ARTICLE IN PRESS

Engineering Applications of Artificial Intelligence 21 (2008) 691– 697

Contents lists available at ScienceDirect

Engineering Applications of Artificial Intelligence

0952-19

doi:10.1

� Corr

E-m

(J.C. Bec

journal homepage: www.elsevier.com/locate/engappai

A global constraint for total weighted completion time forcumulative resources

Andras Kovacs a,�, J. Christopher Beck b

a Computer and Automation Research Institute, Hungarian Academy of Sciences, Budapest, Hungaryb Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Canada

a r t i c l e i n f o

Article history:

Received 29 January 2008

Accepted 1 March 2008Available online 28 April 2008

Keywords:

Constraint programming

Scheduling

Total weighted completion time

Global constraint

Container loading

76/$ - see front matter & 2008 Elsevier Ltd. A

016/j.engappai.2008.03.004

esponding author.

ail addresses: akovacs@sztaki.hu (A. Kovacs),

k).

a b s t r a c t

The criterion of total weighted completion time occurs as a sub-problem of combinatorial optimization

problems in such diverse areas as scheduling, container loading and storage assignment in warehouses.

These applications often necessitate considering a rich set of requirements and preferences, which

makes constraint programming (CP) an effective modeling and solving approach. On the other hand,

basic CP techniques can be inefficient in solving models that require inference over sum type

expressions. In this paper, we address increasing the solution efficiency of constraint-based approaches

to cumulative resource scheduling with the above criterion. Extending previous results for unary

capacity resources, we define the COMPLETIONm global constraint for propagating the total weighted

completion time of activities that require the same cumulative resource. We present empirical results in

two different problem domains: scheduling a single cumulative resource, and container loading with

constraints on the location of the center of gravity. In both domains, the proposed constraint

propagation algorithm out-performs existing propagation techniques.

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Much of the success of constraint programming (CP) arises forthe ability to declaratively model and solve a given problem with acombination of global constraints. A global constraint captures arecurrent and important relation amongst a set of variables (vanHoeve and Katriel, 2006) and, importantly, has an associatedalgorithm for efficiently removing inconsistent values from thedomains of the variables. The classic example of a globalconstraint is the all-different constraint which imposes that aset of variables must each take a unique value and which has aninference algorithm based on matchings in bi-partite graph(Regin, 1994). As CP has been increasingly used to addresscombinatorial optimization problems, the need to explicitlyincorporate reasoning about costs within global constraints hasbeen recognized (Focacci et al., 2002b). In this paper, we continuethis line of work by addressing the objective function of the totalweighted completion time of a set of jobs that must share a singlecumulative resource. While such a constraint is directly applicableto and, indeed, most clearly explained in the context of,scheduling problems, there are a wide variety of combinatorialoptimization problems in other domains for which this constraint

ll rights reserved.

jcb@mie.utoronto.ca

is useful. In particular, in this paper, after validating the constrainton scheduling problems, we use it to model and solve containerloading problems. Our empirical results demonstrate significantimprovement in problem solving in both domains as compared tothe standard weighted-sum constraint.

In the next section, we provide background on the unarycapacity COMPLETION constraint and a short discussion ofrelated work. We then present the details of the cumulativeCOMPLETIONm constraint including our proposed propagationalgorithm. Section 4 empirically validates the global constraint byapplying it to single machine scheduling problems. In Section 5,we turn to the container loading problem and demonstrate howsuch a problem can be modeled and solved with the use of theCOMPLETIONm constraint. We then conclude in Section 6.

2. Previous work

A critical component of the success of CP techniques tooptimization problems is the ability to design a model thatexhibits significant back propagation. Back propagation is thereduction in search space through pruning of the domains ofdecision variables as a result of a new bound on the objectivefunction. Models that exhibit a high degree of back propagationtend to be successful as each new (sub-optimal) solution willresult in a tighter bound on the cost and, in turn, a smaller

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A. Kovacs, J.C. Beck / Engineering Applications of Artificial Intelligence 21 (2008) 691–697692

subsequent search space. In contrast, without back propagation,the full search space will need to be explored, suggesting that CPwill not result in better performance than any other solutiontechnique.

The significance of cost-based global constraints for strongback propagation has been emphasized by Focacci et al. (2002b).The same authors define a global constraint for the path cost intraveling salesman problems in Focacci et al. (2002a, b). Furtherexamples of cost-based constraints with efficient propagationalgorithms are the global cardinality constraint with costs (Regin,1999), the minimum weight all-different constraint (Sellmann,2002), the cost-regular constraint (Demassey et al., 2006), and theinequality-sum constraint (Regin and Rueher, 2005). The latterconstraint propagates objective functions of the form

Pi xi where

variables xi are subject to binary inequality constraints of the formxj � xipc. The inequality-sum constraint can be applied topropagate the total (non-weighted) completion time or total(non-weighted) tardiness objective functions in a special family ofscheduling problems, the simple temporal problems (STP)(Dechter et al., 1991). An STP involves finding start times foractivities subject to minimum and maximum time lags betweenactivities, but it provides no means for representing limitedcapacity resources.

The COMPLETION constraint is a recently proposed globalconstraint to propagate the total weighted completion time ofactivities on a single unary capacity resource (Kovacs and Beck,2007). Formally, the COMPLETION constraint is defined as follows:

COMPLETIONð½S1; . . . ; Sn�; ½p1; . . . ; pn�; ½w1; . . . ;wn�;CÞ,

where there are n activities, Ai, to be executed without preemp-tion on a single, unary resource. Each activity is characterizedby its processing time, pi, and a non-negative weight, wi. Thestart time variable of Ai is denoted by Si with Ci ¼ Si þ pi beingthe activity’s completion time in the schedule. The totalweighted completion time of the activities will be denoted by C.We assume that all data are integral. The constraint enforcesC ¼

Pi wiðSi þ piÞ.

The COMPLETION constraint can be viewed as requiring asolution to a scheduling problem denoted as 1jri; dij

PwiCi in the

classical scheduling notation. Though this problem is NP-hard, apre-emptive variant with a slightly modified objective function issolvable in polynomial time and can serve as a lower bound. LetMi be the mean-busy time of activity Ai, i.e., the mean point intime at which the machine is busy processing activity Ai. Theproblem 1jri; pmtnj

PwiMi can be solved in Oðn log nÞ where n is

the number of activities (Goemans et al., 2002). The COMPLETIONconstraint filters the domains of the start time variables bycomputing the cost of the optimal preemptive mean-busy timerelaxation for each activity Ai and each possible start time t ofactivity Ai, with the added constraint that activity Ai must start attime t. If the cost of the relaxed solution is greater than the upperbound on the cost, then t is removed from the domain of Si. Inpractice, the relaxation is not actually re-solved for each start timein the domain of each activity. Instead, a much faster algorithmtransforms an initial relaxed solution into a series of preemptiveschedules with given start time assignments. For a detailedpresentation of this algorithm and the COMPLETION constraintreaders are referred to Kovacs and Beck (2007).

Fig. 1. The variable-intensity relaxation of the cumulative resource scheduling

problem.

3. Propagating total weighted completion time on acumulative resource

Extending the COMPLETION constraint to a cumulativeresource, we introduce the global constraint COMPLETIONm,which states that given a set of non-preemptive activities

fA1; . . . ;Ang that require the same cumulative resource withcapacity R, the total weighted completion time of these activitiesis C. The constraint takes the form

COMPLETIONmð½S1; . . . ; Sn�; ½p1; . . . ; pn�; ½R1; . . . ; Rn�,

½w1; . . . ;wn�;R;CÞ,

where finite-domain variables Si stand for the start time of Ai,while pi, Ri, and wi denote the duration, the capacity requirement,and the weight of Ai, respectively. The cost variable C is also afinite-domain variable. We assume that pi, Ri, wi, and R are non-negative integer constants, however, our approach can be easilyadapted to reasoning with the lower bounds of pi, Ri, and wi, andthe upper bound of R. The minimum and maximum values in thecurrent domain of a variable X will be denoted by �X and X,respectively. When appropriate, we call the current lower boundof a start time variable, �Si, the release time of the activity, anddenote it by ri. For brevity, we denote the relative weight of anactivity by mi ¼ wi=piRi.

The COMPLETIONm constraint tightens the lower and theupper bounds of variables Si, but only the lower bound of C. Thelatter is sufficient in problems where the cost is to be minimized.In other applications, the upper bound of C can be tightened byposting a COMPLETIONm constraint on a flipped schedule.

3.1. The relaxed problem

A standard approach to the effective propagation of a globalconstraint is to embed a polynomially solvable relaxation into theconstraint. For the COMPLETIONm constraint, we seek a relaxationof the single cumulative resource total weighted completion timeproblem that simultaneously considers capacity constraints,resource requirements, and release times. Unlike in the unarycase, such a relaxation does not appear to have been presented inthe literature. We therefore propose a novel relaxation of thisscheduling problem.

In the relaxed problem, defined in Fig. 1, we assume thatactivities Ai can be executed with a varying intensity over time.That is, in each time period ½t; t þ 1Þ; t ¼ 0; . . . ; T � 1, an intensityxi

t of Ai can be chosen from ½0;R�. As an extremity of varyingintensity, preemption is allowed. The sum of the intensities overtime has to match the original volume of the activity (3), and thecapacity constraint must be respected (4). The release timeconstraint (5) now states that Ai cannot be processed before ri,and for tXri, the maximum volume of Ai processed in ½0; t� growslinearly with t. The objective is to minimize the total weightedmean-busy time of the activities (

Pi;t tmix

it), shifted by a constant

(12

Pi wi) that compensates for the difference between the mean-

busy time and the total weighted completion time criteria.Below, we differentiate between the original problem and thevariable-intensity relaxation by denoting the former as P, and the

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A. Kovacs, J.C. Beck / Engineering Applications of Artificial Intelligence 21 (2008) 691–697 693

latter as P0. C0 will stand for the cost of the optimal relaxedsolution to P0.

Proposition 1. C0 is a valid lower bound on the original problem P.

Proof. Let us denote the optimal schedule for P by s and its totalweighted completion time by C. Since each activity is processedwithout preemption in s, the difference between the totalweighted completion time (weighted sum of the end times,P

i wiCi) and the mean-busy time (weighted sum of the mid-

points,P

i wiMi) equals 12

Pi wipi. In the variable intensity

representation, the mean-busy time of s can be calculated asPi;t tmix

it . Note that s is a feasible solution of the relaxed problem

P0 as well, and its relaxed solution cost equals

Xi;t

tmixit þ

1

2

Xi

wipi ¼X

i

wiMi þ1

2

Xi

wipi

¼X

i

wiCi ¼ C.

Since s is a possibly sub-optimal relaxed solution, its cost, C, is

greater or equal to the cost of the optimal relaxed solution C0.

Hence, C0 is a lower bound on C. &

The optimal solution of the variable-intensity mean-busy timerelaxation can be computed using the following procedure, calledPrepareRelaxed(), which constructs the schedule chron-ologically. At each point in time t when a scheduling decision hasto be made, the algorithm assigns intensities to activities in theorder of non-increasing mi. The intensity of Ai will be theminimum of

�

the intensity allowed by constraint (5),maxð0; ðt � ri þ 1ÞRiÞ �

Pt�1t0¼0 xi

t; P

� the remaining volume of Ai, piRi �

t�1t0¼0 xi

t;

� the remaining capacity for the subsequent time period.

These intensity values are applied until an activity can no longerbe processed at this rate or the algorithm reaches the release timeof another activity. The algorithm finishes when all activities arecompletely processed. Since the number of scheduling decisions isat most Oðn2Þ and intensities can be assigned in OðnÞ time, theoverall time complexity of the algorithm is Oðn3Þ. Note that thecomplexity is independent of the length of the schedulinghorizon. The pseudo-code of the algorithm is presented in Fig. 2.

Fig. 2. Algorithm for solving the variable-intensity relaxed problem.

Proposition 2. The above algorithm builds an optimal schedule for

the variable-intensity mean-busy time problem.

Proof. Let s be an arbitrary feasible schedule that differs fromschedule s� built by our algorithm, such that the difference cannotbe characterized by an interchange of intensities betweenactivities with identical relative weights. Let t1 be the earliestpoint in time and Ai1 be the activity with the highest mi1 such thatxi1

t1½s�axi1t1½s

��. The construction of the algorithm ensures thatxi1

t1½s�oxi1t1½s

��. Then, there exists a time t2 and activity Ai2 witht1ot2 and mi14mi2 such that increasing xi1

t1½s� and xi2t2½s� and

decreasing xi2t1½s� and xi1

t2½s� preserves feasibility and improves theobjective value of s. Therefore a schedule that differs essentiallyfrom the one built by the algorithm cannot be optimal. &

3.2. From relaxed solutions to bounds tightening

The above presented algorithm can easily be modified toPrepareRelaxed(Ai; t), which computes optimal relaxedsolutions with the additional constraint that activity Ai must startat t. This can be achieved by assigning ri ¼ t and mi ¼ 1, whichgives activity Ai the largest relative weight among all theactivities. This ensures that Ai starts at t and it is processed atintensity Ri throughout its duration. This restricted relaxedproblem will be denoted by P0hSi; ti, and the value of its optimalsolution by C0hSi; ti. Our approach to tightening the bounds of thestart time variables Si exploits the following proposition.

Proposition 3. If C0hSi; ti4C, then t can be removed from the

domain of Si.

Unlike in the unary case, we are not able to define methods forcomputing C0hSi; ti for all values of i and t in low-degreepolynomial time. Instead, we apply PrepareRelaxed tocompute two relaxed solutions for each activity, for the situationswhen Ai starts at �Si and when it starts at Si. If either of theserelaxed solutions violate the current upper bound on the cost,then we estimate how the current bounds of Si must be modifiedto achieve consistency. Since our methods can under-estimatethe change that is necessary, this procedure must be iterated untilconsistent bounds are found for Si (see Fig. 3). We note that insome uncommon cases the number of adjustment cycles neededto achieve consistent bounds can be large, which is computation-ally costly. For this reason, we limited the number of cycles to 5.

3.2.1. Adjusting the earliest start time

In order to adjust �Si, procedure AdjustEST(s, Ai) departsfrom the relaxed solution s prepared for P0hSi ¼

�Sii. It investigateshow the relaxed cost changes as the start of Ai is increased from �Si

to t4�Si.Since the exact computation of the relaxed cost for all possible

values of t is computationally expensive, our procedure exploits afurther relaxation. We take release times rj such that jai and�Siorjot þ pi, and relax rj to r0j ¼

�Si. This leads to relaxed solutions

Fig. 3. Algorithm for tightening the bounds of the start time variables.

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A. Kovacs, J.C. Beck / Engineering Applications of Artificial Intelligence 21 (2008) 691–697694

in which intensities xjt0 with t0o�Si or t0Xt þ pi equal the

corresponding intensities in s. Activity Ai is processed from t

until t þ pi at intensity Ri. However, in interval ½�Si; t þ pi� activitiesother than Ai will be processed in the non-increasing order of mj.

Solutions to this further relaxed problem can be easily foundfor all relevant values of t. Only a small section of the schedule,namely the section in interval ½�Si; t þ pi� varies over differentvalues of t. However, this section can be represented as a queue.The queue, as well as the cost of the schedule, is updatedincrementally for subsequent values of t at time OðnÞ for each step.This step is iterated until the cost of st decreases below thecurrent upper bound cost C. The earliest start time of Ai is thenupdated to this value of t.

3.2.2. Adjusting the latest start time

Given an optimal relaxed solution s with cost C0 for P0hSi ¼ Sii,let t� denote the earliest point in time with t�XSi þ pi in thisrelaxed solution such that all the volume of the activities that hasbeen released before t� is processed before t�:

8jXt��1

t¼0

xjt ¼minðpjRj; ðt

� � rjÞRjÞ.

Furthermore, let W denote the total weight of activity fragmentsprocessed between Si and t�, including activity Ai:

W ¼Xn

j¼1

Xt��1

t0¼Si

mjxjt0 .

Procedure AdjustLST(s, Ai)computes these values t� and W,and adjust the latest start time of Ai according to the followingproposition.

Proposition 4. Activity Ai cannot start later than S0i ¼ Si�

dðC0 � CÞ=We.

Proof. Let us divide the scheduling horizon to four intervals,

I1 : ½0; S0

iÞ, I2 : ½S0

i; SiÞ, I3 : ½Si; t�Þ, and I4 : ½t

�;1Þ. The construction of

the PrepareRelaxed algorithm ensures that decreasing ri

from Si to S0i does not affect activity intensities in intervals I1 and

I4. Activity fragments in I2 may be swapped later, while activity

fragments in I3 will be swapped earlier by at most Si � S0i. Since the

total weight of activities in I3 is exactly W, this operation

decreases the cost of the relaxed solution by at most WðSi � S0iÞ

pC0 � C. &

3.2.3. Computational complexity

For a given activity Ai, the time and space complexity ofcomputing optimal relaxed solutions for P0hSi ¼

�Sii and P0hSi ¼ Sii

is Oðn3Þ, which equals the maximum size of a relaxed solution. Asingle run of the functions AdjustEST(.) and Ad-justLST(.) takes at most the same amount of time. However,we cannot give an estimation of the number of recomputationcycles that are necessary to achieve consistent bounds. With aconstant limit on the number of cycles (as it is done in ourimplementation), the time complexity of the propagation algo-rithm is Oðn3Þ for a single activity, and Oðn4Þ for a complete run onn activities.

4. Scheduling a single cumulative resource

To evaluate the COMPLETIONm constraint in isolation, we rancomputational experiments on a set of single cumulative resourcescheduling problems with release times, with the objective ofminimizing total weighted completion time. The variables in the

constraint-based model of the problem are the start times of theactivities. They are subject to inequality constraints expressingrelease times, and a single cumulative resource constraint. Wecompared the performance of the SUM and the COMPLETIONm

models. The former uses the standard weighted-sum constraint topropagate the objective function, the latter uses the COMPLETIONm

constraint. The proposed propagation algorithms have been im-plemented in Cþþ and embedded into ILOG Solver and Schedulerversions 6.1. The resource capacity constraint is enforced using thetimetable and the disjunctive constraints (Scheduler, 2002).

We applied a depth-first search with a customized version ofthe SetTimes branching heuristic (Le Pape et al., 1994; Scheduler,2002): in each search node we identify the set of activities thathas not been either scheduled or postponed. We then select theactivity with the minimum start time, breaking ties by choosingthe activity with greatest mi. If any postponed activity can bescheduled before the selected activity, search backtracks imme-diately as such a decision would lead to a state that we havealready searched over. Otherwise a choice point is createdassigning the selected activity to its minimum start time and,on the other branch, postponing the activity. An activity is nolonger postponed when its minimum start time is modified viaconstraint propagation.

Our set of benchmark instances consisted of problems with thenumber of activities, n, taken from f15;20;25;30g, the resourcerequirement range a from f0:5;1:0g, and the release time range bfrom f0;0:2;0:6;1:0g. For each combination of the above values wegenerated 10 instances giving 320 problem instances in total. Thecapacity of the resource was fixed to R ¼ 10. Activity durations, pi,were randomized from ½1;100�with a discrete uniform distribution,weights, wi, from ½1;10�, and resource requirements, Ri, from ½1; aR�.This leads to instances where approximately k ¼ 2

a �12 activities are

processed in parallel on the resource. Hence, the release times wererandomized from ½0;50:5nb=k�. The experiments were run on a1.86 GHz Intel Xeon computer with 2 GB of RAM under a MSWindows Server 2003, with a time limit of 1200 s.

The experimental results are presented in Table 1. Each row ofthe table contains combined results for given values of n and b,achieved with the SUM and the COMPLETIONm models. The resultsdo not depend significantly on the value of a. For each model, Opt isthe number of instances out of 20 for which the optimal solution isfound and proved. Columns MRE and XRE contain the mean andmaximum relative error compared to the best solution known.Column Nodes shows the average number of search nodes, whileTime presents the average search time, including the proof ofoptimality, or 1200 s where the solver hit the time limit.

Problem instances with few activities (n ¼ 15) or release timesvarying substantially (b ¼ 1:0) were easily solved by both models.In contrast, neither model could find and prove optimal solutionsfor the larger and tighter problems. The results confirm that theCOMPLETIONm constraint reduced the search space efficiently forall the instances: it helped find optimal solutions quicker for theeasy problems, prove optimality more often, and construct bettersolutions for the hard instances. The latter is the most significantadvantage of the COMPLETIONm model, since the differencebetween the solutions found reached up to 19% in favor of theCOMPLETIONm model. Notably, the COMPLETIONm model alwaysfound at least as good solution as the SUM model for eachproblem, except for a single problem instance (with n ¼ 30 andb ¼ 0:6, the difference was 0.74%).

5. Using COMPLETIONm to solve the container loading problem

Our primary interest in developing the COMPLETIONm con-straint is to use it as a component in a variety of combinatorial

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Table 1Experimental results: number of instances (out of 20) solved to optimality (Opt), mean and maximum relative error in percent (MRE and XRE), average number of search

nodes (Nodes) and average search time in seconds (Time) with the SUM and the COMPLETIONm model

n b SUM COMPLETIONm

Opt MRE XRE Nodes Time Opt MRE XRE Nodes Time

15 0.0 20 0.00 0.00 1221819 37.8 20 0.00 0.00 41022 16.2

0.2 20 0.00 0.00 80897 2.2 20 0.00 0.00 3359 1.6

0.6 20 0.00 0.00 5175 0.1 20 0.00 0.00 413 0.0

1.0 20 0.00 0.00 5714 0.2 20 0.00 0.00 198 0.1

20 0.0 0 0.25 1.35 33724 243 1200.0 15 0.00 0.00 583550 589.9

0.2 15 0.91 13.84 14223 048 494.5 20 0.00 0.00 48554 50.0

0.6 20 0.00 0.00 813644 36.0 20 0.00 0.00 6654 7.0

1.0 20 0.00 0.00 3225 0.0 20 0.00 0.00 549 0.2

25 0.0 0 2.40 8.61 30768 029 1200.0 2 0.00 0.00 1136 717 1149.0

0.2 0 3.11 11.30 31261661 1200.0 11 0.00 0.00 942258 763.2

0.6 14 1.03 10.48 10424189 401.1 20 0.00 0.00 77950 124.1

1.0 20 0.00 0.00 410177 14.8 20 0.00 0.00 4058 4.5

30 0.0 0 4.01 14.71 31718 440 1200.0 0 0.00 0.00 1550 706 1200.0

0.2 0 3.71 18.72 27453 645 1200.0 3 0.00 0.00 1273 294 1093.9

0.6 6 0.42 3.81 21736 970 946.8 15 0.04 0.74 249753 457.0

1.0 20 0.00 0.00 3132 597 157.1 20 0.00 0.00 50870 77.1

A. Kovacs, J.C. Beck / Engineering Applications of Artificial Intelligence 21 (2008) 691–697 695

optimization problems. In order to evaluate its wider applicability,in this section we use the COMPLETIONm constraint to model andsolve the container loading problem.

The container loading problem involves the placement of a setof items in a container. While the core of the problem correspondsto bin packing with the objective of high volumetric utilization, inpractical applications there are various further requirements.These include stacking conditions, cargo stability, and visibilityand accessibility considerations. The rich set of requirementsmakes CP an attractive modeling approach in this domain (Martinand Fages, 2007). Among the additional requirements, Davies andBischoff (1999) highlight the importance of the weight distribu-tion of the loaded container, focusing on the location of the centerof gravity (COG). The exact requirements depend on the specificapplication, especially on the means of transport. For example, inaircraft loading, or when loading a container that will be lifted bya crane, the COG of the cargo has to be located in the center of thecontainer. Since the length of the container (or the aircraft hull) ismuch greater than its width or height, the longitudinal balance isthe most important issue. In contrast, in road transport, it is oftenpreferred to have the COG above the axles of the vehicle.

For container loading with weight distribution considerations,Davies and Bischoff (1999) proposed a heuristic that achieves highspace utilization combined with an even weight distribution, andclaim that the latter leads to a COG located near to the center ofthe container. Wodziak and Fadel (1994) apply a genetic algorithmto minimize the distance of the COG from the desired location inone, two, and two and a half dimensions. Various authors havefocused on the longitudinal balance, and approached the problemfrom a one-dimensional perspective (e.g. Mathur, 1998). Fasano(2004) proposed a mixed-integer programming approach tosolving the three-dimensional single bin packing problem withorthogonal rotation allowed and the COG location constrained inall three dimensions.

We address the problem of loading box-shaped items into arectangular container, with rotation disallowed. The location ofthe COG can be constrained to an arbitrary rectangular region ofthe container. For simplicity, we present our results on the two-dimensional variant of the problem, although it is straightforwardto extend the model to k dimensions.

5.1. Modeling the problem

Let there be given a set of two-dimensional boxes B1; . . . ;Bn

that have to be placed in a rectangular container of length L andwidth W. Box Bi is characterized by its length ai, width bi, andweight wi. The placement of Bi is described by two integervariables xi and yi that stand for the horizontal and vertical originof the box, respectively. Assuming homogeneous boxes, the COGof the cargo is then located at

ðxCOG; yCOGÞ ¼

Pi wi xi þ

ai

2

� �P

i wi;

Pi wi yi þ

bi

2

� �P

i wi

0BB@

1CCA.

The objective is to find a placement of the boxes in the containersuch that the COG of the cargo is situated in the rectangular areadefined by xCOG

min , xCOGmax, yCOG

min , and yCOGmax. In order to avoid fractional

variables and facilitate simpler computation, our model workswith scaled variables x� and y� as follows:

x� ¼X

i

wiðxi þ aiÞ ¼ axCOG þ bx,

y� ¼X

i

wiðyi þ biÞ ¼ ayCOG þ by,

where a ¼P

i wi, bx ¼12

Pi wiai and by ¼

12

Pi wibi. The bounds on

the COG location in the scaled representation can be described bythe following constants:

x�min ¼ axCOGmin þ bx; y�min ¼ ayCOG

min þ by,

x�max ¼ axCOGmax þ bx; y�max ¼ ayCOG

max þ by.

Then, the container loading problem can be stated as displayed inFig. 4.

Constraints (6) and (7) state that the boxes have to be locatedinside the container. The boxes must not overlap (8). Finally,inequalities (9)–(12) determine the location of the COG andconstrain it in both dimensions. Strong propagation is trivialfor the inequality constraint, and several current constraintsolvers offer efficient global constraints for propagating the

ARTICLE IN PRESS

b i ≈ ρ iB i ≈ A i

COGxmaxCOGxmin

COGymax

COGymin

ai ≈ pi

bi ≈ ρiBi ≈ Ai

Fig. 5. Similarities of a container loading and a cumulative resource scheduling

problem.

Table 2Corresponding notions in container loading and scheduling

Container loading Scheduling

Container length Scheduling horizon

Container width Resource capacity

Box Activity

Box length Activity duration

Box width Activity’s resource requirement

COG location Total weighted completion time (scaled)

Fig. 4. The SUM model of container loading problem.

A. Kovacs, J.C. Beck / Engineering Applications of Artificial Intelligence 21 (2008) 691–697696

non-overlapping relation of a set of boxes (Beldiceanu andCarlsson, 2001; Clautiaux et al., 2008). The weakness of thismodel is the propagation of the weighted-sum constraints inequalities (9) and (10). This model will be referred to as the SUMmodel. In the sequel we demonstrate how the COMPLETIONm

constraint can be used to strengthen the propagation of theconstraints on the COG location.

Observe that x� equals the total weighted completion time in asingle cumulative resource scheduling problem, where thecontainer corresponds to the hull of the schedule, defined by thescheduling horizon (horizontal axis) and the resource capacity(vertical axis). Boxes correspond to activities, with box lengthstanding for activity duration and box width for the resourcerequirement of the activity. The physical weight of the boxcorresponds to the weight assigned to activity Ai.

1 Fig. 5 illustratesthe similarities of the container loading and the schedulingproblems, while the corresponding notions in the two problemdomains are summarized in Table 2. The same relation holdsbetween y� and the total weighted completion time in theschedule that is received by rotating Fig. 5 by 901. Furthermore,since the COMPLETIONm constraint propagates only the lowerbound of the cost but we need both lower and upper bounds on x�

and y�, we post the COMPLETIONm constraint on flipped schedulesas well. Hence, we define the COMPLETIONm model of thecontainer loading problem by adding the following four globalconstraints to the basic SUM model:

COMPLETIONmð½x1; . . . ; xn�; ½a1; . . . ; an�,

½b1; . . . ;bn�; ½w1; . . . ;wn�;W ; x�Þ,

COMPLETIONm ½L� x1 � a1; . . . ; L� xn � an�;

½a1; . . . ; an�; ½b1; . . . ; bn�,

½w1; . . . ;wn�;W ; x� þX

i

wiðLþ 2aiÞ

!,

COMPLETIONmð½y1; . . . ; yn�; ½b1; . . . ; bn�,

½a1; . . . ; an�; ½w1; . . . ;wn�; L; y�Þ,

COMPLETIONm ½W � y1 � b1; . . . ;W � yn � bn�;

½b1; . . . ;bn�; ½a1; . . . ; an�,

½w1; . . . ;wn�; L; y� þ

Xi

wiðW þ 2biÞ

!.

1 Cumulative resource scheduling is a relaxation of container loading, rather

than an equivalent problem. The difference is that in scheduling it is allowed to

assign any two units of the resource to an activity, while in container loading,

adjacent units of the container surface must be assigned to a box. For instance, it is

impossible to place one half of the box to the lowermost, the other half to the

uppermost region of the container.

5.2. Computational experiments

The performance of the COMPLETIONm and the SUM modelsof the container loading problem was compared in computa-tional experiments. We addressed the satisfiability problempresented above, hence the outcome of the solution process waseither a feasible solution or a proof that the boxes could not beloaded with an appropriate weight distribution. A depth-firstsearch was implemented for solving the problem using a simplelabeling heuristic (called IloGenerate in Ilog) that creates twochildren of a search node: (xi ¼ �xi) vs. (xi4�xi) or (yi ¼ �yi) vs.(yi4�yi). We experimented with more sophisticated branchingheuristics as well that made decisions on the relative positions oftwo boxes, but the simple strategy proved to be the best with bothmodels.

Problem instances have been generated with the number ofboxes, n, varying between 10 and 45 with increments of 5, and thecontainer length, L, varying between 20 and 60 with increments of10. The container width was fixed to 8. Box sizes ai and bi weretaken randomly from ½1;5�, while weights wi from ½1;10� with adiscrete uniform distribution. We focused on the longitudinalbalance, and imposed xCOG

min ¼ L=2� 1 and xCOGmax ¼ L=2þ 1. For each

value of n and L, we generated 10 instances, which resulted in 400instances altogether.

The models were implemented in ILOG Solver and Scheduler6.1, using the earlier presented COMPLETIONm constraint, writtenin Cþþ. The experiments were run on a 1.86 GHz Intel Xeoncomputer with 2 GB of RAM under a MS Windows Server 2003operating system, with a time limit of 1200 CPU seconds.

The experimental results are displayed in Table 3. Each rowcontains results for a given number of boxes. Columns Solved,Nodes, and Time show the percentage of solved instances, theaverage number of search nodes, and the average search time foreither of the SUM and the COMPLETIONm models. Nodes and Timealso contain the elapsed effort and time for instances where thetime limit was reached. Instances in which the boxes did not fitinto the container, independent of the COG location constraints,have been excluded from the experimental results.

ARTICLE IN PRESS

Table 3Experimental results on the container loading problem, for the SUM and the

COMPLETIONm models

n SUM COMPLETIONm

Solved (%) Nodes Time (s) Solved (%) Nodes Time (s)

10 100.0 27 0.0 100.0 16 0.0

15 100.0 71530 30.5 100.0 314 0.1

20 97.6 195948 102.1 97.6 25645 28.8

25 68.6 638110 415.3 100.0 4190 6.3

30 40.0 517817 840.5 90.0 62148 133.7

35 28.6 484507 869.1 92.9 32192 139.6

40 31.6 301704 875.6 78.9 71021 301.5

45 8.3 177966 1154.3 75.0 79416 486.5

A. Kovacs, J.C. Beck / Engineering Applications of Artificial Intelligence 21 (2008) 691–697 697

The results illustrate that propagation by the COMPLETIONm

constraint efficiently reduced the search space in the containerloading problem, resulting in an order of magnitude reduction insolution times. Furthermore, the COMPLETIONm model scaledmuch better with n. It solved 75% of the largest instancesexperimented as well, whereas those were practically intractablewith the SUM model.

6. Conclusions

In this paper we investigated constraint-based scheduling withthe total weighted completion time criterion. This schedulingproblem is interesting because it appears as a sub-problem ofcombinatorial optimization problems in diverse areas, such ascontainer loading or storage assignment in warehouses. Extendingthe earlier defined COMPLETION constraint, we introduced theCOMPLETIONm global constraint for propagating the totalweighted completion time of activities on a cumulative resource.Our propagation algorithm is based on a novel variable-intensityrelaxation of the single cumulative resource scheduling problem.

To evaluate the applicability of the COMPLETIONm constraintwith different sets of side constraints, we performed computa-tional experiments in two different problem domains: singlecumulative resource scheduling and container loading withweight distribution considerations. We showed that the locationof the COG of the cargo in a container corresponds to the totalweighted completion time of the activities in a cumulativeresource scheduling problem, and hence, can be propagated bythe COMPLETIONm constraint. In both domains, the introductionof the novel global constraint led to significant improvementcompared to the classical constraint-based model of the problem:better solutions, or the same solutions found often an order ofmagnitude faster.

Acknowledgments

This research was supported in part by the Natural Sciencesand Engineering Research Council of Canada, ILOG, S.A., and theOTKA Grant K 73376. A. Kovacs acknowledges the support of theJanos Bolyai scholarship of the Hungarian Academy of Sciences.

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