

81.6

03,2

,1,1

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bj=

5

66

05

3>

4279353

>58318807

>7225.1

52.0

0ti

me

lim

it>

3653479

>56560514

>7211.8

32.0

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me

lim

it(B

.2.2

36)

66

05

4215070

2200788

334.5

32.0

03,3

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bj=

5>

4600345

>57313289

>7215.6

92.0

0ti

me

lim

it(B

.2.2

32),

(B.2

.234),

(B.2

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66

05

5>

3833969

>56401722

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lim

it>

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it

66

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697307

1252818

193.5

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5

66

05

7>

3456657

>49717792

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12.0

0ti

me

lim

it>

3382748

>42879149

>7211.5

42.0

0ti

me

lim

it(B

.2.2

36)

66

05

83768980

55255821

6582.5

72.2

03,2

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j=6

3132899

28424627

5176.7

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66

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5

66

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10

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49525

7.7

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542489

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.2.2

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44

05

y53418

439493

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9

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015

y12173248

197585735

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5615.5

2

55

05

y5592297

52886828

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913295065

134041235

11289.2

6

66

05

y188917168

2226706110

276014.4

5144886950

1746802341

284078.8

1

4.10. Conclusion 101

4.10 Conclusion

The aim of this chapter was to further improve the computation time of the best polynomialsized exact model considered in Chapter 3, the Johnston and Sadinlija (JS) model, for solving theBeam-on Time Constrained Minimum Cardinality problem. The technique of polyhedral analysiswas explored in an attempt to reduce the search space for the JS model and hence to improvecomputational efficiency.

This work has investigated the Equality Switch Polytope, ESP , which is derived from particularconstraints relating the aijk variables of the JS model. We have investigated facets of ESP , andfacets of an extension of ESP , ESP -C1, which incorporates the strict consecutive-1-constraint.(The strict consecutive-1-constraint is necessary to model valid shape matrices within the JSmodel). We determined the effect of the facets of small support on models of ESP and ESP -C1,and the JS model, and provided numerical results. We note that whilst we solve the Beam-onTime Constrained Minimum Cardinality problem with the JS model in this chapter, the ESPand ESP -C1 polytopes are derived independently of the constraint to minimise total beam-ontime and the objective which minimises shape matrices within the JS model. Furthermore, thetechniques investigated here are applicable to the unconstrained and constrained versions of theBeam-on Time Constrained Minimum Cardinality problem.

The facets of ESP we considered numerically improve the search space for our model of ESPhowever they do not produce any real improvement when applied to the JS model. Whilst thischapter has concentrated on ESP facets of small support, we believe that investigation of ESPfacets of large support in the context of the JS model is not warranted given that those with smallsupport did not improve computation time for the JS model.

The facets of ESP -C1 also improved the search space for our model of ESP -C1, for particular datasets considered. However, in contrast to the ESP facets, the ESP -C1 facets also result in someimprovement when applied to the JS model. We considered a subset of the facets of small supportof ESP -C1, where the I parameters are equal. We found that where there was opportunity toapply these facets to problems, (where the appropriate number of I values were equal in a row of anintensity matrix) they slightly improved at least the number of branch and bound nodes searchedand number of iterations performed. Less often they also slightly improved the computation time.We were unable to test the equal I facets on further batches of problems on the JS model, whereintensity values had a greater chance of being equal in an intensity matrix, since computation timesbecame prohibitively large, particularly since the AMPL and CPLEX preprocessing options wereswitched off. Future work could investigate more general facets for this polytope though perhapsonly if additional measures are also applied to reduce computation time for the JS model. Furtherinvestigation into the properties of the ESP -C1 equal I facets may also prove beneficial.

The ESP and ESP -C1 polytopes considered in this chapter do not appear to have previouslybeen investigated in the mathematical programming literature and may have application outsidethe context of this thesis. Hence our study of ESP and ESP -C1 also contributes to the widerfield of integer programming.


Finally, in general, the ESP and ESP -C1 facets considered in this work do not improve theoverall efficiency of the JS model. Therefore, it is clear that alternative approaches should beconsidered for improving computation time, whether this be via the determination of facets ofother significant polytopes relevant to the JS model or via other methods. In the following chapterwe determine improved bounds for the variables of the JS model (and for the Cumulative Countermodel of Chapter 3, Section 3.8.1) and demonstrate that the application of the bounds results insubstantial computational improvement.

103CHAPTER 5

Novel Bounds on the Beam-on Time Related Variables in

Exact Integer Programming Models for the Modulation of

Intensity Beams in Cancer Radiotherapy Using Multileaf

Collimators

5.1 Introduction

In this final chapter we investigate further new techniques for reducing the computation time ofexact integer programming models for optimising the delivery of Intensity Modulated RadiationTherapy. Our focus in this work is to improve upper and lower bounds on the beam-on time relatedvariables, of the best polynomial sized integer programming model considered in Chapter 3, theJS model, and of our new pseudo-polynomial sized integer programming model, the CumulativeCounter model with simple bounds, also defined in Chapter 3. After the Counter model of Baataret al. [4] (extended to incorporate simple bounds), the Cumulative Counter model with simplebounds is the best pseudo-polynomial sized model we have previously considered. We hereafterrefer to the Cumulative Counter model with simple bounds as the CC model. (We note that theJS model is the best polynomial sized model of Chapter 3 when the Step-up Method is applied tothe formulation).

In this work, ‘beam-on time related variables’ are defined as the integer beam-on time variables bk ofthe JS model, for k = 1, . . . ,K, and, the integer Nb variables of the CC model, for b = 1, . . . , bmax,where the Nb variables count the number of shape matrices given a particular level of beam-on timeor more in a solution. The polynomial sized JS model, indexing on shape matrices k, increases incomputation time with increasing problem size and maximum intensity level within an intensitymatrix. On the other hand, the CC model indexes on radiation level b, and therefore increasesin computation time predominantly with increasing maximum intensity level rather than problemsize. Whilst in Chapter 3 the pseudo-polynomial sized models were shown to outperform thepolynomial sized models, we continue to study the properties, benefits and limitations of bothmodel types in this chapter; particularly how each responds to improved variable bounds and alsosince ‘Counter type’ models may not have direct application within the cutting stock context.Finally, we investigate the CC model in this study, rather than the Counter model with simplebounds, since the CC model shares similar properties with the JS model (which we discuss in thefollowing two sections) and since we can directly apply the bounds we determine in this chapter tothe variables of the CC model. The bounds we determine in this chapter would be applied in theform of constraints in the Counter model with simple bounds and since bounds are handled moreefficiently than constraints in the Simplex solving method, we choose to focus on the CC model.

In this chapter we again consider the unconstrained Beam-on Time Constrained Minimum Car-dinality problem, however the majority of results presented here apply to any upper bound ontotal beam-on time and therefore any objective for the leaf sequencing problem discussed in theIntroduction (Chapter 1). Furthermore, the majority of the bounds work we present can also beapplied to the constrained version of the leaf sequencing problem under consideration. To handle

104Chapter 5. Novel Bounds on the Beam-on Time Related Variables in Exact Integer Programming Models

for the Modulation of Intensity Beams in Cancer Radiotherapy Using Multileaf Collimators

the constrained case, minor amendments would need to be made to our bounds results, for example,calculating Beammin for the constrained problem, i.e. without the closed form solution, and in-corporating more than one closed leaf position within shape matrices. There is also one additionalconstraint we apply to the JS and CC models, in the applications section of this chapter, whichis specific to the unconstrained BTCMC problem. However this constraint is not fundamental toeither model and can be left out of the formulations to solve the constrained case. Of course tosolve the constrained problems, the relevant MLC mechanical constraints would also need to beformulated and included in the model under consideration.

As mentioned in the Introduction (Chapter 1), even if an ‘allowable intensity multiset’ is known(e.g. a set of bk or Nb variable bounds), it is still a strongly NP-hard problem to determine a feasibledecomposition of even a single row of an intensity matrix, [43]. However, the determination of lowerupper bounds and higher lower bounds on variables in an integer program decreases the feasibleregion to be searched and thus our aim in this chapter is to investigate improved bounds for thebeam-on time related variables of the JS and CC models and therefore to reduce the computationtime for the solution of the strongly NP-hard problem. This work is further motivated by resultsin Chapter 3 which demonstrate that even simple bounds on variables can have a dramatic effecton computational efficiency.

In the sections to follow, we initially define the Sum Constrained Sorted Multiset System (SCSMS)which comprises two constraints: the first bounds the total beam-on time of a decomposition andthe second orders the beam-on time related variables. The SCSMS is a property of both theJS and CC formulations and we use the SCSMS to iteratively improve bounds on the beam-ontime related variables of each model. Following this we demonstrate that the bounds on thebeam-on time variables of the JS model can be converted to bounds on the Nb variables of theCC model, and vice versa, via a simple relationship. Since this is the case, in the remainder ofthe chapter, we determine bounds for variables bk of the JS model and simply convert these tobounds on the Nb variables when we numerically test the effect of bounds on the CC model inour applications section. Therefore, in Sections 5.4 to 5.7 we extend the ideas of Baatar et al.[1] on properties of intensity matrices and decompositions, utilise the integrality of variables andderive integer and linear programs to improve bounds on variables bk. Finally, in Section 5.8, wesystematically apply the bk bounds to the JS model, and the Nb bounds to the CC model, todetermine computational improvement. We solve the unconstrained Beam-on Time ConstrainedMinimum Cardinality problem. We also apply the Step-up Method to the JS model, togetherwith our bounds algorithms, to further decrease overall computation time. (It has been previouslyshown that the application of the Step-up Method to the CC model does not improve computationalefficiency, see Chapter 3).

All the bounds algorithms we consider in this chapter are calculated via preprocessing routines.The computation time for the preprocessing is recorded separately and also included in overallcomputation time for each model. Finally, the randomly generated data sets we consider for thiswork are again batches of 100 problems, the smallest of which is of size 4 × 4 with intensitiesranging from 0 and 5 inclusive and the largest of which is of size 18× 18 with intensities rangingfrom 0 and 5 inclusive. In general, the maximum intensity levels we consider are 5, 10 and 15.We test the best algorithms of this chapter to their limits. Once a batch of a particular size and

5.2. The Sum Constrained Sorted Multiset System 105

maximum intensity level exceeds approximately 100,000 seconds of computation time, no furtherbatches of increased size for the same maximum intensity level are computed. We demonstrate inSection 5.8 that the application of improved bounds to the JS and CC models significantly reducescomputation time.

5.2 The Sum Constrained Sorted Multiset System

As discussed in the Introduction to this chapter, we are interested in bounds that can be derivedfor ‘beam-on time related’ variables. We now focus on a particular structure which is common toboth the bk variables of the JS model and the Nb variables of the CC model. As mentioned, weterm this structure a Sum Constrained Sorted Multiset System (SCSMS) and we use properties ofthe SCSMS to tighten bounds on our ‘beam-on time related’ variables bk and Nb.

To describe the Sum Constrained Sorted Multiset System we first define a variable t ∈ ZD+ , whereD is the dimension of t, t ∈ ZD+ to be an upper bound on t and t ∈ ZD+ to be a lower bound on t.We also define B to be the sum over all variables td for d = 1, . . . , D, B to be an upper bound on

B and B to be a lower bound on B. Thus B ≤D∑d=1

td ≤ B and td ≤ td ≤ td for all d = 1, . . . , D.

The Sum Constrained Sorted Multiset System considers non-increasing t variables and is given by

t1 ≥ t2 ≥ · · · ≥ tD ≥ 0 (5.2.1)

and

B ≤D∑d=1

td ≤ B. (5.2.2)

Within the context of Constraint Programming, constraints (5.2.1) and (5.2.2) can be consideredin conjunction, for example via the single constraint ‘ordered sum’ of ECLiPSe, [54].

The Sum Constrained Sorted Multiset System can be written equivalently with the beam-on timevariables, bk, of the JS model or the ‘beam-on time related’ variables, Nb, of the CC model. Wehave

b1 ≥ b2 ≥ · · · ≥ bK ≥ 0 (5.2.3)

and

B ≤K∑k=1

bk ≤ B, (5.2.4)

or,N1 ≥ N2 ≥ · · · ≥ Nbmax ≥ 0 (5.2.5)

and

B ≤bmax∑b=1

Nb ≤ B (5.2.6)

respectively. For B = B = B = Beammin, constraints (5.2.3) and (5.2.4) follow from constraints(2.2.14) and (2.3.2) respectively from the JS model, where the beam-on time variables are inte-ger, and constraints (5.2.5) and (5.2.6) follow from constraints (3.8.3) and (3.8.13), and (3.8.14)respectively of the CC model.



We now demonstrate how the SCSMS, given by constraints (5.2.1) and (5.2.2), can be used toimprove bounds on variables td for all d = 1, . . . , D.

Given a non-increasing sequence of variables (as we have in the Sum Constrained Sorted MultisetSystem) we require the upper bounds on the variables to also be non-increasing. That is, we require

td ≥ td+1, ∀ d = 1, . . . , D − 1. (5.2.7)

Therefore, given any upper bounds, they can be ordered using the following iterative procedure:

Algorithm 5.2.1 t :=UB Consistency(D,t)for h = 2, . . . , D doth := min{th−1, th}

end forreturn t

Similarly, given lower bounds on variables td, for all d = 1, . . . , D, in a non-increasing sequence ofvariables we know that

td ≥ td+1, ∀ d = 1, . . . , D − 1 (5.2.8)

and hence we can again use an iterative procedure to order the lower bounds appropriately:

Algorithm 5.2.2 t :=LB Consistency(D,t)for h = D − 1, D − 2, . . . , 1 doth := max{th, th+1}

end forreturn t

Furthermore, given upper (resp. lower) bounds on variables td, for all d = 1, . . . , D, we can improvethe lower (resp. upper) bounds on the variables again considering our non-increasing sequence andalso using the fact that our variables are integer. We have the following two propositions.

Proposition 5.2.1. Let td, for all d = 1, . . . , D, be a non-increasing sequence with td ≤ td, for all

d = 1, . . . , D andD∑d=1

td ≥ B. Then let

βsg

=

B −

s−1∑d=1

td −D∑d=g

td

g − s

, ∀ s = 1, . . . , D, g = s+ 1, . . . , D + 1,

and a lower bound on each ts is

ts = maxg=s+1,...,D+1

βsg, ∀ s = 1, . . . , D.

Proof. Proof by Contradiction: Suppose that a sequence using variables t1, . . . , tD exists satisfyingthe conditions of Proposition 5.2.1, with ts < β

sgand therefore ts ≤ βsg−1. Now since the sequence


is non-increasing we have tg−1 ≤ tg−2 ≤ · · · ≤ ts ≤ βsg − 1. So

D∑d=1

td ≤s−1∑d=1

td +g−1∑d=s

(βsg− 1) +

D∑d=g

td

=s−1∑d=1

td + (g − s)(βsg− 1) +

D∑d=g

td.

Using the fact that

dye − 1 < y, ∀ y ∈ R, (5.2.9)

we have

D∑d=1

td <

s−1∑d=1

td +

B − s−1∑d=1

td −D∑d=g

td

+D∑d=g

td

= B

by the definition of βsg

, which contradicts the condition that the total sum of the sequence ofvariables td, for all d = 1, . . . , D, is at least B. Hence ts cannot be less than β

sg, i.e. β

sgis a lower

bound on ts, for all g = s + 1, . . . , D + 1. Clearly if βsg< 0 for all g we should take zero as the

lower bound.

Proposition 5.2.2. Let td, for all d = 1, . . . , D, be a non-increasing sequence of variables with

td ≥ td, for all d = 1, . . . , D andD∑d=1

td ≤ B. Then let

βsg =

B −

g∑d=1

td −D∑

d=s+1

td

s− g

, ∀ s = 1, . . . , D, g = 0, . . . , s− 1,

and an upper bound on each ts is

ts = ming=0,...,(s−1)

βsg, ∀ s = 1, . . . , D.

Proof. Proof by Contradiction: Suppose that a sequence using variables t1, . . . , tD exists satisfyingthe conditions of Proposition 5.2.2, with ts > βsg and therefore ts ≥ βsg+1. Now since the sequenceis non-increasing we have tg+1 ≥ tg+2 ≥ · · · ≥ ts ≥ βsg + 1. So

D∑d=1

td ≥g∑d=1

td +s∑

d=g+1

(βsg + 1) +D∑

d=s+1

td

=g∑d=1

td + (s− g)(βsg + 1) +D∑

d=s+1

td.

Using the fact that

byc+ 1 > y, ∀ y ∈ R, (5.2.10)



we have

D∑d=1

td >

g∑d=1

td +

(B −

g∑d=1

td −D∑

d=s+1

td

)+

D∑d=s+1

td

= B

by the definition of βsg, which contradicts the condition that the total sum of the sequence ofvariables td, for all d = 1, . . . , D, is no more than B. Hence ts cannot exceed βsg, i.e. βsg is anupper bound on ts, for all g = 0, . . . , s− 1. Clearly if βsg < 0 for any g we have deduced that theproblem is infeasible.

Considering a non-increasing sequence of variables td, for all d = 1, . . . , D, Proposition 5.2.1 statesthat if we know upper bounds on td for d ≤ D we can determine a lower bound on variable tsfor s = 1, . . . , D by subtracting all upper bounds for variable t with d < s and any number withd > s from the lower bound on B and dividing the result by the number of t variable values whichhave not been included in the calculation. Since we are subtracting upper bounds and since wehave a lower bound on B, we are subtracting more than is necessary, from a total that is lessthan necessary for an optimal solution and hence we will obtain a lower bound on variable ts fors = 1, . . . , D. Furthermore, we can take the least integer greater than or equal to this quantitysince we are dealing with integer variables. Then because we will have multiple lower boundson a single variable ts, we can take the maximum of these as our lower bound. The argumentfor Proposition 5.2.2 is analogous to this argument for Proposition 5.2.1. Finally, where the highindex upper bound inputs for Proposition 5.2.1 are not expected to be good we can just considerg = D + 1 and where the low index lower bound inputs for Proposition 5.2.2 are not expected tobe good we can just consider g = 0.

Propositions 5.2.1 and 5.2.2 improve the bounds on our variables because we can utilise roundingtechniques. Without rounding the propositions are implied by the constraints of the Sum Con-strained Sorted Multiset System. Furthermore, it should be noted that, similar ‘propagation’ ideasto Propositions 5.2.1 and 5.2.2 are well known in the context of Constraint Programming [54].

We apply Propositions 5.2.1 and 5.2.2 iteratively, and recursively, to get as good bounds on theindividual ‘beam-on time related’ values, as possible. This is done formally in an algorithm laterin this section, where we also give an example illustrating the propositions.

In the meantime we show that Propositions 5.2.1 and 5.2.2 return new non-increasing lower boundsand upper bounds respectively on our t variables for increasing s, if the original lower and upperbounds are non-increasing.

Proposition 5.2.3. Proposition 5.2.1 returns non-increasing lower bounds on variables ts for in-creasing s, provided the upper bounds, ts, are non-increasing.

Proof. We wish to show

ts = maxg∈{s+1,...,D+1}

βsg≥ maxg∈{s+2,...,D+1}

βs+1,g

= ts+1, ∀ s = 1, . . . , D − 1. (5.2.11)


We therefore show that βsg≥ β

s+1,g+1for g = s + 1, . . . , D. Note that there is one more β term

for ts than for ts+1.

We let V = B −D∑d=1

td. Then

βsg

=

V +

g−1∑d=s

td

g − s

and β

s+1,g+1=

V +

g∑d=s+1

td

g − s

.

As the td are non-increasing it is clear that βsg≥ β

s+1,g+1, given that each expression has an

equal number of terms in the numerator and the same denominator.

The value of the remaining extra term on the left hand side of expression (5.2.11), βs,D+1

=V +

D∑d=s

td

D + 1− s

, does not detract from the result.

Hence Proposition 5.2.1 returns non-increasing lower bounds on variables ts for increasing s, pro-vided the upper bounds, ts, are non-increasing.

Proposition 5.2.4. Proposition 5.2.2 returns non-increasing upper bounds on variables ts for in-creasing s, provided the lower bounds, ts, are non-increasing.

Proof. We wish to show

ts−1 = ming∈{0,...,s−2}

βs−1,g ≥ ming∈{0,...,s−1}

βsg = ts, ∀ s = 2, . . . , D. (5.2.12)

We therefore show that βs−1,g ≥ βs,g+1 for g = 0, . . . , s − 2. Note that there is one more β termfor ts than for ts−1.

We let V = B −D∑d=1

td. Then

βs−1,g =

V +

s−1∑d=g+1

td

s− g − 1

and βs,g+1 =

V +

s∑d=g+2

td

s− g − 1

.

As the td are non-increasing it is clear that βs−1,g ≥ βs,g+1, given that each expression has anequal number of terms in the numerator and the same denominator.

The value of the remaining extra term on the right hand side of expression (5.2.12), βs,0 =



V +

s∑d=1

td

s

, does not detract from the result.

Hence Proposition 5.2.2 returns non-increasing upper bounds on variables ts for increasing s,provided the lower bounds, ts, are non-increasing.

We now give an example using beam-on time variables bk, to show the use of Propositions 5.2.1 and5.2.2. Suppose we are given B = 12, B = 13, K = 4, upper bounds b = (8, 7, 5, 4), lower boundsb = (0, 0, 0, 0) and a particular I whose minimal beam-on time, non-increasing, decompositionyields b = (6, 4, 1, 1).

Proposition 5.2.1 returns: b1 = max(−4, 2, 3, 3) = 3, b2 = max(−5, 0, 2) = 2, b3 = max(−7,−1, 0) =0 and b4 = max(−8, 0) = 0. (As mentioned previously, where Proposition 5.2.1 returns only neg-ative values for a lower bound, we take zero as the lower bound). We compare these new lowerbounds against the initialised lower bound values for bk and take the maximum value in each case.In the above example, Proposition 5.2.1 improves or matches the initialised lower bounds for eachk = 1, . . . ,K.

The new lower bound values can now be used as input for Proposition 5.2.2. We obtain: b1 = 11,b2 = min(6, 10) = 6, b3 = min(4, 5, 8) = 4 and b4 = min(3, 3, 4, 8) = 3. Again we comparethese upper bounds against the initialised upper bound values and obtain: b1 = min(11, 8) = 8,b2 = min(6, 7) = 6, b3 = min(4, 5) = 4 and b4 = min(3, 4) = 3.

Therefore, using Propositions 5.2.1 and 5.2.2 we improve our lower bound values from b = (0, 0, 0, 0)to b = (3, 2, 0, 0) and our upper bound values from b = (8, 7, 5, 4) to b = (8, 6, 4, 3).

For use in our applications section, Section 5.8, we refer to Propositions 5.2.1 and 5.2.2 via a singlealgorithm which we name Bounds Propagate.

5.3. Bounds Conversion Between Models 111

Algorithm 5.2.3 (t, t) :=Bounds Propagate(D,B,B, t, t)repeat until no change in t

for s = 1, . . . , D dofor g = s+ 1, . . . , D + 1 do

βsg

:=

B −

s−1∑d=1

td −D∑d=g

td

g − s

end forts := max

(max

g=s+1,...,D+1βsg, 0, ts

)end forfor s = 1, . . . , D do

for g = 0, . . . , s− 1 do

βsg :=

B −

g∑d=1

td −D∑

d=s+1

td

s− g

if βsg < 0 then

return (−,−)end if

end forts := min

(min

g=0,...,s−1βsg, ts

)end for

end repeatreturn (t, t)

In practice, we generally do not see improvement in lower and upper bounds after the first timethrough the repeat loop in the Bounds Propagate algorithm.

Finally, the simple upper bounds on the Nb variables that we previously used in the CC model,constraint (3.8.17), see Chapter 3, Section 3.8.1, are a special case of Proposition 5.2.2. If welet b = 1, . . . , bmax, s = b, g = 0, B = Beammin, D = bmax, t = N and if all Nd = 0 for d =b+ 1, . . . , bmax we obtain (3.8.17). Hence the simple bounds on the Nb variables are incorporatedinto the algorithms we test on the CC model via algorithms involving Bounds Propagate.

5.3 Bounds Conversion Between Models

Given that the bk variables and bk bounds of the JS model and the Nb variables and Nb boundsof the CC model are ordered in non-increasing sequences, as illustrated by constraints (2.2.14),and (3.8.3) and (3.8.13), respectively, and algorithms UB Consistency and LB Consistency of theprevious section, we now present two expressions which describe the relationship between the Nbvariables and the bk variables:

bk ≤ bk ⇔ Nbk+1 ≤ k − 1, ∀ k = 1, . . . ,K (5.3.1)



andbk ≥ bk ⇔ Nbk ≥ k, ∀ k = 1, . . . ,K. (5.3.2)

Hence any bounds on variables bk can be converted to bounds on variables Nb and vice versa.Specifically, expression (5.3.1) says that if we know an upper bound on the beam-on time for aparticular shape matrix k in a non-increasing decomposition then the number of shape matricesthat can be given more than this amount of radiation has to be at most one less than k, wherewe have also used the integrality of the beam-on time variables. Expression (5.3.2) says that ifwe know a lower bound on the beam-on time that can be applied to shape matrix k then thenumber of shape matrices that can be given that amount of radiation or more must be at least k.Algorithmically we have:

Algorithm 5.3.1(bmax, N , N

):=Consistent b N Bounds(K,Klb, b, b, b

max, N , N)

bmax := min(bmax, b1)for k = 1, . . . ,K such that 0 ≤ bk ≤ bmax − 1 doNbk+1 := min

(Nbk+1, k − 1

)end forN1 := min

(N1,K

)for k = 1, . . . ,K such that bmax ≥ bk ≥ 1 doNbk := max

(Nbk , k

)end forN1 := max

(N1,Klb

)return

(bmax, N , N

)and

Algorithm 5.3.2 (K, b, b) :=Consistent N b Bounds(K,Klb, b, b, bmax, N , N)

K := min(K, N1,min{k ∈ {1, . . . ,K} : bk = 0} − 1

)for b = 1, . . . , bmax such that 0 ≤ Nb ≤ K − 1 dobNb+1

:= min(bNb+1

, b− 1)

end forb1 := min

(b1, b

max)

for b = 1, . . . , bmax such that K ≥ Nb ≥ 1 do

bNb:= max

(bNb

, b

)end forfor v = 1, . . . ,Klb dobv := max (bv, 1)

end forreturn (K, b, b)

where the upper and lower bounds on the Nb variables in Consistent b N Bounds are subscriptedby upper and lower bounds on the bk variables respectively and the upper and lower bounds onthe bk variables in Consistent N b Bounds are subscripted by upper and lower bounds on the Nbvariables respectively.

5.4. Bounds Arising from Properties of the Intensity Matrix 113

In Consistent b N Bounds, if bk = 0 for any k ∈ {1, . . . ,K} then in a non-increasing decompositionwe can reduce the value of K and set N1 equal to the minimum value of k ∈ {1, . . . ,K} forwhich this occurs, minus 1. If K is equal to the number of shape matrices to be used in anoptimal solution then no bk will equal zero for k = 1, . . . ,K and N1 = N1 = N1 = K. Also, inConsistent b N Bounds, we set bmax to the minimum of its current value and b1, as both quantitiesrepresent the most radiation that can be applied to any shape matrix when we consider a non-increasing decomposition. Furthermore, as mentioned previously, Klb is a known lower bound onthe minimum number of shape matrices that can be used in a solution. (We provide a formulafor calculating Klb, as determined by Baatar et al. [1], in Chapter 2, Section 2.3.1). Hence, theinitialisation of N1 in Consistent b N Bounds must incorporate Klb, since it is, by definition, alsoa lower bound on the number of shape matrices that can be given radiation level 1 or more in asolution.

Consistent b N Bounds may not return a full set of upper and lower bounds on the Nb variablesfor b = 1, . . . , bmax. The particular bounds returned for variables Nb will be dependent on thevalues of the bk bounds input into the algorithm. Where there are terms missing in the set ofNb and/or Nb bounds, so that the elements returned are non-consecutive, the missing terms canbe determined using algorithms UB Consistency and LB Consistency respectively of the previoussection. This observation is also true for Consistent N b Bounds, where again any missing termsin the bk bounds returned can be determined.

In Consistent N b Bounds, if Nb equals zero for any b ∈ {1, . . . , bmax}, then in a non-increasingdecomposition, the size of bmax can be reduced and we set b1 equal to the minimum value ofb ∈ {1, . . . , bmax} for which Nb = 0, minus 1. Otherwise, Nb is not equal to zero for any b ∈{1, . . . , bmax}. In this case, bmax is the appropriate bound for b1. In this same algorithm, weinitialise K to the minimum of, the minimum value of k for which bk is zero, minus 1, and theupper bound we have for N1, the number of shape matrices given radiation level 1 or more. Finally,lower bounds on variables bk must be set greater than or equal to 1 for k = 1, . . . ,Klb, in additionto our other initialisations for bk, since we must apply radiation of at least 1 unit to at least Klb

shape matrices in a solution.

Therefore, as a result of this relationship between the bk bounds and the Nb bounds, in the followingsections, we simply focus on the determination of bounds for the beam-on time variables, bk, of theJS model. Then, in Section 5.8, we apply the bounds algorithms we develop to both the JS andCC models, where, in the case of the CC model, we first convert the bk bounds we have determinedto Nb bounds using the Consistent b N Bounds algorithm of this section.

5.4 Bounds Arising from Properties of the Intensity Matrix

5.4.1 Initialisation of Bounds on Beam-on Time Variables. In this section, we deter-mine bounds for the beam-on time variables, bk, of the JS model using properties of the intensitymatrix under investigation.

Considering our non-increasing decomposition, we can immediately write some very simple bounds



for beam-on time variables bk:

bk =⌊B

k

⌋, ∀ k = 1, . . . ,K, (5.4.1)

b1 = max(⌈

B

K

⌉, 1), (5.4.2)

bk = 1, ∀ k = 2, . . . ,Klb (5.4.3)

andbk = 0, ∀ k = Klb + 1, . . . ,K. (5.4.4)

In the case of equation (5.4.2), we would expect K to be less than B in practice and therefore thatthe first term in the bracket should dominate.

We improve on this simple initialisation by now considering the entries in our intensity matrix.If there is an entry equal to one in an intensity matrix, then we know that there must be atleast 1 shape matrix with a beam-on time of one unit to satisfy this entry. Therefore we can say

that b1 ≥⌈B − 1K − 1

⌉and bK ≤ 1. If there is an entry equal to two, and an entry equal to one,

in an intensity matrix, then we know that b1 ≥⌈B − 3K − 2

⌉, bK ≤ 1 and bK−1 ≤ 2. (We do not

extend these ideas beyond the smallest two entries in the intensity matrix since the number ofcombinations for solutions grows combinatorially as we consider larger elements). Denoting thesmallest non-zero intensity value in our intensity matrix by I(1) and the second smallest, different,non-zero intensity value by I(2), we generalise this result as follows:

b1 ≥ max(⌈

B − I(1)

K − 1

⌉,

⌈B − (I(1) + I(2))

K − 2

⌉),

bK ≤ I(1) and bK−1 ≤ I(2).

We initialise these bounds on variables bk using the following algorithm:


Algorithm 5.4.1 (b, b) :=Initialise Multiset Bounds(K,Klb, B,B, b1, I(1), I(2))

for k = 2, . . . ,K − 2 do

bk :=⌊B

k

⌋end for

bK−1 := min(⌊

B

K − 1

⌋, I(2)

)bK := min

(⌊B

K

⌋, I(1)

)b1 := max

(⌈B

K

⌉,

⌈B − I(1)

K − 1

⌉,

⌈B − (I(1) + I(2))

K − 2

⌉, 1)

for k = 2, . . . ,Klb dobk := 1

end forfor k = Klb + 1, . . . ,K dobk := 0

end forreturn (b, b)

We have purposefully not included the initialisation of b1 in this procedure but have instead passedb1 as an argument to Initialise Multiset Bounds. The reason for this is that we continue to improvethe value for b1, in particular, in the sections to follow.

5.4.2 Extension of the Work of Baatar et al. [1, 2]: Properties of IntensityMatrices and Decompositions. We now extend the work of Baatar et al. [1] on propertiesof intensity matrices and decompositions. A number of definitions in this section are taken fromBaatar et al. [1], however the notation has been changed, where appropriate, to be consistent withthe current work. As mentioned in the Introduction (Section 5.1), we focus on the unconstrainedcase in this chapter (and in fact in this entire thesis). We note again that only minor amendmentsare necessary to the results of this section, and to those of this thesis generally, to solve theconstrained problem, provided the appropriate MLC mechanical constraints can also be formulated.For details of the constrained versions of the Baatar et al. [1] results presented here, see Baatar etal. [1].

Baatar et al. [1] define the following matrices:

Iij =

Iij − Ii(j−1), 1 < j ≤ nIi1, j = 1−Iin, j = n+ 1

, ∀ i = 1, . . . ,m, (5.4.5)

where I is the m× n intensity matrix,

Lij = max{

0, Iij}, ∀ i = 1, . . . ,m, j = 1, . . . , n+ 1 (5.4.6)

andRij = max

{0,−Iij

}, ∀ i = 1, . . . ,m, j = 1, . . . , n+ 1, (5.4.7)

such that L− R = I. For example:



I =

0 7 7 9 71 3 5 8 01 0 6 3 81 8 4 5 75 6 5 3 2

, I =

0 7 0 2 −2 −71 2 2 3 −8 01 −1 6 −3 5 −81 7 −4 1 2 −75 1 −1 −2 −1 −2

,

L =

0 7 0 2 0 01 2 2 3 0 01 0 6 0 5 01 7 0 1 2 05 1 0 0 0 0

and R =

0 0 0 0 2 70 0 0 0 8 00 1 0 3 0 80 0 4 0 0 70 0 1 2 1 2

.

Using these definitions, Baatar et al. [1] prove that the minimum total beam-on time for thedelivery of row i of intensity matrix I is given by:

Bi =n+1∑j=1

Lij =n+1∑j=1

Rij , ∀ i = 1, . . . ,m. (5.4.8)

In our example: (Bi) = (9, 8, 12, 11, 6)′. This result implies that any decomposition of I must applyat least Bi units of radiation to shape matrices which expose some cell in row i. We have:∑

k=1,...,K s.t.∑nj=1 xijk> 0

bk ≥ Bi, ∀ i = 1, . . . ,m. (5.4.9)

Furthermore, there exists a decomposition of row i of I with total beam-on time equal to Bi.

Baatar et al. [1] also prove that the minimum total beam-on time, Beammin, for the decompositionof I (for the unconstrained case) is given by:

Beammin = maxi=1,...,m

Bi, (5.4.10)

demonstrating that the problem of determining the minimum total beam-on time of a decompo-sition is solvable in polynomial time, as discussed in the Introduction to this thesis (Chapter 1).In our example Beammin = max(9, 8, 12, 11, 6) = 12. Engel [27] independently proves (5.4.8) and(5.4.10).

Additionally, as previously mentioned, Baatar et al. [1] also define a lower bound on the minimumnumber of shape matrices that can be used in any decomposition of I, Klb (see Chapter 2, Section2.3.1).

Referring to our example, a non-optimal decomposition of I in terms of total beam-on time (15)and number of shape matrices used (7) is:

Example 1:

I =

0 7 7 9 71 3 5 8 01 0 6 3 81 8 4 5 75 6 5 3 2

= 4

0 1 1 1 10 0 1 1 00 0 1 0 00 1 1 1 11 1 1 0 0

+3

0 1 1 1 10 1 0 0 00 0 0 1 10 1 0 0 00 0 0 1 0

+2

0 0 0 1 00 0 0 1 00 0 1 0 00 0 0 0 10 1 0 0 0

+


2

0 0 0 0 00 0 0 1 00 0 0 0 10 0 0 0 00 0 0 0 1

+ 2

0 0 0 0 00 0 0 0 00 0 0 0 10 0 0 0 00 0 0 0 0

+

0 0 0 0 01 0 0 0 01 0 0 0 01 1 0 0 01 0 0 0 0

+

0 0 0 0 00 0 1 0 00 0 0 0 10 0 0 1 10 0 1 0 0

.

We see that the total beam-on time applied to each row, where at least one cell is open, doesindeed satisfy (5.4.9):

b1 + b2 + b3 = 9 ≥ B1 = 9,b1 + b2 + b3 + b4 + b6 + b7 = 13 ≥ B2 = 8,b1 + b2 + b3 + b4 + b5 + b6 + b7 = 15 ≥ B3 = 12,b1 + b2 + b3 + b6 + b7 = 11 ≥ B4 = 11,b1 + b2 + b3 + b4 + b6 + b7 = 13 ≥ B5 = 6,

and that the decomposition of row 1 and row 4 of I are examples of attaining the minimumbeam-on time for the decomposition of the row, B1 and B4 respectively. Klb for Example 1 ismax(2, 4, 3, 4, 4) = 4.

We can then solve for a minimum total beam-on time decomposition of I. We have:

Example 2:

I =

0 7 7 9 71 3 5 8 01 0 6 3 81 8 4 5 75 6 5 3 2

= 3

0 1 1 1 10 0 0 1 00 0 0 0 10 1 0 0 00 1 1 0 0

+3

0 1 1 1 10 1 1 1 00 0 1 0 00 1 1 1 11 0 0 0 0

+2

0 0 0 0 00 0 0 0 00 0 0 0 10 0 0 0 10 0 0 1 1

+

2

0 0 0 1 00 0 1 1 00 0 1 1 10 0 0 1 11 1 0 0 0

+

0 0 0 0 10 0 0 0 00 0 1 1 10 1 0 0 00 0 1 0 0

+

0 1 1 1 01 0 0 0 01 0 0 0 01 1 1 0 00 1 1 1 0

.

In this case only 6 shape matrices are used. Again (5.4.9) is satisfied and row 3 attains B3 whichequals Beammin. Klb is independent of the decomposition itself and is therefore the same as forExample 1, equaling 4.

In any decomposition, every shape matrix has left and right multileaf collimator leaf positionscorresponding to each row. We define leftik(x) to be the left leaf position in row i of shape matrixk and rightik(x) to be the right leaf position in row i of shape matrix k. We have:

leftik(x) = minj=1,...,n

{j : xijk = 1} − 1, ∀ i = 1, . . . ,m, k = 1, . . . ,K (5.4.11)

andrightik(x) = max

j=1,...,n{j : xijk = 1}+ 1, ∀ i = 1, . . . ,m, k = 1, . . . ,K (5.4.12)

for non-zero row i. Since we consider the unconstrained case, if row i is a zero row in shape



matrix k, leftik(x) and rightik(x) can be chosen arbitrarily in {0, 1, . . . , n} and {1, 2, . . . , n + 1}respectively, so that rightik(x) = leftik(x) + 1. In Example 2, left(x) and right(x) are as follows:

left(x) =

1 1 5 3 4 13 1 5 2 5 04 2 4 2 2 01 1 4 3 1 01 0 3 0 2 1

and right(x) =

6 6 6 5 6 55 5 6 5 6 26 4 6 6 6 23 6 6 6 3 44 2 6 3 4 5

where we have arbitrarily chosen the closed leaf position in all shape matrices to be left(x) = n,right(x) = n+ 1. (For the constrained case, closed leaf positions compatible with MLC mechanicalconstraints must be considered; a single closed leaf position will in general not suffice).

Utilising these definitions to extend their arguments further, Baatar et al. [1] prove that if thereexist m × (n + 1) matrices Lij =

∑k=1,...,K s.t.leftik(x)=j−1

bk, for all i = 1, . . . ,m, j = 1, . . . , n + 1 and Rij =

∑k=1,...,K s.t.rightik(x)=j

bk, for all i = 1, . . . ,m, j = 1, . . . , n+1, with non-negative elements, such that L−R = I ,

then the total beam-on time used in any decomposition of I into shape matrices will equal

K∑k=1

bk =n+1∑j=1

Lpj =n+1∑j=1

Rqj , ∀ p, q = 1, . . . ,m (5.4.13)

where the elements of L and R represent the total beam-on time applied to shape matrices withleft leaf in position j − 1 and right leaf in position j respectively in row i. Therefore, using thenon-negativity of matrices L and R, the fact that L − R = I = L − R and (5.4.6) and (5.4.7),Baatar et al. [1] conclude that, for any decomposition of I, the relationship between matrices L,R, L and R is simply:

Lij ≥ Lij , ∀ i = 1, . . . ,m, j = 1, . . . , n+ 1 (5.4.14)

andRij ≥ Rij , ∀ i = 1, . . . ,m, j = 1, . . . , n+ 1 (5.4.15)

and so we can use a single variable, Wij ≥ 0, integer, for all i = 1, . . . ,m, j = 1, . . . , n+ 1, to relatematrices L and L, and R and R respectively. We have

Lij = Lij +Wij , ∀ i = 1, . . . ,m, j = 1, . . . , n+ 1 (5.4.16)

andRij = Rij +Wij , ∀ i = 1, . . . ,m, j = 1, . . . , n+ 1. (5.4.17)

According to equations (5.4.8), (5.4.13) and (5.4.16) we now have:

K∑k=1

bk =n+1∑j=1

Lij =n+1∑j=1

Lij +n+1∑j=1

Wij = Bi +n+1∑j=1

Wij , ∀ i = 1, . . . ,m. (5.4.18)

(We have an equivalent expression involving R and R if we use equations (5.4.8), (5.4.13) and(5.4.17)). This result demonstrates that only a single variable is necessary to describe the differencebetween the minimum total beam-on time of a decomposition and the total beam-on time used in


any decomposition of I. Furthermore, in a minimal beam-on time decomposition, where row i ofI is decomposed with minimal beam-on time Bi, we have:

Lij = Lij , ∀ j = 1, ..., n+ 1, i ∈ {1, . . . ,m} such thatK∑k=1

bk = Bi = Beammin (5.4.19)

and

Rij = Rij , ∀ j = 1, ..., n+ 1, i ∈ {1, . . . ,m} such thatK∑k=1

bk = Bi = Beammin. (5.4.20)

Referring back to our example of a non-optimal decomposition of I, Example 1, using the definitionfor matrices L and R we have:

L =

0 7 0 2 0 61 3 5 4 0 21 0 6 3 5 01 7 0 1 2 45 2 1 3 2 2

and R =

0 0 0 0 2 130 1 3 1 8 20 1 0 6 0 80 0 4 0 0 110 1 2 5 3 4

,

where we have arbitrarily chosen the closed leaf position to be left(x) = n and right(x) = n + 1.L − R does indeed equal I, equation (5.4.13) is satisfied with the sum of each row in L and R

equaling the total beam-on time used, 15, (5.4.14) and (5.4.15) are clearly satisfied and the Wvariables equal:

W =

0 0 0 0 0 60 1 3 1 0 20 0 0 3 0 00 0 0 0 0 40 1 1 3 2 2

.

Finally, (5.4.18) yields: (9, 8, 12, 11, 6)′ + (6, 7, 3, 4, 9)′ = (15, 15, 15, 15, 15)′, as required.

In Example 2, our minimal beam-on time decomposition, matrices L and R are:

L =

0 7 0 2 1 21 3 2 3 0 31 0 6 0 5 01 7 0 2 2 05 4 1 2 0 0

and R =

0 0 0 0 3 90 1 0 0 8 30 1 0 3 0 80 0 4 1 0 70 3 2 4 1 2

,

respectively, and L3j = L3j and R3j = R3j , for all j = 1, . . . , n + 1, as expected, since row 3of I achieves B3 in the minimal beam-on time decomposition. Conditions (5.4.13), (5.4.14) and(5.4.15) are satisfied and

W =

0 0 0 0 1 20 1 0 0 0 30 0 0 0 0 00 0 0 1 0 00 3 1 2 0 0

.



Condition (5.4.18) yields: (9, 8, 12, 11, 6)′ + (3, 4, 0, 1, 6)′ = (12, 12, 12, 12, 12)′, as required.

Furthermore, if we let the total beam-on time used in a decomposition be B, then from (5.4.18)we have

n+1∑j=1

Wij = B −Bi, ∀ i = 1, . . . ,m. (5.4.21)

It is clear from the above equation that if a decomposition of I uses total beam-on time B, thenin rows i with B > Bi, there is an ‘excess’ radiation delivery of B −Bi. In general, for any upperbound, B, on B, we therefore have:

Lij ≤ Lij +B −Bi, ∀ i = 1, . . . ,m, j = 1, ..., n+ 1 (5.4.22)

andRij ≤ Rij +B −Bi, ∀ i = 1, . . . ,m, j = 1, ..., n+ 1. (5.4.23)

We extend this work of Baatar et al. [1] in Section 5.7, using constraints (5.4.22) and (5.4.23) todetermine the maximum amount of radiation that can be applied to shape matrices exposing cell(i, j), and appropriate bounds. In the meantime we focus on the amount of radiation that can beapplied to shape matrices with a particular leaf position in row i of shape matrix k.

The work of Baatar et al. [1] further implies that an upper bound on the beam-on time that canbe applied to any shape matrix with a particular left and right leaf position in row i is given by

bk ≤ Li(leftik(x)+1) +B −Bi, ∀ i = 1, . . . ,m, k = 1, . . . ,K (5.4.24)

andbk ≤ Ri(rightik(x)) +B −Bi, ∀ i = 1, . . . ,m, k = 1, . . . ,K, (5.4.25)

where shape matrix k has non-zero ith row. Of course, we also require

bk ≤ minj=leftik(x)+1,...,rightik(x)−1

Iij , ∀ i = 1, . . . ,m, k = 1, . . . ,K, (5.4.26)

where shape matrix k has non-zero ith row, and if shape matrix k has zero ith row then

bk ≤ B −Bi,∀ k = 1, . . . ,K, i ∈ {1, . . . ,m} such that shape matrix k has zero ith row.

(5.4.27)

We generalise this work of Baatar et al. [1] as follows.

We first consider all possible ‘row intervals’, [p, q], which expose cells p, p + 1, . . . , q in a shapematrix. A ‘row interval’ is defined for p = 1, . . . , n, q = 0, 1, . . . , n where [p, q] = [1, 0] is a singlearbitrarily chosen closed leaf interval and otherwise p, q ≥ 1 and q ≥ p. Row i of shape matrix k

has corresponding row interval [leftik(x) + 1, rightik(x)− 1] if leftik(x) < rightik(x)− 1 and is theclosed-leaf interval otherwise. We define

P = {(p, q) : p = 1, . . . , n, q = 1, . . . , n, q ≥ p} ∪ {(1, 0)} (5.4.28)

to be the set of row intervals possible for a shape matrix with n columns. (Again for the constrainedcase, all closed leaf positions should be considered). For each row i in an m × n intensity matrixI, we can now calculate an upper bound, U ipq, on the beam-on time that can be applied to any


shape matrix, in a decomposition of I, with row interval [p,q] in row i. We make use of expressions(5.4.24), (5.4.25), (5.4.26) and (5.4.27) to obtain:

U ipq =

B −Bi if p = 1, q = 0min{Lip +B −Bi, Ri(q+1) +B −Bi, min

j=p,p+1,...,qIij} otherwise,

∀ i = 1, . . . ,m, (p, q) ∈ P.

(5.4.29)

We take the minimum of the three upper bounds expressed in (5.4.24), (5.4.25) and (5.4.26) todetermine our upper bound for shape matrices having row interval [p, q] in row i where row i is anon-zero row, since each row has a left and right leaf position and is limited by the correspondingintensity of that row.

The multisetU i def= {U ipq : (p, q) ∈ P}, for each row i = 1, . . . ,m, (5.4.30)

gives upper bounds on the set of possible beam-on times that could apply to row i. This multisetcan be calculated quite efficiently since relatively few of the values of L and R are non-zero. Themultisets for our minimal beam-on time decomposition example, Example 2, are:

U1 = {3, 0, 0, 0, 0, 0, 3, 3, 5, 7, 3, 3, 3, 5, 5, 3}

U2 = {4, 1, 1, 1, 1, 0, 3, 3, 3, 0, 4, 5, 0, 7, 0, 0}

U3 = {0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 3, 0, 0, 5}

U4 = {1, 1, 1, 1, 1, 1, 5, 1, 1, 4, 1, 1, 1, 1, 2, 3}

and

U5 = {6, 5, 5, 5, 3, 2, 6, 5, 3, 2, 5, 3, 2, 3, 2, 2},

where the elements, U ipq, are given in order of increasing p and q indices, with the closed leaf positionoccurring first, i.e. (p, q) = (1, 0), (1, 1), (1, 2), . . . , (1, 5), (2, 2), (2, 3), . . . , (2, 5), (3, 3), . . . , (5, 5).We see that the optimal solution corresponding to Example 2 is indeed in keeping with the upperbound beam-on time values given by our parameters U ipq for all i = 1, . . . ,m, (p, q) ∈ P.

For a given m × n intensity matrix I, and for a given row i, we now define a one-to-one functionπi such that {1, . . . , |P|} → P to sort the multiset U i such that U iπi(h) ≥ U

iπi(h′) whenever h ≥ h′.

Thenb1 = min

i=1,...,mU iπi(1), (5.4.31)

is an upper bound on the largest intensity that can be applied to the first shape matrix in anon-increasing decomposition and hence on every shape matrix in the decomposition. That is

mini=1,...,m

U iπi(1) is an upper bound on bk in a non-increasing decomposition for k = 1, . . . ,K. In our

minimal beam-on time decomposition example, Example 2,

bk = mini=1,...,m

U iπi(1) = min(7, 7, 5, 5, 6) = 5, ∀ k = 1, . . . ,K.



So far in this work, the upper bounds we have determined for b1 are⌊B

1

⌋and min

i=1,...,mU iπi(1).

As discussed in the Introduction to this chapter, we are interested in solving the Beam-on TimeConstrained Minimum Cardinality problem and therefore in this case B = B = B = Beammin. Inthe remainder of this chapter we consider this case and therefore set B = B = B = Beammin.

We now show that mini=1,...,m

U iπi(1) is at least as good as Beammin as an upper bound for b1 when

solving BTCMC.

Proposition 5.4.1. The upper bound b1 = mini=1,...,m

U iπi(1) is less than or equal to Beammin when

solving BTCMC.

Proof. We consider our definition for U ipq and in particular, a row which yields Beammin in ourintensity matrix which we denote by i∗. We have

U i∗

pq =

0 if p = 1, q = 0min{Li∗p, Ri∗(q+1), min

j=p,p+1,...,qIi∗j} otherwise ,

∀ (p, q) ∈ P.

(5.4.32)

This expression comes about since B −Bi∗ = 0 in a row yielding B = B = Beammin.

Now Lij ≤n+1∑y=1

Liy for all i = 1, . . . ,m, j = 1, . . . , n + 1 and further Lzj ≤ maxi=1,...,m

n+1∑y=1

Liy for all

z = 1, . . . ,m, j = 1, . . . , n+ 1 and we have equivalent expressions using R. Recalling our definitionfor Beammin,

Beammin = maxi=1,...,m

n+1∑j=1

Lij = maxi=1,...,m

n+1∑j=1

Rij ,

we can therefore say that the first two non-zero components of U i∗

pq, for (p, q) 6= (1, 0), are lessthan or equal to Beammin and since we are taking the minimum of three quantities in U i

∗

pq, for(p, q) 6= (1, 0), then the third term only effects U i

∗

pq if it is smaller than the first two. ThereforeU i∗

pq ≤ Beammin for all (p, q) ∈ P (since we also have 0 ≤ Beammin) and furthermore everyelement in the set U i∗

πi∗ (1)is less than or equal to Beammin. Finally min

i=1,...,mU iπi(1) is less than

or equal to every element in the set U i∗πi∗ (1)

which are less than or equal to Beammin. Thus

b1 = mini=1,...,m

U i∗

πi∗ (1) ≤ Beammin when B = B = Beammin.

Hence it is sufficient to initialise b1 to mini=1,...,m

U iπi(1) when solving the BTCMC problem.

We incorporate this new upper bound on b1 into the following algorithm, where we are now focusedon solving the BTCMC problem and have set B = B = B = Beammin:


Algorithm 5.4.2 (b, b, {U ipq}) :=Leaf Pair Bounds Initialise(K,Klb, Beammin, {Bi}, I, I(1), I(2), L, R,P)

for i = 1, . . . ,m, (p, q) ∈ P doif p = 1 and q = 0 thenU ipq := Beammin −Bi

elseU ipq := min{Lip +Beammin −Bi, Ri(q+1) +Beammin −Bi, min

j=p,p+1,...,qIij}

end ifend forfor i = 1, . . . ,m doU i def= {U ipq : (p, q) ∈ P}πi such that {1, . . . , |P|} → P sort the multiset U i such that U iπi(h) ≥ U

iπi(h′) whenever h ≥ h′

end forb1 := min

i=1,...,mU iπi(1)

(b, b) := Initialise Multiset Bounds(K,Klb, Beammin, b1, I(1), I(2))

return (b, b, {U ipq})

Algorithm Leaf Pair Bounds Initialise is the best initialisation of bounds on variables bk, fork = 1, . . . ,K, that we have determined thus far. Initialise Multiset Bounds initialises boundson variables bk for k = 2, . . . ,K, and is defined in Section 5.4.1.

Computational results for the application of the bounds algorithms developed in Sections 5.2 to5.4 inclusive, on the JS and CC models, are given in Section 5.8.

To end this section we briefly discuss the recent work of Baatar et al. [39]. As mentioned in the In-troduction (Chapter 1), some of the ideas for this thesis with regard to bounds were simultaneouslydetermined by Baatar et al. [39].

In particular, within their study of an heuristic for the constrained BTCMC problem, Baatar etal. [39] use expressions (5.4.14), (5.4.15), (5.4.22) and (5.4.23) to bound the elements of matricesL and R, however they also write the bounds in terms of upper and lower bounds on the elementsof the W matrices, where 0 ≤ Wij ≤ B − Bi, for all i = 1, . . . ,m, j = 1, . . . , n + 1, follows fromexpression (5.4.21). They then seek to improve the bounds on the individual Wij values, andhence on Lij and Rij , for all i = 1, . . . ,m, j = 1, . . . , n+ 1, by solving an integer program with theintegrality property such that the W matrices themselves correspond to an ‘L-R representation’of the intensity matrix, see Theorem 3.1 in Baatar et al. [1]. (The integrality property is whenthe linear relaxation of an integer program is guaranteed to yield an optimal integer solution).Baatar et al. [39] computationally test the upper bounds, (5.4.22) and (5.4.23) on L and R,as they relate to the variables in their Simple Extraction Optimisation-Integer Program. Theythen computationally test improved upper bounds on L and R based on the improved boundsfor the W variables returned by the former integer program. The original upper bounds on L

and R reduce the computation time of the ‘sequential maximising extraction heuristic’ algorithmby approximately 34.4% and the improved upper bounds reduce computation time by a further32.6%. Baatar et al. [39] solve the constrained BTCMC problem (heuristically) and hence requireB = B = Beammin in all the bounds they utilise.



Baatar et al. [39] also determine b1 = mini=1,...,m

U iπi(1) as an upper bound on any beam-on time value

in a decomposition. They utilise this value, naming the parameter cmax, with B = B = Beammin,in their ‘sequential maximising extraction heuristic’ algorithm. They do not consider a singleclosed leaf position in the calculation of U ipq, for all i = 1, . . . ,m, (p, q) ∈ P, since they solve theconstrained BTCMC problem.

5.5 Improving Bounds Using the Integrality of Variables

In this section we continue to investigate our parameters U ipq, for all i = 1, . . . ,m, (p, q) ∈ P,which describe the most radiation that can be given to any shape matrix having leaf position (p, q)in row i. However now we also utilise the integrality of variables bk, and of our shape matrices,to develop further bounds. For a particular row i, if we wish to determine the number of shapematrices k that can be given c or more units of radiation, we can divide the maximum amount ofradiation that can be applied to each individual leaf position by c, round down since the numberof shape matrices must be integer, and then sum over all possible leaf positions. This gives us anupper bound on the number of shape matrices that can be given radiation level c or more in row i.Then since we are concerned with a non-increasing integer decomposition, the beam-on time valueassociated with the (k + 1)th shape matrix must have beam-on time less than or equal to c − 1.Mathematically we have the following:

If for some k ∈ {1, . . . ,K} and for some i ∈ {1, . . . ,m}⌊U i1,0c

⌋+

n∑p=1

n∑q=p

⌊U ipqc

⌋≤ k, (5.5.1)

then at most k shape matrices can be given radiation intensity of c or more. Furthermore, if (5.5.1)holds, for any i ∈ {1, . . . ,m}, it must be that bk+1 ≤ c − 1, i.e. c − 1 is a valid upper bound onbk+1 for k < K.

We can use this result iteratively to improve any given upper bounds on the bk variables.

Proposition 5.5.1. Given parameters U ipq for i = 1, . . . ,m, (p, q) ∈ P, and b1 for an intensity matrixI, we can determine new upper bounds on b1, . . . , bK as follows:

set donei = false for all i = 1, . . . ,mfor c = b1, . . . , 1 and while donei = false for some i ∈ {1, . . . ,m} do

set k =

⌊U i1,0c

⌋+

n∑p=1

n∑q=p

⌊U ipqc

⌋if k ≥ K then set donei = true

else set bk+1 = min{bk+1, c− 1}endfor

Proposition 5.5.1 iterates through the row values i = 1, . . . ,m, for each c value, from b1 down to 1in steps of 1 (since beam-on time is integer), calculating a value for k. (We use b1 as the largestthat parameter c can be as this is the most radiation that can be given to any shape matrix k ina non-increasing decomposition). If k is less than K then we can set bk+1 to the minimum of its

5.5. Improving Bounds Using the Integrality of Variables 125

existing value and c − 1, since if k corresponds to the maximum number of shape matrices thatcan receive c or more units of radiation, then the (k + 1)th shape matrix can have beam-on timevalue c − 1 or less, in a non-increasing, integer decomposition. If k is greater than or equal tothe maximum number of shape matrices we know the model can use, K, then the correspondingbk+1 value does not exist in a decomposition and the row for which this occurs must no longer bepart of the iteration. A smaller c value means a larger k value so again for this reason no furtheriterations should be completed for this row value for decreasing c. Finally, iterations using thesame value of c over the remaining i values must be completed to determine the extent to whicheach row allows c or more units of radiation to be applied to shape matrices.

When we consider the BTCMC problem, b1 can be set to mini=1,...,m

U iπi(1) and since c = b1 is the

largest c can be, no improvement can be made to b1 using Proposition 5.5.1. That is, the left handside of (5.5.1) can never be zero since at least one term must be at least 1 over all (p, q) ∈ P ineach row. Therefore if k can never be zero then b1 can not be improved using Proposition 5.5.1.If however we initially set b1 to be greater than min

i=1,...,mU iπi(1), Proposition 5.5.1 can obviously

improve the value of b1.

In practical examples, Proposition 5.5.1 can be shown to yield very strong upper bounds onthe beam-on time variables. For our minimal beam-on time decomposition example, Example2, K = 6 and Beammin = 12. Using algorithms Leaf Pair Bounds Initialise (of Section 5.4.2) andUB Consistency (of Section 5.2) we can calculate the multiset U i for i = 1, . . . ,m for Example2, as given previously in Section 5.4.2, and hence initialise b = (5, 5, 4, 3, 2, 1). The application ofProposition 5.5.1 to this initialisation of bounds is given in Table 5.5.1.



Table 5.5.1: Application of Proposition 5.5.1 to upper bounds on the beam-on time variables ofExample 2, initialised using Leaf Pair Bounds Initialise and ordered with UB Consistency; K = 6.

c i k bk+1 donei, i = 1, . . . , m

Initial bounds for bk for

k = 1, . . . , K and ini-

tial values for donei for

i = 1, . . . , m

b = (5, 5, 4, 3, 2, 1) done1 = done2 =

done3 = done4 =

done5 = false

5 1 4 b5 := min(2, 4) = 2

5 2 2 b3 := min(4, 4) = 4

5 3 1 b2 := min(5, 4) = 4

5 4 1 b2 := min(4, 4) = 4

5 5 7 done5 = true

4 1 4 b5 := min(2, 3) = 2

4 2 4 b5 := min(2, 3) = 2

4 3 1 b2 := min(4, 3) = 3

4 4 2 b3 := min(4, 3) = 3

3 1 12 done1 = true

3 2 8 done2 = true

3 3 3 b4 := min(3, 2) = 2

3 4 3 b4 := min(2, 2) = 2

2 3 4 b5 := min(2, 1) = 1

2 4 6 done4 = true

1 3 12 done3 = true

Final bounds b = (5, 3, 3, 2, 1, 1)

Hence Proposition 5.5.1 improves the upper bounds on the beam-on time variables in Example 2from b = (5, 5, 4, 3, 2, 1) to b = (5, 3, 3, 2, 1, 1).

We now write a formal algorithm for applying Proposition 5.5.1. We have:

5.6. Flow Models for Improving Bounds 127

Algorithm 5.5.1 b :=Leaf Pair Bounds Improve(K, b, {U ipq})for i = 1, . . . ,m dodonei := false

end forwhile donei = false for some i ∈ {1, . . . ,m} do

for c = b1, . . . , 1 do

k :=

⌊U i1,0c

⌋+

n∑p=1

n∑q=p

⌊U ipqc

⌋if k ≥ K thendonei := true

elsebk+1 := min{bk+1, c− 1}

end ifend for

end whilereturn b

In Section 5.8 we apply initialisation and bounds propagation procedures to algorithmLeaf Pair Bounds Improve and numerically compare the resulting algorithm with other combina-tions of algorithms considered in this chapter.

5.6 Flow Models for Improving Bounds

In this section we return to more general ways of finding upper bounds on the total beam-on timethat can be applied to shape matrices with ith row having a certain left and right leaf position.

Recall the definitions for matrices L, R, left(x) and right(x) from Section 5.4.2. Recall also thatmatrix L is defined as Lij =

∑k=1,...,K s.t.leftik(x)=j−1

bk, for all i = 1, . . . ,m, j = 1, . . . , n+1, and hence represents

the total beam-on time applied to shape matrices with left leaf in position j−1 in row i. Similarlymatrix R is defined as Rij =

∑k=1,...,K s.t.rightik(x)=j

bk, for all i = 1, . . . ,m, j = 1, . . . , n + 1, and therefore

represents the total beam-on time applied to shape matrices with right leaf in position j in rowi. We know from our discussion of the work of Baatar et al. [1] in Section 5.4.2 that if L and R

represent a decomposition of I having total beam-on time B then they must satisfy Lij = Lij+Wij

and Rij = Rij +Wij , for some Wij ≥ 0, integer, for all i = 1, . . . ,m, j = 1, . . . , n+ 1. Given suchan L and R, we must be able to find variables Fi(j−1)h, for all i = 1, . . . ,m, j = 1, . . . , n + 1, h =j, . . . , n + 1, and variables Wij , for all i = 1, . . . ,m, j = 1, . . . , n + 1, such that F represents thebeam-on time applied to shape matrices with left leaf in position j − 1 and right leaf in positionh in row i, and W describes the ‘excess’ beam-on time applied to shape matrices with left leafpositioned at j−1 (or equivalently right leaf positioned at j) in row i. Thus F and W must satisfy



the following for each row i = 1, . . . ,m:n+1∑h=j

Fi(j−1)h = Lij +Wij , ∀ j = 1, . . . , n+ 1

h∑j=1

Fi(j−1)h = Rih +Wih, ∀ h = 1, . . . , n+ 1

n+1∑j=1

Wij = B −Bi

Fi(j−1)h ≤ mins=j,...,h−1

Iis, ∀ j = 1, . . . , n+ 1, h = j, . . . , n+ 1

Fi(j−1)h,Wij ≥ 0, integer, ∀ j = 1, . . . , n+ 1, h = 1, . . . , n+ 1.

(5.6.1)

Variable Fi(j−1)h, for all i = 1, . . . ,m, j = 1, . . . , n+ 1, h = 1, . . . , n+ 1, must also be bounded bythe minimum intensity value exposed by (left, right) leaf position (j − 1, h).

We can interpret (5.6.1) as a network flow in a Bipartite Graph, where we have two disjoint setsof nodes, and arcs connecting nodes in one set to nodes in the other. A total flow of B units mustpass from one set of nodes, which we denote by {λ1, λ2, . . . , λn, λn+1}, to another, denoted by{κ1, κ2, . . . , κn, κn+1}, where for the particular row, the total flow out of node λj cannot exceedLij + B − Bi, for j = 1, . . . , n + 1, the total flow into node κh cannot exceed Rih + B − Bi, forh = 1, . . . , n+ 1, and the only arcs available to carry flow are those of the form (λj , κh) for h ≥ j.The flow on any arc must also be bounded by the minimum intensity value exposed by (left,right)leaf position (λj , κh), where h ≥ j. The variable Fi(j−1)h, as defined previously, represents the flowon this arc. An example of a bipartite graph, for a particular row i in an intensity matrix withcolumn dimension 3, exhibiting the characteristics described, is given in Figure 5.6.1. We havearbitrarily assigned the extra B −Bi radiation to nodes λ2 and κ2 in the example.

Figure 5.6.1: An Example Bipartite Graph, for a particular Row i in an Intensity Matrix withcolumn dimension 3, which exhibits the characteristics of the Flow Model (5.6.1)

We now define new variables, Aci(j−1)h, for c = 1, . . . , b1, i = 1, . . . ,m, j = 1, . . . , n + 1, h =


j, . . . , n+ 1, to represent the number of shape matrices using leaf position (j − 1, h) in row i thatcan be given radiation level c or more. The relationship between variables Fi(j−1)h and Aci(j−1)h

is Fi(j−1)h =b1∑c=1

Aci(j−1)h ≥v∑c=1

Aci(j−1)h ≥v∑c=1

Avi(j−1)h = vAvi(j−1)h for v ≤ b1, i = 1, . . . ,m, j =

1, . . . , n+ 1, h = 1, . . . , n+ 1, since Aci(j−1)h is non-increasing with increasing c. Therefore, for anyfeasible decomposition of I, variables Aci(j−1)h, for c = 1, . . . , b1, i = 1, . . . ,m, j = 1, . . . , n+ 1, h =j, . . . , n+ 1, must satisfy constraints (5.6.3) to (5.6.7) inclusive, of the following integer programs,for some Wij ≥ 0 integer, i = 1, . . . ,m, j = 1, . . . , n+ 1.

For c = 1, . . . , b1 and i = 1, . . . ,m we define IP(c, i) as follows:

IP OBJ(c, i) := maxn+1∑j=1

n+1∑h=j

Aci(j−1)h (5.6.2)

subject to

c

n+1∑h=j

Aci(j−1)h ≤ Lij +Wij , ∀ j = 1, . . . , n+ 1, (5.6.3)

c

h∑j=1

Aci(j−1)h ≤ Rih +Wih, ∀ h = 1, . . . , n+ 1, (5.6.4)

n+1∑j=1

Wij = B −Bi, (5.6.5)

Aci(j−1)h ≤

mins=j,...,h−1

Iis

c

, ∀ j = 1, . . . , n+ 1, h = j, . . . , n+ 1 (5.6.6)

andAci(j−1)h,Wij ≥ 0, integer, ∀ j = 1, . . . , n+ 1, h = 1, . . . , n+ 1. (5.6.7)

To find an upper bound on the number of shape matrices that can receive radiation level c ormore, we first seek for each row i an upper bound on the number that can receive radiation levelc or more in any decomposition of that row. Thus we maximise (5.6.2). Then, to obtain an upperbound on the number of shape matrices that can receive radiation level c or more for each valueof c = 1, . . . , b1 we calculate the minimum of the objective values over all i. We have:

N c = mini=1,...,m

IP OBJ(c, i) for all c = 1, . . . , b1.

We convert the values of N c, for c = 1, . . . , b1, to bounds on variables bk of the JS model fork ∈ {1, . . . ,K} using algorithm Consistent N b Bounds from Section 5.3. Finally, we can stopthe solution of our integer programs when IP OBJ(c, i) = 0 for any i ∈ {1, . . . ,m} and anyc ∈ {1, . . . , b1}, and set, a potentially new value for, b1, since the minimum over all i = 1, . . . ,mof IP OBJ(c, i) for the particular value of c will also be zero in this case and the objective valuesfor each row are decreasing for increasing c. (We note that to solve the constrained case, theappropriate MLC mechanical constraints would need to be formulated and included in the integerprogram, and the objective function would need to be amended to also sum over all rows of theintensity matrix. The stopping condition and calculation of N c values, for all c = 1, . . . , b1, would



use Aci(j−1)h variable values, for all c = 1, . . . , b1, i = 1, . . . ,m, j = 1, . . . , n + 1, h = 1, . . . , n + 1,adjusted to the constrained case).

Integer programs generally take longer to solve than their corresponding linear relaxations. Hencevalues for bk for k ∈ {1, . . . ,K} may be found more quickly by solving a related linear program.

The first linear program we trial is simply the integer program, IP(c, i), for c = 1, . . . , b1 andi = 1, . . . ,m, with the integrality condition on the variables, (5.6.7), removed. We name this linearprogram LP1(c, i) and rename the objective function in LP1(c, i) to be LP1 OBJ(c, i).

In practice, for LP1(c, i), we maximise the sum over all rows i = 1, . . . ,m of objective LP1 OBJ(c, i)and hence solve a single linear program, LP1(c), for each c = 1, . . . , b1, rather than treating eachrow individually. This increases the size of the linear program, however this is more efficient inpractice than solving m linear programs of slightly smaller size. The objective we implement istherefore

LP1 OBJ(c) = maxm∑i=1

n+1∑j=1

n+1∑h=j

Aci(j−1)h, ∀ c = 1, . . . , b1, (5.6.8)

which achieves an equivalent optimal result when compared with the solution of row problems.We do not consider a similar objective for our integer program, because, as mentioned, doing soincreases the size of the program. In the case of an integer program increased size has a moredramatic effect on computation time and in practice this is not as efficient to solve.

The calculation of N c for each c = 1, . . . , b1 for LP1(c) is equivalent to that determined for IP(c, i)other than we now also take the greatest integer less than or equal to the returned value. We have

N c =

mini=1,...,m

n+1∑j=1

n+1∑h=j

Aci(j−1)h

for all c = 1, . . . , b1,

since the minimum value forn+1∑j=1

n+1∑h=j

Aci(j−1)h over all i = 1, . . . ,m may be fractional and since

N c must be integer for all c = 1, . . . , b1. We again cease the solution of our linear programs and

set a, potentially new, value for b1 when

n+1∑j=1

n+1∑h=j

Aci(j−1)h

= 0 for some i ∈ {1, . . . ,m} and

c ∈ {1, . . . , b1}.

In the second linear program we consider, LP2(c, i), for c = 1, . . . , b1, i = 1, . . . ,m, we replacevariables W in the original program, IP(c, i), with a constant, which allows us to divide throughby c and round the right hand sides of the expressions bounding the total flow into and out of nodes,constraints (5.6.3) and (5.6.4). (We also remove constraint (5.6.5) and the integrality conditionson variables Aci(j−1)h, for all c = 1, . . . , b1, i = 1, . . . ,m, j = 1, . . . , n+1, h = 1, . . . , n+1). LP2(c, i)for each c = 1, . . . , b1 and each row i = 1, . . . ,m, therefore has the integrality property, as it has theform of a network flow in a Bipartite Graph, and therefore IP(c, i) can be solved by implementingLP2(c, i).


For each c = 1, . . . , b1 and each row i = 1, . . . ,m, LP2(c, i) is defined as:

LP2 OBJ(c, i) := maxn+1∑j=1

n+1∑h=j

Aci(j−1)h (5.6.9)

subject ton+1∑h=j

Aci(j−1)h ≤

⌊Lij +B −Bi

c

⌋, ∀ j = 1, . . . , n+ 1, (5.6.10)

h∑j=1

Aci(j−1)h ≤

⌊Rih +B −Bi

c

⌋, ∀ h = 1, . . . , n+ 1, (5.6.11)

Aci(j−1)h ≤

mins=j,...,h−1

Iis

c

, ∀ j = 1, . . . , n+ 1, h = j, . . . , n+ 1 (5.6.12)

andAci(j−1)h ≥ 0, ∀ j = 1, . . . , n+ 1, h = 1, . . . , n+ 1. (5.6.13)

however again, in practice, when solving LP2(c, i), we maximise the sum over all rows i = 1, . . . ,mof LP2 OBJ(c, i) instead of solving each row individually. This objective for LP2(c) is the same asfor LP1(c) and is given by:

LP2 OBJ(c) = maxm∑i=1

n+1∑j=1

n+1∑h=j

Aci(j−1)h, ∀ c = 1, . . . , b1. (5.6.14)

Consequently the same stopping conditions and method for setting bounds on beam-on time vari-ables apply as described for LP1(c) for c = 1, . . . , b1.

In LP2(c, i), the ‘excess’ radiation, B−Bi, is added to each arc of the network whereas in LP1(c, i),this quantity is applied only once for each row i via the W variables. Using B − Bi as an upperbound on W in LP2(c, i) instead of the W variables themselves reduces the total number ofvariables in the linear program though at the same time constraints (5.6.10) and (5.6.11) becomeless restricted. We computationally test LP2(c) against LP1(c) and IP(c, i) in Section 5.8.5 for theJS model and Section 5.8.9 for the CC model to determine the difference in computation time andbounds returned. We set B = Beammin in each program since we ultimately wish to solve theBTCMC problem.

The algorithm we apply for solving our integer and linear programs, and to calculate upper boundson variables bk for k = 1, . . . ,K of the JS model, is given below. The general algorithm we con-sider is written to solve the integer and linear programs for each row i = 1, . . . ,m and eachc = 1, . . . , b1 however as mentioned, in practice, we solve the linear programs for each c only asdescribed in this section. (The algorithm we present also returns lower bound values for vari-ables bk, for k = 1, . . . ,K, however these are simply lower bounds initialised with algorithmInitialise Multiset Bounds of Section 5.4.1. The application of Consistent N b Bounds (of Sec-tion 5.3) in the algorithm does not improve the initialised lower bounds. Consistent N b Boundssets bk ≥ 1 for k ≤ Klb and the only non-zero lower bound input on the N b variables forConsistent N b Bounds in the algorithm is N1 = Klb which implies that bKlb ≥ 1.Initialise Multiset Bounds initialises lower bounds on variables bk, for k = 1, . . . ,K, of at least



these values. The linear and integer programs we consider do not apply to lower bounds, how-ever for use in our applications section, we calculate and return lower bounds in this way in thisalgorithm). We set a parameter Z in the algorithm to indicate which of models IP(c, i) (Z=IP),LP1(c, i) (Z=LP1) or LP2(c, i) (Z=LP2), for c = 1, . . . , b1 and i = 1, . . . ,m, we wish to solve.

Algorithm 5.6.1 (b, b) :=Z Bounds Initialise(K,Klb, Beammin, {Bi}, I, I(1), I(2), L, R)

1: b1 := Beammin

2: (b, b) :=Initialise Multiset Bounds(K,Klb, Beammin, b1, I(1), I(2))

3: bmax := Beammin

4: N1 := K

5: N1 := Klb

6: for b = 2, . . . , bmax do7: Nb := K

8: Nb := 09: end for

10: set Z:=IP, LP1 or LP211: for c = 1, . . . , b1 do12: for i = 1, . . . ,m do13: Solve Z(c, i)14: if Z OBJ(c, i) = 0 then15: b1 := min(b1, c− 1)16: goto step 2117: end if18: end for19: N c := min

i=1,...,mZ OBJ(c, i)

20: end for21: (K, b, b) :=Consistent N b Bounds(K,Klb, b, b, b

max, N , N)22: return (b, b)

We now give an example to illustrate how the upper bounds returned by each linear programand the integer program may differ. We again consider Example 2 from Section 5.4.2 and thecorresponding L and R matrices and B parameters. The values for Klb and Beammin for Example2 are also determined in Section 5.4.2 to be Klb = 4 and Beammin = 12, and we can see frommatrix I in Example 2 that I(1) = 1 and I(2) = 2. Suppose also that we are given a value for thenumber of shape matrices that can be used in a solution, K = 6.

We wish to determine improved upper bounds on variables bk for k ∈ {1, . . . ,K} of the JS modelusing IP(c, i), for c = 1, . . . , b1, i = 1, . . . ,m, LP1(c) and LP2(c), for c = 1, . . . , b1, respectively,and to compare the bounds returned by each algorithm. Using algorithm Z Bounds Initialise, wefirst initialise upper bounds for Example 2. The initialised bounds are given in the first row ofTable 5.6.1. As mentioned, the lower bounds for variables bk for k = 1, . . . ,K are not affectedby the integer and linear programs and hence we do not consider these bounds in our example.The remaining rows of the table apply the appropriate integer or linear program, and set upperbounds on variables bk for k ∈ {1, . . . ,K}, for values of c ∈ {1, . . . , b1}. The final row of the

5.7. Bounds On Radiation Delivered To Cell (i, j) 133

Table 5.6.1: Application of Z Bounds Initialise, with Z=IP, Z=LP1 and Z=LP2 respectively, toExample 2; bounds ordered with UB Consistency; K = 6.

Nc bk+1

c IP LP1 LP2 IP LP1 LP2

Initial bounds for bk for b = (12, 6, 4, 3, 2, 1) b = (12, 6, 4, 3, 2, 1) b = (12, 6, 4, 3, 2, 1)

k = 1, . . . , K

1 12 12 12

2 4 5 4 b5 = min(2, 1) = 1 b6 = min(1, 1) = 1 b5 = min(2, 1) = 1

3 2 3 3 b3 = min(4, 2) = 2 b4 = min(3, 2) = 2 b4 = min(3, 2) = 2

4 1 2 1 b2 = min(6, 3) = 3 b3 = min(4, 3) = 3 b2 = min(6, 3) = 3

5 1 1 1 b2 = min(3, 4) = 3 b2 = min(6, 4) = 4 b2 = min(3, 4) = 3

6 1 b1 =min(12, 5) = 5 b2 = min(4, 5) = 4 b1 =min(12, 5) = 5

7 b1 =min(12, 6) = 6

Final bounds b = (5, 3, 2, 2, 1, 1) b = (6, 4, 3, 2, 2, 1) b = (5, 3, 3, 2, 1, 1)

table applies the UB Consistency algorithm of Section 5.2 to the upper bounds determined byZ Bounds Initialise.

Hence, for this example, as we expect, IP Bounds Initialise obtains the lowest upper bounds for allk = 1, . . . ,K. LP2 Bounds Initialise returns the next best bounds followed by LP1 Bounds Initialise.If we compare the bounds returned by each of the linear and integer programs of this section withthe bounds returned by Proposition 5.5.1 of Section 5.5, for this particular example, see Table5.5.1, we see that although the initialisation procedures are different in the two cases, the boundsreturned by Proposition 5.5.1 are equal to the bounds returned by LP2 Bounds Initialise.

In Section 5.8.5, we consider a small number of problems, each containing 100 examples, and we pro-duce some statistics on which of LP1 Bounds Initialise and/or LP2 Bounds Initialise determinesthe best bounds in general for these batches of problems, compared with IP Bounds Initialise. Theexperiments apply UB Consistency to the initialised bounds as per our example above. Table 5.8.5presents these results. As mentioned, each of the algorithms, IP Bounds Initialise,LP1 Bounds Initialise and LP2 Bounds Initialise, is also applied to the JS and CC models, with thebest of the ordering, bounds propagation and bounds improvement algorithms we have previouslyconsidered in this work. Details of the algorithms applied are given in Section 5.8.1.

5.7 Bounds On Radiation Delivered To Cell (i, j)

We now use properties (5.4.22) and (5.4.23) from Section 5.4.2 to determine the maximum amountof radiation that can be applied to shape matrices exposing cell (i, j).

By definition, an upper bound on the total beam-on time applied to shape matrices with left leafin position h in row i is Li(h+1) + B − Bi, an upper bound on the total beam-on time applied to



shape matrices with right leaf in position s in row i is Ris + B − Bi and mint=(h+1),...,(s−1)

Iit is the

minimum intensity value between positions h+ 1 and s− 1 inclusive in row i, which are the cellsexposed when the left leaf is in position h and the right leaf is in position s in row i. Therefore anupper bound on the total beam-on time applied to shape matrices with left leaf in position h andright leaf in position s in row i is found by taking the minimum of these three quantities. Now,cell (i, j) is exposed in a number of ways. If we fix the left leaf at h = 0, . . . , j − 1, then the rightleaf position can be in any of positions s = j+ 1, . . . , n+ 1. We are interested in the most radiationthat can be applied to cell (i, j), so we take the maximum over all these exposed positions. Hencewe have the following expression:

Buij(B) = maxh=0,...,(j−1),

s=(j+1),...,(n+1)

(min(Li(h+1) +B −Bi, Ris +B −Bi, mint=(h+1),...,(s−1)

Iit))

∀ i = 1, . . . ,m, j = 1, . . . , n,

(5.7.1)

where the superscript u refers to an upper bound.

If B is not too much larger than Beammin then it is not hard to construct examples where someof the values of Bu(B) are strictly less than I. To illustrate this we use our example intensitymatrix, with B equal to the total beam-on time for Example 1, 15. We indicate the entries ofBu(15) which are less than I with bold face in the matrix below:

Bu(15) =

0 7 7 8 71 3 5 8 01 0 6 3 81 8 4 5 65 6 5 3 2

.

If we decrease the value of B to Beammin as is necessary for Example 2, we obtain the following:

Bu(12) =

0 7 7 7 71 3 5 7 01 0 3 3 51 5 4 4 45 6 5 3 2

.

Here we see that as B approaches Beammin, the Bu(B) values become progressively better. Allvalues in bold in the above matrix are smaller than the corresponding components of I. Thisproperty is particularly useful since we wish to solve the BTCMC problem. We define the followingalgorithm for determining Bu(B).

5.7. Bounds On Radiation Delivered To Cell (i, j) 135

Algorithm 5.7.1 Bu(B) :=Cell Based Bounds Initialise(B, {Bi}, I, L, R)for i = 1, . . . ,m, j = 1, . . . , n doBuij(B) := max

h=0,...,(j−1),s=(j+1),...,(n+1)

(min(Li(h+1) +B −Bi, Ris +B −Bi,

mint=(h+1),...,(s−1)

Iit))

end forreturn Bu(B)

We discuss how we apply parameters Buij(B), for all i = 1, . . . ,m, j = 1, . . . , n, within the JS modelin Section 5.8.1 and we provide the results of numerical tests using Cell Based Bounds Initialisein Tables 5.8.2 - 5.8.7.

5.7.1 Returning to Possible Initialisations for Upper Bounds on Beam-on TimeVariables. Further information can be deduced from the Bu(B) values of the previous section.Since the matrix, Bu(B), indicates the maximum amount of radiation that can be applied to shapematrices exposing cell (i, j), we see that the maximum value in the Bu(B) matrix itself must be anupper bound on the beam-on time that can be applied to the first shape matrix in a non-increasingdecomposition and hence on each shape matrix k for k = 1, . . . ,K. In other words, the largestbeam-on time value that can be delivered to any (non-zero) shape matrix k, for k = 1, . . . ,K mustbe bounded by

maxi=1,...,m,j=1,...,n

Buij(B). (5.7.2)

In our minimal beam-on time decomposition example, Example 2, bk, for k = 1, . . . ,K, using thisexpression, is 7.

In Section 5.4.2 we demonstrated that when solving the Beam-on Time Constrained MinimumCardinality problem, which is our focus, the best initialisation we have so far for an upper boundon b1 is given by (5.4.31), min

i=1,...,mU iπi(1). We now compare (5.4.31) with (5.7.2) when B = B =

B = Beammin. (We do not consider here the possibility of an improved initialisation of b1 usingthe linear or integer programs given in Section 5.6. We cover this case more generally by comparingalgorithms from Section 5.6 with the best of the remaining algorithms of this work in Sections 5.8.5and 5.8.9).

Proposition 5.7.1. If B = Beammin, bounds on b1 satisfy

mini=1,...,m

U iπi(1) ≤ maxi=1,...,m,j=1,...,n

Buij(B).

Proof. We demonstrate that mini=1,...,m

U iπi(1) ≤ maxi=1,...,m,j=1,...,n

Buij(Beammin).

Buij(B) for all i = 1, . . . ,m, j = 1, . . . , n, can be written in terms of U ipq for all i = 1, . . . ,m, (p, q) ∈P, as follows:

Buij(B) = maxp≤j,q≥j

U ipq, for all i = 1, . . . ,m, j = 1, . . . , n and (p, q) ∈ P, (p, q) 6= (1, 0).

Here we have used the definition of Buij(B) for all i = 1, . . . ,m, j = 1, . . . , n given by expression



(5.7.1), the corresponding relationship between parameters h and p, and s and q, being p = h+ 1and q = s− 1 respectively, and the fact that Buij(B) does not exist for closed leaf positions.

We can also relate the most radiation that can be applied to any cell in row i to the most radiationthat can be applied to any open leaf position in row i as follows:

maxj=1,...,n

Buij(B) = max(p,q)∈Ps.t.(p,q)6=(1,0)

U ipq, for all i = 1, . . . ,m.

Therefore

U iπi(1) = max

{max

(p,q)∈Ps.t.(p,q) 6=(1,0)

U ipq, B −Bi

}= max

{max

j=1,...,nBuij(B), B −Bi

}, for all i = 1, . . . ,m.

Now if we consider B = Beammin and if we denote by i∗ a row in our intensity matrix which yieldsBeammin, we have

U i∗πi∗ (1)

= maxj=1,...,n

Bui∗j(Beammin) since B − Bi∗ = 0 when B = Beammin, and since the Bu(B)

values are intensity values and therefore greater than or equal to zero.

Hence mini=1,...,m

U iπi(1) ≤ Ui∗

πi∗ (1) = maxj=1,...,n

Bui∗j(Beammin) ≤ maxi=1,...,m,j=1,...,n

Buij(Beammin), as required.

Hence it is still best to initialise b1 to mini=1,...,m

U iπi(1) when solving the BTCMC problem. Again,

we consider the possibility of improved initialisations for b1, using the linear and integer programsof Section 5.6, by comparing appropriate algorithms in the sections to follow.

5.8 Application of Improved Bounds

In all the algorithms we trial on both the JS and CC models we use the unconstrained GreedyHeuristic Algorithm (GHA) of Baatar et al. [1] to calculate Beammin and Kbm, where Kbm = K

is an upper bound on the number of shape matrices to be used in a solution. Beammin and Kbm

are calculated in relatively negligible time, however all results given in this section incorporate thistime component in the Total Time columns of each table. The determination of Beammin andKbm, and our intensity matrix I, is not included as a ‘step’ in any of the algorithms presented inthis section: these parameters are presumed known prior to the application of an algorithm.

As for the GHA time component, the total preprocessing time to calculate bounds, for each batchof 100 problems solved, is also included in the Total Time columns in each table presented in thissection. However to highlight the contribution of the preprocessing time to the total computationtime for an algorithm, the preprocessing time is also shown separately in each table. (The pre-processing techniques we use to determine bounds generally require relatively little computationtime). In the preprocessing stage we also test whether any lower bound for a variable is greaterthan its corresponding upper bound and we terminate the calculation citing infeasibility if this is

5.8. Application of Improved Bounds 137

the case. Finally, we do not include the preprocessing time in the 2 hour time limit we imposewhen solving each individual problem instance. The 2 hour time limit is instead applied to thesolution of the JS or CC model once bounds on variables are set.

We now define an algorithm to initialise the common parameters of the JS and CC algorithms ofthis section. Where further parameters specific to either the JS or CC models are necessary foran algorithm to be computed, these are determined within the appropriate JS or CC algorithmrespectively. Parameters Bi, for each i = 1, . . . ,m, are required for all algorithms we test on theJS model and all but the first algorithm we test on the CC model. Set P is required for all butthe first of the JS algorithms and all but the first of the CC algorithms. However for notationalconvenience, and since computation time for calculating these parameters is relatively negligible,we include Bi, for i = 1, . . . ,m, and set P in Initialise Parameters Algorithms given below:

Algorithm 5.8.1 (Klb, {Bi}, I(1), I(2), I, L, R,P) :=Initialise Parameters Algorithms(I)for i = 1, . . . ,m do

Iij :=

Iij − Ii(j−1), 1 < j ≤ nIi1, j = 1−Iin, j = n+ 1

end forfor i = 1, . . . ,m, j = 1, . . . , n+ 1 doLij := max{0, Iij}Rij := max{0,−Iij}

end forKlb := max

i=1,...,m(max{|{j ∈ {1, . . . , n+ 1} : Lij > 0}|, |{j ∈ {1, . . . , n+ 1} : Rij > 0}|})

I(1) := the smallest non-zero value in II(2) := the second smallest non-zero value in Ifor i = 1, . . . ,m do

Bi :=n+1∑j=1

Lij

end forP := {(p, q) : p = 1, . . . , n, q = 1, . . . , n, q ≥ p} ∪ {(1, 0)}return (Klb, {Bi}, I(1), I(2), I, L, R,P)

5.8.1 Bounds Application within the JS Model. In the JS model there are ‘big-M’constraints linking variables bk and sk, and variables aijk, xijk and bk respectively. Constraints(2.2.6) and (2.2.9) are repeated here:

0 ≤ bk ≤ Hksk, ∀ k = 1, . . . ,K (5.8.1)

and

aijk +Gijk(1− xijk) ≥ bk, ∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K. (5.8.2)

The value of Hk and Gijk can be any valid upper bound on bk. In Chapters 2 to 4 inclusive, theupper bound we use is the maximum value in our intensity matrix, max

i=1,...,m,j=1,...,nIij . We now set

Hk = bk, for all k = 1, . . . ,K and Gijk = bk, for all i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K.



There is a further ‘big-M’ in the JS model in the constraint linking variables aijk and xijk. Con-straint (2.2.7) is repeated here:

aijk ≤Mijkxijk, ∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K. (5.8.3)

Mijk represents an upper bound on the most radiation that can be delivered by a shape matrixexposing cell (i, j). We can therefore use Buij(Beammin), for all i = 1, . . . ,m, j = 1, . . . , n, asthis ‘big-M’ value. (Buij(Beammin) for all i = 1, . . . ,m, j = 1, . . . , n is calculated with algorithmCell Based Bounds Initialise of Section 5.7). However, aijk is also bounded by the most radiationthat can be applied to shape matrix k, bk, so we set Mijk = min(Buij(Beammin), bk) for alli = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K. (In Chapters 2 to 4 inclusive, we use Mijk = Iij for alli = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K as this bound).

Finally we apply

bk ≤ bk ≤ bk, ∀ k = 1, . . . ,K, (5.8.4)

to ensure that variable bk in the JS model itself lies between the final values of its upper and lowerbounds after any preprocessing routines.

5.8.2 Additional Constraints Tested on the JS Model. We trial the following addi-tional constraints with the JS model:

k∑p=1

bp + (K − k)bk+1 ≥ Beammin, ∀ k = 1, . . . ,K − 2, (5.8.5)

(K − k)bK−k +K∑

p=K−k+1

bp ≤ Beammin, ∀ k = 1, . . . ,K − 2, (5.8.6)

K−2∑k=1

bk ≥ Beammin − (I(1) + I(2)) (5.8.7)

andK−1∑k=1

bk ≥ Beammin − I(1). (5.8.8)

Constraints (5.8.5) and (5.8.6) apply relationships between total minimal beam-on time and thenumber of beam-on time values considered in a non-increasing decomposition. These constraintsare redundant with respect to the linear relaxation of the JS model (constraints (2.2.14) and (2.3.2)imply constraints (5.8.5) and (5.8.6)) and are therefore not essential to the model. Constraints(5.8.7) and (5.8.8) are implied by our initialised bounds, in Initialise Multiset Bounds, (of Section5.4.1), and constraints (2.2.14) and (2.3.2), and are therefore also redundant with respect to thelinear relaxation of the JS model. The results of the inclusion of constraints (5.8.5) and (5.8.6),and (5.8.7) and (5.8.8) respectively in the JS model are shown in Table 5.8.2.

We finally investigate one other property of the intensity matrix values, considered by Kalinowski[28], which relates to possible solution patterns for the JS model. However, this property onlyholds in the absence of MLC mechanical constraints.


Proposition 5.8.1. For any j ∈ {2, . . . , n} and for row i∗ ∈ {1, . . . ,m}, where i∗ denotes a row ofour intensity matrix yielding Beammin, if Ii∗j−1 = Ii∗j then xi∗,j−1,k = xi∗jk for all k = 1, . . . ,K.For all other rows i, and for j such that Iij−1 = Iij , there exists an optimal solution wherexi,j−1,k = xijk for all k = 1, . . . ,K.

Proof. In a row yielding the minimum beam-on time we have:

Li∗j = Li∗j =∑

k=1,...,K s.t.lefti∗k(x)=j−1

bk, ∀ j = 1, . . . , n+ 1 and

Ri∗j = Ri∗j =∑

k=1,...,K s.t.righti∗k(x)=j

bk, ∀ j = 1, . . . , n+ 1.

When the left leaf is in position lefti∗k(x) = j− 1 in shape matrix k this implies that xi∗,j−1,k = 0and xi∗jk = 1. When the right leaf is in position righti∗k(x) = j in shape matrix k then xi∗jk = 1and xi∗,j+1,k = 0.

Furthermore, if Iij−1 = Iij for a particular i and j, then Lij = Rij = 0.

Then if Ii∗j−1 = Ii∗j in the minimum beam-on time row, Li∗j = Li∗j = 0 and Ri∗j = Ri∗j = 0.This means that there are no shape matrices with xi∗,j−1,k = 0 and xi∗jk = 1 and no shapematrices with xi∗jk = 1 and xi∗,j+1,k = 0. Hence xi∗,j−1,k = xi∗jk when Ii∗j−1 = Ii∗j in a rowyielding Beammin.

In any other row, i ∈ {1, . . . ,m}, if xi,j−1,k = xijk for some j ∈ {2, . . . , n} and for all k = 1, . . . ,K,

thenK∑k=1

bkxi,j−1,k = Ii,j−1 = Iij =K∑k=1

bkxijk, Lij = 0 and Rij = 0, satisfying constraints (5.4.22)

and (5.4.23) of Section 5.4.2. Therefore, for rows i = 1, . . . ,m and j such that Iij−1 = Iij ,xi,j−1,k = xijk for all k = 1, . . . ,K, satisfies the conditions for an optimal solution.

Hence we can apply a constraint to the JS model to enforce this condition. For both cases, rowswhere Bi = Beammin and otherwise, we have:

−|Iij − Iij−1| ≤ xijk − xi,j−1,k ≤ |Iij − Iij−1|,∀ i = 1, . . . ,m, j = 2, . . . , n, k = 1, . . . ,K.

(5.8.9)

We show that constraint (5.8.9) improves computation time for the JS model in Table 5.8.2. (Asmentioned, additional constraint (5.8.9) is specific to the unconstrained problem and would not beused in a formulation of the JS model if the constrained case was being considered).

In the following sections, we solve the BTCMC problem with the JS model and investigate theeffect of the various algorithms of the previous sections and additional constraints, given above,on computational efficiency. The algorithms we test have the following standard form:



Algorithm JS 5.8.2

1: (Klb, {Bi}, I(1), I(2), I, L, R,P) :=Initialise Parameters Algorithms(I)2: set Y := A, B, C, D or E3: begin cases4: Case Y5: end cases6: if not(b ≥ b) then7: return infeasible and exit8: end if9: Bu(Beammin) :=Cell Based Bounds Initialise(Beammin, {Bi}, I, L, R)

10: Hk := bk, ∀ k = 1, . . . ,K in constraint (2.2.6)11: Gijk := bk, ∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K in constraint (2.2.9)12: Mijk := min(Buij(Beammin), bk), ∀ i = 1, . . . ,m, j = 1, . . . , n, k = 1, . . . ,K in constraint

(2.2.7)13: Include constraint (5.8.4) into the JS model14: Include constraint (5.8.9) into the JS model15: Solve the JS model

where Case Y, with Y equal to one of A, B, C, D or E, describes the specific initialisation, and/orbounds propagation and/or bounds improvement steps we wish to trial on the JS model. Table5.8.1 details these steps for each case. We highlight in bold face in Table 5.8.1 the steps thateach case has in common with a previous case to demonstrate how we systematically test thebounds results of this chapter. Following the table we describe each case in detail and presentcorresponding numerical results. Step 14 of Algorithm JS 5.8.2 is where we trial the additionalconstraints of this section. We discuss our choice of constraint (5.8.9) in step 14 and our choice ofIP Bounds Initialise (of Section 5.6) in Algorithm JS 5.8.2 Case E in the relevant sections below.Table 5.8.1 also relates to the CC model. We discuss the specific algorithms we apply to the CCmodel in Sections 5.8.6 to 5.8.9 including our choice of LP1 Bounds Initialise (of Section 5.6) forCase E.

5.8.3 Case A: Application of Initialise Multiset Bounds andCell Based Bounds Initialise to the JS Model. Algorithm JS 5.8.2 Case A initialises boundson variables bk for k = 2, . . . ,K according to Initialise Multiset Bounds (of Section 5.4.1) andsets b1 = Beammin. The algorithm then applies these bounds, and the bounds determined byCell Based Bounds Initialise (of Section 5.7), to the appropriate ‘big-M’ constraints in the JSmodel. Cell Based Bounds Initialise appears in the standard form of the algorithm applied to theJS model, Algorithm JS 5.8.2, and is therefore applied in this way in all cases. Furthermore,the only additional constraint which shows a significant improvement in computation time for theJS model is constraint (5.8.9) and therefore Algorithm JS 5.8.2 is written to include only thisadditional constraint in step 14. The results for the application of all additional constraints areshown in Table 5.8.2 and discussed in more detail below.

Table 5.8.2 demonstrates the effect on computation time and number of branch and bound nodes(BB) searched when we apply Algorithm JS 5.8.2 Case A to the JS model, solving the batches


Table 5.8.1: The specific initialisation, and/or bounds propagation and/or bounds improvementsteps trialled on the JS and CC models with Case Y, where Y=A, B, C, D or E. Steps shown inbold are those that are in common with a previous case.

Case Algorithmic Steps

A b1 := Beammin

(b, b) := Initialise Multiset Bounds(K, Klb, Beammin, b1, I(1), I(2))

B (b, b, {U ipq}) := Leaf Pair Bounds Initialise(K, Klb, Beammin, {Bi}, I, I(1), I(2), L, R,P)

C (b, b, {Uipq}) := Leaf Pair Bounds Initialise(K, Klb, Beammin, {Bi}, I, I(1), I(2), L, R, P)

b :=UB Consistency(K,b)

b :=LB Consistency(K,b)

(b, b) := Bounds Propagate(K, Beammin, b,b)

D (b, b, {Uipq}) := Leaf Pair Bounds Initialise(K, Klb, Beammin, {Bi}, I, I(1), I(2), L, R, P)

b :=UB Consistency(K, b)

b :=LB Consistency(K, b)

(b, b) := Bounds Propagate(K, Beammin, b, b)

b := Leaf Pair Bounds Improve(K, b, {U ipq})

b :=UB Consistency(K,b)

(b, b) := Bounds Propagate(K, Beammin, b,b)

E JS:Z=IP (b, b) := Z Bounds Initialise(K, Klb, Beammin, {Bi}, I, I(1), I(2), L, R)

CC:Z=LP1 b :=UB Consistency(K,b)


for i = 1, . . . , m, (p, q) ∈ P do

if p = 1 and q = 0 then

U ipq := Beammin −Bi

else

U ipq := min{Lip + Beammin −Bi, Ri(q+1) + Beammin −Bi, min

j=p,p+1,...,qIij}

end if

end for

b := Leaf Pair Bounds Improve(K, b, {Uipq})

b :=UB Consistency(K, b)




of 100 problems of the particular sizes and maximum intensity levels shown. (Data set 7 7 0 10is only tested on Algorithm JS 5.8.2 Case A with and without constraint (5.8.9) to clarify thatthis constraint improves total computation time). Comparing column 4 with column 9 in Table5.8.2, we see that approximately 55 hours of computation time are saved when we add constraint(5.8.9). The addition of constraints (5.8.5) and (5.8.6) without constraint (5.8.9) (column 5) andthe addition of constraints (5.8.7) and (5.8.8) without constraint (5.8.9) (column 6) marginallyworsens computation times overall, (by approximately 4 hours and by approximately 45 minutesrespectively) when compared with column 4, and where both versions of the algorithm were com-puted.

Finally, Algorithm JS 5.8.2 Case A improves computation time over the standard JS model by ap-proximately 256 hours and improves the total number of branch and bound nodes searched by ap-proximately 1.82×108 nodes, where both versions are computed. Table 5.8.2 also demonstrates thatthe preprocessing time associated with Algorithm JS 5.8.2 Case A is negligible compared with therun time for the model itself. Therefore Algorithm JS 5.8.2 Case A, which applies a basic initialisa-tion of bounds to variables bk for k = 1, . . . ,K and Cell Based Bounds Initialise to the JS model, re-sults in computational improvement over the JS model. As mentioned, Cell Based Bounds Initialise,appears in the standard form of the algorithm applied to the JS model and hence as computationtime improves for Case A we apply Cell Based Bounds Initialise to all cases we trial.

Table 5.8.2: Numerical results for Algorithm JS 5.8.2 Case A applied to the JS model solving theBTCMC problem, using CPLEX version 8.1 and AMPL version 8.1 on a 2GHz AMD 64 3000+:time in seconds, 2-hour time limit on individual problem instances.

Batches JS model Algorithm Algorithm Algorithm Algorithm JS 5.8.2 Case A

of 100 defined in JS 5.8.2 Case JS 5.8.2 Case JS 5.8.2 Case

problems Chapter 3, A w/o (5.8.9) A with (5.8.5) A with (5.8.7)

Section 3.2.1 and (5.8.6)

and w/o

(5.8.9)

and (5.8.8)

and w/o

(5.8.9)

Total

BB

Nodes

Total

Time

Total Time Total Time Total Time Prep.

Time

Total

BB

Nodes

Total

Time

4 4 0 15 366929 329.33 159.16 187.62 165.75 0.63 129456 129.07

5 5 0 5 150639 205.35 137.34 25.64 143.87 0.76 47348 220.59

5 5 0 10 18702767 43832.30 1208.74 681.48 1228.44 0.92 704496 1147.97

5 5 0 15 80348775 260609.65 56346.51 68141.50 57857.47 1.03 31724204 69619.72

6 6 0 5 31104907 96619.01 1120.08 1308.22 1100.14 1.30 411988 510.64

6 6 0 10 84019936 447870.26 205978.35 196854.81 205990.28 1.76 33131301 128329.00

7 7 0 5 55648233 342394.16 95790.10 109199.27 95795.69 2.47 21900810 70863.62

7 7 0 10 511191.57 3.07 60946316 403371.12


5.8.4 Cases B, C and D: Application of Leaf Pair Bounds Algorithms,UB Consistency, LB Consistency, Bounds Propagate and Cell Based Bounds Initialiseto the JS Model. Having determined the most efficient form of Algorithm JS 5.8.2 Case A, withregard to the inclusion of additional constraints, we now test Cases B, C and D, which introducethe Leaf Pair Bounds ideas of Sections 5.4.2 and 5.5, and the ordering and bounds propagationideas of Section 5.2.

Algorithm JS 5.8.2 Case B is equivalent to Algorithm JS 5.8.2 Case A other than improving theinitialisation of the upper bound on variable b1 of the JS model via the Leaf Pair Bounds Initialisealgorithm of Section 5.4.2. Algorithm JS 5.8.2 Case C extends Algorithm JS 5.8.2 Case B testing theeffect of the ordering procedures UB Consistency and LB Consistency, and the Bounds Propagatealgorithm (of Section 5.2) on the bounds initialised by Leaf Pair Bounds Initialise. The final casewe consider in this section, Case D, investigates the effect of the Leaf Pair Bounds Improve algo-rithm (of Section 5.5), which utilises Leaf Pair Bounds and the integrality of shape matrices andbeam-on time variables to iteratively improve upper bounds on variables bk for k ∈ {1, . . . ,K}.Algorithm JS 5.8.2 Case D is an extension of Algorithm JS 5.8.2 Case C in that the initialised, or-dered and propagated bounds resulting in Case C are then improved, re-ordered and re-propagatedin Case D.

The numerical results of tests on Algorithm JS 5.8.2 Cases B, C and D applied to the JS modelare given in Table 5.8.3, where we present total computation time and total branch and boundnodes searched for each algorithm and each batch of problems considered. We compare the resultsof Algorithm JS 5.8.2 Cases B, C and D against the results for Algorithm JS 5.8.2 Case A and theJS model. Again the preprocessing time required for each algorithm is negligible compared withsolving the model itself. We show in bold face in the table the best Total Time value and the bestTotal BB Nodes value for each batch of problems investigated. We see that Algorithm JS 5.8.2Case D obtains the lowest values for each of these indicators in 5 out of the 7 batches tested andthat these batches are the larger sized matrices. This is significant since in practice we wish tosolve larger sized matrices potentially with larger maximum intensity levels.

We now provide a further table, Table 5.8.4, which presents a summary of the overall total com-putation time saved and reduction in total numbers of branch and bound nodes searched whenwe compare each of Algorithm JS 5.8.2 Cases A, B, C and D with each other and the JS model.Table 5.8.4 clearly shows that of the algorithms tested thus far, Algorithm JS 5.8.2 Case D is thebest performing, followed by Algorithm JS 5.8.2 Case A, Algorithm JS 5.8.2 Case B and finallyAlgorithm JS 5.8.2 Case C. All algorithms investigated improve on the JS model, and for Algo-rithm JS 5.8.2 Case D, this is by approximately 291 hours and approximately 2.28×108 nodes. Ourexpectation for the algorithms was that with each level of improvement in bounds on variables bk,the size of the feasible region for a given problem would decrease and we would see a correspond-ing further improvement in the computation time and total number of branch and bound nodessearched for the JS model. The fact that Algorithm JS 5.8.2 Case D is the best performing ofthe algorithms we have currently tested is consistent with this hypothesis, however we would haveexpected the order for decreased efficiency for the remaining algorithms to have been AlgorithmJS 5.8.2 Case C followed by Algorithm JS 5.8.2 Case B and finally Algorithm JS 5.8.2 Case A.We instead see that Algorithm JS 5.8.2 Case A outperforms Algorithm JS 5.8.2 Case B which



in turn outperforms Algorithm JS 5.8.2 Case C (though in this instance only by a small mar-gin). Therefore Algorithm JS 5.8.2 Case D, which applies algorithms Leaf Pair Bounds Improve,UB Consistency and Bounds Propagate to the best upper and lower bounds we have determinedthus far, (previously initialised with Leaf Pair Bounds Initialise, ordered and propagated) is thebest performing of the algorithms we have currently tested on the JS model. Therefore, whilstimproving the initialisation of b1 alone, Case B, and applying ordering and bounds propagationto our best initialisation, Case C, did not further improve computational efficiency over our first,in general, higher initialisation of upper bounds and lower initialisation of lower bounds via CaseA, the application of Leaf Pair Bounds Improve (which utilises Leaf Pair Bounds and integralityof shape matrices and beam-on time variables) in conjunction with this improved initialisation,ordering and propagation, via Algorithm JS 5.8.2 Case D, results in a significant improvement intotal computation time and total number of branch and bound nodes searched when applied tothe JS model and when compared with our other algorithms.


Tab

le5.

8.3:

Num

eric

alre

sults

for

Alg

orit

hmJS

5.8.

2C

ases

A,

B,

Can

dD

appl

ied

toth

eJS

mod

elso

lvin

gth

eB

TC

MC

prob

lem

,us

ing

CP

LE

Xve

rsio

n8.

1an

dA

MP

Lve

rsio

n8.

1on

a2G

Hz

AM

D64

3000

+:

tim

ein

seco

nds,

2-ho

urti

me

limit

onin

divi

dual

prob

lem

inst

ance

s.

Batc

hes

JS

mod

elA

lgori

thm

JS

5.8

.2C

ase

AA

lgori

thm

JS

5.8

.2C

ase

BA

lgori

thm

JS

5.8

.2C

ase

CA

lgori

thm

JS

5.8

.2C

ase

D

of

100

defi

ned

in

pro

ble

ms

Ch

ap

ter

3,

Sec

tion

3.2

.1

Tota

l

BB

Nod

es

Tota

l

Tim

e

Pre

p.

Tim

e

Tota

l

BB

Nod

es

Tota

l

Tim

e

Pre

p.

Tim

e

Tota

l

BB

Nodes

Tota

l

Tim

e

Pre

p.

Tim

e

Tota

l

BB

Nod

es

Tota

l

Tim

e

Pre

p.

Tim

e

Tota

l

BB

Nod

es

Tota

l

Tim

e

44

015

366929

329.3

30.6

3129456

129.0

70.7

3339932

269.7

21.1

3355842

347.7

61.4

5239533

257.2

9

55

05

150639

205.3

50.7

647348

220.5

90.9

16045

20.7

21.1

75859

19.9

21.3

61477

11.6

5

55

010

18702767

43832.3

00.9

2704496

1147.9

71.1

6473769

1195.4

51.4

4615743

927.6

21.7

9688999

2019.1

7

55

015

80348775

260609.6

51.0

331724204

69619.7

21.3

623716333

58097.3

51.7

328145264

74713.1

92.3

317260557

43872.9

5

66

05

31104907

96619.0

11.3

0411988

510.6

41.6

3101458

177.2

41.9

7184860

319.7

02.1

76985

38.0

2

66

010

84019936

447870.2

61.7

633131301

128329.0

02.0

748007572

201064.6

62.5

649974175

189297.1

73.0

122241122

92137.2

7

77

05

55648233

342394.1

62.4

721900810

70863.6

22.9

521747362

47294.2

53.2

425325311

51376.0

43.6

82220673

5529.2

3



Table 5.8.4: Summary of numerical results for Algorithm JS 5.8.2 Cases A, B, C and D applied to the

JS model solving the BTCMC problem, using CPLEX version 8.1 and AMPL version 8.1 on a 2GHz

AMD 64 3000+: 2-hour time limit on individual problem instances. The column entitled Hours gives

the approximate total time saved in hours over the JS model. The column entitled BB Nodes gives the

approximate reduction in total number of branch and bound nodes searched compared with the JS model.

Description of Algorithm Hours BB Nodes

Algorithm JS

5.8.2 Case A

basic initialisation of

bounds on bk for

k = 2, . . . , K using

Initialise Multiset Bounds

of Section 5.4.1, b1

set to Beammin

and application of

Cell Based Bounds Initialise

of Section 5.7

256 1.82×108

Algorithm JS

5.8.2 Case B

improved value

for b1, using

Leaf Pair Bounds Initialise

of Section 5.4.2 o/w equiv-

alent to Algorithm JS

5.8.2 Case A

245 1.76×108

Algorithm JS

5.8.2 Case C

application of the

UB Consistency,

LB Consistency and

Bounds Propagate algo-

rithms of Section 5.2 to the

bounds set in Algorithm

JS 5.8.2 Case B

243 1.66×108

Algorithm JS

5.8.2 Case D

application of the

Leaf Pair Bounds Improve

algorithm of Section

5.5, UB Consistency and

Bounds Propagate to

the bounds resulting in

Algorithm JS 5.8.2 Case C

291 2.28×108


In the following section we compare algorithms IP Bounds Initialise, LP1 Bounds Initialise andLP2 Bounds Initialise of Section 5.6 applied to the JS model within the framework of our bestperforming JS algorithm thus far, Algorithm JS 5.8.2 Case D. The algorithm we apply is AlgorithmJS 5.8.2 Case E. In Table 5.8.1 we indicate that Algorithm JS 5.8.2 Case E should be computedwith IP Bounds Initialise in the appropriate step. We discuss this choice in the following section.It is at this step where we also substitute LP1 Bounds Initialise and LP2 Bounds Initialise fortesting.

5.8.5 Case E: Application of IP Bounds Initialise, LP1 Bounds Initialise orLP2 Bounds Initialise, UB Consistency, LB Consistency, Bounds Propagate,Leaf Pair Bounds Improve and Cell Based Bounds Initialise to the JS Model. In thissection we test Algorithm JS 5.8.2 Case E with IP Bounds Initialise, LP1 Bounds Initialise andLP2 Bounds Initialise respectively inserted at the appropriate step in the algorithm. AlgorithmJS 5.8.2 Case E first sets b1 = Beammin and applies Initialise Multiset Bounds (of Section 5.4.1).The algorithm then determines new upper bounds for variables bk for k ∈ {1, . . . ,K} with IP(c, i),LP1(c) or LP2(c), for c = 1, . . . , b1 and i = 1, . . . ,m, (of Section 5.6) by converting Nc upperbounds, for c ∈ {1, . . . , b1}, to bk upper bounds, for k ∈ {1, . . . ,K}, using Consistent N b Bounds(of Section 5.3). The resulting bounds are then ordered with UB Consistency, and theBounds Propagate algorithm is applied (Section 5.2). We then implement Leaf Pair Bounds Improve(Section 5.5) and again apply UB Consistency and Bounds Propagate. Algorithm JS 5.8.2 CaseE utilises the Leaf Pair Bounds Improve algorithm since its inclusion has been shown to improvecomputational efficiency for the JS model, (Algorithm JS 5.8.2 Case D).

Table 5.8.5 compares the numerical results for Algorithm JS 5.8.2 Case E with each ofIP Bounds Initialise, LP1 Bounds Initialise and LP2 Bounds Initialise included at the appropriatestep in the algorithm. The lowest Total Time value and Total Branch and Bound (BB) Nodes valuefor each batch of problems is highlighted in bold face in the table. We also investigate the qualityof the upper bounds returned by each version of the algorithm in three batches of 100 problems ofsize 5×5 with intensities ranging from 0 to 5, 0 to 10 and 0 to 15 respectively. The upper bounds wecompare are the bounds returned after all initialisation procedures and after the UB Consistencyalgorithm has been applied. The table indicates the percentage of cases where the algorithm withLP1 Bounds Initialise or LP2 Bounds Initialise achieves the minimum upper bound for a beam-ontime variable in comparison to the other algorithm and the algorithm with IP Bounds Initialiseapplied, which of course yields the smallest upper bounds.



Tab

le5.

8.5:

Com

pari

son

ofco

mpu

tati

onti

me

and

uppe

rbo

unds

retu

rned

byIP

Bou

nds

Init

ialis

e,L

P1

Bou

nds

Init

ialis

ean

dL

P2

Bou

nds

Init

ialis

e,of

Sect

ion

5.6.

Col

umns

2to

7in

clus

ive

and

16to

18in

clus

ive

show

num

eric

alre

sults

for

Alg

orit

hmJS

5.8.

2C

ase

Ew

ith

each

ofIP

Bou

nds

Init

ialis

e,L

P1

Bou

nds

Init

ialis

ean

dL

P2

Bou

nds

Init

ialis

ere

spec

tive

ly,

appl

ied

toth

eJS

mod

el,

solv

ing

the

BT

CM

Cpr

oble

m,

usin

gC

PL

EX

vers

ion

8.1

and

AM

PL

vers

ion

8.1

ona

2GH

zA

MD

6430

00+

:ti

me

inse

cond

s,2-

hour

tim

elim

iton

indi

vidu

alpr

oble

min

stan

ces.

Batc

hes

IPL

P1

LP

2

of

100

Pre

p.

Tota

lT

ota

lP

rep

.T

ota

lT

ota

lP

erce

nta

ge

of

case

sw

her

eL

P1

Pre

p.

Tota

lT

ota

lP

erce

nta

ge

of

case

sw

her

eL

P2

Tim

eB

BT

ime

Tim

eB

BT

ime

ach

ieves

the

min

imu

mu

pp

erb

ou

nd

Tim

eB

BT

ime

ach

ieves

the

min

imu

mu

pp

erb

ou

nd

Nod

esN

od

esfo

rb k

com

pare

dto

IPan

dL

P2

Nod

esfo

rb k

com

pare

dto

IPan

dL

P1

k=

1k=

2k=

3k=

4k=

5k=

6k=

7k=

8k=

1k=

2k=

3k=

4k=

5k=

6k=

7k=

8

44

015

24.8

1113886

135.0

512.3

6195238

387.9

87.1

4118142

140.9

3

55

05

12.2

71593

18.2

67.1

2936

22.1

570

81

84

86

99

100

4.0

51466

13.7

793

84

88

95

99

100

55

010

23.1

962395

142.4

711.0

685004

313.9

447

57

73

80

94

98

100

6.3

8659896

1962.8

090

83

88

90

96

100

100

55

015

44.8

39664445

27104.5

510.3

013712385

35817.8

042

52

45

69

74

92

98

100

8.9

715055289

37228.4

793

80

90

83

84

99

99

100

66

05

16.8

67541

40.8

89.6

76159

65.8

75.3

46916

40.1

9

66

010

39.5

81720600175294.0

19.5

019495839

79724.0

47.8

721407795

85194.4

1

77

05

24.2

7581212

3084.3

810.9

62222492

8046.7

37.1

02219547

5081.5

4


For the batches of problems of size 5 × 5 tested, LP2 Bounds Initialise yields the next best up-per bounds after IP Bounds Initialise, improving on or matching LP1 Bounds Initialise in everyelement of bk for k = 1, . . . ,K. Furthermore, the improvement of LP2 Bounds Initialise overLP1 Bounds Initialise becomes more marked as the maximum intensity level increases from 5 to15 for the batches of 5 × 5 problems tested. We cannot conclude on the basis of these aggregateupper bound results whether LP1 Bounds Initialise or LP2 Bounds Initialise will yield the greaterreduction in computation time for the JS model, since it could be that one algorithm returns betterbounds for a smaller number of more difficult problems within a batch, and hence reduces overallcomputation time for the JS model by a greater margin, than another algorithm which in generalmay return smaller bounds though for easier problems. Hence we test each of the algorithmsIP Bounds Initialise, LP1 Bounds Initialise and LP2 Bounds Initialise, within Algorithm JS 5.8.2Case E applied to the JS model using the batches given in Table 5.8.5.

In Table 5.8.5 we see that preprocessing time is greatest for IP Bounds Initialise, as we expect,followed by LP1 Bounds Initialise and LP2 Bounds Initialise. We also see that whilst preprocess-ing time is greater for IP Bounds Initialise, it is not prohibitively large and in fact it is AlgorithmJS 5.8.2 Case E with IP Bounds Initialise in the relevant step, that is the best performing of thealgorithms we test in this section when applied to the JS model, taking approximately 5 hoursless overall computation time than the algorithm with LP1 Bounds Initialise and approximately7 hours less overall computation time than the algorithm with LP2 Bounds Initialise. The to-tal number of branch and bound nodes saved when using IP Bounds Initialise is approximately8.08×106 nodes compared with LP1 Bounds Initialise and approximately 1.18×107 nodes whencompared with LP2 Bounds Initialise. This is significant in that the numbers of branch and boundnodes for Algorithm JS 5.8.2 Case E with IP Bounds Initialise, are the branch and bound nodesto run IP Bounds Initialise and for the JS model, whereas, for the linear program versions ofthe algorithm, the number of branch and bound nodes are only for solving the JS model. Fur-thermore, whilst it appears that the bounds returned by LP2 Bounds Initialise are superior toLP1 Bounds Initialise, and again this may not be true in general, we see that when applied to theJS model for the batches of problems tested, Algorithm JS 5.8.2 Case E with LP1 Bounds Initialiseperforms better than Algorithm JS 5.8.2 Case E with LP2 Bounds Initialise overall and in par-ticular for problems with larger maximum intensity level. Therefore Algorithm JS 5.8.2 Case E,which initialises bounds on variables bk, for k = 1, . . . ,K, via IP Bounds Initialise, and appliesUB Consistency, Bounds Propagate and Leaf Pair Bounds Improve to the resulting bounds, is themost efficient of the Case E algorithms we have investigated on the JS model.

We hereafter refer to Algorithm JS 5.8.2 Case E with IP Bounds Initialise as Algorithm JS 5.8.2Case E and we now compare Algorithm JS 5.8.2 Case E with Algorithm JS 5.8.2 Case D, the bestperforming algorithm we have previously tested on the JS model. The results for the comparisonof Algorithm JS 5.8.2 Cases D and E are given in Table 5.8.6, where we also present the resultsfor the JS model itself and the Step-up method applied to each algorithm and model we considerto determine the combined effect of the procedures. The algorithmic difference in Algorithm JS5.8.2 Cases D and E when applied with the Step-up method is that step 10 is no longer necessarysince constraint (2.2.6) is no longer in the JS model, and all lower bounds on bk are initialised toat least 1, since we know the value of K in the model is less than or equal to the number of shapematrices used in an optimal solution to BTCMC. There are also other changes to the JS model



itself with the application of the Step-up method as described in Chapter 3, Section 3.2.1. Theentries in bold face in Table 5.8.6 are again the best Total Time values and best Total BB Nodesvalues for each batch of 100 problems tested.

Table 5.8.6 demonstrates that Algorithm JS 5.8.2 Case E applied with the Step-up Method tothe JS model is the most efficient of the algorithms we now consider, yielding computationalimprovement over the JS model of approximately 328 hours and 2.65×108 branch and boundnodes. We provide further confirmation of these results in Table 5.8.7, where we summarise Table5.8.6 giving approximate total time saved and approximate reduction in numbers of branch andbound nodes searched, again when Algorithm JS 5.8.2 Cases D and E are applied to the JS modelwith and without the Step-up Method. Table 5.8.7 clearly shows that Algorithm JS 5.8.2 Case Ewith the Step-up Method outperforms Algorithm JS 5.8.2 Case D with the Step-up Method andthat the algorithms with the Step-up Method are far superior to the ‘standard’ versions of thealgorithms applied to the JS model. Of the ‘standard’ algorithms, Algorithm JS 5.8.2 Case E ismore efficient than Algorithm JS 5.8.2 Case D.

Finally, it should be mentioned that the preprocessing time associated with Algorithm JS 5.8.2Case E with and without the Step-up Method applied is greater than for the other algorithms wehave tested on the JS model, however the time is not significantly greater when compared withthe run time for solving the JS model itself. The time saved in solving the JS model, with thebounds returned by Algorithm JS 5.8.2 Case E, and with the Step-up Method applied, considerablyoutweighs the preprocessing time gained in determining the improved bounds.

In the following section, we apply our bounds techniques to the CC model. We again solve theBTCMC problem.


Tab

le5.

8.6:

Num

eric

alre

sults

for

Alg

orit

hmJS

5.8.

2C

ases

Dan

dE

appl

ied

toth

eJS

mod

elso

lvin

gth

eB

TC

MC

prob

lem

,us

ing

CP

LE

Xve

rsio

n8.

1an

dA

MP

Lve

rsio

n8.

1on

a2G

Hz

AM

D64

3000

+:

tim

ein

seco

nds,

2-ho

urti

me

limit

onin

divi

dual

prob

lem

inst

ance

s.

Batc

hes

JS

mod

elA

lgori

thm

JS

5.8

.2C

ase

DA

lgori

thm

JS

5.8

.2C

ase

ES

tep

-up

met

hod

Ste

p-u

pm

eth

od

an

dS

tep

-up

met

hod

an

d

of

100

defi

ned

inap

plied

toth

eA

lgori

thm

JS

5.8

.2C

ase

DA

lgori

thm

JS

5.8

.2C

ase

E

pro

ble

ms

Ch

ap

ter

3,

JS

mod

eld

efin

edap

plied

toth

eJS

mod

elap

plied

toth

eJS

mod

el

Sec

tion

3.2

.1in

Ch

ap

ter

3,

Sec

tion

3.2

.1

Tota

l

BB

Nod

es

Tota

l

Tim

e

Pre

p.

Tim

e

Tota

l

BB

Nod

es

Tota

l

Tim

e

Pre

p.

Tim

e

Tota

l

BB

Nod

es

Tota

l

Tim

e

Tota

l

BB

Nod

es

Tota

l

Tim

e

Pre

p.

Tim

e

Tota

l

BB

Nod

es

Tota

l

Tim

e

Pre

p.

Tim

e

Tota

l

BB

Nod

es

Tota

l

Tim

e

44

015

366929

329.3

31.4

5239533

257.2

924.8

1113886

135.0

5220120

107.2

11.5

627982

33.5

523.7

037914

45.1

1

55

05

150639

205.3

51.3

61477

11.6

512.2

71593

18.2

618833

18.7

30.9

1191

3.4

911.2

81080

13.2

3

55

010

18702767

43832.3

01.7

9688999

2019.1

723.1

962395

142.4

71866861

1788.2

21.9

01291887

1016.3

621.9

324578

42.6

8

55

015

80348775

260609.6

52.3

317260557

43872.9

544.8

39664445

27104.5

565593439

83603.0

63.3

32460642

4441.9

943.3

52456571

4495.4

9

66

05

31104907

96619.0

12.1

76985

38.0

216.8

67541

40.8

811639444

17023.7

11.7

94261

10.0

115.3

93611

19.4

1

66

010

84019936

447870.2

63.0

122241122

92137.2

739.5

817206001

75294.0

1120222080

268304.8

33.6

514405017

18638.6

740.4

32362822

4501.0

7

77

05

55648233

342394.1

63.6

82220673

5529.2

324.2

7581212

3084.3

871846279

226664.0

33.4

0196425

224.8

622.2

3137599

145.8

1



Table 5.8.7: Summary of numerical results for Algorithm JS 5.8.2 Cases D and E with and without the

Step-up Method applied solving the BTCMC problem, using CPLEX version 8.1 and AMPL version 8.1 on

a 2GHz AMD 64 3000+: 2-hour time limit on individual problem instances. The column entitled Hours

gives the approximate total time saved in hours over the JS model. The column entitled BB Nodes gives the

approximate reduction in total number of branch and bound nodes searched compared with the JS model.


Algorithm JS

5.8.2 Case D

application of the


(Section 5.5),

UB Consistency and


rithms (Section 5.2) to

the bounds resulting in

Algorithm JS 5.8.2 Case C

291 2.28×108

Algorithm JS

5.8.2 Case E

initialisation of bounds

on variables bk, for

k = 1, . . . , K, via

IP Bounds Initialise of

Section 5.6, applica-

tion of UB Consistency,

Bounds Propagate and

Leaf Pair Bounds Improve.

New initialisation of

bounds within the frame-

work of Algorithm JS 5.8.2

Case D

302 2.43×108

Algorithm JS

5.8.2 Case D and

Step-up Method

324 2.52×108

Algorithm JS

5.8.2 Case E and

Step-up Method

328 2.65×108


5.8.6 Bounds Application within the CC Model. In the CC model we have variablesNb defined for b = 1, . . . , bmax. In Chapter 3, we set bmax equal to the maximum value in ourintensity matrix. We can now set bmax equal to b1, the maximum amount of radiation that anyshape matrix can receive in a non-increasing decomposition. Setting bmax = b1, using our improvedinitialisations for b1, decreases the number of variables and constraints in the CC model.

As mentioned earlier in this chapter, the upper bounds we previously set for variables Nb inChapter 3 are a special case of Proposition 5.2.2 and hence are applied to the CC model wheneveran algorithm involves the Bounds Propagate algorithm of Section 5.2 (which utilises Proposition5.2.2). We therefore do not specifically apply constraint (3.8.17) as an upper bound on the Nbvariables for b = 1, . . . , b1 in any of the algorithms of this section. Furthermore, to include thefinal upper and lower bound values we calculate for Nb in our algorithms we introduce constraint

Nb ≤ Nb ≤ Nb, ∀ b = 1, . . . , b1 (5.8.10)

into the CC model.

Due to the successful application of additional constraint (5.8.9) to the JS model, we now con-sider an equivalent property to apply to the CC model. Again this property only applies to theunconstrained case.

Corollary 5.8.1. For any j ∈ {2, . . . , n} and for row i∗ ∈ {1, . . . ,m}, where i∗ denotes a row ofour intensity matrix yielding Beammin, if Ii∗j−1 = Ii∗j then Qi∗,j−1,b = Qi∗jb for all b = 1, . . . , b1.For all other rows i, and for j such that Iij−1 = Iij , there exists an optimal solution whereQi,j−1,b = Qijb for all b = 1, . . . , b1.

The proof of Corollary 5.8.1 follows from the definition of variable Qijb of the CC model.

We apply the following constraint to the CC model to implement this property:

−K|Iij − Iij−1| ≤ Qijb −Qi,j−1,b ≤ K|Iij − Iij−1|,∀ i = 1, . . . ,m, j = 2, . . . , n, b = 1, . . . , b1.

(5.8.11)

Constraint (5.8.11) for the CC model differs from constraint (5.8.9) for the JS model in that werequire a ‘big M’ in constraint (5.8.11), since we are now concerned with integer Qijb variablesrather than binary xijk variables. K, the number of shape matrices that can be used in a solution,is an appropriate value for ‘big M’ in this constraint, since variables Qijb represent the number ofshape matrices exposing cell (i, j) that can be given radiation level b or more. Again, to solve theconstrained case, additional constraint (5.8.11) would be left out of the CC model formulation.

Finally, we do not apply the Step-up method to the CC model since we have previously shown inChapter 3, Section 3.8.1 that its application does not improve computation time for this model.

In the following sections, we solve the BTCMC problem with the CC model and investigate theeffect of the various bounds algorithms on computational efficiency. In each of the CC algorithms,we initialise the N b and N b bounds, for all b = 1, . . . , bmax, to the ‘best known’ values we havewithout doing any bounds improvement work. (If the ‘best known’ values are not known in advancewe replace the initialisations with appropriate values). Then once we apply Consistent b N Bounds



(of Section 5.3) in each CC algorithm we obtain the ‘best known’ values for the variables if thesehave not already been set. The algorithms we test on the CC model have the following standardform:

Algorithm CC 5.8.3

1: (Klb, {Bi}, I(1), I(2), I, L, R,P) :=Initialise Parameters Algorithms(I)2: bmax := Beammin

3: N1 := K

4: N1 := Klb

5: for b = 2, . . . , bmax do6: Nb := K

7: Nb := 08: end for9: set Y:= A, B, C, D or E

10: begin cases11: Case Y12: end cases13: if not(b ≥ b) then14: return infeasible and exit15: end if16: (bmax, N , N) :=Consistent b N Bounds(K,Klb, b, b, b

max, N , N)17: N :=UB Consistency(bmax, N)18: N :=LB Consistency(bmax, N)19: Include constraint (5.8.10) into the CC model20: Include constraint (5.8.11) into the CC model21: Solve the CC model

Cases Y, where Y equals one of A, B, C, D or E, for Algorithm CC 5.8.3, are equivalent to Cases Yfor Algorithm JS 5.8.2 given in Table 5.8.1, other than Algorithm CC 5.8.3 Case E which utilisesLP1 Bounds Initialise rather than IP Bounds Initialise in step 1 of Case E. Therefore, the onlydifference between the algorithms we apply to the JS model and those we apply to the CC modelis that, in the latter case, the bk bounds are converted to Nb bounds using Consistent b N Bounds(of Section 5.3) and ordered using UB Consistency and LB Consistency (of Section 5.2) priorto application within the CC model. We systematically apply equivalent bounds initialisation,propagation and improvement techniques to the CC model for consistency with the JS model andease of comparison.

It should be mentioned that Algorithm CC 5.8.3 Case E does not require the initialisation ofparameters bmax, Nb or Nb, for b = 1, . . . , bmax, which appears in Algorithm CC 5.8.3. Theseinitialisations already occur within Case E as they are also necessary when Case E is applied to theJS model. However, since all other cases for the CC model require this initialisation, for notationalconvenience we set these parameters in Algorithm CC 5.8.3. Furthermore, we discuss our inclusionof constraint (5.8.11) in step 20 of Algorithm CC 5.8.3, and our choice of LP1 Bounds Initialise inAlgorithm CC 5.8.3 Case E, in the relevant sections below.


For all cases of Algorithm CC 5.8.3, we compute numerical results for many more batches ofproblems than for the JS algorithms and model, since the CC model and Algorithm CC 5.8.3 aremore computationally efficient than the corresponding JS versions. For completeness however, wecompare Algorithm JS 5.8.2 Case E applied with the Step-up Method (the best of the algorithmswe apply to the JS model) to the best of the algorithms tested on the CC model, in Section 5.8.9.In the following sections we now present each case of Algorithm CC 5.8.3 for the CC model andcorresponding numerical results.

5.8.7 Case A: Application of Initialise Multiset Bounds, Consistent b N Bounds,UB Consistency and LB Consistency to the CC Model. As discussed, Algorithm CC 5.8.3Case A is equivalent to Algorithm JS 5.8.2 Case A other than the conversion of bounds on variablesbk, for k = 1, . . . ,K, to bounds on variables Nb, for b = 1, . . . , bmax, and ordering of the final boundsvalues prior to application within the CC model.

Table 5.8.8 shows the effect on computation time and number of branch and bound nodes searchedwhen we apply Algorithm CC 5.8.3 Case A to the CC model. We test the algorithm with andwithout additional constraint (5.8.11) in step 20. The main result demonstrated in Table 5.8.8is that the application of our most basic algorithm does not improve computation time or thenumber of branch and bound nodes searched in general for the CC model. Total computationtime increases by approximately 10 hours and total number of branch and bound nodes searchedincreases by approximately 8.54×107 nodes over the batches of problems considered in Table 5.8.8.

The results in Table 5.8.8 also demonstrate that there is some improvement in computation timewhen constraint (5.8.11) is included in step 20 of Algorithm CC 5.8.3 Case A. We see that totalcomputation time reduces by approximately 9 hours, however the total number of branch andbound nodes searched increases by approximately 1.83×106 nodes, with the application of con-straint (5.8.11). Since computation time improves, we choose to incorporate constraint (5.8.11)into the standard version of the algorithm we apply to the CC model, Algorithm CC 5.8.3, towhich we now trial our remaining bounds algorithms.

Finally, in comparison to the run time for the CC model itself, the preprocessing time for AlgorithmCC 5.8.3 Case A with and without constraint (5.8.11) is significant for small problem sizes andsmall maximum intensity levels (as is the time to find Kbm using the Greedy Heuristic Algorithmof Baatar et al. [1], which is incorporated into the Total Time columns for the algorithms). Forlarger problem sizes and maximum intensity levels however, preprocessing time and the GHA timecomponent are relatively negligible in comparison to the time taken to solve the CC model in eachcase. This is an important result since it is our aim to solve larger problems with larger maximumintensity levels, as they more closely resemble real world problems.

5.8.8 Cases B, C and D: Application of Leaf Pair Bounds Algorithms,Bounds Propagate, Consistent b N Bounds, UB Consistency and LB Consistency tothe CC Model. Again, Algorithm CC 5.8.3 Cases B, C and D are equivalent to Algorithm JS5.8.2 Cases B, C and D other than the conversion of bounds on variables bk, for k = 1, . . . ,K,to bounds on variables Nb, for b = 1, . . . , bmax, and ordering of the final bounds values prior to



application within the CC model. We have now also demonstrated that constraint (5.8.11) shouldbe included in the standard CC algorithm, Algorithm CC 5.8.3.

Table 5.8.9 shows the total number of branch and bound nodes searched and total computationtime for the CC model itself and Algorithm CC 5.8.3 Cases B, C and D. We also include thenumerical results for Algorithm CC 5.8.3 Case A in Table 5.8.9 for ease of comparison. We againhighlight entries in bold face in the table if they are the lowest values of Total Time or Total BBNodes for a particular batch of 100 problems. Where a bold face entry is superscripted by an

Table 5.8.8: Numerical results for Algorithm CC 5.8.3 Case A applied to the CC model solving theBTCMC problem, using CPLEX version 8.1 and AMPL version 8.1 on a 2GHz AMD 64 3000+:time in seconds, 2-hour time limit on individual problem instances.

Batches CC model Algorithm CC 5.8.3 Case A Algorithm CC 5.8.3 Case A

of 100 defined in w/o (5.8.11)

problems Chapter 3,

Section 3.8.1

Total

BB

Nodes

Total

Time

Prep.

Time

Total

BB

Nodes

Total

Time

Prep.

Time

Total

BB

Nodes

Total

Time

4 4 0 15 15862 35.91 0.74 23488 40.23 0.78 23752 40.52

5 5 0 5 332 2.20 0.50 417 4.51 0.58 348 4.34

5 5 0 10 17009 25.00 0.85 13809 26.12 0.94 15880 26.69

5 5 0 15 315226 355.84 1.14 426766 446.16 1.37 422655 402.49

6 6 0 5 603 3.46 0.72 951 7.02 0.76 1303 7.07

6 6 0 10 192786 151.41 1.15 492616 336.50 1.39 581870 316.57

6 6 0 15 11565292 15483.70 1.72 32374442 36842.98 2.02 14341202 18660.35

7 7 0 5 1831 6.23 0.97 2593 11.83 1.12 1454 10.28

7 7 0 10 22021523 17745.40 1.78 11228026 9415.71 2.12 33259179 24177.10

7 7 0 15 98850536 177213.00 2.59 127721653 198554.84 3.20 132725015 183190.31

8 8 0 5 11180 13.27 1.35 6340 17.72 1.60 16787 18.56

8 8 0 10 64706187 66811.50 2.43 88219710 90465.87 2.95 98880611 85228.02

9 9 0 5 6016 14.58 1.81 7362 24.72 2.25 16011 24.32

10 10 0 5 154512 87.25 2.33 10253513 5014.80 2.94 1285011 1811.77

11 11 0 5 135611 1620.69 3.02 10748165 6669.22 3.86 1783140 1088.16

asterisk, this means that the CC model achieves the lowest value of Total Time or Total BB Nodes(whichever is applicable) for the batch under consideration.

Comparing Algorithm CC 5.8.3 Cases A, B, C and D, Table 5.8.9 immediately shows that AlgorithmCC 5.8.3 Cases B and D achieve the majority of the lowest values for Total Time and Total BBNodes. Algorithm CC 5.8.3 Case B obtains the lowest Total Time value in 4 out of the 15 batchestested and Algorithm CC 5.8.3 Case D obtains the lowest Total Time value in 8 out of the 15batches. Algorithm CC 5.8.3 Case B achieves the lowest Total BB nodes value in 6 out of the 15batches and Algorithm CC 5.8.3 Case D achieves the lowest Total BB Nodes value in 5 out of the 15


batches. Both algorithms achieve lowest values over a range of matrix sizes and maximum intensitylevels. (Here we compare just the algorithms and do not consider whether the CC model itselfachieved the lowest value for an indicator). If we now also compare the total time and total numberof branch and bound nodes searched over all batches tested we see that Algorithm CC 5.8.3 Case Dimproves on Algorithm CC 5.8.3 Case B by approximately 15 hours and approximately 6.66×107

nodes overall. We also see that where Algorithm CC 5.8.3 Case D did not achieve the lowest TotalTime or Total BB Nodes value, it achieves the second lowest value in 5 of the remaining 7 batchesfor Total Time and in 8 of the remaining 10 batches for Total BB Nodes. (Algorithm CC 5.8.3 CaseB does not achieve comparable results using this test). Finally whilst Algorithm CC 5.8.3 Case Dincreases the number of branch and bound nodes searched by approximately 2.40×107 nodes whencompared with the CC model itself, it also reduces overall computation time for the CC model byapproximately 20 hours. Therefore, we conclude that Algorithm CC 5.8.3 Case D, which appliesalgorithms Leaf Pair Bounds Improve (of Section 5.5), UB Consistency and Bounds Propagate (ofSection 5.2) to bk bounds, for k = 1, . . . ,K, previously initialised (with Leaf Pair Bounds Initialiseof Section 5.4.2), ordered and propagated, and finally converted to bounds on variables Nb, forb = 1, . . . , bmax, is the best performing of the algorithms tested thus far on the CC model.

Table 5.8.9 also demonstrates that in general, for all algorithms, preprocessing time is relativelynegligible for larger problem sizes with larger maximum intensity levels of which we are mostinterested. Given the computational benefits of Algorithm CC 5.8.3 Case D in particular, we canconclude that the time savings achieved by including this preprocessing outweigh the increase intime associated with the preprocessing itself.



Tab

le5.

8.9:

Num

eric

alre

sults

for

Alg

orit

hmC

C5.

8.3

Cas

esA

,B

,C

and

Dap

plie

dto

the

CC

mod

elso

lvin

gth

eB

TC

MC

prob

lem

,us

ing

CP

LE

Xve

rsio

n8.

1an

dA

MP

Lve

rsio

n8.

1on

a2G

Hz

AM

D64

3000

+:

tim

ein

seco

nds,

2-ho

urti

me

limit

onin

divi

dual

prob

lem

inst

ance

s.

Batc

hes

CC

mod

elA

lgori

thm

CC

5.8

.3C

ase

AA

lgori

thm

CC

5.8

.3C

ase

BA

lgori

thm

CC

5.8

.3C

ase

CA

lgori

thm

CC

5.8

.3C

ase

D

of

100

defi

ned

in

pro

ble

ms

Ch

ap

ter

3,

Sec

tion

3.8

.1

Tota

l

BB

Nod

es

Tota

l

Tim

e

Pre

p.

Tim

e

Tota

l

BB

Nod

es

Tota

l

Tim

e

Pre

p.

Tim

e

Tota

l

BB

Nod

es

Tota

l

Tim

e

Pre

p.

Tim

e

Tota

l

BB

Nodes

Tota

l

Tim

e

Pre

p.

Tim

e

Tota

l

BB

Nod

es

Tota

l

Tim

e

44

015

15862

35.9

10.7

823752

40.5

20.7

317616

23.4

51.0

214554

22.2

61.4

116309

20.9

7

55

05

332

2.2

00.5

8348

4.3

40.6

9266

4.2

2∗

1.0

0171

4.5

21.1

8157

4.6

0

55

010

17009

25.0

00.9

415880

26.6

90.8

113708

20.3

81.1

19796

17.5

61.5

117353

19.2

9

55

015

315226

355.8

41.3

7422655

402.4

90.9

8226686

233.9

71.4

4311059

216.8

62.0

5241843

200.6

5

66

05

603

3.4

60.7

61303

7.0

70.9

4553

6.2

71.1

2750

5.9

51.4

2378

5.8

6∗

66

010

192786

151.4

11.3

9581870

316.5

71.1

7251910

144.4

81.5

3329471

199.7

22.1

5103753

70.6

7

66

015

11565292

15483.7

02.0

214341202

18660.3

51.3

513543282∗

9176.9

91.9

214936117

17006.9

62.8

514269774

9868.5

7

77

05

1831

6.2

31.1

21454

10.2

81.2

81141

9.0

01.5

31490

9.2

61.8

31164

8.1

1∗

77

010

22021523

17745.4

02.1

233259179

24177.1

01.6

216107338

9912.1

72.1

018605741

10610.2

22.8

726074332

12420.3

4

77

015

98850536

177213.0

03.2

0132725015

183190.3

11.8

6155858376

164712.2

92.7

0148473475

159640.8

63.9

092065991

115667.1

5

88

05

11180

13.2

71.6

016787

18.5

61.7

33456

13.5

62.1

311078

15.9

92.5

24219

12.3

1

88

010

64706187

66811.5

02.9

598880611

85228.0

22.0

892773544

70490.3

32.7

485741504∗

62752.9

43.8

788006126

67605.6

4

99

05

6016

14.5

82.2

516011

24.3

22.1

819309

21.7

32.7

04271

18.7

2∗

3.6

67366

19.7

2

10

10

05

154512

87.2

52.9

41285011

1811.7

72.7

284968

51.2

03.4

0689727

250.2

94.3

8596602

226.4

2

11

11

05

135611

1620.6

93.8

61783140

1088.1

63.7

29747173

4320.4

94.3

829908734

12025.4

45.6

0637716∗

268.2

3


For completeness we now provide Table 5.8.10 which compares each CC algorithm, Cases A toD inclusive, and the CC model on the basis of the total hours of computation time saved andthe reduction in the total number of branch and bound nodes searched, over the CC model, forthe batches of problems under consideration. Since all algorithms increase the number of branchand bound nodes searched when compared with the CC model, these entries are given as negativevalues in the table, as is the increase in total time returned by Algorithm CC 5.8.3 Case A overthe CC model. Algorithm CC 5.8.3 Cases B, C and D all improve overall computation time whencompared with the CC model.

Again, as for the JS model, we expect that with each new improvement in bounds on variables bk,for k = 1, . . . ,K, in turn converted to bounds on variables Nb, for b = 1, . . . , bmax, there will be afurther improvement in computational efficiency for the CC model. Our best performing algorithmthus far, Algorithm CC 5.8.3 Case D, is consistent with this hypothesis, and given that there islittle difference in computation time for Algorithm CC 5.8.3 Cases B and C, then our remainingalgorithms perform roughly as we would expect, with Algorithm CC 5.8.3 Case A yielding theleast, and in fact, no improvement over the CC model. See Table 5.8.10 for these comparativeresults.

Finally, since Algorithm CC 5.8.3 Case D is the best algorithm we have considered thus farfor the CC model, in the following section we apply each of algorithms IP Bounds Initialise,LP1 Bounds Initialise and LP2 Bounds Initialise of Section 5.6 within the framework of AlgorithmCC 5.8.3 Case D to the CC model. The algorithm we apply is Algorithm CC 5.8.3 Case E. Algo-rithm CC 5.8.3 Case E tests IP Bounds Initialise, LP1 Bounds Initialise and LP2 Bounds Initialiseby substituting these algorithms, in turn, in step 1 of Case E, where Case E is as described inTable 5.8.1. The best version of Algorithm CC 5.8.3 Case E is then compared with Algorithm CC5.8.3 Case D and as a final test, we numerically compare the best of Algorithm CC 5.8.3 Cases Dand E against the Counter model of Baatar et al. [4], with simple bounds, as described in Chapter3, Section 3.8. The Counter model with simple bounds is the best performing of the models wehave previously considered and therefore for completeness we provide comparative results at theend of the following section.

5.8.9 Case E: Application of IP Bounds Initialise, LP1 Bounds Initialise orLP2 Bounds Initialise, Bounds Propagate, Leaf Pair Bounds Improve,Consistent b N Bounds, UB Consistency and LB Consistency to the CC Model. Inthis section we test each of the algorithms of Section 5.6, IP Bounds Initialise, LP1 Bounds Initialiseand LP2 Bounds Initialise respectively, within Algorithm CC 5.8.3 Case E, which extends our bestperforming algorithm thus far for the CC model, Algorithm CC 5.8.3 Case D. Again the only dif-ference between Algorithm JS 5.8.2 Case E and Algorithm CC 5.8.3 Case E, apart from the factthat we are trialling the initialisation procedure in this section, is that bounds are re-converted toNb bounds, for b = 1, . . . , bmax, in the case of the CC algorithm.

Table 5.8.11 shows preprocessing time, total computation time and total number of branch andbound nodes searched for Algorithm CC 5.8.3 Case E with each of IP Bounds Initialise,LP1 Bounds Initialise and LP2 Bounds Initialise respectively. We also include in the table theresults for Algorithm CC 5.8.3 Case D and for the CC model itself. Again, entries in bold face



Table 5.8.10: Summary of numerical results for Algorithm CC 5.8.3 Cases A, B, C and D applied to the CC

model solving the BTCMC problem, using CPLEX version 8.1 and AMPL version 8.1 on a 2GHz AMD 64 3000+:

2-hour time limit on individual problem instances. The column entitled Hours gives the approximate total time

saved in hours over the CC model. The column entitled BB Nodes gives the approximate reduction in total number

of branch and bound nodes searched compared with the CC model.


Algorithm CC

5.8.3 Case A

basic initialisation

of bounds on bk for

k = 2, . . . , K using

Initialise Multiset Bounds

of Section 5.4.1, b1 set to

Beammin, conversion to

bounds on variables Nb,

for b = 1, . . . , bmax, via

Consistent b N Bounds

of Section 5.3, applica-

tion of UB Consistency

and LB Consistency of

Section 5.2

-10 -8.54×107

Algorithm CC

5.8.3 Case B

improved value

for b1, using

Leaf Pair Bounds Initialise

of Section 5.4.2 o/w

equivalent to Algorithm

CC 5.8.3 Case A

6 -9.07×107

Algorithm CC

5.8.3 Case C

application of the

UB Consistency,

LB Consistency and


rithms of Section 5.2 to

the bounds on variables

bk, for k = 1, . . . , K,

set in Algorithm CC

5.8.3 Case B, then

conversion to Nb bounds,

for b = 1, . . . , bmax, via

Consistent b N Bounds,

and further application

of UB Consistency and

LB Consistency

5 -1.01×108

Algorithm CC

5.8.3 Case D

application of


of Section 5.5,

UB Consistency and

Bounds Propagate to

the bk bounds, for

k = 1, . . . , K, resulting

in Algorithm CC 5.8.3

Case C, then conver-

sion to Nb bounds, for

b = 1, . . . , bmax, via




LB Consistency

20 -2.40×107


indicate the lowest value for Total Time or Total BB Nodes when comparing Algorithm CC 5.8.3Cases D and E and the CC model, where bold face entries superscripted with an asterisk implythat the CC model achieves the lowest value of Total Time or Total BB Nodes for the batch underconsideration.

Table 5.8.11 indicates that Algorithm CC 5.8.3 Case E, utilising LP1 Bounds Initialise, is the bestperforming of the Case E algorithms tested on the CC model. Algorithm CC 5.8.3 Case E, withLP1 Bounds Initialise, yields time savings of approximately 12 hours over Algorithm CC 5.8.3 CaseE, with IP Bounds Initialise and of approximately 6 hours over Algorithm CC 5.8.3 Case E, withLP2 Bounds Initialise, over the batches of problems where both versions of the algorithm are calcu-lated. Over the same batches of problems, Algorithm CC 5.8.3 Case E, with LP1 Bounds Initialisereduces the total number of branch and bound nodes searched by approximately 1.43×108 nodescompared with Algorithm CC 5.8.3 Case E, with IP Bounds Initialise, and by approximately2.60×107 nodes compared with Algorithm CC 5.8.3 Case E, with LP2 Bounds Initialise. Algo-rithm CC 5.8.3 Case E, with LP1 Bounds Initialise, achieves the lowest Total Time value of theCase E algorithms investigated in 10 out of the 15 common batches tested and the lowest TotalBB Nodes value in 9 out of the 15 common batches tested. The batches for which Algorithm CC5.8.3 Case E, with LP1 Bounds Initialise achieves the lowest values cover a range of problem sizesand maximum intensity levels.

Table 5.8.11 also demonstrates that, for larger problem sizes with larger maximum intensity levels,the preprocessing time for all Algorithm CC 5.8.3 Case E versions is relatively insignificant, thoughthe preprocessing time associated with Algorithm CC 5.8.3 Case E, with IP Bounds Initialise, is sig-nificantly larger than the preprocessing time for the other Case E algorithms, as expected. We notethat in contrast to Algorithm JS 5.8.2 Case E, Algorithm CC 5.8.3 Case E with IP Bounds Initialisetakes longer to solve the CC model itself than the linear programming versions of Case E, in ad-dition to also having the longer preprocessing times.

We now compare the best of the Case E algorithms when applied to the CC model, Algorithm CC5.8.3 Case E, with LP1 Bounds Initialise, (hereafter referred to as Algorithm CC 5.8.3 Case E),with the best of the algorithms we have previously tested on the CC model, Algorithm CC 5.8.3Case D. Additional batches of problems are considered for Algorithm CC 5.8.3 Cases D and E asboth algorithms yield similar overall computational results.

Considering the extended range of problems given in Table 5.8.11 we see that total computationtime, for all batches, reduces by approximately 5 hours using Algorithm CC 5.8.3 Case E, comparedwith using Algorithm CC 5.8.3 Case D, whereas overall numbers of branch and bounds nodessearched increase with Algorithm CC 5.8.3 Case E, by approximately 6.33 ×105 nodes, whencompared with Algorithm CC 5.8.3 Case D. (Over the restricted batches of problems previouslyconsidered, the comparative figures are approximately 45 minutes and 8.88×106 nodes). If we nowfocus on the ‘hardest’ problems solved by each algorithm (we consider batches 6 6 0 15, 7 7 0 10,7 7 0 15, 8 8 0 10, 9 9 0 10, 13 13 0 5, 14 14 0 5 and 15 15 0 5) and determine the percentageimprovement of each algorithm over the other for these problems, we see that Algorithm CC 5.8.3Case E, yields significantly greater percentage improvement over these ‘harder’ problems thanAlgorithm CC 5.8.3 Case D.



We therefore conclude that Algorithm CC 5.8.3 Case E is the most efficient of the algorithms wehave investigated for application on the CC model. Table 5.8.12 summarises the numerical resultsfor Algorithm CC 5.8.3 Case D and Algorithm CC 5.8.3 Case E, compared with the CC model. Wesee that over the extended range of problems, overall computation time reduces by approximately 64hours and total number of branch and bound nodes searched increases by approximately 2.15×108

nodes when using Algorithm CC 5.8.3 Case E, compared with the CC model.

It is interesting to note that Algorithm CC 5.8.3 Case E, achieves lower overall computationtime when compared with Algorithm CC 5.8.3 Case D, considering the former algorithm hasgreater associated preprocessing time. We conclude that it is worthwhile calculating bounds in thepreprocessing stages of Algorithm CC 5.8.3 Case E, given that overall computation time for theCC model is significantly improved.

In general the algorithms we have tested on the CC model increase the total number of branch andbound nodes searched when compared with the CC model, however all but Algorithm CC 5.8.3Case A reduce the total computation time for the batches of problems tested. This is in contrast tothe algorithms applied to the JS model, earlier in this section, where all algorithms tested decreasetotal computation time and total numbers of branch and bound nodes searched when comparedwith the JS model.

Finally, we compare Algorithm CC 5.8.3 Case E (with LP1 Bounds Initialise), the most efficient ofthe algorithms tested on the CC model, with Algorithm JS 5.8.2 Case E (with IP Bounds Initialise)applied with the Step-up Method, the most efficient of the algorithms tested on the JS model.Comparing Tables 5.8.6 and 5.8.11 we see that Algorithm CC 5.8.3 Case E, applied to the CCmodel is far superior to Algorithm JS 5.8.2 Case E applied with the Step-up Method to the JSmodel. (Computation time reduces by approximately 2 hours and branch and bound nodes searchedreduces by approximately 4.48×106 nodes considering just the 7 batches of 100 problems tested onthe JS model. The very fact that we test only 7 batches of 100 problems (of relatively small sizes)on all cases of Algorithm JS 5.8.2 is indicative of the fact that the JS algorithms on the JS modeldo not perform as well as the CC algorithms on the CC model for solving the BTCMC problem).Therefore Algorithm CC 5.8.3 Case E, which initialises bounds on variables bk, for k = 1, . . . ,K,via LP1 Bounds Initialise (of Section 5.6), applies UB Consistency, Bounds Propagate (Section5.2) and Leaf Pair Bounds Improve (Section 5.5), converts the resulting bk bounds to bounds onvariables Nb, for b = 1, . . . , bmax, and applies these to the CC model, is the best algorithm we haveconsidered in this work for solving the BTCMC problem.


Tab

le5.

8.11

:N

um

eric

alre

sultsfo

rA

lgori

thm

CC

5.8

.3C

ase

sD

and

Eapplied

toth

eC

Cm

odel

solv

ing

the

BT

CM

Cpro

blem

,usi

ng

CPLEX

vers

ion

8.1

and

AM

PL

vers

ion

8.1

on

a2G

HzA

MD

64

3000+

:tim

ein

seco

nds,

2-h

ourtim

elim

iton

indiv

idualpro

blem

inst

ance

s.A

lgori

thm

CC

5.8

.3C

ase

Etr

ials

each

ofIP

Bounds

Initia

lise

,LP1

Bounds

Initia

lise

and

LP2

Bounds

Initia

lise

resp

ective

lyin

the

appro

pri

ate

step

ofth

ealg

ori

thm

.

Batc

hes

of

CC

mod

eld

efin

edin

Alg

ori

thm

CC

5.8

.3C

ase

DA

lgori

thm

CC

5.8

.3C

ase

E

100

pro

ble

ms

Ch

ap

ter

3,

Sec

tion

3.8

.1

IPL

P1

LP

2

Tota

lT

ota

lP

rep

.T

ota

lT

ota

lP

rep

.T

ota

lT

ota

lP

rep

.T

ota

lT

ota

lP

rep

.T

ota

lT

ota

l

BB

Nod

esT

ime

Tim

eB

BN

od

esT

ime

Tim

eB

B

Nod

es

Tim

eT

ime

BB

Nod

es

Tim

eT

ime

BB

Nod

es

Tim

e

44

05

139

1.5

10.7

917

2.7

4∗

3.1

217

5.2

0

44

010

2533

7.3

51.1

41574

7.3

25.6

71514

11.4

3

44

015

15862

35.9

11.4

116309∗

20.9

725.9

137416

44.8

97.8

417431

27.4

37.4

817610

27.4

4

55

05

332

2.2

01.1

8157

4.6

0∗

12.1

01219

14.9

13.9

8133

6.9

93.6

4154

6.8

3

55

010

17009

25.0

01.5

117353

19.2

923.1

425846

36.6

66.8

810312

21.3

56.1

116135

23.4

4

55

015

315226

355.8

42.0

5241843

200.6

543.4

6511357

271.1

510.1

0248641

203.6

99.3

3241588

205.4

2

66

05

603

3.4

61.4

2378

5.8

6∗

16.0

13782

19.7

04.9

1404

8.9

74.9

3380

9.3

6

66

010

192786

151.4

12.1

5103753

70.6

740.9

3255396

99.3

38.6

6121206

83.8

68.2

8105005

82.3

0

66

015

11565292

15483.7

02.8

514269774∗

9868.5

7451.8

226087848

29851.3

312.6

415865893

10931.5

511.1

728060810

16404.6

9

77

05

1831

6.2

31.8

31164

8.1

1∗

22.0

714961

28.1

36.3

51136

13.0

96.1

21138

13.5

0

77

010

22021523

17745.4

02.8

726074332

12420.3

483.9

55407295

2441.6

111.6

618815999

9439.9

19.9

624427787

12338.8

9

77

015

98850536

177213.0

03.9

092065991

115667.1

57931.0

7243173911

148419.9

117.3

598590202

113639.5

713.7

9116292855

134035.2

4

88

05

11180

13.2

72.5

24219

12.3

129.5

435036

38.9

27.8

52054

17.5

77.9

23926

19.3

8

88

010

64706187

66811.5

03.8

788006126

67605.6

4276.9

389632767

65824.3

613.7

295937129

69731.8

512.0

586514336∗

63242.7

4

99

05

6016

14.5

83.6

67366∗

19.7

2∗

37.9

448495

54.0

710.1

67374

25.0

39.4

07441

25.2

6

99

010

213638702

297533.0

04.8

9290365763∗268069.8

617.9

2307936569

275462.0

7

10

10

05

154512

87.2

54.3

8596602∗

226.4

2∗

83.1

97601591

1503.5

012.4

5664710

242.7

011.5

1604771

239.6

9

11

11

05

135611

1620.6

95.6

0637716∗

268.2

379.8

9869562

361.1

015.4

1639550

283.2

614.0

2642132

285.8

7

12

12

05

449724

20846.0

07.0

86136168

2595.1

220.6

76129043∗

2602.4

5

13

13

05

663137

63777.8

08.2

830196203

18959.5

624.1

916961594∗

10935.7

8

14

14

05

74653962

83977.0

09.8

968886761

42172.8

728.9

862925235

37460.8

3

15

15

05

105870031

138135.0

012.2

6190297169

132072.2

634.3

6183683814∗

121822.0

0



Table 5.8.12: Summary of numerical results for Algorithm CC 5.8.3 Cases D and E applied to the CC

model solving the BTCMC problem, using CPLEX version 8.1 and AMPL version 8.1 on a 2GHz AMD 64

3000+: 2-hour time limit on individual problem instances. The column entitled Hours gives the approximate

total time saved in hours over the CC model. The column entitled BB Nodes gives the approximate reduction

in total number of branch and bound nodes searched compared with the CC model.


Algorithm CC 5.8.3

Case D

application of


of Section 5.5,

UB Consistency and

Bounds Propagate (of

Section 5.2) to the bk

bounds, for k = 1, . . . , K,

resulting in Algorithm

CC 5.8.3 Case C, then

conversion to Nb bounds,

for b = 1, . . . , bmax, via


(of Section 5.3) and

further application of

UB Consistency and

LB Consistency

20 - over the batches

of problems consid-

ered for all CC al-

gorithms, allowing a

direct comparison

-2.40×107

59 - over the ex-

tended batches of

problems considered

for Algorithm CC

5.8.3 Cases D and E

and the CC model

-2.15×108

Algorithm CC 5.8.3

Case E

initialisation of bounds

on variables bk, for

k = 1, . . . , K, via

LP1 Bounds Initialise of

Section 5.6, application

of UB Consistency,

Bounds Propagate and

Leaf Pair Bounds Improve.

(New initialisation of

bounds within the frame-

work of Algorithm CC

5.8.3 Case D). Conver-

sion to Nb bounds for

b = 1, . . . , bmax, via




LB Consistency

21 - restricted -3.29×107

64 - extended -2.15×108

5.9. Conclusion 165

As a final comparison, we compare the results for the Counter model of Baatar et al. [4], with simplebounds, as described in Chapter 3, Section 3.8, against Algorithm CC 5.8.3 Case E, applied to theCC model. Results are given in Table 5.8.13 where bold face entries indicate which model/algorithmreturns the lowest Total Time value or Total BB Nodes value for a particular batch. Batches of 100problems of increasing size and maximum intensity level are tested until total computation timefor a batch exceeds 100,000 seconds. At this point no further increases of size for the particularmaximum intensity level are tested.

Table 5.8.13 shows that the Counter model [4], with simple bounds as described in Chapter 3,Section 3.8, is still superior to all algorithms we have tested in this chapter, achieving time savingsof approximately 105 hours over Algorithm CC 5.8.3 Case E, considering the batches of problemswhere both the model and the algorithm were solved. The total number of branch and boundnodes searched also decreases using the Counter model [4] with simple bounds by approximately4.98×108 nodes overall when compared with Algorithm CC 5.8.3 Case E. Algorithm CC 5.8.3 CaseE obtains the lowest number of branch and bound nodes for a few of the smaller sized matriceswhen maximum intensity level is also small, otherwise, the Counter model [4] with simple boundsyields the smallest numbers of branch and bound nodes and smallest Total Time values for all theremaining batches.

5.9 Conclusion

In this chapter we have further investigated the Cumulative Counter model with simple bounds(CC) and the Johnston and Sadinlija (JS) model, as defined in Chapter 3. We have developednew techniques for reducing the computation time for these exact integer programming models. Inparticular, in this chapter, we determine improved bounds for the ‘beam-on time related’ variablesof each model. We apply the techniques developed in this work to both model types (pseudo-polynomial sized and polynomial sized) to determine how each is affected by improved bounds, andconsidering their benefits and limitations (for example, whilst relatively computationally efficient,CC-type models may not have direct use within the cutting stock application). We investigatethe CC model rather than the Counter model with simple bounds (the best performing pseudo-polynomial sized model previously considered) since the bounds determined in this chapter aredirectly applied within the CC model and would require additional constraints to be appliedwithin the Counter model. Bounds are handled more efficiently than constraints in the Simplexsolving method and furthermore, the ‘beam-on time related’ variables of the JS and CC modelshave similar properties which we are able exploit in this work. The CC model is the second bestperforming pseudo-polynomial sized model we have previously considered and the JS model (withthe Step-up method applied) is the best of the polynomial sized models.

As mentioned in the Introduction to this chapter, given a set of bounds on the ‘beam-on timerelated’ variables of a model, it is still a strongly NP-hard problem to determine a feasible decom-position of even a single row of an intensity matrix, [43]. However, as demonstrated by preliminaryresults in Chapter 3 and the results of this chapter, improving bounds on variables reduces thesize of the feasible region to be searched and can decrease the solution computation time for thestrongly NP-hard problem. All numerical results we have presented in this chapter solve the un-



Table 5.8.13: Numerical results for Algorithm CC 5.8.3 Case E applied to the CC model and theCounter model of Baatar et al. [4], with simple bounds, as described in Chapter 3, Section 3.8,solving the BTCMC problem, using CPLEX version 8.1 and AMPL version 8.1 on a 2GHz AMD64 3000+: time in seconds, 2-hour time limit on individual problem instances.

Batches Algorithm CC 5.8.3 Case E Counter model

of 100 with simple bounds

problems defined in Chapter 3,

Section 3.8

Prep.

Time

Total BB

Nodes

Total

Time

Total BB

Nodes

Total

Time

4 4 0 5 3.12 17 5.20 66 1.42

4 4 0 10 5.67 1514 11.43 2074 5.44

4 4 0 15 7.84 17431 27.43 15862 24.87

5 5 0 5 3.98 133 6.99 305 2.01

5 5 0 10 6.88 10312 21.35 8929 14.92

5 5 0 15 10.10 248641 203.69 197777 137.73

6 6 0 5 4.91 404 8.97 1077 3.43

6 6 0 10 8.66 121206 83.86 88122 61.41

6 6 0 15 12.64 15865893 10931.55 2900843 2560.16

7 7 0 5 6.35 1136 13.09 1615 5.41

7 7 0 10 11.66 18815999 9439.91 11408232 5140.46

7 7 0 15 17.35 98590202 113639.57 57492134 55119.10

8 8 0 5 7.85 2054 17.57 2500 8.07

8 8 0 10 13.72 95937129 69731.85 33359965 24041.30

8 8 0 15 247440624 328547.00

9 9 0 5 10.16 7374 25.03 5603 12.39

9 9 0 10 17.92 307936569 275462.07 144416873 128140.00

10 10 0 5 12.45 664710 242.70 15188 21.60

11 11 0 5 15.41 639550 283.26 288472 221.55

12 12 0 5 20.67 6129043 2602.45 53131 59.62

13 13 0 5 24.19 16961594 10935.78 12249296 7384.16

14 14 0 5 28.98 62925235 37460.83 17456477 17090.40

15 15 0 5 34.36 183683814 121822.00 30235934 33738.30

16 16 0 5 56126480 59622.90

17 17 0 5 16075638 30842.70

18 18 0 5 64270416 100525.00

5.9. Conclusion 167

constrained BTCMC problem, however the majority of the bounds results we determine apply toany upper bound on the total beam-on time of a decomposition and hence any objective for theleaf sequencing problem. Furthermore, with minor amendments the results of this chapter can beapplied to solve the constrained case. However, it should be noted again that MLC mechanicalconstraints may not be simple to formulate and that, in general, the numerical computation timesand solutions returned when solving the constrained BTCMC problem will be different from thoseof this chapter for solving the unconstrained BTCMC problem.

As stated, the aim for the current study was to improve the computational efficiency of the JS andCC models by determining reduced upper bounds and increased lower bounds on the ‘beam-ontime related’ variables of each model. To achieve this we considered a structure common to boththe JS and CC models, which we termed the Sum Constrained Sorted Multiset System (SCSMS),and demonstrated how properties of the structure could be used to tighten bounds on the ‘beam-ontime related’ variables of each model. The properties considered utilised the non-increasing natureof the ‘beam-on time related’ variables of the models, the fact that the variables sum to a fixedtotal beam-on time for the decomposition, the integrality of the variables and recursive proceduresto iteratively tighten bounds.

It was then shown that the bounds determined for the ‘beam-on time related’ variables of eitherthe JS or CC models can be converted to bounds on the ‘beam-on time related’ variables of theother model using a simple relationship. We therefore focused on developing improved bounds forthe beam-on time variables, bk for k = 1, . . . ,K, of the JS model only. These bounds were applieddirectly to the JS model for numerical testing, and were converted to bounds on the ‘beam-on timerelated’ variables of the CC model, Nb for b = 1, . . . , bmax, using the conversion relationship, fornumerical testing on the CC model.

In addition to the relationships and improvements for bounds determined using the SCSMS, weextended the work of Baatar et al. [1] on properties and decompositions of intensity matrices.This involved development of new initialisations of bounds and focused in particular on improvingthe upper bound on the first beam-on time variable, b1, of the JS model. We considered boundsbased on the most radiation that can be applied to shape matrices having a particular (left,right)leaf position in a particular row and then considered the maximum of these bounds over all leafpositions and finally the minimum of these over all rows to determine an improved value for b1.We also utilised the integrality of shape matrix variables to determine an upper bound on thenumber of shape matrices that can be given a particular radiation level or more in a particularrow. This in turn was used to iteratively improve bounds on variables bk by noticing that if thenumber of shape matrices that can be given radiation level c or more is k then the beam-on timefor the (k + 1)th shape matrix is bounded above by c− 1.

We then investigated a different approach, considering the beam-on time applied to shape matriceswith a particular (left,right) leaf position in a particular row as a variable. We developed an integerprogram and two linear relaxations of the integer program to maximise flow through a BipartiteGraph and again we calculated an upper bound on the number of shape matrices that can be givena particular level of radiation or more. These bounds were converted to bounds on variables bkas described previously. We numerically tested the integer program and two linear programs to



determine the trade-off between the computation time for determining bounds and the quality ofthe upper bounds themselves. We found that whilst preprocessing time is greater when using theinteger program to calculate bounds, it is not prohibitively large, and in the case of the JS modelit is the integer program which results in the greatest time savings for this model. For the CCmodel, the first linear program we trial, which is simply the integer program with the integralityconditions on the variables removed, results in the greatest computational improvement.

We finally also determined upper bounds on the most radiation that can be applied to shapematrices exposing a particular cell. We again considered the most radiation that can be appliedto shape matrices having a certain leaf position in a particular row and maximised over the manydifferent leaf positions which expose the same cell in a shape matrix. This bound was utilised asa ‘big-M’ value in a constraint of the JS model.

All other upper bounds we determined for variables bk for k = 1, . . . ,K were applied in variousconstraints within the JS model and of course to bound the beam-on time variables themselves. Wealso tested a number of additional constraints on the JS model however only one such constraintyielded computational improvement. This was incorporated into the standard JS model on whichour bounds algorithms were trialled.

The best bounds algorithm applied to the JS model was Algorithm JS 5.8.2 Case E applied with theStep-up Method, which yielded a reduction in total computation time over the JS model of approxi-mately 328 hours. Algorithm JS 5.8.2 Case E applied with the Step-up Method initialises bounds onvariables bk using IP Bounds Initialise (Section 5.6), applies UB Consistency, Bounds Propagate,(Section 5.2), Leaf Pair Bounds Improve (Section 5.5) and Cell Based Bounds Initialise (Section5.7) and steps up K, in the model, from a known lower bound on K, till a feasible solution isfound. Algorithm JS 5.8.2 Case E uses all the bounds improvement ideas we have developed inthis chapter.

The remaining bounds algorithms applied to the JS model all improved overall computation timeand the number of branch and bound nodes searched when compared with the JS model. In orderof decreased efficiency, the remaining algorithms are: Algorithm JS 5.8.2 Case D applied with theStep-up Method, Algorithm JS 5.8.2 Case E, Algorithm JS 5.8.2 Case D, Algorithm JS 5.8.2 CaseA, Algorithm JS 5.8.2 Case B and Algorithm JS 5.8.2 Case C. The Step-up Method was onlytrialled on the best of the standard bounds algorithms, Algorithm JS 5.8.2 Case E and AlgorithmJS 5.8.2 Case D.

The application of bounds to the CC model reduced the number of variables and also the numberof constraints in the model, when the upper bound on variable b1 was used to bound the b rangefor variables Nb. The bk bounds, converted to bounds on variables Nb, were of course also appliedto the Nb variables themselves. We applied one additional constraint to the standard CC model,the equivalent of that applied to the JS model, after determining that its application improvedcomputation time for the CC model.

The bounds algorithms applied to the CC model were equivalent to those applied to the JS modelother than the bk bounds being converted to bounds on variables Nb, and ordered, in the last stepof each CC bounds algorithm.

5.9. Conclusion 169

The best bounds algorithm applied to the CC model was Algorithm CC 5.8.3 Case E, which re-sulted in time savings of approximately 64 hours over the CC model, considering a larger number ofbatches of problems than trialled on the JS model (since the CC model is more computationally ef-ficient). Algorithm CC 5.8.3 Case E, initialises bounds on variables bk using LP1 Bounds Initialise(Section 5.6), applies UB Consistency, Bounds Propagate and Leaf Pair Bounds Improve, prior toconversion to Nb bounds. Therefore the best bounds algorithm for the CC model also utilises allthe bounds improvement techniques we consider in this work.

All bounds algorithms tested on the CC model increase the total number of branch and boundnodes searched, however, all but Algorithm CC 5.8.3 Case A decrease total computation time whencompared with the CC model. The remaining bounds algorithms in order of decreasing efficiencyfor the CC model are: Algorithm CC 5.8.3 Case D, Algorithm CC 5.8.3 Case B, Algorithm CC5.8.3 Case C and Algorithm CC 5.8.3 Case A. The Step-up Method is not applied to the CCmodel since our previous work, in Chapter 3, Section 3.8.1, has shown that this does not improvecomputational efficiency.

The preprocessing time (and time to calculate Kbm with the Greedy Heuristic Algorithm of Baataret al. [1]) associated with all bounds algorithms applied to the JS and CC models was relativelysmall. Therefore the preprocessing time did not significantly affect total computation time forthe JS model and only contributed to small problem sizes with small maximum intensity levelssolved with the CC model. Since we are interested in solving larger problem sizes with possiblylarge maximum intensity levels, the preprocessing time itself is not a factor. Even the algorithmswhich solved the integer/linear programs first to initialise bounds did not produce prohibitivelylarge preprocessing times for larger matrices. Therefore, the time improvements achieved usingthe preprocessing techniques are worth the small increase in time associated with the calculationof bounds.

We did expect that with each new improvement in bounds, implemented in each new JS or CCalgorithm, there would be a reduction in the size of the feasible region for each problem and afurther increase in computational efficiency for the JS and CC models respectively. The two bestperforming algorithms for the JS model are consistent with this hypothesis, although the remainingalgorithms for the JS model did not conform exactly to this idea. With regard to the CC model,the algorithms generally perform the way we expect.

We also conclude that Algorithm CC 5.8.3 Case E (with LP1 Bounds Initialise) applied to theCC model is computationally superior to Algorithm JS 5.8.2 Case E (with IP Bounds Initialise)applied with the Step-up Method to the JS model for solving the BTCMC problem. However,it is clear from our results that the improvement in computation time for the JS model with theapplication of the best of the JS bounds algorithms, approximately 328 hours over the batches ofproblems tested, is much greater than the improvement for the best of the CC bounds algorithmsapplied to the CC model, approximately 64 hours over the larger set of problems tested. Onereason for this may be that the JS model had much more ‘room to improve’ than the CC model,which already performed significantly better than the JS model itself.

One final comparison was made against our best performing algorithm overall, Algorithm CC



5.8.3 Case E applied to the CC model. We determined that the Counter model [4], extendedto incorporate simple bounds, as described in Chapter 3, Section 3.8, is still superior to all themodels/algorithms we have tested in the current work, taking approximately 105 hours less thanAlgorithm CC 5.8.3 Case E again over the larger set of problems tested where both the model andalgorithm were solved.

Clearly our bounds work has resulted in much improvement for the CC and JS models, howeverthe best algorithm developed using these techniques does not outperform the Counter model withsimple bounds. Our previous work, in Chapter 3, Section 3.9, indicated that whilst the Countermodel with simple bounds solves large problem sizes, at least with small maximum intensity levels,it is still unable to solve the complete spectrum of medical data to which we have access inreasonable time. This being the case we must still look for new ways to improve the CC and JSmodels and the Counter model in particular.

171CHAPTER 6

Conclusion

In this thesis, we have developed new exact models, in particular mixed integer and integer pro-gramming formulations, for solving the Total Treatment Time (TTT) and Beam-on Time Con-strained Minimum Cardinality (BTCMC) problems, for the delivery of Intensity Modulated Radi-ation Therapy (IMRT) using Multileaf Collimators (MLC). Within the framework of exact mixedinteger and integer programming formulations, we have investigated the possibilities for computa-tional improvement.

During the application of Intensity Modulated Radiation Therapy patients must remain as sta-tionary as possible and hence the smaller the treatment time the less chance of patient movement.Minimising treatment time may therefore improve radiation delivery to the cancer and increasethe chance of a successful outcome for the patient. The TTT and BTCMC problems we examinein this thesis minimise two different definitions of treatment time.

The study of exact models for these strongly NP-hard ‘step and shoot’ leaf sequencing problemsis important since previous approaches to the TTT and BTCMC problems in the literature aremainly heuristics. The few existing exact formulations either solve only small problem instancesto optimality with excessive computation times, or their solution method prevents the inclusion ofMLC mechanical constraints. It is therefore necessary to further investigate exact approaches andmany alternatives are possible within the context of integer programming, since the features ofthe TTT and BTCMC problems are naturally described using such formulations. Furthermore, asmost currently used multileaf collimators have a tongue-and-groove design and prohibit interleafmotion it is clear that solution methods and models must be able to incorporate MLC mechanicalconstraints. Exact approaches which can handle MLC mechanical constraints are also necessary tovalidate heuristic methods. Therefore, the objective for this thesis was to develop new exact mixedinteger and integer programming formulations, for the TTT and BTCMC problems, using solutionmethods which do not prevent the inclusion of MLC mechanical constraints, and to examineand apply techniques for reducing the computation time of the exact models. We did not applyMLC mechanical constraints to the models in this thesis, however the relevant constraints can beincorporated into all the models we present.

In Chapter 2 of this study we introduced a new exact mixed integer programming formulation forthe TTT problem, which explicitly optimises the trade-off between the total beam-on time of adecomposition and the total number of shape matrices used, for different values of constant set-uptime. The model is based on a formulation for solving cutting stock problems [6], which does notrequire beam-on times to be integer. We observed that the TTT model solved quickly when thenumber of shape matrices in the model, K, was set to an infeasible value. We thus formulatedour Step-up algorithm, exploiting the bicriteria nature of the TTT objective, to successively solvefeasibility problems for increasing values of K, and therefore solve the TTT problem. Direct so-lution of the TTT model, with K set to an upper bound on the number of shape matrices in anoptimal solution, yielded relatively large computation times. Utilising the Step-up algorithm com-putational improvement was achieved. We also considered restricting the beam-on time variablesof our TTT formulation to take integer values. We noticed that in practice the ‘integer’ version

172 Chapter 6. Conclusion

of the TTT model always returned the optimal solution for the ‘real’ TTT model. For T = 0it is known there is no integrality gap however whether this is true for T > 0 requires furtherinvestigation. Numerical results comparing the ‘real’ and ‘integer’ TTT models and the ‘real’ and‘integer’ Step-up algorithms demonstrated that the ‘integer’ formulations outperformed the ‘real’formulations and that the ‘integer’ Step-up algorithm was the best approach considered, yield-ing significant computational savings. The ‘integer’ TTT model outperformed the Langer et al.[3] model, modified to solve the TTT problem, by approximately three orders of magnitude. Ourresults also illustrated that for the (small) sizes of problems tested there were few examples demon-strating a trade-off between the minimum cardinality solution and the beam-on time constrainedminimum cardinality solution, with cardinality differing by only one unit where a trade-off waspossible. Further investigation is required into the potential trade-off when techniques are foundand applied, to further reduce computation times for exact solutions to the TTT problem. Finally,we mention again that Chapter 2, with minor amendments for consistency with this thesis, hasbeen published in the Journal of Computers and Operations Research, see [8].

In Chapter 3 we investigated different approaches for improving the computational efficiency of fourexact integer programming formulations for the BTCMC problem (three of which are new). Wepresented the BTCMC version of the integer TTT formulation of Chapter 2, and many variationsusing this model as a basis. We investigated alternative formulations of, what is now termed, theJohnston and Sadinlija (JS) model, using different variables to describe valid shape matrices andtrialling various symmetry breaking constraints. The JS model is of polynomial size and indexesits variables on shape matrices. We also developed an improved formulation of the Langer et al.[3] model, the Unit Radiation Pattern (URP) model, which is also of polynomial size, though inthis case indexes its variables on individual monitor units of radiation. Finally, we developed a newexact formulation, the Binary Expansion (BE) model, of pseudo-polynomial size, which indexeson radiation level. Again, we investigated numerous variations of symmetry breaking constraintson the URP and BE models, and in the case of the BE model, we also tested a number of addi-tional constraints. We compared our models against the Counter model of Baatar et al. [4], whichis also pseudo-polynomial in size indexing on radiation level, and we developed another pseudo-polynomial sized model, the Cumulative Counter model, which utilises the cumulative version ofthe variables of the Counter model. We explored the effect of simple bounds on the variables ofthe Counter and Cumulative Counter models and found that simple bounds significantly improveoverall computation time for both models. This result motivated our continued study of boundsin Chapter 5 of this thesis. We also applied the Step-up Method (the ‘step-up’ idea of Chapter2 adapted to the solution of the BTCMC problem) to our exact formulations, with numericalexperiments demonstrating that the application of the Step-up Method improves computationalefficiency for the JS and BE models but does not decrease overall computation time for the URP,Counter and Cumulative Counter models. Finally we compared the ‘best’ versions of each modelinvestigated, considering the particular variables used, additional, and symmetry breaking, con-straints and simple bounds. Of these final versions, the best formulation overall was the Countermodel of Baatar et al. [4], with our extension to incorporate simple bounds, followed by the Cu-mulative Counter model with simple bounds (the CC model). The Counter model with simplebounds and the CC model (both pseudo-polynomial sized models indexing on radiation level) aresuperior to the other models tested in Chapter 3 by a substantial margin. The JS model with theStep-up Method applied is the best of the polynomial sized models considered. Whilst the Counter

Chapter 6. Conclusion 173

model with simple bounds is the best exact integer programming model investigated in Chapter3, it was still unable to solve half the available medical data within a 2 hour individual problemtime limit. Hence continued study was necessary to further decrease the computation time ofthe better performing models of Chapter 3, of each type (pseudo-polynomial and polynomial), toexplore the benefits and limitations of such models when new techniques are applied for improvingcomputational efficiency.

In Chapter 4 therefore, we focused on the best polynomial sized model of Chapter 3 and consideredthe technique of polyhedral analysis for reducing the feasible region and in turn the computationtime of the JS model. The linear relaxation of the JS model ‘cheats’ by allowing the aijk variablevalues of the model, within shape matrix k, to take on different positive values, rather than thebeam-on time for shape matrix k or zero. Consequently we investigated a polyhedral structurewithin the JS model, the Equality Switch Polytope (ESP ), which captures the property of theaijk variables we wish to enforce. We then incorporated the strict consecutive-1-constraint (C1)into our definition of ESP to consider a polytope more representative of the feasible region ofthe JS model. We call this new polytope ESP -C1. To the best of our knowledge, neither theESP nor the ESP -C1 polytopes have previously been studied, though the C1 polytope is wellknown in the mathematical programming literature, as the strict consecutive-1-property occurs inother applications. Our analysis of the ESP and ESP -C1 polytopes, which may have broaderapplication outside the context of this thesis, therefore also contributes to the wider field of integerprogramming. We determined facets of small support of ESP and of ESP -C1 (under specialconditions) and numerically tested the facets on models of ESP and ESP -C1 respectively, and onthe JS model. For numerical testing we solved the BTCMC problem with the JS model, though thedefinitions of ESP and ESP -C1 do not involve the BTCMC objective nor the minimum beam-ontime constraint. The ESP and ESP -C1 facets improved the search space for the particular ESPand ESP -C1 models (respectively) investigated. With regard to the JS model, neither the facetsof ESP nor those of ESP -C1 yielded much, if any, improvement over the standard JS model. TheESP -C1 facets, which were applied if there was an appropriate number of equal I values in a row ofan intensity matrix, showed some slight improvement in total numbers of branch and bound nodessearched and total iterations in some of the batches tested, though very rarely did the applicationalso reduce computation time. To completely test the effect of the ESP -C1 facets on the JS model,large problem sizes (with low maximum intensity levels) would need to be considered, though thiswould only be possible if computation times for the JS model were first reduced via other methods.It is clear that different approaches for reducing the computation time for the JS model must beconsidered and hence we further investigated the JS model (as well as the Cumulative Countermodel with simple bounds of Chapter 3) in Chapter 5, where we again considered new techniques.

In Chapter 5, we developed and applied new, improved bounds to the integer ‘beam-on timerelated’ variables of the JS and CC models of Chapter 3, to further reduce the computation timefor these exact formulations. We were interested in how each model type might respond to theapplication of bounds to variables and the extent to which computation time might be improved.We investigated the CC model rather than the Counter model with simple bounds (which was thebest of the models studied in Chapter 3) since the JS and CC models share a common structure,which we termed the Sum Constrained Sorted Multiset System (SCSMS), and since the boundsdetermined could be utilised directly in the CC model. The BTCMC problem remains strongly


NP-hard even when bounds on ‘beam-on time related’ variables are known [43], however the sizeof the feasible region to be searched decreases as better bounds are determined. The numericalexperiments given in Chapter 5 again solved the BTCMC problem, however most of our boundsresults hold for any upper bound on total beam-on time for a decomposition. In Chapter 5 we alsoderived a relationship between the beam-on time variables of the JS model and the ‘beam-on timerelated’ variables of the CC model, such that we needed only concentrate on the determination ofbounds for the JS model, since these bounds were easily converted to bounds for the CC model.To determine improved bounds we considered properties of the SCSMS, of intensity matrices anddecompositions (building on some of the ideas of Baatar et al. [1]), integrality conditions, andsolution of integer and linear programs. We focused in particular on decreasing the upper boundon the first, largest, beam-on time value of the JS model. Other bounds determined describedthe most radiation that can be applied to shape matrices with a particular leaf position in aparticular row, the most radiation that can be applied to shape matrices exposing a particular cell,and bounds for the number of shape matrices that can be given a particular level of radiation ormore. All bounds results were systematically tested on the JS and CC models, where in the lattercase, the bounds were simply converted prior to application. In general preprocessing time fordetermining bounds was relatively negligible compared to the run time for the models themselves,in particular for larger problem sizes. The significant improvement in computation time experiencedby both models with the application of new improved bounds greatly outweighs the small additionalcomputation time necessary to calculate bounds. In particular we found that the preprocessingtime associated with the solution of the integer or linear programs (for determining bounds) wasnot prohibitively large. In fact the bounds returned by the solution of the integer program yieldedthe greatest computational savings for the JS model, when the Step-up Method was also applied.(Algorithm JS 5.8.2 Case E, with the Step-up Method). Furthermore, the simple linear relaxationof the integer program we considered yielded the greatest improvement for the CC model of thebounds algorithms trialled. (Algorithm CC 5.8.3 Case E). The best bounds algorithms appliedto the JS and CC models, Algorithm JS 5.8.2 Case E with the Step-up Method and AlgorithmCC 5.8.3 Case E respectively, used all of the bounds ideas we considered in Chapter 5, with theJS model achieving greater overall computational improvement with the application of boundswhen compared with the improvement experienced by the CC model. We also noted that all thebounds algorithms tested on the JS model improved total numbers of branch and bound nodessearched whereas none of the bounds algorithms improved total numbers of branch and boundnodes searched for the CC model. However, Algorithm CC 5.8.3 Case E applied to the CC modelwas the best of all algorithms tested in Chapter 5 for solving the BTCMC problem, and was farsuperior computationally to Algorithm JS 5.8.2 Case E, with the Step-up Method, applied to theJS model. Finally, the Counter model with simple bounds of Chapter 3 outperforms the bestalgorithm of Chapter 5, Algorithm CC 5.8.3 Case E applied to the CC model. Therefore whilstmuch has been achieved with regard to improving the computational efficiency of the JS and CCmodels in Chapter 5, the best of our algorithms and the best model overall, the Counter modelwith simple bounds, still require further improvement to be able to solve the entire range of medicaldata. In other words, even the best exact algorithms and models we have examined in this thesisstill show relatively large computation times for large problem sizes (and in some cases even forsmaller problem sizes). Hence, we must continue to investigate new methods and exact models forsolving the strongly NP-hard leaf sequencing problems.

Chapter 6. Conclusion 175

Future research in this area could perhaps investigate:

1. Similar ideas to those of the relatively successful ‘tailored branch and bound type searchmethods’ currently in the literature, though within the context of exact models able to solve theconstrained case. One could consider implementing variable ordering within the branch and boundtree so that particular variables are chosen to undergo branching before others. For example, thelast shape matrix variable in the JS model (given an upper bound on the number of shape matricesthat can be used in an optimal solution) could be considered one of the ‘most important’ variablesin the model and could be chosen to be branched on first in the branch and bound tree.

2. Equal I facets of the ESP -C1 polytope applied to the JS model in conjunction with the boundswork of Chapter 5 of this thesis.

3. Combining a similar approach to that considered in Taskin et al. [43] to solve the unconstrainedTTT problem, with the bounds ideas we have presented in Chapter 5 of this thesis. One could alsoconsider a row wise approach to the Counter model with simple bounds to solve unconstrainedproblems.

In conclusion, this thesis has presented a comprehensive study of new exact integer programmingmodels for the strongly NP-hard leaf sequencing problems, the Total Treatment Time and Beam-onTime Constrained Minimum Cardinality problems. The methods and models investigated allowthe incorporation of MLC mechanical constraints. This thesis makes a contribution to the widerfield of integer programming through the examination of an interesting substructure of an exactinteger programming model. This thesis has also developed and applied many new and existingtechniques for reducing the computation time of the exact models presented and has demonstratedthat significant computational improvement can be achieved via the application of ‘step-up’ ideas,additional constraints, in particular symmetry breaking constraints and novel bounds on variables.Within the context of exact integer programming formulations, this thesis contributes to the fieldof leaf sequencing for the application of Intensity Modulated Radiation Therapy using MultileafCollimators.


177APPENDIX A

Modified Langer et al. [3] Model

Here we give in detail our modification of the model of Langer et al. [3].

The model focuses on individual units of radiation. It is based on the assumption that the totalbeam-on time is fixed. We take the fixed value to be the best upper bound we know on thebeam-on time in any optimal TTT solution. It is not hard to see that this is given by B =Beammin + T (Kbm −Klb), since any larger value would of necessity have a higher TTT objectivevalue.

What is not known is: for each unit of radiation, what shape matrix should be used for the deliveryof that unit? In the model, binary variables dtij are used to indicate whether the element (i, j) isexposed in the tth shape matrix corresponding to the tth unit of radiation, for t = 1, . . . , B. Theyare linked to the intensity matrix by

Iij =B∑t=1

dtij ,

for all i = 1, . . . ,m, j = 1, . . . , n.

The leaf structure in the shape matrix is captured by binary variables ptij indicating that the rightleaf in row i of shape matrix t covers column j, and `tij indicating that the left leaf in row i ofshape matrix t covers column j, for all t = 1, . . . , B, i = 1, . . . ,m, j = 1, . . . , n. The relationshipbetween these three sets of binary variables is given by

ptij + `tij = 1− dtij ,

for all t = 1, . . . , B, i = 1, . . . ,m, j = 1, . . . , n and

ptij ≤ pti(j+1),

`ti(j+1) ≤ `tij ,

for all t = 1, . . . , B, i = 1, . . . ,m, j = 1, . . . , n− 1. These constraints ensure that the dt induce amatrix which has the strict consecutive-1-property.

To count the number of different shape matrices in a solution, the number of times adjacent shapematrices are different can be recorded; since these are minimised in the objective no two shapematrices that are the same would be non-adjacent in an optimal solution. That shape matrices tand t + 1 differ is reflected in the binary variable gt, where we ask that gB = 1 if and only if Bunits of radiation are required in the solution. (We show how this is enforced later.) [3] tally valuesfor g using additional binary variables: ctij which holds if dtij = 1 and d(t+1)ij = 0, utij whichholds if dtij = 0 and d(t+1)ij = 1, and stij which holds if dtij 6= d(t+1)ij , for all t = 1, . . . , B − 1,i = 1, . . . ,m, and j = 1, . . . , n. The relationship of these variables is established by the linear

178 Appendix A. Modified Langer et al. [3] Model

constraints

− ctij ≤ d(t+1)ij − dtij ≤ utij ,

utij + ctij = stij ,m∑i=1

n∑j=1

stij ≤ mngt,

for all t = 1, . . . , B − 1, i = 1, . . . ,m, j = 1, . . . , n.

Since the beam-on time in the optimal solution may not be as large as B, it may be that someunits of radiation are not needed. We use binary variable zt to indicate whether or not shapematrix t is needed in the solution, and link it to the d variables via

m∑i=1

n∑j=1

dijt ≤ mnzt,

for all t = 1, . . . , B.

To ensure the number of different shape matrices is counted correctly, we also ask that

gB = zB .

Now the number of different shape matrices needed is given byB∑t=1

gt and the total beam-on time

used is given byB∑t=1

zt, so the objective function is:

minB∑t=1

(zt + Tgt).

The original model in [3] does not contain symmetry-breaking constraints. We added symmetrybreaking constraints as follows:

zt ≥ zt+1, ∀ t = 1, . . . , B − 1,

y∑t=1

(gt + 1− zt) ≤2y∑

t=y+1

(gt + 1− zt), ∀ y = 1, . . . ,⌊B

2

⌋,

r+y∑t=r

(gt + 1− zt)− 1 ≤r+2y∑

t=r+y+1

(gt + 1− zt), ∀ r = 1, . . . , B − 2, y = 1, . . . ,⌊B − r

2

⌋.

These are modifications of constraints that appeared in [4], adapted to this situation where not allradiation units may be required: in place of gt in [4] we have gt + 1− zt, and we extend the indexranges.

We also include the redundant constraintB∑t=1

zt ≥ Beammin,

Appendix A. Modified Langer et al. [3] Model 179

for consistency with our TTT model, and since it reduces computation time overall.

180 Appendix A. Modified Langer et al. [3] Model

181APPENDIX B

Additional Facets, Examples and Numerical Results for

Polytopes ESP and ESP -C1

B.1 Facets of ESP of Small Support

B.1.1 Constraint ‘1a-big-x-small-x’ is a Facet of ESP .

Proposition B.1.1. For any s, t = 1, . . . n, s 6= t such that Is < It the constraint

(It − Is)(1− xs) + Isxt − at ≥ 0 (B.1.1)

is facet defining for ESP . We call this constraint the ‘1a-big-x-small-x’ facet of ESP . The ‘1a-big-x-small-x’ facet is a generalisation of the constraints given in Section 4.3.1.7.

Proof. Without loss of generality, we take t = 1 and s = 2. Thus we assume I2 < I1. Constraint(B.1.1) becomes:

(I1 − I2)(1− x2) + I2x1 − a1 ≥ 0. (B.1.2)

First we must check the validity of constraint (B.1.2). Let (x, a) ∈ ESP and define S = {p ∈{1, . . . , n} : xp = 1}. We show that (x, a) satisfies (B.1.2) by checking all possible cases for valuesof x1 and x2 (and hence for a1 and a2).

Case (i): x1 = 0 and x2 = 0. In this case, a1 = a2 = 0. Since I1 ≥ I2 by assumption, (B.1.2) issatisfied.

Case (ii): x1 = 0 and x2 = 1. In this case, a1 = 0. Hence (B.1.2) is satisfied.

Case (iii): x1 = 1 and x2 = 0. In this case, a2 = 0. Since 1 ∈ S and Lemma 4.2.1 gives a1 ≤minp∈S

Ip ≤ I1, then (B.1.2) is satisfied.

Case (iv): x1 = 1 and x2 = 1. In this case, from the definition of ES it must be that a1 = a2.Since 2 ∈ S and Lemma 4.2.1 gives a2 ≤ min

p∈SIp ≤ I2, then (B.1.2) is satisfied.

Case of Equal I’s: For the case that I1 = I2, constraint (B.1.2) becomes I1x1 ≥ a1, which is ofcourse a condition of ESP .

We now show that (B.1.2) defines a facet of ESP . Let F = {(x, a) ∈ ESP : (I1 − I2)(1 − x2) +I2x1 − a1 = 0}. We will show that F is a facet of ESP using Theorem 3.6 in Nemhauser andWolsey [53]. Thus we note that constraint (B.1.2) can be equivalently expressed as

(−I2

I1 − I2e1 + e2,

1I1 − I2

e1

)(x, a) ≤ 1,

182 Appendix B. Additional Facets, Examples and Numerical Results for Polytopes ESP and ESP -C1

since(I1 − I2)− (I1 − I2)x2 + I2x1 − a1 ≥ 0

⇔ −I2x1 + (I1 − I2)x2 + a1 ≤ I1 − I2⇔ −I2

I1 − I2x1 + x2 +

1I1 − I2

a1 ≤ 1

⇔(−I2

I1 − I2e1 + e2,

1I1 − I2

e1

)(x, a) ≤ 1.


(µ, λ) = µ0

(−I2

I1 − I2e1 + e2,

1I1 − I2

e1

)and so are able to conclude that F is a facet. We do this by considering six classes of points in F

and observing what each implies about the values of µ, λ and µ0. Again, the points take the formof those in ES′ and so are easily seen to be in ESP . In each case we assert that they are also inF ; this is readily checked by observation.


2. (e2, I2e2) ∈ F . Now (µ, λ)(e2, I2e2) = µ0, so µ2 + I2λ2 = µ0 and thus λ2 = 0.

3. (e1, I1e1) ∈ F . Now (µ, λ)(e1, I1e1) = µ0, so µ1 + I1λ1 = µ0.

4. (e1 +e2, I2(e1 +e2)) ∈ F . Now (µ, λ)(e1 +e2, I2(e1 +e2)) = µ0, so µ1 +µ2 +I2λ1 +I2λ2 = µ0

and thus µ1 + I2λ1 = 0.

Solving with (3) simultaneously we have: µ0 − I1λ1 + I2λ1 = 0 which yields λ1 =1

I1 − I2µ0

and µ1 =−I2

I1 − I2µ0.

5. (e2 + ep, 0) ∈ F for p = 3, . . . , n. Now for each p ∈ {3, . . . , n} it must be that (µ, λ)(e2 +ep, 0) = µ0 and so µ2 + µp = µ0, implying that µp = 0.

6. (e2 + ep,min(I2, Ip)(e2 + ep)) ∈ F for p = 3, . . . , n. Now for each p ∈ {3, . . . , n} it must bethat (µ, λ)(e2 + ep,min(I2, Ip)(e2 + ep)) = µ0 and so µ2 + µp + min(I2, Ip)(λ2 + λp) = µ0.Hence min(I2, Ip)λp = 0, implying that λp = 0.

Thus (µ, λ) = µ0

(−I2

I1 − I2e1 + e2,

1I1 − I2

e1

)as required.

Therefore the ‘1a-big-x-small-x’ constraint, (B.1.1), is facet defining for ESP .

B.1.2 Constraint ‘3a-any-x’ is a Facet of ESP .

Proposition B.1.2. For any s, t, u = 1, . . . n, s 6= t, t 6= u, s 6= u such that Is < It the constraint

ItIs(1− xu) + Itau − (It − Is)as − Isat ≥ 0 (B.1.3)

is facet defining for ESP . We call this constraint the ‘3a-any-x’ facet of ESP . The ‘3a-any-x’facet is a generalisation of the constraints given in Section 4.3.1.5.

B.1. Facets of ESP of Small Support 183

Proof. Without loss of generality, we take t = 1, s = 2 and u = 3. Thus we assume I2 < I1.Constraint (B.1.3) becomes:

I1I2(1− x3) + I1a3 − (I1 − I2)a2 − I2a1 ≥ 0. (B.1.4)

First we must check the validity of constraint (B.1.4). Let (x, a) ∈ ESP and define S = {p ∈{1, . . . , n} : xp = 1}. We show that (x, a) satisfies (B.1.4) by checking all possible cases for valuesof x1, x2 and x3 (and hence for a1, a2 and a3).

Case (i): x1 = 0, x2 = 0 and x3 = 0. In this case, a1 = a2 = a3 = 0. Since I1 ≥ 0 and I2 ≥ 0 byassumption, (B.1.4) is satisfied.

Case (ii): x1 = 0, x2 = 0 and x3 = 1. In this case, a1 = a2 = 0. Since I1 ≥ 0 by assumption anda3 ≥ 0 by definition, (B.1.4) is satisfied.

Case (iii): x1 = 0, x2 = 1 and x3 = 0. In this case, a1 = a3 = 0. Since I1 ≥ 0 and I2 ≥ 0 byassumption and a2 ≥ 0 by definition, and further, since 2 ∈ S and Lemma 4.2.1 gives a2 ≤minp∈S

Ip ≤ I2, then (B.1.4) is satisfied.

Case (iv): x1 = 1, x2 = 0 and x3 = 0. In this case, a2 = a3 = 0. Since I2 ≥ 0 by assumption andsince 1 ∈ S and Lemma 4.2.1 gives a1 ≤ min


Case (v): x1 = 0, x2 = 1 and x3 = 1. In this case, a1 = 0 and from the definition of ES it must bethat a2 = a3. Since I2 ≥ 0 by assumption and a2 ≥ 0 by definition, then (B.1.4) is satisfied.

Case (vi): x1 = 1, x2 = 0 and x3 = 1. In this case, a2 = 0 and from the definition of ES it mustbe that a1 = a3. Since I1 ≥ I2 by assumption and a1 ≥ 0 by definition, then (B.1.4) is satisfied.

Case (vii): x1 = 1, x2 = 1 and x3 = 0. In this case, a3 = 0 and from the definition of ES it must bethat a1 = a2. Since I1 ≥ 0 by assumption and since 2 ∈ S and Lemma 4.2.1 gives a2 ≤ min

p∈SIp ≤ I2,

then (B.1.4) is satisfied.

Case (viii): x1 = 1, x2 = 1 and x3 = 1. In this case, from the definition of ES it must be thata1 = a2 = a3. Hence (B.1.4) is satisfied.

Case of Equal I’s: For the case that I1 = I2, constraint (B.1.4) becomes I21 (1−x3)+I1a3−I1a1 ≥ 0

which simplifies to I1(1−x3)−a1 +a3 ≥ 0, which corresponds to constraint (4.4.1), the ‘2a-any-x’facet of ESP .

We now show that (B.1.4) defines a facet of ESP . Let F = {(x, a) ∈ ESP : I1I2(1− x3) + I1a3 −(I1 − I2)a2 − I2a1 = 0}. We will show that F is a facet of ESP using Theorem 3.6 in Nemhauserand Wolsey [53]. Thus we note that constraint (B.1.4) can be equivalently expressed as

(e3,

1I1e1 +

(I1 − I2)I1I2

e2 +−1I2e3

)(x, a) ≤ 1,


sinceI1I2 − I1I2x3 + I1a3 − (I1 − I2)a2 − I2a1 ≥ 0

⇔ I1I2x3 − I1a3 + (I1 − I2)a2 + I2a1 ≤ I1I2

⇔ x3 −I1I1I2

a3 +(I1 − I2)I1I2

a2 +I2I1I2

a1 ≤ 1

⇔ x3 −1I2a3 +

(I1 − I2)I1I2

a2 +1I1a1 ≤ 1

⇔(e3,

1I1e1 +

(I1 − I2)I1I2

e2 +−1I2e3

)(x, a) ≤ 1.


(µ, λ) = µ0

(e3,

1I1e1 +

(I1 − I2)I1I2

e2 +−1I2e3

)and so are able to conclude that F is a facet. We do this by considering six classes of points in F

and observing what each implies about the values of µ, λ and µ0. Again, the points take the formof those in ES′ and so are easily seen to be in ESP . In each case we assert that they are also inF ; this is readily checked by observation.


2. (e3 + ep, 0) ∈ F for p = 1, . . . , n, p 6= 3. Now for each p ∈ {1, . . . , n}, p 6= 3, it must be that(µ, λ)(e3 + ep, 0) = µ0 and so µ3 + µp = µ0, implying that µp = 0.

3. (e1, I1e1) ∈ F . Now (µ, λ)(e1, I1e1) = µ0, so µ1 + λ1I1 = µ0 and thus λ1 =1I1µ0.

4. (e1 +e2, I2(e1 +e2)) ∈ F . Now (µ, λ)(e1 +e2, I2(e1 +e2)) = µ0, so µ1 +µ2 +I2(λ1 +λ2) = µ0,

implying thatI2I1µ0 + I2λ2 = µ0 and thus λ2 =

(I1 − I2)I1I2

µ0.

5. (e1 + e2 + e3,min(I2, I3)(e1 + e2 + e3)) ∈ F . Now (µ, λ)(e1 + e2 + e3,min(I2, I3)(e1 + e2 + e3)) = µ0, so µ1 + µ2 + µ3 + min(I2, I3)(λ1 + λ2 + λ3) = µ0, implying that

min(I2, I3)(λ1 + λ2 + λ3) = 0. Therefore we have1I1µ0 +

(I1 − I2)I1I2

µ0 + λ3 = 0 which

simplifies to λ3 = −µ0

(1I1

+(I1 − I2)I1I2

)and further to λ3 =

−1I2µ0.

6. (e1 + e2 + e3 + ep,min(I2, I3, Ip)(e1 + e2 + e3 + ep)) ∈ F for p = 4, . . . , n. Now for eachp ∈ {4, . . . , n} it must be that (µ, λ)(e1 + e2 + e3 + ep,min(I2, I3, Ip)(e1 +e2 +e3 +ep)) = µ0. So µ1 +µ2 +µ3 +µp+ min(I2, I3, Ip)(λ1 +λ2 +λ3 +λp) = µ0 whichsimplifies to min(I2, I3, Ip)(λ1 + λ2 + λ3 + λp) = 0 and further to λ1 + λ2 + λ3 + λp = 0. We

therefore have1I1µ0+

(I1 − I2)I1I2

µ0+−1I2µ0+λp = 0 and hence

(I2 + (I1 − I2)− I1)I1I2

µ0+λp = 0,

implying that λp = 0.

Thus (µ, λ) = µ0

(e3,

1I1e1 +

(I1 − I2)I1I2

e2 +−1I2e3

)as required.

Therefore the ‘3a-any-x’ constraint, (B.1.3), is facet defining for ESP .


B.1.3 Constraint ‘3a-big-x-any-x’ is a Facet of ESP .

Proposition B.1.3. For any s, t, u = 1, . . . , n, s 6= t, t 6= u, s 6= u such that Is < It the constraint

(It − Is)Is(1− xu) + I2sxt − (It − Is)(as − au)− Isat ≥ 0 (B.1.5)

is facet defining for ESP . We call this constraint the ‘3a-big-x-any-x’ facet of ESP . The ‘3a-big-x-any-x’ facet is a generalisation of constraints (4.3.47), (4.3.48), (4.3.49), (4.3.50), (4.3.51),(4.3.52), (4.3.53), (4.3.54), (4.3.55), (4.3.56), (4.3.57) and (4.3.58) given in Section 4.3.1.8.

Proof. Without loss of generality we take t = 1, s = 2 and u = 3. Thus we assume I2 < I1.Constraint (B.1.5) becomes:

(I1 − I2)I2(1− x3) + I22x1 − (I1 − I2)(a2 − a3)− I2a1 ≥ 0. (B.1.6)

First we must check the validity of constraint (B.1.6). Let (x, a) ∈ ESP and define S = {p ∈{1, . . . , n} : xp = 1}. We show that (x, a) satisfies (B.1.6) by checking all possible values of x1, x2

and x3 (and hence for a1, a2 and a3).

Case (i): x1 = 0, x2 = 0 and x3 = 0. In this case, a1 = a2 = a3 = 0. Since I1 ≥ I2 ≥ 0 byassumption, (B.1.6) is satisfied.

Case (ii): x1 = 0, x2 = 0 and x3 = 1. In this case, a1 = a2 = 0. Since I1 ≥ I2 by assumption anda3 ≥ 0 by definition, (B.1.6) is satisfied.

Case (iii): x1 = 0, x2 = 1 and x3 = 0. In this case, a1 = a3 = 0. Since I1 ≥ I2 by assumption andsince 2 ∈ S and Lemma 4.2.1 gives a2 ≤ min




Case (v): x1 = 0, x2 = 1 and x3 = 1. In this case, a1 = 0 and from the definition of ES it must bethat a2 = a3. Hence (B.1.6) is satisfied.

Case (vi): x1 = 1, x2 = 0 and x3 = 1. In this case, a2 = 0 and from the definition of ES it mustbe that a1 = a3. Substituting into (B.1.6) gives I2

2 + (I1 − I2)a1 − I2a1 ≥ 0 and simplifying givesI22 +I1a1−2I2a1 ≥ 0. Rearranging this expression gives I2

2 ≥ a1(2I2−I1). Assume to the contrarythat I2

2 < a1(2I2 − I1) < I1(2I2 − I1), where we have also used that fact that 1 ∈ S and Lemma4.2.1 gives a1 ≤ min

p∈SIp ≤ I1. Rearranging this expression gives 0 < 2I1I2 − I2

1 − I22 which when

simplified gives 0 < −(I1 − I2)2, which is a contradiction. Therefore I22 ≥ a1(2I2 − I1) is true and

(B.1.6) is satisfied.

Case (vii): x1 = 1, x2 = 1 and x3 = 0. In this case, a3 = 0 and from the definition of ES it must bethat a1 = a2. Since I1 ≥ 0 by assumption and since 2 ∈ S and Lemma 4.2.1 gives a2 ≤ min

p∈SIp ≤ I2,


Case (viii): x1 = 1, x2 = 1 and x3 = 1. In this case, from the definition of ES it must be that


a1 = a2 = a3. Since I2 ≥ 0 by assumption and since 2 ∈ S and Lemma 4.2.1 gives a2 ≤ minp∈S

Ip ≤ I2,


Case of Equal I’s: For the case that I1 = I2, constraint (B.1.6) becomes I21x1 − I1a1 ≥ 0 which

simplifies to I1x1 ≥ a1, which is of course a condition of ESP .

We now show that (B.1.6) defines a facet of ESP . Let F = {(x, a) ∈ ESP : (I1 − I2)I2(1− x3) +I22x1− (I1− I2)(a2−a3)− I2a1 = 0}. We will show that F is a facet of ESP using Theorem 3.6 in

Nemhauser and Wolsey [53]. Thus we note that constraint (B.1.6) can be equivalently expressedas

(e3 +

−I2(I1 − I2)

e1,1

(I1 − I2)e1 +

1I2e2 +

−1I2e3

)(x, a) ≤ 1,

since

(I1 − I2)I2 − (I1 − I2)I2x3 + I22x1 − (I1 − I2)a2 + (I1 − I2)a3 − I2a1 ≥ 0

⇔ (I1 − I2)I2x3 − I22x1 + (I1 − I2)a2 − (I1 − I2)a3 + I2a1 ≤ (I1 − I2)I2

⇔ x3 −I2

(I1 − I2)x1 +

1I2a2 −

1I2a3 +

1(I1 − I2)

a1 ≤ 1

⇔(e3 +

−I2(I1 − I2)

e1,1

(I1 − I2)e1 +

1I2e2 +

−1I2e3

)(x, a) ≤ 1.


(µ, λ) = µ0

(−I2

(I1 − I2)e1 + e3,

1(I1 − I2)

e1 +1I2e2 +

−1I2e3

)and so are able to conclude that F is a facet. We do this by considering seven classes of pointsin F and observing what each implies about the values of µ, λ and µ0. Again, the points take theform of those in ES′ and so are easily seen to be in ESP . In each case we assert that they arealso in F ; this is readily checked by observation.


2. (e3 + ep, 0) ∈ F for p = 2, . . . , n, p 6= 3. Now for each p ∈ {2, . . . , n}, p 6= 3 it must be that(µ, λ)(e3 + ep, 0) = µ0 and so µ3 + µp = µ0, implying that µp = 0.

3. (e1, I1e1) ∈ F . Now (µ, λ)(e1, I1e1) = µ0, so µ1 + λ1I1 = µ0.

4. (e1 + e2, I2(e1 + e2)) ∈ F . Now (µ, λ)(e1 + e2, I2(e1 + e2)) = µ0, so µ1 + I2(λ1 + λ2) = µ0.

5. (e2 + e3,min(I2, I3)(e2 + e3)) ∈ F . Now (µ, λ)(e2 + e3,min(I2, I3)(e2 + e3)) = µ0, somin(I2, I3)(λ2 + λ3) = 0 and thus λ2 = −λ3.

6. (e2, I2e2) ∈ F . Now (µ, λ)(e2, I2e2) = µ0, so λ2 =1I2µ0 and therefore λ3 =

−1I2µ0.

Substituting λ2 =1I2µ0 into (4) and simultaneously solving (3) and (4) we have µ1 = −I2λ1

and µ1 = µ0 − I1λ1 which implies λ1 =1

(I1 − I2)µ0 and µ1 =

−I2(I1 − I2)

µ0.


7. (e2 +e3 +ep,min(I2, I3, Ip)(e2 +e3 +ep)) ∈ F for p = 4, . . . , n. Now for each p ∈ {4, . . . , n} itmust be that (µ, λ)(e2+e3+ep,min(I2, I3, Ip)(e2+e3+ep)) = µ0 and so min(I2, I3, Ip)λp = 0,implying that λp = 0.

Thus (µ, λ) = µ0

(−I2

(I1 − I2)e1 + e3,

1(I1 − I2)

e1 +1I2e2 +

−1I2e3

)as required.

Therefore the ‘3a-big-x-any-x’ constraint, (B.1.5), is facet defining for ESP .

B.1.4 Constraint ‘3a-middle-x-small-x’ is a Facet of ESP .

Proposition B.1.4. For any s, t, u = 1, . . . , n, s 6= t, t 6= u, s 6= u such that Is < It < Iu theconstraint

(It − Is)Iu(1− xs) + (Iu − It)Isxt − (Iu − It)at + (It − Is)(as − au) ≥ 0 (B.1.7)

is facet defining for ESP . We call this constraint the ‘3a-middle-x-small-x’ facet of ESP . The‘3a-middle-x-small-x’ facet is a generalisation of constraints (4.3.59), (4.3.60), (4.3.61) and (4.3.62)given in Section 4.3.1.8.

Proof. Without loss of generality we take t = 1, s = 2 and u = 3. Thus we assume I2 < I1 < I3.Constraint (B.1.7) becomes:

(I1 − I2)I3(1− x2) + (I3 − I1)I2x1 − (I3 − I1)a1 + (I1 − I2)(a2 − a3) ≥ 0. (B.1.8)

First we must check the validity of constraint (B.1.8). Let (x, a) ∈ ESP and define S = {p ∈{1, . . . , n} : xp = 1}. We show that (x, a) satisfies (B.1.8) by checking all possible values of x1, x2

and x3 (and hence for a1, a2 and a3).

Case (i): x1 = 0, x2 = 0 and x3 = 0. In this case, a1 = a2 = a3 = 0. Since I1 ≥ I2 and I3 ≥ 0 byassumption, (B.1.8) is satisfied.

Case (ii): x1 = 0, x2 = 0 and x3 = 1. In this case, a1 = a2 = 0. Since I1 ≥ I2 by assumption andsince 3 ∈ S and Lemma 4.2.1 gives a3 ≤ min


Case (iii): x1 = 0, x2 = 1 and x3 = 0. In this case, a1 = a3 = 0. Since I1 ≥ I2 by assumption anda2 ≥ 0 by definition, (B.1.8) is satisfied.

Case (iv): x1 = 1, x2 = 0 and x3 = 0. In this case, a2 = a3 = 0. Substituting into (B.1.8) gives(I1 − I2)I3 + (I3 − I1)I2 − (I3 − I1)a1 ≥ 0. Simplifying and rearranging this expression gives(I3 − I2)I1 ≥ (I3 − I1)a1. Since I3 ≥ I1 ≥ I2 by assumption and since 1 ∈ S and Lemma 4.2.1gives a1 ≤ min

p∈SIp ≤ I1, then (I3 − I2)I1 ≥ (I3 − I1)I1 ≥ (I3 − I1)a1 and (B.1.8) is satisfied.



Case (vi): x1 = 1, x2 = 0 and x3 = 1. In this case, a2 = 0 and from the definition of ES it must bethat a1 = a3. Since I3 ≥ I2 by assumption and since 1 ∈ S and Lemma 4.2.1 gives a1 ≤ min

p∈SIp ≤ I1,


Case (vii): x1 = 1, x2 = 1 and x3 = 0. In this case, a3 = 0 and from the definition of ES it mustbe that a1 = a2. Since I3 ≥ I1 ≥ I2 by assumption and a2 ≥ 0 by definition, and since 2 ∈ S andLemma 4.2.1 gives a2 ≤ min


Case (viii): x1 = 1, x2 = 1 and x3 = 1. In this case, from the definition of ES it must be that a1 =a2 = a3. Since I3 ≥ I1 by assumption and since 2 ∈ S and Lemma 4.2.1 gives a2 ≤ min

p∈SIp ≤ I2,


Cases of Equal I’s

Case (a): I1 = I2, I1 < I3. For the case that I1 = I2, I1 < I3, constraint (B.1.8) becomes (I3 −I1)I1x1 − (I3 − I1)a1 ≥ 0 which simplifies to (I3 − I1)(I1x1 − a1) ≥ 0 or I1x1 ≥ a1, which is ofcourse a condition of ESP .

Case (b): I1 = I3, I1 > I2. For the case that I1 = I3, I1 > I2, constraint (B.1.8) becomes(I1 − I2)I3(1− x2) + (I1 − I2)(a2 − a3) ≥ 0 which simplifies to (I1 − I2)(I3(1− x2)− a3 + a2) ≥ 0or I3(1− x2)− a3 + a2 ≥ 0, which corresponds to constraint (4.4.1), the ‘2a-any-x’ facet of ESP .

Case (c): I3 = I2 = I1. For the case that I3 = I2 = I1, constraint (B.1.8) reduces to 0 ≥ 0.

We now show that (B.1.8) defines a facet of ESP . Let F = {(x, a) ∈ ESP : (I1 − I2)I3(1− x2) +(I3 − I1)I2x1 − (I3 − I1)a1 + (I1 − I2)(a2 − a3) = 0}. We will show that F is a facet of ESPusing Theorem 3.6 in Nemhauser and Wolsey [53]. Thus we note that constraint (B.1.8) can beequivalently expressed as

(−(I3 − I1)I2(I1 − I2)I3

e1 + e2,(I3 − I1)

(I1 − I2)I3e1 +

−1I3e2 +

1I3e3

)(x, a) ≤ 1,

since

(I1 − I2)I3 − (I1 − I2)I3x2 + (I3 − I1)I2x1 − (I3 − I1)a1 + (I1 − I2)(a2 − a3) ≥ 0⇔ (I1 − I2)I3x2 − (I3 − I1)I2x1 + (I3 − I1)a1 − (I1 − I2)(a2 − a3) ≤ (I1 − I2)I3

⇔ x2 −(I3 − I1)I2(I1 − I2)I3

x1 +(I3 − I1)

(I1 − I2)I3a1 −

1I3a2 +

1I3a3 ≤ 1

⇔(−(I3 − I1)I2(I1 − I2)I3

e1 + e2,(I3 − I1)

(I1 − I2)I3e1 +

−1I3e2 +

1I3e3

)(x, a) ≤ 1.


(µ, λ) = µ0

(−(I3 − I1)I2(I1 − I2)I3

e1 + e2,(I3 − I1)

(I1 − I2)I3e1 +

−1I3e2 +

1I3e3

)and so are able to conclude that F is a facet. We do this by considering seven classes of pointsin F and observing what each implies about the values of µ, λ and µ0. Again, the points take the


form of those in ES′ and so are easily seen to be in ESP . In each case we assert that they arealso in F ; this is readily checked by observation.


2. (e2 + ep, 0) ∈ F for p = 3, . . . , n. Now for each p ∈ {3, . . . , n} it must be that (µ, λ)(e2 +ep, 0) = µ0 and so µ2 + µp = µ0, implying that µp = 0.

3. (e3, I3e3) ∈ F . Now (µ, λ)(e3, I3e3) = µ0, so µ3 + I3λ3 = µ0 and thus λ3 =1I3µ0.

4. (e2 +e3, I2(e2 +e3)) ∈ F . Now (µ, λ)(e2 +e3, I2(e2 +e3)) = µ0, so µ2 +µ3 + I2(λ2 +λ3) = µ0

and thus λ2 = −λ3 =−1I3µ0.


and thus µ1 + I1λ1 +I1I3µ0 = µ0, implying that µ1 + I1λ1 =

(I3 − I1)I3

µ0.

6. (e1 + e2 + e3, I2(e1 + e2 + e3)) ∈ F . Now (µ, λ)(e1 + e2 + e3, I2(e1 + e2 + e3)) = µ0, soµ1 + µ2 + µ3 + I2(λ1 + λ2 + λ3) = µ0 and thus µ1 = −I2λ1.

Substituting (6) into (5) yields λ1(I1 − I2) =(I3 − I1)

I3µ0

or λ1 =(I3 − I1)I3(I1 − I2)

µ0, and µ1 =−I2(I3 − I1)I3(I1 − I2)

µ0.

7. (e2 + e3 + ep,min(I2, Ip)(e2 + e3 + ep)) ∈ F for p = 4, . . . , n. Now for each p ∈ {4, . . . , n}it must be that (µ, λ)(e2 + e3 + ep,min(I2, Ip)(e2 + e3 + ep)) = µ0 and so µ2 + µ3 + µp +min(I2, Ip)(λ2 + λ3 + λp) = µ0, implying that min(I2, Ip)λp = 0 or λp = 0.

Thus (µ, λ) = µ0

(−I2(I3 − I1)I3(I1 − I2)

e1 + e2,(I3 − I1)I3(I1 − I2)

e1 +−1I3e2 +

1I3e3

)as required.

Therefore the ‘3a-middle-x-small-x’ constraint, (B.1.7), is facet defining for ESP .

B.1.5 Constraint ‘3a-middle-x-big-x’ is a Facet of ESP .


(It − Is)It(1− xu) + ItIsxt − Itat − (It − Is)(as − au) ≥ 0 (B.1.9)

is facet defining for ESP . We call this constraint the ‘3a-middle-x-big-x’ facet of ESP . The‘3a-middle-x-big-x’ facet is a generalisation of constraints (4.3.63), (4.3.64), (4.3.65) and (4.3.66)given in Section 4.3.1.8.


(I1 − I2)I1(1− x3) + I1I2x1 − I1a1 − (I1 − I2)(a2 − a3) ≥ 0. (B.1.10)


First we must check the validity of constraint (B.1.10). Let (x, a) ∈ ESP and define S = {p ∈{1, . . . , n} : xp = 1}. We show that (x, a) satisfies (B.1.10) by checking all possible values of x1,x2 and x3 (and hence for a1, a2 and a3).

Case (i): x1 = 0, x2 = 0 and x3 = 0. In this case, a1 = a2 = a3 = 0. Since I1 ≥ I2 ≥ 0 byassumption, (B.1.10) is satisfied.

Case (ii): x1 = 0, x2 = 0 and x3 = 1. In this case, a1 = a2 = 0. Since I1 ≥ I2 by assumption anda3 ≥ 0 by definition, then (B.1.10) is satisfied.

Case (iii): x1 = 0, x2 = 1 and x3 = 0. In this case, a1 = a3 = 0. Since I1 ≥ I2 by assumption andsince 2 ∈ S and Lemma 4.2.1 gives a2 ≤ min





Case (vi): x1 = 1, x2 = 0 and x3 = 1. In this case, a2 = 0 and from the definition of ES it must bethat a1 = a3. Since I2 ≥ 0 by assumption and since 1 ∈ S and Lemma 4.2.1 gives a1 ≤ min

p∈SIp ≤ I1,


Case (vii): x1 = 1, x2 = 1 and x3 = 0. In this case, a3 = 0 and from the definition of ES it mustbe that a1 = a2. Substituting into (B.1.10) gives (I1 − I2)I1 + I1I2 − I1a2 − (I1 − I2)a2 ≥ 0.Simplifying this expression gives I2

1 − I1a2− (I1− I2)a2 ≥ 0. Rearranging gives I21 ≥ 2I1a2− I2a2,

which simplifies to I21 ≥ a2(2I1− I2). Assume to the contrary that I2

1 < a2(2I1− I2) < I2(2I1− I2)where we have also used that fact that 2 ∈ S and Lemma 4.2.1 gives a2 ≤ min

p∈SIp ≤ I2. Rearranging

this expression gives 0 < 2I1I2 − I22 − I2

1 which when simplified gives 0 < −(I1 − I2)2, which is acontradiction. Therefore I2

1 ≥ a2(2I1 − I2) is true and (B.1.10) is satisfied.

Case (viii): x1 = 1, x2 = 1 and x3 = 1. In this case, from the definition of ES it must be thata1 = a2 = a3. Since I1 ≥ 0 by assumption and since 2 ∈ S and Lemma 4.2.1 gives a2 ≤ min

p∈SIp ≤ I2,


Case of Equal I’s: For the case that I1 = I2, I1 < I3, constraint (B.1.10) becomes I21x1 − I1a1 ≥ 0

which simplifies to I1x1 ≥ a1, which is of course a condition of ESP .

We now show that (B.1.10) defines a facet of ESP . Let F = {(x, a) ∈ ESP : (I1− I2)I1(1− x3) +I1I2x1−I1a1−(I1−I2)(a2−a3) = 0}. We will show that F is a facet of ESP using Theorem 3.6 inNemhauser and Wolsey [53]. Thus we note that constraint (B.1.10) can be equivalently expressedas

(−I2

(I1 − I2)e1 + e3,

1(I1 − I2)

e1 +1I1e2 +

−1I1e3

)(x, a) ≤ 1,


since

(I1 − I2)I1 − (I1 − I2)I1x3 + I1I2x1 − I1a1 − (I1 − I2)(a2 − a3) ≥ 0⇔ (I1 − I2)I1x3 − I1I2x1 + I1a1 + (I1 − I2)(a2 − a3) ≤ (I1 − I2)I1

⇔ x3 −I2

(I1 − I2)x1 +

1(I1 − I2)

a1 +1I1a2 −

1I1a3 ≤ 1

⇔(−I2

(I1 − I2)e1 + e3,

1(I1 − I2)

e1 +1I1e2 +

−1I1e3

)(x, a) ≤ 1.


(µ, λ) = µ0

(−I2

(I1 − I2)e1 + e3,

1(I1 − I2)

e1 +1I1e2 +

−1I1e3




3. (e1, I1e1) ∈ F . Now (µ, λ)(e1, I1e1) = µ0, so µ1 + λ1I1 = µ0.


and thus µ1 + I1(λ1 + λ3) = 0.


and thus I2(λ2 + λ3) = 0, implying that λ2 = −λ3.

6. (e1 + e2 + e3, I2(e1 + e2 + e3)) ∈ F . Now (µ, λ)(e1 + e2 + e3, I2(e1 + e2 + e3)) = µ0, soµ1 + µ2 + µ3 + I2(λ1 + λ2 + λ3) = µ0 and thus µ1 + I2λ1 = 0.Simultaneously solving (3) and (6) yields µ0 − λ1I1 + I2λ1 = 0 or −λ1(I1 − I2) = −µ0,

implying that λ1 =1

(I1 − I2)µ0, and µ1 = µ0 −

I1(I1 − I2)

µ0 =−I2

(I1 − I2)µ0. Substituting µ1

and λ1 into (4) yields−I2

(I1 − I2)µ0+

I1(I1 − I2)

µ0 = −I1λ3, implying that λ3 =−1I1µ0 and (5) yields λ2 =

1I1µ0.

7. (e2 + e3 + ep,min(I2, Ip)(e2 + e3 + ep)) ∈ F for p = 4, . . . , n. Now for each p ∈ {4, . . . , n}it must be that (µ, λ)(e2 + e3 + ep,min(I2, Ip)(e2 + e3 + ep)) = µ0 and so µ2 + µ3 + µp +min(I2, Ip)(λ2 + λ3 + λp) = µ0, implying that λp = 0.

Thus (µ, λ) = µ0

(−I2

(I1 − I2)e1 + e3,

1(I1 − I2)

e1 +1I1e2 +

−1I1e3

)as required.

Therefore the ‘3a-middle-x-big-x’ constraint, (B.1.9), is facet defining for ESP .


B.1.6 Constraint ‘3a-big-x-small-x’ is a Facet of ESP .


(Iu − Is)It(1− xs) + ItIsxu − (Iu − It)(at − as)− Itau ≥ 0 (B.1.11)

is facet defining for ESP . We call this constraint the ‘3a-big-x-small-x’ facet of ESP . The ‘3a-big-x-small-x’ facet is a generalisation of constraints (4.3.67), (4.3.68), (4.3.69) and (4.3.70) givenin Section 4.3.1.8.


(I3 − I2)I1(1− x2) + I1I2x3 − (I3 − I1)(a1 − a2)− I1a3 ≥ 0. (B.1.12)



Case (ii): x1 = 0, x2 = 0 and x3 = 1. In this case, a1 = a2 = 0. Since I1 ≥ 0 by assumption andsince 3 ∈ S and Lemma 4.2.1 gives a3 ≤ min


Case (iii): x1 = 0, x2 = 1 and x3 = 0. In this case, a1 = a3 = 0. Since I3 ≥ I1 by assumption anda2 ≥ 0 by definition, then (B.1.12) is satisfied.

Case (iv): x1 = 1, x2 = 0 and x3 = 0. In this case, a2 = a3 = 0. Substituting into (B.1.12) gives(I3 − I2)I1 − (I3 − I1)a1 ≥ 0. Rearranging this expression gives (I3 − I2)I1 ≥ (I3 − I1)a1. SinceI3 ≥ I1 ≥ I2 by assumption and since 1 ∈ S and Lemma 4.2.1 gives a1 ≤ min

p∈SIp ≤ I1, then

(I3 − I2)I1 ≥ (I3 − I1)I1 ≥ (I3 − I1)a1 and (B.1.12) is satisfied.

Case (v): x1 = 0, x2 = 1 and x3 = 1. In this case, a1 = 0 and from the definition of ES it mustbe that a2 = a3. Since I3 ≥ I1 ≥ 0 by assumption and a2 ≥ 0 by definition, and since 2 ∈ S andLemma 4.2.1 gives a2 ≤ min



p∈SIp ≤ I1,


Case (vii): x1 = 1, x2 = 1 and x3 = 0. In this case, a3 = 0 and from the definition of ES it mustbe that a1 = a2. Hence (B.1.12) is satisfied.


p∈SIp ≤ I2,



Cases of Equal I’s:

Case (a): I1 = I2, I1 < I3. For the case that I1 = I2, I1 < I3, constraint (B.1.12) becomes(I3 − I1)I1(1− x2) + I2

1x3 − (I3 − I1)(a1 − a2)− I1a3 ≥ 0 which simplifies to

(I3 − I1)(I1(1− x2)− a1 + a2) + I1(I1x3 − a3) ≥ 0. (B.1.13)

We now check the validity of constraint (B.1.13). We show that (x, a) satisfies (B.1.13) by checkingall possible values of x1, x2 and x3 (and hence for a1, a2 and a3).

Case (a) (i): x1 = 0, x2 = 0 and x3 = 0. In this case, a1 = a2 = a3 = 0. Since I3 ≥ I1 ≥ 0 byassumption, (B.1.13) is satisfied.

Case (a) (ii): x1 = 0, x2 = 0 and x3 = 1. In this case, a1 = a2 = 0. Since I1 ≥ 0 by assumptionand since 3 ∈ S and Lemma 4.2.1 gives a3 ≤ min


Case (a) (iii): x1 = 0, x2 = 1 and x3 = 0. In this case, a1 = a3 = 0. Since I3 ≥ I1 by assumptionand a2 ≥ 0 by definition, then (B.1.13) is satisfied.

Case (a) (iv): x1 = 1, x2 = 0 and x3 = 0. In this case, a2 = a3 = 0. Since I3 ≥ I1 by assumptionand since 1 ∈ S and Lemma 4.2.1 gives a1 ≤ min


Case (a) (v): x1 = 0, x2 = 1 and x3 = 1. In this case, a1 = 0 and from the definition of ES it mustbe that a2 = a3. Substituting into (B.1.13) gives (I3 − I1)a2 + I1(I1 − a2) ≥ 0. Simplifying thisexpression gives I2

1 ≥ a2(2I1 − I3). Assume to the contrary that I21 < a2(2I1 − I3) < I3(2I1 − I3)

where we have also used that fact that 2 ∈ S and Lemma 4.2.1 gives a2 ≤ minp∈S

Ip ≤ I2, and I3 ≥ I2by assumption. Rearranging this expression gives 0 < 2I1I3 − I2

3 − I21 which when simplified gives

0 < −(I1 − I3)2, which is a contradiction. Therefore I21 ≥ a2(2I1 − I3) is true and (B.1.13) is

satisfied.

Case (a) (vi): x1 = 1, x2 = 0 and x3 = 1. In this case, a2 = 0 and from the definition of ES itmust be that a1 = a3. Since I3 ≥ 0 by assumption and since 1 ∈ S and Lemma 4.2.1 givesa1 ≤ min


Case (a) (vii): x1 = 1, x2 = 1 and x3 = 0. In this case, a3 = 0 and from the definition of ES itmust be that a1 = a2. Hence (B.1.13) is satisfied.

Case (a) (viii): x1 = 1, x2 = 1 and x3 = 1. In this case, from the definition of ES it must be thata1 = a2 = a3. Since I1 ≥ 0 by assumption and since 1 ∈ S and Lemma 4.2.1 gives a1 ≤ min

p∈SIp ≤ I1,


Case (b): I1 = I3, I1 > I2. For the case that I1 = I3, I1 > I2, constraint (B.1.12) becomes

(I1 − I2)(1− x2) + I2x3 − a3 ≥ 0. (B.1.14)



Case (b) (i): x1 = 0, x2 = 0 and x3 = 0. In this case, a1 = a2 = a3 = 0. Since I1 ≥ I2 by assump-tion, (B.1.14) is satisfied.

Case (b) (ii): x1 = 0, x2 = 0 and x3 = 1. In this case, a1 = a2 = 0. Since 3 ∈ S and Lemma 4.2.1gives a3 ≤ min

p∈SIp ≤ I3, and I3 = I1, then (B.1.14) is satisfied.

Case (b) (iii): x1 = 0, x2 = 1 and x3 = 0. In this case, a1 = a3 = 0. Hence (B.1.14) is satisfied.

Case (b) (iv): x1 = 1, x2 = 0 and x3 = 0. In this case, a2 = a3 = 0. Since I1 ≥ I2 by assumption,then (B.1.14) is satisfied.

Case (b) (v): x1 = 0, x2 = 1 and x3 = 1. In this case, a1 = 0 and from the definition of ES it mustbe that a2 = a3. Since 2 ∈ S and Lemma 4.2.1 gives a2 ≤ min


Case (b) (vi): x1 = 1, x2 = 0 and x3 = 1. In this case, a2 = 0 and from the definition of ES itmust be that a1 = a3. Since 1 ∈ S and Lemma 4.2.1 gives a1 ≤ min

p∈SIp ≤ I1, then (B.1.14) is

satisfied.

Case (b) (vii): x1 = 1, x2 = 1 and x3 = 0. In this case, a3 = 0 and from the definition of ES itmust be that a1 = a2. Hence (B.1.14) is satisfied.

Case (b) (viii): x1 = 1, x2 = 1 and x3 = 1. In this case, from the definition of ES it must be thata1 = a2 = a3. Since 2 ∈ S and Lemma 4.2.1 gives a2 ≤ min


Case (c): I3 = I2 = I1. For the case that I3 = I2 = I1, constraint (B.1.12) becomes I23x3−I3a3 ≥ 0


We now show that (B.1.12) defines a facet of ESP . Let F = {(x, a) ∈ ESP : (I3− I2)I1(1− x2) +I1I2x3−(I3−I1)(a1−a2)−I1a3 = 0}. We will show that F is a facet of ESP using Theorem 3.6 inNemhauser and Wolsey [53]. Thus we note that constraint (B.1.12) can be equivalently expressedas

(e2 −

I2(I3 − I2)

e3,(I3 − I1)

(I3 − I2)I1e1 −

(I3 − I1)(I3 − I2)I1

e2 +1

(I3 − I2)e3

)(x, a) ≤ 1,

since

(I3 − I2)I1 − (I3 − I2)I1x2 + I1I2x3 − (I3 − I1)(a1 − a2)− I1a3 ≥ 0⇔ (I3 − I2)I1x2 − I1I2x3 + (I3 − I1)(a1 − a2) + I1a3 ≤ (I3 − I2)I1

⇔ x2 −I2

(I3 − I2)x3 +

(I3 − I1)(I3 − I2)I1

a1 −(I3 − I1)

(I3 − I2)I1a2 +

1(I3 − I2)

a3 ≤ 1

⇔(e2 −

I2(I3 − I2)

e3,(I3 − I1)

(I3 − I2)I1e1 −

(I3 − I1)(I3 − I2)I1

e2 +1

(I3 − I2)e3

)(x, a) ≤ 1.


(µ, λ) = µ0

(e2 −

I2(I3 − I2)

e3,(I3 − I1)

(I3 − I2)I1e1 −

(I3 − I1)(I3 − I2)I1

e2 +1

(I3 − I2)e3

)


and so are able to conclude that F is a facet. We do this by considering seven classes of pointsin F and observing what each implies about the values of µ, λ and µ0. Again, the points take theform of those in ES′ and so are easily seen to be in ESP . In each case we assert that they arealso in F ; this is readily checked by observation.







and thus µ3 + I1(λ1 + λ3) = µ0.

6. (e1 + e2 + e3, I2(e1 + e2 + e3)) ∈ F . Now (µ, λ)(e1 + e2 + e3, I2(e1 + e2 + e3)) = µ0, soµ1 + µ2 + µ3 + I2(λ1 + λ2 + λ3) = µ0 and thus µ3 + I2λ3 = 0.

Simultaneously solving (3) and (6) yields −I2λ3 + I3λ3 = µ0 or λ3 =1

(I3 − I2)µ0, and µ3 =

−I2(I3 − I2)

µ0. Substituting µ3 and λ3 into (5) yields−I2

(I3 − I2)µ0 + I1λ1 +

I1(I3 − I2)

µ0 = µ0

which simplifies to λ1 =I3 − I2 − I1 + I2

(I3 − I2)I1µ0 or λ1 =

(I3 − I1)(I3 − I2)I1

µ0. Then (4) yields

λ2 =−(I3 − I1)(I3 − I2)I1

µ0

7. (e1 + e2 + ep,min(I2, Ip)(e1 + e2 + ep)) ∈ F for p = 4, . . . , n. Now for each p ∈ {4, . . . , n}it must be that (µ, λ)(e1 + e2 + ep,min(I2, Ip)(e1 + e2 + ep)) = µ0 and so µ1 + µ2 + µp +min(I2, Ip)(λ1 + λ2 + λp) = µ0 and hence λp = 0.

Thus (µ, λ) = µ0

(e2 +

−I2(I3 − I2)

e3,(I3 − I1)I1(I3 − I2)

e1 +−(I3 − I1)I1(I3 − I2)

e2 +1

(I3 − I2)e3

)as required.

For completeness, referring back to Case (a), where I2 = I1 and I1 < I3, constraint (B.1.12)becomes constraint (B.1.13). Rearranging constraint (B.1.13) we obtain(

e2 +−I1

(I3 − I1)e3,

1I1e1 +

−1I1e2 +

1(I3 − I1)

e3

)(x, a) ≤ 1.

That constraint (B.1.13) is a facet of ESP follows directly from our expression for (µ, λ) withI2 = I1.

Again, referring back to Case (b), where I3 = I1 and I1 > I2, constraint (B.1.12) becomes constraint(B.1.14). Rearranging constraint (B.1.14) we obtain(

e2 +−I2

(I1 − I2)e3,

1(I1 − I2)

e3

)(x, a) ≤ 1.



Therefore the ‘3a-big-x-small-x’ constraint, (B.1.11), is facet defining for ESP .

B.1.7 Constraint ‘3a-big-x-middle-x’ is a Facet of ESP .


(Iu − Is)It(1− xt) + ItIsxu + (It − Is)(at − as)− Itau ≥ 0 (B.1.15)

is facet defining for ESP . We call this constraint the ‘3a-big-x-middle-x’ facet of ESP . The‘3a-big-x-middle-x’ facet is a generalisation of constraints (4.3.71), (4.3.72), (4.3.73) and (4.3.74)given in Section 4.3.1.8.


(I3 − I2)I1(1− x1) + I1I2x3 + (I1 − I2)(a1 − a2)− I1a3 ≥ 0. (B.1.16)



Case (ii): x1 = 0, x2 = 0 and x3 = 1. In this case, a1 = a2 = 0. Since I1 ≥ 0 by assumption andsince 3 ∈ S and Lemma 4.2.1 gives a3 ≤ min


Case (iii): x1 = 0, x2 = 1 and x3 = 0. In this case, a1 = a3 = 0. Substituting into (B.1.16) gives(I3 − I2)I1 − (I1 − I2)a2 ≥ 0. Rearranging this expression gives (I3 − I2)I1 ≥ (I1 − I2)a2. SinceI3 ≥ I1 ≥ I2 by assumption and since 2 ∈ S and Lemma 4.2.1 gives a2 ≤ min


(I3 − I2)I1 ≥ (I1 − I2)I1 ≥ (I1 − I2)I2 ≥ (I1 − I2)a2 and (B.1.16) is satisfied.

Case (iv): x1 = 1, x2 = 0 and x3 = 0. In this case, a2 = a3 = 0. Since I1 ≥ I2 by assumption anda1 ≥ 0 by definition, then (B.1.16) is satisfied.

Case (v): x1 = 0, x2 = 1 and x3 = 1. In this case, a1 = 0 and from the definition of ES it must bethat a2 = a3. Substituting into (B.1.16) gives (I3−I2)I1 +I1I2−(I1−I2)a2−I1a2 ≥ 0. Simplifyingthis expression gives I3I1 − (I1 − I2)a2 − I1a2 ≥ 0 and rearranging gives I1(I3 − a2) ≥ (I1 − I2)a2.Since I3 ≥ I1 ≥ I2 by assumption and since 2 ∈ S and Lemma 4.2.1 gives a2 ≤ min


I1(I3 − a2) ≥ I2(I3 − a2) ≥ I2(I1 − a2) ≥ I2(I1 − I2) ≥ a2(I1 − I2) and (B.1.16) is satisfied.


p∈SIp ≤ I1,


Case (vii): x1 = 1, x2 = 1 and x3 = 0. In this case, a3 = 0 and from the definition of ES it mustbe that a1 = a2. Hence (B.1.16) is satisfied.



p∈SIp ≤ I2,


Cases of Equal I’s:

Case (a): I1 = I2, I1 < I3. For the case that I1 = I2, I1 < I3, constraint (B.1.16) becomes

(I3 − I1)(1− x1) + I1x3 − a3 ≥ 0. (B.1.17)


Case (a) (i): x1 = 0, x2 = 0 and x3 = 0. In this case, a1 = a2 = a3 = 0. Since I3 ≥ I1 by assump-tion, (B.1.17) is satisfied.

Case (a) (ii): x1 = 0, x2 = 0 and x3 = 1. In this case, a1 = a2 = 0. Since 3 ∈ S and Lemma 4.2.1gives a3 ≤ min


Case (a) (iii): x1 = 0, x2 = 1 and x3 = 0. In this case, a1 = a3 = 0. Since I3 ≥ I1 by assumption,then (B.1.17) is satisfied.

Case (a) (iv): x1 = 1, x2 = 0 and x3 = 0. In this case, a2 = a3 = 0. Hence (B.1.17) is satisfied.

Case (a) (v): x1 = 0, x2 = 1 and x3 = 1. In this case, a1 = 0 and from the definition of ES it mustbe that a2 = a3. Since 3 ∈ S and Lemma 4.2.1 gives a3 ≤ min


Case (a) (vi): x1 = 1, x2 = 0 and x3 = 1. In this case, a2 = 0 and from the definition of ES itmust be that a1 = a3. Since 1 ∈ S and Lemma 4.2.1 gives a1 ≤ min

p∈SIp ≤ I1, then (B.1.17) is

satisfied.

Case (a) (vii): x1 = 1, x2 = 1 and x3 = 0. In this case, a3 = 0 and from the definition of ES itmust be that a1 = a2. Hence (B.1.17) is satisfied.

Case (a) (viii): x1 = 1, x2 = 1 and x3 = 1. In this case, from the definition of ES it must be thata1 = a2 = a3. Since 1 ∈ S and Lemma 4.2.1 gives a1 ≤ min


Case (b): I1 = I3, I1 > I2. For the case that I1 = I3, I1 > I2, constraint (B.1.16) becomes(I1 − I2)I1(1− x1) + I1I2x3 + (I1 − I2)(a1 − a2)− I1a3 ≥ 0 which simplifies to

(I1 − I2)(I1(1− x1) + a1 − a2) + I1(I2x3 − a3) ≥ 0. (B.1.18)


Case (b) (i): x1 = 0, x2 = 0 and x3 = 0. In this case, a1 = a2 = a3 = 0. Since I1 ≥ I2 ≥ 0 byassumption, (B.1.18) is satisfied.


Case (b) (ii): x1 = 0, x2 = 0 and x3 = 1. In this case, a1 = a2 = 0. Since I1 ≥ 0 by assumptionand since 3 ∈ S and Lemma 4.2.1 gives a3 ≤ min

p∈SIp ≤ I3, and I3 = I1, then (B.1.18) is satisfied.

Case (b) (iii): x1 = 0, x2 = 1 and x3 = 0. In this case, a1 = a3 = 0. Since I1 ≥ I2 by assumptionand since 2 ∈ S and Lemma 4.2.1 gives a2 ≤ min


Case (b) (iv): x1 = 1, x2 = 0 and x3 = 0. In this case, a2 = a3 = 0. Since I1 ≥ I2 by assumptionand a1 ≥ 0 by definition, then (B.1.18) is satisfied.

Case (b) (v): x1 = 0, x2 = 1 and x3 = 1. In this case, a1 = 0 and from the definition of ES it mustbe that a2 = a3. Substituting into (B.1.18) gives (I1 − I2)I1 + I1I2 − (I1 − I2)a2 − I1a2 ≥ 0 whichsimplifies to I2

1 + a2(I2 − 2I1) ≥ 0. Rearranging this expression gives I21 ≥ a2(2I1 − I2). Assume

to the contrary that I21 < a2(2I1− I2) < I2(2I1− I2) where we have also used that fact that 2 ∈ S

and Lemma 4.2.1 gives a2 ≤ minp∈S

Ip ≤ I2. Rearranging this expression gives 0 < 2I1I2 − I22 − I2

1

which when simplified gives 0 < −(I1− I2)2, which is a contradiction. Therefore I21 ≥ a2(2I1− I2)

is true and (B.1.18) is satisfied.

Case (b) (vi): x1 = 1, x2 = 0 and x3 = 1. In this case, a2 = 0 and from the definition of ES itmust be that a1 = a3. Since I2 ≥ 0 by assumption and since 1 ∈ S and Lemma 4.2.1 givesa1 ≤ min


Case (b) (vii): x1 = 1, x2 = 1 and x3 = 0. In this case, a3 = 0 and from the definition of ES itmust be that a1 = a2. Hence (B.1.18) is satisfied.

Case (b) (viii): x1 = 1, x2 = 1 and x3 = 1. In this case, from the definition of ES it must be thata1 = a2 = a3. Since I1 ≥ 0 by assumption and since 2 ∈ S and Lemma 4.2.1 gives a2 ≤ min

p∈SIp ≤ I2,


Case (c): I3 = I2 = I1. For the case that I3 = I2 = I1, constraint (B.1.16) becomes I23x3−I3a3 ≥ 0


We now show that (B.1.16) defines a facet of ESP . Let F = {(x, a) ∈ ESP : (I3− I2)I1(1− x1) +I1I2x3 +(I1−I2)(a1−a2)−I1a3 = 0}. We will show that F is a facet of ESP using Theorem 3.6 inNemhauser and Wolsey [53]. Thus we note that constraint (B.1.16) can be equivalently expressedas

(e1 +

−I2(I3 − I2)

e3,−(I1 − I2)I1(I3 − I2)

e1 +(I1 − I2)I1(I3 − I2)

e2 +1

(I3 − I2)e3

)(x, a) ≤ 1,

since

(I3 − I2)I1 − (I3 − I2)I1x1 + I1I2x3 + (I1 − I2)(a1 − a2)− I1a3 ≥ 0⇔ (I3 − I2)I1x1 − I1I2x3 − (I1 − I2)(a1 − a2) + I1a3 ≤ (I3 − I2)I1

⇔ x1 +−I2

(I3 − I2)x3 +

−(I1 − I2)I1(I3 − I2)

a1 +(I1 − I2)I1(I3 − I2)

a2 +1

(I3 − I2)a3 ≤ 1

⇔(e1 +

−I2(I3 − I2)

e3,−(I1 − I2)I1(I3 − I2)

e1 +(I1 − I2)I1(I3 − I2)

e2 +1

(I3 − I2)e3

)(x, a) ≤ 1.



(µ, λ) = µ0

(e1 +

−I2(I3 − I2)

e3,−(I1 − I2)I1(I3 − I2)

e1 +(I1 − I2)I1(I3 − I2)

e2 +1

(I3 − I2)e3








and thus µ3 + I1(λ1 + λ3) = 0.

6. (e1 + e2 + e3, I2(e1 + e2 + e3)) ∈ F . Now (µ, λ)(e1 + e2 + e3, I2(e1 + e2 + e3)) = µ0, soµ1 + µ2 + µ3 + I2(λ1 + λ2 + λ3) = µ0 and thus µ3 + I2λ3 = 0.

Simultaneously solving (3) and (6) yields µ0−I3λ3+I2λ3 = 0 which implies λ3 =1

(I3 − I2)µ0,

and µ3 =−I2

(I3 − I2)µ0. Substituting µ3 and λ3 into (5) yields I1λ1+

I1(I3 − I2)

µ0 =I2

(I3 − I2)µ0,

which simplifies to I1(I3 − I2)λ1 + I1µ0 = I2µ0 and hence λ1 =−(I1 − I2)I1(I3 − I2)

µ0 and by (4),

λ2 =(I1 − I2)I1(I3 − I2)

µ0.

7. (e1 + e2 + ep,min(I2, Ip)(e1 + e2 + ep)) ∈ F for p = 4, . . . , n. Now for each p ∈ {4, . . . , n}it must be that (µ, λ)(e1 + e2 + ep,min(I2, Ip)(e1 + e2 + ep)) = µ0 and so µ1 + µ2 + µp +min(I2, Ip)(λ1 + λ2 + λp) = µ0, implying that λp = 0.

Thus (µ, λ) = µ0

(e1 +

−I2(I3 − I2)

e3,−(I1 − I2)I1(I3 − I2)

e1 +(I1 − I2)I1(I3 − I2)

e2 +1

(I3 − I2)e3

)as required.

For completeness, referring back to Case (a), where I2 = I1 and I1 < I3, constraint (B.1.16)becomes constraint (B.1.17). Rearranging constraint (B.1.17) we obtain(

e1 +−I1

(I3 − I1)e3,

1(I3 − I1)

e3

)(x, a) ≤ 1.



Again, referring back to Case (b), where I3 = I1 and I1 > I2, constraint (B.1.16) becomes constraint(B.1.18). Rearranging constraint (B.1.18) we obtain(

e1 +−I2

(I1 − I2)e3,−1I1e1 +

1I1e2 +

1(I1 − I2)

e3

)(x, a) ≤ 1.


Therefore the ‘3a-big-x-middle-x’ constraint, (B.1.15), is facet defining for ESP .

B.1.8 Proof that Constraint ‘na-any-x’ of Section 4.5 is a Facet of ESP .

Proof. Consider any j ∈ {1, . . . , n}. We first show that (4.5.2) is valid for ESP .

Let (x, a) ∈ ESP . We consider two cases: xj = 1 or xj = 0.

In the first case, it must be that aσj(y) ≤ aj for all y = 1, . . . , n, since either aσj(y) = 0 oraσj(y) = aj by the definition of ESP . Thus by Lemma 4.5.1 it must be that

xj −1

Iσj(n−1)aj +

1Iσj(1)

aσj(1) +n−1∑y=2

(1


)aσj(y)

≤ xj −1

Iσj(n−1)aj +

1Iσj(n−1)

aj

= 1

as required.

In the second case, that xj = 0, choose r to maximise y ∈ {1, . . . , n − 1} such that aσj(y) > 0, soIσj(r) is the minimal Ik value such that ak > 0, and aσj(y) = 0 for all y = r + 1, . . . , n − 1. Thensince aσj(r) > 0, by the definition of ESP , it must be that aσj(y) ≤ aσj(r) for all y = 1, . . . , r andby Lemma 4.5.1 we have that

xj −1

Iσj(n−1)aj +

1Iσj(1)

aσj(1) +n−1∑y=2

(1


)aσj(y)

=1

Iσj(1)aσj(1) +

r∑y=2

(1


)aσj(y)

≤ 1Iσj(r)

aσj(r)

≤ 1.

If any consecutive I’s are equal, the summation term on the left-hand side of each case we considerabove will be smaller and hence the ‘equal I’s’ case is also valid. Thus (4.5.2) is valid for ESP .

We now show that (4.5.2) defines a facet of ESP . Again let j ∈ {1, . . . , n}, and let F j denote the

set of points in ESP satisfying (4.5.2) at equality. That is F j = {(x, a) ∈ ESP : xj−1

Iσj(n−1)aj +


1Iσj(1)

aσj(1) +n−1∑y=2

(1


)aσj(y) = 1} for j ∈ {1, . . . , n}. We will show that F j for

j ∈ {1, . . . , n} is a facet of ESP using Theorem 3.6 in Nemhauser and Wolsey [53]. Thus we notethat constraint (4.5.2) can be equivalently expressed as

(ej ,

−1Iσj(n−1)

ej +1

Iσj(1)eσj(1) +

n−1∑y=2

(1


)eσj(y)

)(x, a) ≤ 1,

since

xj −1

Iσj(n−1)aj +

1Iσj(1)

aσj(1) +n−1∑y=2

(1


)aσj(y) ≤ 1

⇔

(ej ,

−1Iσj(n−1)

ej + 1Iσj(1)

eσj(1) +n−1∑y=2

(1


)eσj(y)

)(x, a) ≤ 1.

We begin by supposing that for some µ0 ∈ R and µ, λ ∈ Rn, (µ, λ)(x, a) = µ0 for all (x, a) ∈ F j

for j ∈ {1, . . . , n}. In what follows, we deduce that

(µ, λ) =

µ0

(ej ,

−1Iσj(n−1)

ej +1

Iσj(1)eσj(1) +

n−1∑y=2

(1


)eσj(y)

)

for j ∈ {1, . . . , n} and so are able to conclude that F j is a facet. We do this by considering sevenclasses of points in F j for j ∈ {1, . . . , n} and observing what each implies about the values of µ, λand µ0. Again, the points take the form of those in ES′ and so are easily seen to be in ESP . Ineach case we assert that they are also in F j ; this is readily checked by observation.

1. (ej , 0) ∈ F j for j = 1, . . . , n. Now by our supposition it must be that (µ, λ)(ej , 0) = µ0,implying that µj = µ0 for j = 1, . . . , n.

2. (ej + ep, 0) ∈ F j for j = 1, . . . , n, p = 1, . . . , n, p 6= j. Now for each j and each p it must bethat (µ, λ)(ej + ep, 0) = µ0 and so µj +µp = µ0, implying that µp = 0 for p = 1, . . . , n, p 6= j.

3. (eσj(1), Iσj(1)eσj(1)) ∈ F j for j = 1, . . . , n. Now (µ, λ)(eσj(1), Iσj(1)eσj(1)) = µ0, so µσj(1) +

Iσj(1)λσj(1) = µ0 and thus λσj(1) =1

Iσj(1)µ0 for j = 1 . . . , n.

4. (eσj(1) + eσj(2), Iσj(2)(eσj(1) + eσj(2))) ∈ F j for j = 1 . . . , n. Now (µ, λ)(eσj(1) + eσj(2), Iσj(2)(eσj(1) + eσj(2))) = µ0, so µσj(1) + µσj(2) + Iσj(2)(λσj(1) + λσj(2)) = µ0

and thus λσj(2) =(

1Iσj(2)

− 1Iσj(1)

)µ0 for j = 1, . . . , n.

5. (eσj(1)+eσj(2)+eσj(3), Iσj(3)(eσj(1)+eσj(2)+eσj(3))) ∈ F j for j = 1, . . . , n. Now (µ, λ)(eσj(1)+eσj(2) +eσj(3), Iσj(3)(eσj(1) +eσj(2) +eσj(3))) = µ0, so µσj(1) +µσj(2) +µσj(3) +Iσj(3)(λσj(1) +

λσj(2)+λσj(3)) = µ0 and thus λσj(3) =(

1Iσj(3)

− 1Iσj(1)

− 1Iσj(2)

+1

Iσj(1)

)µ0 =

(1

Iσj(3)− 1Iσj(2)

)µ0

for j = 1, . . . , n.


6. (eσj(1) + eσj(2) + eσj(3) + eσj(4), Iσj(4)(eσj(1) + eσj(2) + eσj(3) + eσj(4))) ∈ F j for j = 1, . . . , n.Now (µ, λ)(eσj(1) + eσj(2) + eσj(3) + eσj(4), Iσj(4)(eσj(1) + eσj(2) + eσj(3) + eσj(4))) = µ0, soµσj(1) + µσj(2) + µσj(3) + µσj(4) + Iσj(4)(λσj(1) + λσj(2) + λσj(3) + λσj(4)) = µ0 and thus

λσj(4) =(

1Iσj(4)

− 1Iσj(1)

− 1Iσj(2)

+1

Iσj(1)− 1Iσj(3)

+1

Iσj(2)

)µ0 =(

1Iσj(4)

− 1Iσj(3)

)µ0 for j = 1 . . . , n.

Continuing in this manner, we see that λσj(2), . . . , λσj(n−1) all have the same structure:

λσj(2) =(

1Iσj(2)

− 1Iσj(1)

)µ0,

λσj(3) =(

1Iσj(3)

− 1Iσj(2)

)µ0, λσj(4) =

(1

Iσj(4)− 1Iσj(3)

)µ0, . . . ,

λσj(n−1) =(

1Iσj(n−1)

− 1Iσj(n−2)

)µ0 for j = 1, . . . , n.

7. (eσj(1) + · · ·+ eσj(n−1) + ej , Iσj(n−1)(eσj(1) + · · ·+ eσj(n−1) + ej)) ∈ F j for j = 1, . . . , n. Now(µ, λ)(eσj(1) + · · ·+ eσj(n−1) + ej , Iσj(n−1)(eσj(1) + · · ·+ eσj(n−1) + ej)) = µ0, so µσj(1) + · · ·+µσj(n−1) + µj + Iσj(n−1)(λσj(1) + · · · + λσj(n−1) + λj) = µ0 and thus λj = −(λσj(1) + · · · +

λσj(n−1)), implying that λj =−1

Iσj(n−1)µ0 for j = 1, . . . , n.

Thus (µ, λ) =

µ0

(ej ,

−1Iσj(n−1)

ej +1

Iσj(1)eσj(1) +

(1

Iσj(2)− 1Iσj(1)

)eσj(2)+

(1

Iσj(3)− 1Iσj(2)

)eσj(3) +

(1

Iσj(4)− 1Iσj(3)

)eσj(4) + · · ·+

(1

Iσj(n−1)− 1Iσj(n−2)

)eσj(n−1)

)for j ∈ {1, . . . , n}, as required.

Therefore the ‘na-any-x’ constraint, (4.5.2), is facet defining for ESP .

Considering Example 1, of Chapter 4, Section 4.3.1, we see that (4.5.2) becomes

x1 −1I2a1 +

1I4a4 +

(1I3− 1I4

)a3 +

(1I2− 1I3

)a2 ≤ 1, (B.1.19)

x2 −1I1a2 +

1I4a4 +

(1I3− 1I4

)a3 +

(1I1− 1I3

)a1 ≤ 1, (B.1.20)

x3 −1I2a3 +

1I4a4 +

(1I1− 1I4

)a1 +

(1I2− 1I1

)a2 ≤ 1, (B.1.21)

and

x4 −1I2a4 +

1I3a3 +

(1I1− 1I3

)a1 +

(1I2− 1I1

)a2 ≤ 1, (B.1.22)

since I4 > I3 > I1 > I2. Substituting in the values for I1, I2, I3 and I4 from Example 1 yieldsconstraints (4.3.38), (4.3.37), (4.3.39) and (4.3.40) respectively from Section 4.3.1.6 as required.

B.2. Examples of Points Satisfying ESP -C1 and Facets of ESP -C1 of Small Support for the Equal I’sCase 203

Table B.2.1: Example 2: points satisfying ESP -C1

x1 x2 x3 x4 a1 a2 a3 a4

0 0 0 0 0 0 0 01 0 0 0 0 0 0 01 0 0 0 7 0 0 00 1 0 0 0 0 0 00 1 0 0 0 4 0 00 0 1 0 0 0 0 00 0 1 0 0 0 12 00 0 0 1 0 0 0 00 0 0 1 0 0 0 211 1 0 0 0 0 0 01 1 0 0 4 4 0 00 1 1 0 0 0 0 00 1 1 0 0 4 4 00 0 1 1 0 0 0 00 0 1 1 0 0 12 121 1 1 0 0 0 0 01 1 1 0 4 4 4 00 1 1 1 0 0 0 00 1 1 1 0 4 4 41 1 1 1 0 0 0 01 1 1 1 4 4 4 4

B.2 Examples of Points Satisfying ESP -C1 and Facets of ESP -C1 of Small Supportfor the Equal I’s Case

B.2.1 Example 2. We again consider Example 1 of Section 4.3.1, which has dimension 4and intensity values I1 = 7, I2 = 4, I3 = 12 and I4 = 21. The corresponding complete set of pointssatisfying ESP -C1 are given in Table B.2.1. Table B.2.1 shows that we can never have a zero xvariable value surrounded by non-zero x variable values.

Again utilising the web-based tool Polymake [52], the facets of ESP -C1 resulting from the pointsin Example 2 are:

B.2.1.1 Facets containing just one a variable

a1 ≥ 0, (B.2.1)

a2 ≥ 0, (B.2.2)

a3 ≥ 0, (B.2.3)

a4 ≥ 0, (B.2.4)


B.2.1.2 Facets containing just one x variable

1− x1 ≥ 0, (B.2.5)

1− x2 ≥ 0, (B.2.6)

1− x3 ≥ 0, (B.2.7)

1− x4 ≥ 0, (B.2.8)

B.2.1.3 Facets containing one x variable and one a variable

7x1 − a1 ≥ 0, (B.2.9)

4x2 − a2 ≥ 0, (B.2.10)

12x3 − a3 ≥ 0, (B.2.11)

21x4 − a4 ≥ 0, (B.2.12)

B.2.1.4 Facets containing one x variable and two a variables

4− 4x1 + a1 − a2 ≥ 0, (B.2.13)

12− 12x1 + a2 − a3 ≥ 0, (B.2.14)

21− 21x1 + a3 − a4 ≥ 0, (B.2.15)

7− 7x2 − a1 + a2 ≥ 0, (B.2.16)

12− 12x2 + a2 − a3 ≥ 0, (B.2.17)

21− 21x2 + a3 − a4 ≥ 0, (B.2.18)

7− 7x3 − a1 + a2 ≥ 0, (B.2.19)

4− 4x3 − a2 + a3 ≥ 0, (B.2.20)

21− 21x3 + a3 − a4 ≥ 0, (B.2.21)

7− 7x4 − a1 + a2 ≥ 0, (B.2.22)

4− 4x4 − a2 + a3 ≥ 0, (B.2.23)

4− 4x4 − a2 + a4 ≥ 0, (B.2.24)

12− 12x4 − a3 + a4 ≥ 0, (B.2.25)


B.2.1.5 Facets containing one x variable and three a variables

12− 12x1 + 3a1 − 2a2 − a3 ≥ 0, (B.2.26)

28− 28x1 +73a2 − a3 −

43a4 ≥ 0, (B.2.27)

12− 12x2 −127a1 +

197a2 − a3 ≥ 0, (B.2.28)

28− 28x2 +73a2 − a3 −

43a4 ≥ 0, (B.2.29)

283− 28

3x3 −

43a1 − a2 +

73a3 ≥ 0, (B.2.30)

21− 21x3 −214a2 +

254a3 − a4 ≥ 0, (B.2.31)

283− 28

3x4 −

43a1 − a2 +

73a3 ≥ 0, (B.2.32)

283− 28

3x4 −

43a1 − a2 +

73a4 ≥ 0, (B.2.33)

12− 12x4 − 2a2 − a3 + 3a4 ≥ 0, (B.2.34)

B.2.1.6 Facets containing one x variable and four a variables

28− 28x1 + 7a1 −143a2 − a3 −

43a4 ≥ 0, (B.2.35)

21− 21x1 +214a1 −

214a2 + a3 − a4 ≥ 0, (B.2.36)

28− 28x2 − 4a1 +193a2 − a3 −

43a4 ≥ 0, (B.2.37)

21− 21x2 − 3a1 + 3a2 + a3 − a4 ≥ 0, (B.2.38)

21− 21x3 − 3a1 + 3a2 + a3 − a4 ≥ 0, (B.2.39)

21− 21x3 − 3a1 −94a2 +

254a3 − a4 ≥ 0, (B.2.40)

12− 12x4 −127a1 +

127a2 − a3 + a4 ≥ 0, (B.2.41)

42− 42x4 − 6a1 − a2 −72a3 +

212a4 ≥ 0, (B.2.42)

B.2.1.7 Facets containing two x variables and one a variable

3 + 4x1 − 3x2 − a1 ≥ 0, (B.2.43)

3 + 4x1 − 3x3 − a1 ≥ 0, (B.2.44)

8− 8x1 + 4x3 − a3 ≥ 0, (B.2.45)

3 + 4x1 − 3x4 − a1 ≥ 0, (B.2.46)

17− 17x1 + 4x4 − a4 ≥ 0, (B.2.47)

8− 8x2 + 4x3 − a3 ≥ 0, (B.2.48)

17− 17x2 + 4x4 − a4 ≥ 0, (B.2.49)

9− 9x3 + 12x4 − a4 ≥ 0, (B.2.50)


B.2.1.8 Facets containing two x variables and three a variables

12 + 16x1 − 12x2 − 4a1 + a2 − a3 ≥ 0, (B.2.51)

21 + 28x1 − 21x2 − 7a1 + a3 − a4 ≥ 0, (B.2.52)

8− 8x1 + 4x3 + 2a1 − 2a2 − a3 ≥ 0, (B.2.53)

4 +163x1 − 4x3 −

43a1 − a2 + a3 ≥ 0, (B.2.54)

21 + 28x1 − 21x3 − 7a1 + a3 − a4 ≥ 0, (B.2.55)

4 +163x1 − 4x4 −

43a1 − a2 + a3 ≥ 0, (B.2.56)

17− 17x1 + 4x4 +174a1 −

174a2 − a4 ≥ 0, (B.2.57)

4 +163x1 − 4x4 −

43a1 − a2 + a4 ≥ 0, (B.2.58)

12 + 16x1 − 12x4 − 4a1 − a3 + a4 ≥ 0, (B.2.59)

12− 12x1 + 16x4 + a2 − a3 −43a4 ≥ 0, (B.2.60)

683− 68

3x1 +

163x4 + a2 − a3 −

43a4 ≥ 0, (B.2.61)

8− 8x2 + 4x3 −87a1 +

87a2 − a3 ≥ 0, (B.2.62)

17− 17x2 + 4x4 −177a1 +

177a2 − a4 ≥ 0, (B.2.63)

12− 12x2 + 16x4 + a2 − a3 −43a4 ≥ 0, (B.2.64)

683− 68

3x2 +

163x4 + a2 − a3 −

43a4 ≥ 0, (B.2.65)

9− 9x3 + 12x4 −97a1 +

97a2 − a4 ≥ 0, (B.2.66)

17− 17x3 + 4x4 −174a2 +

174a3 − a4 ≥ 0, (B.2.67)

512− 51

2x3 + 6x4 − a2 + a3 −

32a4 ≥ 0, (B.2.68)

B.2.1.9 Facets containing two x variables and four a variables

28 +1123x1 − 28x2 −

283a1 +

73a2 − a3 −

43a4 ≥ 0, (B.2.69)

21 + 28x1 − 21x3 − 7a1 −214a2 +

254a3 − a4 ≥ 0, (B.2.70)

12− 12x1 + 16x4 + 3a1 − 2a2 − a3 −43a4 ≥ 0, (B.2.71)

12 + 16x1 − 12x4 − 4a1 − 2a2 − a3 + 3a4 ≥ 0, (B.2.72)683− 68

3x1 +

163x4 +

173a1 −

143a2 − a3 −

43a4 ≥ 0, (B.2.73)

12− 12x2 + 16x4 −127a1 +

197a2 − a3 −

43a4 ≥ 0, (B.2.74)

683− 68

3x2 +

163x4 −

6821a1 +

8921a2 − a3 −

43a4 ≥ 0, (B.2.75)

512− 51

2x3 + 6x4 −

5114a1 +

3714a2 + a3 −

32a4 ≥ 0, (B.2.76)

17− 17x3 + 4x4 −177a1 −

5128a2 +

174a3 − a4 ≥ 0, (B.2.77)


B.2.1.10 Facets containing just three x variables

1− x1 + x2 − x3 ≥ 0, (B.2.78)

1− x1 + x2 − x4 ≥ 0, (B.2.79)

1− x1 + x3 − x4 ≥ 0, (B.2.80)

1− x2 + x3 − x4 ≥ 0, (B.2.81)

B.2.1.11 Facets containing three x variables and one a variable

8− 8x1 + 12x3 − 8x4 − a3 ≥ 0, (B.2.82)

8− 8x2 + 12x3 − 8x4 − a3 ≥ 0, (B.2.83)

B.2.1.12 Facets containing three x variables and two a variables

4− 4x1 + 4x2 − 4x3 + a1 − a2 ≥ 0, (B.2.84)

8 +323x1 − 8x2 + 4x3 −

83a1 − a3 ≥ 0, (B.2.85)

4− 4x1 + 4x2 − 4x3 − a2 + a3 ≥ 0, (B.2.86)

21− 21x1 + 21x2 − 21x3 + a3 − a4 ≥ 0, (B.2.87)

4− 4x1 + 4x2 − 4x4 + a1 − a2 ≥ 0, (B.2.88)

17 +683x1 − 17x2 + 4x4 −

173a1 − a4 ≥ 0, (B.2.89)

4− 4x1 + 4x2 − 4x4 − a2 + a3 ≥ 0, (B.2.90)

4− 4x1 + 4x2 − 4x4 − a2 + a4 ≥ 0, (B.2.91)

12− 12x1 + 12x2 − 12x4 − a3 + a4 ≥ 0, (B.2.92)

4− 4x1 + 4x3 − 4x4 + a1 − a2 ≥ 0, (B.2.93)

9 + 12x1 − 9x3 + 12x4 − 3a1 − a4 ≥ 0, (B.2.94)

12− 12x1 + 12x3 − 12x4 + a2 − a3 ≥ 0, (B.2.95)

6− 6x1 + 6x3 − 6x4 − a3 + a4 ≥ 0, (B.2.96)

8− 8x1 + 4x3 − 4x4 − a3 + a4 ≥ 0, (B.2.97)

12− 12x1 + 18x3 − 12x4 −52a3 + a4 ≥ 0, (B.2.98)

9− 9x1 +92x3 + 12x4 −

98a3 − a4 ≥ 0, (B.2.99)

17− 17x1 +92x3 + 4x4 −

98a3 − a4 ≥ 0, (B.2.100)

7− 7x2 + 7x3 − 7x4 − a1 + a2 ≥ 0, (B.2.101)

12− 12x2 + 12x3 − 12x4 + a2 − a3 ≥ 0, (B.2.102)

6− 6x2 + 6x3 − 6x4 − a3 + a4 ≥ 0, (B.2.103)

8− 8x2 + 4x3 − 4x4 − a3 + a4 ≥ 0, (B.2.104)

12− 12x2 + 18x3 − 12x4 −52a3 + a4 ≥ 0, (B.2.105)

9− 9x2 +92x3 + 12x4 −

98a3 − a4 ≥ 0, (B.2.106)

17− 17x2 +92x3 + 4x4 −

98a3 − a4 ≥ 0, (B.2.107)


B.2.1.13 Facets containing three x variables and three a variables

4− 4x1 + 4x2 − 4x3 + a1 − 2a2 + a3 ≥ 0, (B.2.108)

21− 21x1 + 21x2 − 21x3 −214a2 +

254a3 − a4 ≥ 0, (B.2.109)

4− 4x1 + 4x2 − 4x4 + a1 − 2a2 + a3 ≥ 0, (B.2.110)

4− 4x1 + 4x2 − 4x4 + a1 − 2a2 + a4 ≥ 0, (B.2.111)

12− 12x1 + 12x2 − 12x4 − 2a2 − a3 + 3a4 ≥ 0, (B.2.112)

12− 12x1 + 12x3 − 12x4 + 3a1 − 2a2 − a3 ≥ 0, (B.2.113)

8− 8x1 + 12x3 − 8x4 + 2a1 − 2a2 − a3 ≥ 0, (B.2.114)

12− 12x1 + 12x3 − 12x4 + a2 − 2a3 + a4 ≥ 0, (B.2.115)

8− 8x2 + 12x3 − 8x4 −87a1 +

87a2 − a3 ≥ 0, (B.2.116)

12− 12x2 + 12x3 − 12x4 −127a1 +

197a2 − a3 ≥ 0, (B.2.117)

12− 12x2 + 12x3 − 12x4 + a2 − 2a3 + a4 ≥ 0, (B.2.118)

8− 4x2 + 4x3 − 8x4 − a2 − a3 + 2a4 ≥ 0, (B.2.119)

B.2.1.14 Facets containing three x variables and four a variables

21− 21x1 + 21x2 − 21x3 +214a1 −

212a2 +

254a3 − a4 ≥ 0, (B.2.120)

21− 21x1 + 21x2 − 21x3 +214a1 −

214a2 + a3 − a4 ≥ 0, (B.2.121)

12 + 16x1 − 12x2 + 16x4 − 4a1 + a2 − a3 −43a4 ≥ 0, (B.2.122)

12− 12x1 + 12x2 − 12x4 + 3a1 − 3a2 − a3 + a4 ≥ 0, (B.2.123)

683

+2729x1 −

683x2 +

163x4 −

689a1 + a2 − a3 −

43a4 ≥ 0, (B.2.124)

12− 12x1 + 12x2 − 12x4 + 3a1 − 5a2 − a3 + 3a4 ≥ 0, (B.2.125)

9− 9x1 +92x3 + 12x4 +

94a1 −

94a2 −

98a3 − a4 ≥ 0, (B.2.126)

12− 12x1 + 12x3 − 12x4 + 3a1 − 2a2 − 2a3 + a4 ≥ 0, (B.2.127)

17 +683x1 − 17x3 + 4x4 −

173a1 −

174a2 +

174a3 − a4 ≥ 0, (B.2.128)

17− 17x1 +92x3 + 4x4 +

174a1 −

174a2 −

98a3 − a4 ≥ 0, (B.2.129)

512

+ 34x1 −512x3 + 6x4 −

172a1 − a2 + a3 −

32a4 ≥ 0, (B.2.130)

12− 12x1 + 18x3 − 12x4 + 3a1 − 3a2 −52a3 + a4 ≥ 0, (B.2.131)

6− 6x1 + 6x3 − 6x4 +32a1 −

32a2 − a3 + a4 ≥ 0, (B.2.132)

8− 8x1 + 4x3 − 4x4 + 2a1 − 2a2 − a3 + a4 ≥ 0, (B.2.133)


9− 9x2 +92x3 + 12x4 −

97a1 +

97a2 −

98a3 − a4 ≥ 0, (B.2.134)

12− 12x2 + 18x3 − 12x4 −127a1 +

127a2 −

52a3 + a4 ≥ 0, (B.2.135)

17− 17x2 +92x3 + 4x4 −

177a1 +

177a2 −

98a3 − a4 ≥ 0, (B.2.136)

56− 28x2 + 28x3 − 56x4 − 8a1 + a2 − 7a3 + 14a4 ≥ 0, (B.2.137)

12− 12x2 + 12x3 − 12x4 −127a1 +

197a2 − 2a3 + a4 ≥ 0, (B.2.138)

7− 7x2 + 7x3 − 7x4 − a1 + a2 −76a3 +

76a4 ≥ 0, (B.2.139)

8− 8x2 + 4x3 − 4x4 −87a1 +

87a2 − a3 + a4 ≥ 0, (B.2.140)

B.2.1.15 Facets containing four x variables and one a variable

3 + 4x1 − 3x2 + 3x3 − 3x4 − a1 ≥ 0, (B.2.141)

9− 9x1 + 9x2 − 9x3 + 12x4 − a4 ≥ 0, (B.2.142)

B.2.1.16 Facets containing four x variables and two a variables

8 +323x1 − 8x2 + 12x3 − 8x4 −

83a1 − a3 ≥ 0, (B.2.143)

B.2.1.17 Facets containing four x variables and three a variables

12 + 16x1 − 12x2 + 12x3 − 12x4 − 4a1 + a2 − a3 ≥ 0, (B.2.144)

9− 9x1 + 9x2 − 9x3 + 12x4 +94a1 −

94a2 − a4 ≥ 0, (B.2.145)

9 + 12x1 − 9x2 +92x3 + 12x4 − 3a1 −

98a3 − a4 ≥ 0, (B.2.146)

8 +323x1 − 8x2 + 4x3 − 4x4 −

83a1 − a3 + a4 ≥ 0, (B.2.147)

17 +683x1 − 17x2 +

92x3 + 4x4 −

173a1 −

98a3 − a4 ≥ 0, (B.2.148)

12 + 16x1 − 12x2 + 18x3 − 12x4 − 4a1 −52a3 + a4 ≥ 0, (B.2.149)

6 + 8x1 − 6x2 + 6x3 − 6x4 − 2a1 − a3 + a4 ≥ 0, (B.2.150)

512− 51

2x1 +

512x2 −

512x3 + 6x4 − a2 + a3 −

32a4 ≥ 0, (B.2.151)

17− 17x1 + 17x2 − 17x3 + 4x4 −174a2 +

174a3 − a4 ≥ 0, (B.2.152)

8− 8x1 + 4x2 + 4x3 − 8x4 − a2 − a3 + 2a4 ≥ 0, (B.2.153)


B.2.1.18 Facets containing four x variables and four a variables

12 + 16x1 − 12x2 + 12x3 − 12x4 − 4a1 + a2 − 2a3 + a4 ≥ 0, (B.2.154)

17− 17x1 + 17x2 − 17x3 + 4x4 +174a1 −

172a2 +

174a3 − a4 ≥ 0, (B.2.155)

8 +323x1 − 4x2 + 4x3 − 8x4 −

83a1 − a2 − a3 + 2a4 ≥ 0, (B.2.156)

512− 51

2x1 +

512x2 −

512x3 + 6x4 +

518a1 −

598a2 + a3 −

32a4 ≥ 0, (B.2.157)

8− 8x1 + 4x2 + 4x3 − 8x4 + 2a1 − 3a2 − a3 + 2a4 ≥ 0. (B.2.158)

There are 158 facets for this example (in comparison to 140 for our ESP example, given in Section4.3.1) which specifies one particular ordering of the I parameters. Clearly once all orderings of Iparameters have been considered for ESP -C1 the complete list of facets becomes very large andhence we focus on determining facets for a special case only which is relevant to real world data:equal I parameters.

B.2.2 Example 3. Hence we now consider an example of dimension 4 where the intensityvalues are all the same: I1 = I2 = I3 = I4 = 7. The corresponding complete set of points satisfyingESP -C1 are given in Table B.2.2.

Again utilising the web-based tool Polymake [52], the facets of ESP -C1 resulting from the pointsin Example 3 are:

B.2.2.1 Facets containing just one a variable

a1 ≥ 0, (B.2.159)

a2 ≥ 0, (B.2.160)

a3 ≥ 0, (B.2.161)

a4 ≥ 0, (B.2.162)

B.2.2.2 Facets containing just one x variable

1− x1 ≥ 0, (B.2.163)

1− x2 ≥ 0, (B.2.164)

1− x3 ≥ 0, (B.2.165)

1− x4 ≥ 0, (B.2.166)

B.2.2.3 Facets containing one x variable and one a variable

7x1 − a1 ≥ 0, (B.2.167)

7x2 − a2 ≥ 0, (B.2.168)

7x3 − a3 ≥ 0, (B.2.169)

7x4 − a4 ≥ 0, (B.2.170)


Table B.2.2: Example 3: points satisfying ESP -C1

x1 x2 x3 x4 a1 a2 a3 a4

0 0 0 0 0 0 0 01 0 0 0 0 0 0 01 0 0 0 7 0 0 00 1 0 0 0 0 0 00 1 0 0 0 7 0 00 0 1 0 0 0 0 00 0 1 0 0 0 7 00 0 0 1 0 0 0 00 0 0 1 0 0 0 71 1 0 0 0 0 0 01 1 0 0 7 7 0 00 1 1 0 0 0 0 00 1 1 0 0 7 7 00 0 1 1 0 0 0 00 0 1 1 0 0 7 71 1 1 0 0 0 0 01 1 1 0 7 7 7 00 1 1 1 0 0 0 00 1 1 1 0 7 7 71 1 1 1 0 0 0 01 1 1 1 7 7 7 7


B.2.2.4 Facets containing one x variable and two a variables ‘2a-any-x-eq’

7− 7x1 + a1 − a2 ≥ 0, (B.2.171)

7− 7x1 + a1 − a3 ≥ 0, (B.2.172)

7− 7x1 + a1 − a4 ≥ 0, (B.2.173)

7− 7x2 − a1 + a2 ≥ 0, (B.2.174)

7− 7x2 + a2 − a3 ≥ 0, (B.2.175)

7− 7x2 + a2 − a4 ≥ 0, (B.2.176)

7− 7x3 − a1 + a3 ≥ 0, (B.2.177)

7− 7x3 − a2 + a3 ≥ 0, (B.2.178)

7− 7x3 + a3 − a4 ≥ 0, (B.2.179)

7− 7x4 − a1 + a4 ≥ 0, (B.2.180)

7− 7x4 − a2 + a4 ≥ 0, (B.2.181)

7− 7x4 − a3 + a4 ≥ 0, (B.2.182)

‘2a-1diff-x-eq’7− 7x1 + a2 − a3 ≥ 0, (B.2.183)

7− 7x1 + a2 − a4 ≥ 0, (B.2.184)

7− 7x1 + a3 − a4 ≥ 0, (B.2.185)

7− 7x2 + a3 − a4 ≥ 0, (B.2.186)

7− 7x3 − a1 + a2 ≥ 0, (B.2.187)

7− 7x4 − a1 + a2 ≥ 0, (B.2.188)

7− 7x4 − a1 + a3 ≥ 0, (B.2.189)

7− 7x4 − a2 + a3 ≥ 0, (B.2.190)

B.2.2.5 Facets containing one x variable and three a variables: ‘3a-coeff2-x-eq’

7− 7x2 − a1 + 2a2 − a3 ≥ 0, (B.2.191)

7− 7x2 − a1 + 2a2 − a4 ≥ 0, (B.2.192)

7− 7x3 − a1 + 2a3 − a4 ≥ 0, (B.2.193)

7− 7x3 − a2 + 2a3 − a4 ≥ 0, (B.2.194)

B.2.2.6 Facets containing one x variable and four a variables: ‘4a-any-x-eq’

7− 7x1 + a1 − a2 + a3 − a4 ≥ 0, (B.2.195)

7− 7x2 − a1 + a2 + a3 − a4 ≥ 0, (B.2.196)

7− 7x3 − a1 + a2 + a3 − a4 ≥ 0, (B.2.197)

7− 7x4 − a1 + a2 − a3 + a4 ≥ 0, (B.2.198)


B.2.2.7 Facets containing just three x variables

1− x1 + x2 − x3 ≥ 0, (B.2.199)

1− x1 + x2 − x4 ≥ 0, (B.2.200)

1− x1 + x3 − x4 ≥ 0, (B.2.201)

1− x2 + x3 − x4 ≥ 0, (B.2.202)

B.2.2.8 Facets containing three x variables and two a variables

7− 7x1 + 7x2 − 7x3 + a1 − a2 ≥ 0, (B.2.203)

7− 7x1 + 7x2 − 7x3 − a2 + a3 ≥ 0, (B.2.204)

7− 7x1 + 7x2 − 7x3 + a3 − a4 ≥ 0, (B.2.205)

7− 7x1 + 7x2 − 7x4 + a1 − a2 ≥ 0, (B.2.206)

7− 7x1 + 7x2 − 7x4 − a2 + a3 ≥ 0, (B.2.207)

7− 7x1 + 7x2 − 7x4 − a2 + a4 ≥ 0, (B.2.208)

7− 7x1 + 7x2 − 7x4 − a3 + a4 ≥ 0, (B.2.209)

7− 7x1 + 7x3 − 7x4 + a1 − a2 ≥ 0, (B.2.210)

7− 7x1 + 7x3 − 7x4 + a1 − a3 ≥ 0, (B.2.211)

7− 7x1 + 7x3 − 7x4 + a2 − a3 ≥ 0, (B.2.212)

7− 7x1 + 7x3 − 7x4 − a3 + a4 ≥ 0, (B.2.213)

7− 7x2 + 7x3 − 7x4 − a1 + a2 ≥ 0, (B.2.214)

7− 7x2 + 7x3 − 7x4 + a2 − a3 ≥ 0, (B.2.215)

7− 7x2 + 7x3 − 7x4 − a3 + a4 ≥ 0, (B.2.216)

B.2.2.9 Facets containing three x variables and three a variables

7− 7x1 + 7x2 − 7x3 + a1 − 2a2 + a3 ≥ 0, (B.2.217)

7− 7x1 + 7x2 − 7x3 − a2 + 2a3 − a4 ≥ 0, (B.2.218)

7− 7x1 + 7x2 − 7x4 + a1 − 2a2 + a3 ≥ 0, (B.2.219)

7− 7x1 + 7x2 − 7x4 + a1 − 2a2 + a4 ≥ 0, (B.2.220)

7− 7x1 + 7x3 − 7x4 + a1 − 2a3 + a4 ≥ 0, (B.2.221)

7− 7x1 + 7x3 − 7x4 + a2 − 2a3 + a4 ≥ 0, (B.2.222)

7− 7x2 + 7x3 − 7x4 − a1 + 2a2 − a3 ≥ 0, (B.2.223)

7− 7x2 + 7x3 − 7x4 + a2 − 2a3 + a4 ≥ 0, (B.2.224)


B.2.2.10 Facets containing three x variables and four a variables

7− 7x1 + 7x2 − 7x3 + a1 − a2 + a3 − a4 ≥ 0, (B.2.225)

7− 7x1 + 7x2 − 7x3 + a1 − 2a2 + 2a3 − a4 ≥ 0, (B.2.226)

7− 7x1 + 7x2 − 7x4 + a1 − a2 − a3 + a4 ≥ 0, (B.2.227)

7− 7x1 + 7x3 − 7x4 + a1 − a2 − a3 + a4 ≥ 0, (B.2.228)

7− 7x2 + 7x3 − 7x4 − a1 + a2 − a3 + a4 ≥ 0, (B.2.229)

7− 7x2 + 7x3 − 7x4 − a1 + 2a2 − 2a3 + a4 ≥ 0. (B.2.230)

When we consider our special case of equal I parameters, 72 facets result. In the sections to followwe prove that generalisations of the facets, of small support, of Example 3 are facets of ESP -C1.

B.2.3 Constraint ‘2a-any-x-eq’ is a Facet of ESP -C1.

Proposition B.2.3.1. Constraint (4.4.1), ‘2a-any-x’, with It = Is, is facet defining for ESP -C1. Wecall this constraint the ‘2a-any-x-eq’ facet of ESP -C1. The ‘2a-any-x-eq’ facet is a generalisationof constraints (B.2.171), (B.2.172), (B.2.173), (B.2.174), (B.2.175), (B.2.176), (B.2.177), (B.2.178),(B.2.179), (B.2.180), (B.2.181) and (B.2.182) given in Section B.2.2.4.

Proof. All the points in ESP -C1 are contained within ESP . Since we have checked the validityof constraint (4.4.1) for ESP , and since our checks hold for It = Is, and all points that must beconsidered satisfy the strict consecutive-1-property, then we have checked the validity of constraint‘2a-any-x-eq’ for ESP -C1.

Without loss of generality we take t = 1 and s = 2 and we assume I1 = I2 = I in constraint (4.4.1),with It = Is. We have:

I(1− x2)− a1 + a2 ≥ 0. (B.2.231)

We now show that (B.2.231) defines a facet of ESP -C1. Let F = {(x, a) ∈ ESP -C1 : I(1− x2)−a1 + a2 = 0}. We will show that F is a facet of ESP -C1 using Theorem 3.6 in Nemhauser andWolsey [53]. Thus we note that constraint (B.2.231) can be equivalently expressed as(

e2,1Ie1 −

1Ie2

)(x, a) ≤ 1,

sinceI − Ix2 − a1 + a2 ≥ 0

⇔ Ix2 + a1 − a2 ≤ I

⇔ x2 +1Ia1 −

1Ia2 ≤ 1

⇔(e2,

1Ie1 −

1Ie2

)(x, a) ≤ 1.


(µ, λ) = µ0

(e2,

1Ie1 −

1Ie2

)and so are able to conclude that F is a facet. We do this by considering four classes of points inF and observing what each implies about the values of µ, λ and µ0. We note that in each case the


points take the form of those in ES′-C1 and so are easily seen to be in ESP -C1. In each case weassert that they are also in F ; this is readily checked by observation.


2. (p′∑p=1

ep, 0) ∈ F for p′ = 2, . . . , n. Now for each p′ ∈ {2, . . . , n} it must be that (µ, λ)(p′∑p=1

ep, 0) =

µ0 and sop′∑p=1

µp = µ0. Now if p′ = 2 then µ1 = 0; if p′ = 3 then µ3 = 0; if p′ = 4 then

µ4 = 0. Continuing in this manner yields µp = 0 for p = 1, 3, . . . , n.

3. (e1, Ie1) ∈ F . Now (µ, λ)(e1, Ie1) = µ0, so µ1 + Iλ1 = µ0 and thus λ1 =1Iµ0.

4. (p′∑p=1

ep, I(p′∑p=1

ep)) ∈ F for p′ = 2, . . . , n. Now for each p′ ∈ {2, . . . , n} it must be that

(µ, λ)(p′∑p=1

ep, I(p′∑p=1

ep)) = µ0 and sop′∑p=1

µp + I(p′∑p=1

λp) = µ0. Now if p′ = 2 then λ2 =−1Iµ0;

if p′ = 3 then λ3 = 0 since λ1 = −λ2; if p′ = 4 then λ4 = 0. Continuing in this manner yieldsλp = 0 for p = 3, . . . , n. (We have also used µ2 = µ0 and µp = 0 for p = 1, . . . , n, p 6= 2).

Thus (µ, λ) = µ0

(e2,

1Ie1 −

1Ie2

)as required.

Therefore the ‘2a-any-x-eq’ constraint, (4.4.1) with It = Is, is facet defining for ESP -C1.

B.2.4 Constraint ‘2a-1diff-x-eq’ is a Facet of ESP -C1.

Proposition B.2.4.1. For any s, t, u = 1, . . . , n, s 6= t, t 6= u, s 6= u, s 6= t + 1, t 6= s + 1,min(s, t) <u < max(s, t), (u is between s and t), It = Iu = Is = I the constraint

I(1− xs)− at + au ≥ 0 (B.2.232)

is facet defining for ESP -C1. We call this constraint the ‘2a-1diff-x-eq’ facet of ESP -C1. The‘2a-1diff-x-eq’ facet is a generalisation of constraints (B.2.183), (B.2.184), (B.2.185), (B.2.186),(B.2.187), (B.2.188), (B.2.189) and (B.2.190) given in Section B.2.2.4.

Proof. Without loss of generality we take t = 1, u = 2 and s = 3 and we assume I1 = I2 = I3 = I.Constraint (B.2.232) becomes:

I(1− x3)− a1 + a2 ≥ 0. (B.2.233)

First we must check the validity of constraint (B.2.233). Let (x, a) ∈ ESP -C1 and define S = {p ∈{1, . . . , n} : xp = 1}. We show that (x, a) satisfies (B.2.233) by checking all possible values of x1,x2 and x3 (and hence for a1, a2 and a3).

Case (i): x1 = 0, x2 = 0 and x3 = 0. In this case, a1 = a2 = a3 = 0. Since I ≥ 0 by assumption,(B.2.233) is satisfied.


Case (ii): x1 = 0, x2 = 0 and x3 = 1. In this case, a1 = a2 = 0. Hence (B.2.233) is satisfied.

Case (iii): x1 = 0, x2 = 1 and x3 = 0. In this case, a1 = a3 = 0. Since I ≥ 0 by assumption anda2 ≥ 0 by definition, then (B.2.233) is satisfied.

Case (iv): x1 = 1, x2 = 0 and x3 = 0. In this case, a2 = a3 = 0. Since 1 ∈ S and Lemma 4.2.1gives a1 ≤ min

p∈SIp ≤ I1, and I1 = I, then (B.2.233) is satisfied.

Case (v): x1 = 0, x2 = 1 and x3 = 1. In this case, a1 = 0 and from the definition of ES-C1 it mustbe that a2 = a3. Since a2 ≥ 0 by definition, then (B.2.233) is satisfied.

Case (vi): x1 = 1, x2 = 1 and x3 = 0. In this case, a3 = 0 and from the definition of ES-C1 itmust be that a1 = a2. Since I ≥ 0 by assumption, then (B.2.233) is satisfied.

Case (vii): x1 = 1, x2 = 1 and x3 = 1. In this case, from the definition of ES-C1 it must be thata1 = a2 = a3. Hence (B.2.233) is satisfied.

We now show that (B.2.233) defines a facet of ESP -C1. Let F = {(x, a) ∈ ESP -C1 : I(1− x3)−a1 + a2 = 0}. We will show that F is a facet of ESP -C1 using Theorem 3.6 in Nemhauser andWolsey [53]. Thus we note that constraint (B.2.233) can be equivalently expressed as

(e3,

1Ie1 −

1Ie2

)(x, a) ≤ 1,

sinceI − Ix3 − a1 + a2 ≥ 0

⇔ Ix3 + a1 − a2 ≤ I

⇔ x3 +1Ia1 −

1Ia2 ≤ 1

⇔(e3,

1Ie1 −

1Ie2

)(x, a) ≤ 1.


(µ, λ) = µ0

(e3,

1Ie1 −

1Ie2

)and so are able to conclude that F is a facet. We do this by considering six classes of points inF and observing what each implies about the values of µ, λ and µ0. The points take the form ofthose in ES′-C1 and so are easily seen to be in ESP -C1. In each case we assert that they are alsoin F ; this is readily checked by observation.

1. (e3, 0) ∈ F . Now by our supposition (µ, λ)(e3, 0) = µ0, implying that µ3 = µ0.

2. (e2 + e3, 0) ∈ F . Now (µ, λ)(e2 + e3, 0) = µ0 and so µ2 + µ3 = µ0, implying that µ2 = 0.

3. (p′∑p=1


ep, 0) =




µ5 = 0. Continuing in this manner yields µp = 0 for p = 1, . . . , n, p 6= 3.


5. (e3, Ie3) ∈ F . Now (µ, λ)(e3, Ie3) = µ0, so µ3 + Iλ3 = µ0 and thus λ3 = 0.

6. (p′∑p=1

ep, I(p′∑p=1


(µ, λ)(p′∑p=1

ep, I(p′∑p=1


µp + I(p′∑p=1

λp) = µ0. Now if p′ = 3 then λ2 =−1Iµ0.

If p′ = 4 then λ4 = 0 since λ2 = −λ1 and λ3 = 0. If p′ = 5 then λ5 = 0. Continuing inthis manner yields λp = 0 for p = 3, . . . , n. (We have also used µ3 = µ0 and µp = 0 forp = 1, . . . , n, p 6= 3).

Thus (µ, λ) = µ0

(e3,

1Ie1 −

1Ie2

)as required.

Therefore the ‘2a-1diff-x-eq’ constraint, (B.2.232), is facet defining for ESP -C1.

B.2.5 Constraint ‘3a-coeff2-x-eq’ is a Facet of ESP -C1.

Proposition B.2.5.1. For any s, t, u = 1, . . . , n, s < t, t 6= u, s 6= u, t 6= s + 1,min(s, t) < u <

max(s, t), (u is between s and t), Is = It = Iu = I the constraint

I(1− xu)− as + 2au − at ≥ 0 (B.2.234)

is facet defining for ESP -C1. We call this constraint the ‘3a-coeff2-x-eq’ facet of ESP -C1. The‘3a-coeff2-x-eq’ facet is a generalisation of the constraints given in Section B.2.2.5.

Proof. Without loss of generality we take s = 1, u = 2 and t = 3 and we assume I1 = I2 = I3 = I.Constraint (B.2.234) becomes:

I(1− x2)− a1 + 2a2 − a3 ≥ 0. (B.2.235)

First we must check the validity of constraint (B.2.235). Let (x, a) ∈ ESP -C1 and define S = {p ∈{1, . . . , n} : xp = 1}. We show that (x, a) satisfies (B.2.235) by checking all possible values of x1,x2 and x3 (and hence for a1, a2 and a3).

Case (i): x1 = 0, x2 = 0 and x3 = 0. In this case, a1 = a2 = a3 = 0. Since I ≥ 0 by assumption,(B.2.235) is satisfied.

Case (ii): x1 = 0, x2 = 0 and x3 = 1. In this case, a1 = a2 = 0. Since 3 ∈ S and Lemma 4.2.1gives a3 ≤ min


Case (iii): x1 = 0, x2 = 1 and x3 = 0. In this case, a1 = a3 = 0. Since a2 ≥ 0 by definition, then(B.2.235) is satisfied.


Case (iv): x1 = 1, x2 = 0 and x3 = 0. In this case, a2 = a3 = 0. Since 1 ∈ S and Lemma 4.2.1gives a1 ≤ min


Case (v): x1 = 0, x2 = 1 and x3 = 1. In this case, a1 = 0 and from the definition of ES-C1 it mustbe that a2 = a3. Since a2 ≥ 0 by definition, then (B.2.235) is satisfied.

Case (vi): x1 = 1, x2 = 1 and x3 = 0. In this case, a3 = 0 and from the definition of ES-C1 itmust be that a1 = a2. Since a2 ≥ 0 by definition, then (B.2.235) is satisfied.

Case (vii): x1 = 1, x2 = 1 and x3 = 1. In this case, from the definition of ES-C1 it must be thata1 = a2 = a3. Hence (B.2.235) is satisfied.

We now show that (B.2.235) defines a facet of ESP -C1. Let F = {(x, a) ∈ ESP -C1 : I(1− x2)−a1 + 2a2 − a3 = 0}. We will show that F is a facet of ESP -C1 using Theorem 3.6 in Nemhauserand Wolsey [53]. Thus we note that constraint (B.2.235) can be equivalently expressed as

(e2,

1Ie1 −

2Ie2 +

1Ie3

)(x, a) ≤ 1,

sinceI − Ix2 − a1 + 2a2 − a3 ≥ 0

⇔ Ix2 + a1 − 2a2 + a3 ≤ I

⇔ x2 +1Ia1 −

2Ia2 +

1Ia3 ≤ 1

⇔(e2,

1Ie1 −

2Ie2 +

1Ie3

)(x, a) ≤ 1.


(µ, λ) = µ0

(e2,

1Ie1 −

2Ie2 +

1Ie3

)and so are able to conclude that F is a facet. We do this by considering five classes of points inF and observing what each implies about the values of µ, λ and µ0. The points take the form ofthose in ES′-C1 and so are easily seen to be in ESP -C1. In each case we assert that they are alsoin F ; this is readily checked by observation.

1. (e2, 0) ∈ F , Now by our supposition it must be that (µ, λ)(e2, 0) = µ0, implying that µ2 = µ0.

2. (p′∑p=1


ep, 0) =



µ4 = 0. Continuing in this manner yields µp = 0 for p = 1, . . . , n, p 6= 2.




5. (p′∑p=1

ep, I(p′∑p=1


(µ, λ)(p′∑p=1

ep, I(p′∑p=1


µp + I(p′∑p=1

λp) = µ0. Now if p′ = 3 then λ2 =−2Iµ0

since λ1 = λ3 =1Iµ0; if p′ = 4 then λ4 = 0 since λ1 + λ2 + λ3 = 0; if p′ = 5 then λ5 = 0.

Continuing in this manner yields λp = 0 for p = 4, . . . , n. (We have also used µ2 = µ0 andµp = 0 for p = 1, . . . , n, p 6= 2).

Thus (µ, λ) = µ0

(e2,

1Ie1 +

−2Ie2 +

1Ie3

)as required.

Therefore the ‘3a-coeff2-x-eq’ constraint, (B.2.234), is facet defining for ESP -C1.

B.2.6 Constraint ‘4a-any-x-eq’ is a Facet of ESP -C1.

Proposition B.2.6.1. For any s, t, u, v = 1, . . . , n, s 6= t, s 6= u, s 6= v, t < u < v, Is = It = Iu = Iv =I the constraint

I(1− xs) + as − at + au − av ≥ 0 (B.2.236)

is facet defining for ESP -C1. We call this constraint the ‘4a-any-x-eq’ facet of ESP -C1. The‘4a-any-x-eq’ facet is a generalisation of the constraints given in Section B.2.2.6.

Remark: The conditions of Proposition B.2.6.1 allow the following orderings of cells in any row(not necessarily in consecutive order): stuv, tsuv, tusv, tuvs. That is, tuv must always be orderedand s is allowed to exist anywhere in the order.

Proof. Without loss of generality we take s = 1, t = 2, u = 3 and v = 4 and we assume I1 = I2 =I3 = I4 = I. Constraint (B.2.236) becomes:

I(1− x1) + a1 − a2 + a3 − a4 ≥ 0. (B.2.237)

First we must check the validity of constraint (B.2.237). Let (x, a) ∈ ESP -C1 and define S = {p ∈{1, . . . , n} : xp = 1}. We show that (x, a) satisfies (B.2.237) by checking all possible values of x1,x2, x3 and x4 (and hence for a1, a2, a3 and a4).

Case (i): x1 = 0, x2 = 0, x3 = 0 and x4 = 0. In this case, a1 = a2 = a3 = a4 = 0. Since I ≥ 0 byassumption, (B.2.237) is satisfied.

Case (ii): x1 = 0, x2 = 0, x3 = 0 and x4 = 1. In this case, a1 = a2 = a3 = 0. Since 4 ∈ S andLemma 4.2.1 gives a4 ≤ min


Case (iii): x1 = 0, x2 = 0, x3 = 1 and x4 = 0. In this case, a1 = a2 = a4 = 0. Since I ≥ 0 byassumption and a3 ≥ 0 by definition, then (B.2.237) is satisfied.

Case (iv): x1 = 0, x2 = 1, x3 = 0 and x4 = 0. In this case, a1 = a3 = a4 = 0. Since 2 ∈ S andLemma 4.2.1 gives a2 ≤ min



Case (v): x1 = 1, x2 = 0, x3 = 0 and x4 = 0. In this case, a2 = a3 = a4 = 0. Since a1 ≥ 0 bydefinition, then (B.2.237) is satisfied.

Case (vi): x1 = 0, x2 = 0, x3 = 1 and x4 = 1. In this case, a1 = a2 = 0 and from the definition ofES-C1 it must be that a3 = a4. Since I ≥ 0 by assumption, then (B.2.237) is satisfied.

Case (vii): x1 = 0, x2 = 1, x3 = 1 and x4 = 0. In this case, a1 = a4 = 0 and from the definition ofES-C1 it must be that a2 = a3. Since I ≥ 0 by assumption, then (B.2.237) is satisfied.

Case (viii): x1 = 1, x2 = 1, x3 = 0 and x4 = 0. In this case, a3 = a4 = 0 and from the definitionof ES-C1 it must be that a1 = a2. Hence (B.2.237) is satisfied.

Case (ix): x1 = 0, x2 = 1, x3 = 1 and x4 = 1. In this case, a1 = 0 and from the definition of ES-C1 it must be that a2 = a3 = a4. Since 2 ∈ S and Lemma 4.2.1 gives a2 ≤ min

p∈SIp ≤ I2, and

I2 = I, then (B.2.237) is satisfied.

Case (x): x1 = 1, x2 = 1, x3 = 1 and x4 = 0. In this case, a4 = 0 and from the definition of ES-C1it must be that a1 = a2 = a3. Since a2 ≥ 0 by definition, then (B.2.237) is satisfied.

Case (xi): x1 = 1, x2 = 1, x3 = 1 and x4 = 1. In this case, from the definition of ES-C1 it mustbe that a1 = a2 = a3 = a4. Hence (B.2.237) is satisfied.

We now show that (B.2.237) defines a facet of ESP -C1. Let F = {(x, a) ∈ ESP -C1 : I(1− x1) +a1−a2 +a3−a4 = 0}. We will show that F is a facet of ESP -C1 using Theorem 3.6 in Nemhauserand Wolsey [53]. Thus we note that constraint (B.2.237) can be equivalently expressed as

(e1,−

1Ie1 +

1Ie2 −

1Ie3 +

1Ie4

)(x, a) ≤ 1,

sinceI − Ix1 + a1 − a2 + a3 − a4 ≥ 0

⇔ Ix1 − a1 + a2 − a3 + a4 ≤ I

⇔ x1 −1Ia1 +

1Ia2 −

1Ia3 +

1Ia4 ≤ 1

⇔(e1,−

1Ie1 +

1Ie2 −

1Ie3 +

1Ie4

)(x, a) ≤ 1.


(µ, λ) = µ0

(e1,−

1Ie1 +

1Ie2 −

1Ie3 +

1Ie4

)and so are able to conclude that F is a facet. We do this by considering five classes of points inF and observing what each implies about the values of µ, λ and µ0. The points take the form ofthose in ES′-C1 and so are easily seen to be in ESP -C1. In each case we assert that they are alsoin F ; this is readily checked by observation.

1. (p′∑p=1


ep, 0) =

B.3. Tables of Results for the Application of Facets of ESP to the JS Model 221


µp = µ0. Now if p′ = 1 then µ1 = µ0; if p′ = 2 then µ2 = 0; if p′ = 3 then

µ3 = 0. Continuing in this manner yields µp = 0 for p = 2, . . . , n.



4. (e1 + e2, I(e1 + e2)) ∈ F . Now (µ, λ)(e1 + e2, I(e1 + e2)) = µ0, so µ1 + µ2 + I(λ1 + λ2) = µ0

and thus λ1 = −λ2 =−1Iµ0.

5. (p′∑p=1

ep, I(p′∑p=1


(µ, λ)(p′∑p=1

ep, I(p′∑p=1

ep)) = µ0, sop′∑p=1

µp + I(p′∑p=1

λp) = µ0. Now if p′ = 4 then λ3 =−1Iµ0

since λ1 = −λ2 and λ4 =1Iµ0; if p′ = 5 then λ5 = 0 since λ1 = −λ2 = λ3 = −λ4; if p′ = 6

then λ6 = 0. Continuing in this manner yields λp = 0 for p = 5, . . . , n. (We have also usedµ1 = µ0 and µp = 0 for p = 2, . . . , n).

Thus (µ, λ) = µ0

(e1,−

1Ie1 +

1Ie2 −

1Ie3 +

1Ie4

)as required.

Therefore the ‘4a-any-x-eq’ constraint, (B.2.236), is facet defining for ESP -C1.

B.3 Tables of Results for the Application of Facets of ESP to the JS Model


Tab

leB

.3.1

:R

esul

tsfo

rth

eap

plic

atio

nof

the

‘1a

-big

-x-s

mal

l-x

’Fa

cet

ofESP

,C

onst

rain

t(B

.1.1

),to

the

JSm

odel

.W

eus

eC

PL

EX

vers

ion

8.1

and

AM

PL

vers

ion

8.1

ona

2GH

zA

MD

3000

+:

prep

roce

ssin

gop

tion

soff

,ti

me

inse

cond

s,2-

hour

tim

elim

iton

indi

vidu

alpr

oble

min

stan

ces.

Data

JS

mod

elw

ith

ou

t(B

.1.1

)S

olu

tion

Nu

mb

erof

JS

mod

elw

ith

(B.1

.1)

Solu

tion

Nu

mb

erof

Con

stra

ints

Con

stra

ints

wit

hou

t

(B.1

.1)

wit

h

(B.1

.1)

BB

ITS

Tim

eR

NL

Bb k

’s,o

bj

BB

ITS

Tim

eR

NL

Bb k

’s,o

bj

44

05

11070

6280

0.4

01.8

04,3

,1,1

,ob

j=4

347

2303

29570

2.8

21.8

04,3

,1,1

,ob

j=4

759

44

05

212

148

0.0

71.0

03,1

,1,o

bj=

3264

14

191

0.4

11.0

03,1

,1,o

bj=

3576

44

05

325

240

0.0

71.2

52,2

,1,o

bj=

3264

11

262

0.4

11.2

52,2

,1,o

bj=

3534

44

05

456

507

0.1

01.4

04,1

,1,1

,ob

j=4

347

144

2382

0.8

61.4

03,2

,1,1

,ob

j=4

759

44

05

57099

62782

2.8

81.6

03,2

,1,1

,1,o

bj=

5430

14322

191783

15.1

91.6

03,2

,1,1

,1,o

bj=

5940

44

05

6280

2582

0.2

61.8

03,3

,2,1

,ob

j=4

430

501

9333

2.3

31.8

03,3

,2,1

,ob

j=4

950

44

05

759

571

0.1

11.5

02,2

,1,1

,ob

j=4

347

758

10080

1.7

01.5

02,2

,1,1

,ob

j=4

703

44

05

8131

789

0.1

61.0

02,1

,1,1

,ob

j=4

347

23

598

0.7

01.0

02,1

,1,1

,ob

j=4

755

44

05

9719

5415

0.3

41.8

04,2

,2,1

,ob

j=4

347

1470

16795

2.4

21.8

04,2

,2,1

,ob

j=4

747

44

05

10

72

762

0.1

01.7

53,2

,1,1

,ob

j=4

347

40

932

0.4

81.7

53,2

,1,1

,ob

j=4

735


Tab

leB

.3.2

:R

esul

tsfo

rth

eap

plic

atio

nof

the

‘3a

-any

-x’

Face

tofESP

,C

onst

rain

t(B

.1.3

),to

the

JSm

odel

.W

eus

eC

PL

EX

vers

ion

8.1

and

AM

PL

vers

ion

8.1

ona

2GH

zA

MD

3000

+:

prep

roce

ssin

gop

tion

soff

,ti

me

inse

cond

s,2-

hour

tim

elim

iton

indi

vidu

alpr

oble

min

stan

ces.

Data

JS

mod

elw

ith

ou

t(B

.1.3

)S

olu

tion

Nu

mb

erof

JS

mod

elw

ith

(B.1

.3)

Solu

tion

Nu

mb

erof

Con

stra

ints

Con

stra

ints

wit

hou

t

(B.1

.3)

wit

h

(B.1

.3)

BB

ITS

Tim

eR

NL

Bb k

’s,o

bj

BB

ITS

Tim

eR

NL

Bb k

’s,o

bj

44

05

11070

6280

0.4

01.8

04,3

,1,1

,ob

j=4

347

40

1027

3.8

81.8

04,3

,1,1

,ob

j=4

6115

44

05

212

148

0.0

71.0

03,1

,1,o

bj=

3264

11

446

2.3

51.0

03,1

,1,o

bj=

34632

44

05

325

240

0.0

71.2

52,2

,1,o

bj=

3264

10

400

2.5

41.2

52,2

,1,o

bj=

34044

44

05

456

507

0.1

01.4

04,1

,1,1

,ob

j=4

347

92

3231

6.0

31.4

04,1

,1,1

,ob

j=4

6115

44

05

57099

62782

2.8

81.6

03,2

,1,1

,1,o

bj=

5430

12590

459859

399.1

61.6

03,2

,1,1

,1,o

bj=

57570

44

05

6280

2582

0.2

61.8

03,3

,2,1

,ob

j=4

430

3596

217428

371.8

61.8

03,3

,2,1

,ob

j=4

7710

44

05

759

571

0.1

11.5

02,2

,1,1

,ob

j=4

347

188

4678

6.1

71.5

02,2

,1,1

,ob

j=4

5331

44

05

8131

789

0.1

61.0

02,1

,1,1

,ob

j=4

347

51

2973

6.9

11.0

02,1

,1,1

,ob

j=4

6059

44

05

9719

5415

0.3

41.8

04,2

,2,1

,ob

j=4

347

54

2098

4.8

21.8

04,2

,2,1

,ob

j=4

5947

44

05

10

72

762

0.1

01.7

53,2

,1,1

,ob

j=4

347

143

3156

4.8

81.7

53,2

,1,1

,ob

j=4

5779


Tab

leB

.3.3

:R

esul

tsfo

rth

eap

plic

atio

nof

the

‘3a

-big

-x-a

ny-x

’Fa

cet

ofESP

,C

onst

rain

t(B

.1.5

),to

the

JSm

odel

.W

eus

eC

PL

EX

vers

ion

8.1

and

AM

PL

vers

ion

8.1

ona

2GH

zA

MD

3000

+:

prep

roce

ssin

gop

tion

soff

,ti

me

inse

cond

s,2-

hour

tim

elim

iton

indi

vidu

alpr

oble

min

stan

ces.

Data

JS

mod

elw

ith

ou

t(B

.1.5

)S

olu

tion

Nu

mb

erof

JS

mod

elw

ith

(B.1

.5)

Solu

tion

Nu

mb

erof

Con

stra

ints

Con

stra

ints

wit

hou

t

(B.1

.5)

wit

h

(B.1

.5)

BB

ITS

Tim

eR

NL

Bb k

’s,o

bj

BB

ITS

Tim

eR

NL

Bb k

’s,o

bj

44

05

11070

6280

0.4

01.8

04,3

,1,1

,ob

j=4

347

54

3916

10.7

01.8

04,3

,1,1

,ob

j=4

6115

44

05

212

148

0.0

71.0

03,1

,1,o

bj=

3264

13

527

3.3

71.0

02,2

,1,o

bj=

34632

44

05

325

240

0.0

71.2

52,2

,1,o

bj=

3264

210

15628

13.3

61.2

52,2

,1,o

bj=

34044

44

05

456

507

0.1

01.4

04,1

,1,1

,ob

j=4

347

286

18058

29.1

71.4

04,1

,1,1

,ob

j=4

6115

44

05

57099

62782

2.8

81.6

03,2

,1,1

,1,o

bj=

5430

27784

1665642

2755.1

41.6

03,2

,1,1

,1,o

bj=

57570

44

05

6280

2582

0.2

61.8

03,3

,2,1

,ob

j=4

430

>43116

>3809159

>7229.4

91.8

0ti

mel

imit

7710

44

05

759

571

0.1

11.5

02,2

,1,1

,ob

j=4

347

400

32173

41.9

51.5

02,2

,1,1

,ob

j=4

5331

44

05

8131

789

0.1

61.0

02,1

,1,1

,ob

j=4

347

118

7100

13.0

81.0

02,1

,1,1

,ob

j=4

6059

44

05

9719

5415

0.3

41.8

04,2

,2,1

,ob

j=4

347

95

4595

12.0

41.8

04,2

,2,1

,ob

j=4

5947

44

05

10

72

762

0.1

01.7

53,2

,1,1

,ob

j=4

347

75

4303

11.2

31.7

53,2

,1,1

,ob

j=4

5779


Tab

leB

.3.4

:R

esul

tsfo

rth

esi

mul

tane

ous

appl

icat

ion

ofth

efir

stfo

urfa

cets

dete

rmin

edfo

rESP

(‘2a

-any

-x’,

‘1a

-big

-x-s

mal

l-x

’,‘3a

-any

-x’

and

‘3a

-bi

g-x

-any

-x’)

corr

espo

ndin

gto

Con

stra

ints

(4.4

.1),

(B.1

.1),

(B.1

.3)

and

(B.1

.5)

resp

ecti

vely

,to

the

JSm

odel

.W

eus

eC

PL

EX

vers

ion

8.1

and

AM

PL

vers

ion

8.1

ona

2GH

zA

MD

3000

+:

prep

roce

ssin

gop

tion

soff

,ti

me

inse

cond

s,2-

hour

tim

elim

iton

indi

vidu

alpr

oble

min

stan

ces.

Data

JS

mod

elw

/o

(4.4

.1),

(B.1

.1),

Solu

tion

Nu

mb

erof

JS

mod

elw

ith

(4.4

.1),

Solu

tion

Nu

mb

erof

(B.1

.3)

an

d(B

.1.5

)C

on

stra

ints

(B.1

.1),

(B.1

.3)

an

d(B

.1.5

)C

on

stra

ints

w/o

(4.4

.1),

(B.1

.1),

(B.1

.3)

an

d

(B.1

.5)

wit

h

(4.4

.1),

(B.1

.1),

(B.1

.3)

an

d

(B.1

.5)

BB

ITS

Tim

eR

NL

Bb k

’s,o

bj

BB

ITS

Tim

eR

NL

Bb k

’s,o

bj

44

05

11070

6280

0.4

01.8

04,3

,1,1

,ob

j=4

347

1522

65544

247.3

81.8

04,3

,1,1

,ob

j=4

13255

44

05

212

148

0.0

71.0

03,1

,1,o

bj=

3264

48

2690

13.3

31.0

03,1

,1,o

bj=

310032

44

05

325

240

0.0

71.2

52,2

,1,o

bj=

3264

7543

8.0

61.2

52,2

,1,o

bj=

38814

44

05

456

507

0.1

01.4

04,1

,1,1

,ob

j=4

347

489

27796

122.5

11.4

04,1

,1,1

,ob

j=4

13255

44

05

57099

62782

2.8

81.6

03,2

,1,1

,1,o

bj=

5430

>17165

>1312141

>7219.5

31.6

0ti

mel

imit

16420

44

05

6280

2582

0.2

61.8

03,3

,2,1

,ob

j=4

430

2817

227375

1371.9

31.8

03,3

,2,1

,ob

j=4

16710

44

05

759

571

0.1

11.5

02,2

,1,1

,ob

j=4

347

2181

147133

460.7

01.5

02,2

,1,1

,ob

j=4

11631

44

05

8131

789

0.1

61.0

02,1

,1,1

,ob

j=4

347

79

6139

39.9

81.0

02,1

,1,1

,ob

j=4

13139

44

05

9719

5415

0.3

41.8

04,2

,2,1

,ob

j=4

347

59

3579

35.9

01.8

04,2

,2,1

,ob

j=4

12907

44

05

10

72

762

0.1

01.7

53,2

,1,1

,ob

j=4

347

225

11732

53.4

61.7

53,2

,1,1

,ob

j=4

12559


227

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