An Optimisation Model for Airlift Load Planning:GALAHAD and the Quest for the ‘Holy Grail’

Bohdan L. Kaluzny R.H.A. David Shaw Ahmed Ghanmi Beomjoon Kim

Abstract—This paper presents an aircraft load allocation opti-misation model, which uses a hybrid of simulated annealing andgenetic algorithm methods to solve a multi-objective optimisationproblem associated with allocating a set of cargo items acrossa heterogeneous fleet of available airlift assets. It representscandidate solutions using macrochromosomes comprised of anordered list of available transport assets followed by an orderedlist of cargo items. A bin packing heuristic is used to mapeach individual to a point in asset-utilization space where anovel convex hull based fitness function is used to evaluatethe relative quality of each individual and drive an elitistapplication of genetic operators on the population—includinga novel extinction operation that infrequently culls solutionscomprising of aircraft chalks that cannot be load balanced. Proofof concept computational results are presented.

I. INTRODUCTION

In a military context, force projection means the massive

transportation of equipment and personnel over long distances

under short timelines. Time-critical force projections often

utilize airlift or a combination of airlift and sealift. Planning

an efficient airlift is a difficult problem that combines options

analysis, movement scheduling, routing and packaging deci-

sions with the dual and competing objectives of minimizing

both movement time and cost. This paper considers a restricted

set of military aircraft loading problems associated with plan-

ning deployment and sustainment airlift operations.

Given a manifest of military equipment and supplies with

known dimensions and weights and a transport fleet consisting

of a mixture of different types of aircraft, each with known

cargo bay dimensions and maximum payload restrictions, we

wish to determine the set of loading solutions that make

most efficient use of the available aircraft. While we are only

interested in finding feasible “floor plan” layouts for each of

the utilized aircraft cargo bays, and we disallow stacking of

items, the problem is still in some sense “two-and-a-half”

dimensional—the heights of items may restrict their placement

to certain aircraft types or even to certain locations within a

given aircraft cargo bay. The problem is also an exercise in

multi-objective optimisation, as we are interested in generating

the entire set of Pareto-efficient loading solutions in aircraft

utilization space. We therefore require an algorithm to solve

a multi-objective, two-and-a-half dimensional heterogeneous

Bohdan L. Kaluzny, R.H.A. David Shaw and Ahmed Ghanmi are affiliatedwith the Centre for Operational Research & Analysis, Defence Research &Development Canada, Canadian Department of National Defence, 101 ColonelBy Dr., Ottawa, Ontario, K1A 0K2 (email: {Bohdan.Kaluzny, David.Shaw,Ahmed.Ghanmi}@drdc-rddc.gc.ca). (Albert) Beomjoon Kim is affiliated withthe Department of Mechanical and Mechatronics Engineering, University ofWaterloo, Waterloo, Ontario, Canada (email: b7kim@engmail.uwaterloo.ca).

bin packing problem. To this end we developed a model

named GALAHAD (Genetic Annealing for Loading of Aircraft,

a Heuristic Aiding Deployment).

A. Related Work

General bin packing problems have been demonstrated to

be NP-hard combinatorial optimization problems. We must

therefore use heuristic methods to obtain good solutions in

reasonable amounts of time. Classical heuristics include the

Bottom-Left (BL) and Bottom-Left-Fill (BLF) approaches

(Baker et al., 1980; Chazelle, 1983), in which ordered lists

of items are sequentially packed into the lowest and leftmost

accessible location within the packing layout. The performance

of BL and BLF heuristics is strongly dependent on the ordering

of the list of items to be packed. In recent years, more

sophisticated heuristics have been developed, including the

Best-Fit approach of Burke et al. (2004) and the recursive

approach of Zhang et al. (2006).

Substantial research has also focussed on improving the per-

formance of these basic bin packing heuristics through the use

of metaheuristic search techniques such as simulated annealing

(SA), genetic algorithm (GA) and tabu search (TS). Dowsland

(1993) developed a SA algorithm for two-dimensional bin

packing problems that explored both feasible solutions and

solutions in which some of the items overlap. Jakobs (1996)

proposed a GA approach for bin packing problems based on

a representation of packing pattern by means of a permutation

giving the order in which the items are packed, while the

packing positions are determined using BLF. Hopper and

Turton (2001) further extended this work by analyzing the

perfromance of BL and BLF packing heuristics when coupled

with a wide range of metaheuristic search techniques. They

showed that SA-based hybrids tended to generate the best

packing solutions, albeit at the cost of long running times;

in contrast, GA-based hybrids tended to display the fastest

convergence, but to slightly worse packing solutions than those

found by SA-based hybrids. Lodi et al. (1999) developed a

TS algorithm to solve two-dimensional bin packing problems.

The key aspect of their algorithm is a unified parametric

neighbourhood that is independent of the specific problem

considered, and whose size is dynamically varied during the

search. Burke et al. (2009) have recently developed a SA-based

enhancement of their original Best-Fit heuristic.

With respect to specific aircraft loading applications, the

United States Air Force developed the Automated Air Load

Planning System (AALPS, 1987). AALPS uses rule-based al-

gorithms and models both the packing and balancing aspects of

Proceedings of the 2009 IEEE Symposium on Computational Intelligence in Security and Defense Applications (CISDA 2009)

©2009 Crown Copyright

the aircraft loading problem. The loading algorithm in AALPS

considers two different loading methods: by item priority and

by ratio of the total cargo weight to the available aircraft

cargo bay space. AALPS does not optimise. More recently,

Liu et al. (2007) studied multi-objective two-dimensional

bin packing problems using an evolutionary particle swarm

methodology. The problem combines bin packing optimization

and bin load balancing objectives: the goal is to minimise

the number of bins used and minimise the average deviation

from an ideal centre of gravity (CG). Instead of combining

both objectives into a composite scalar weighting function,

the model incorporates the concept of Pareto optimality to

evolve a family of solutions along the trade-off surface. In the

model the CG of each loaded bin is computed based on the

placement of the items via a bin packing heuristic. No load

balancing algorithms are invoked.

B. Overview of GALAHAD specific features

The GALAHAD algorithm presented attempts to harness the

advantages of both SA and GA methods through use of a

hybrid of the two techniques inspired by Pakhira (2003) and

the local temperature concept of Cho et al. (1998). Local

temperature consists of assigning each individual a fitness-

based temperature, with higher temperatures assigned to less

fit individuals and vice versa. These temperatures, in turn,

influence the probability that an individual will reproduce,

undergo genetic mutations, etc. GALAHAD also breaks new

ground through use of a convex hull based fitness function to

rank solution quality in the multi-dimensional solution space.

GALAHAD addresses a multi-objective two-dimensional

heterogeneous bin packing problem under load balancing

constraints. In contrast to Liu et al., we wish to simultane-

ously minimize the utilization of several distinct aircraft types

subject to the constraint that all items are transported and all

load plans satisfy CG constraints. The goal, or ‘holy grail’,

is to determine the Pareto optimal family of solutions in the

aircraft utilization space.

C. Organization

This paper is organised as follows. In the following section

the methodology of GALAHAD is detailed. Proof of concept

computational results are presented in Section III. Section IV

summarises the paper and discusses future work.

II. METHODOLOGY

The GALAHAD algorithm uses a combination of meta-

heuristic search techniques to solve aircraft load allocation

optimisation problems. The following sections describe, in

turn, the representation of individuals in genetic space, the bin

packing heuristic used to map individuals from genetic space

into aircraft utilization space, the method used to evaluate the

fitness of individuals, a description of the combination of GA

and SA techniques used to evolve the population from one

generation to the next, and concludes with some refinements

that we have applied to the basic GALAHAD algorithm.

A. Genetic representation of individuals

Each individual in the population is represented in genetic

space by two distinct chromosomes joined together to form a

structure we will call a macrochromosome. One chromosome

consists of a list describing the order in which items will be

considered by a sequential packing heuristic, while the other

chromosome consists of a list describing the order in which

individual aircraft will be loaded. Permuting the order of either

of these lists will realize different loading solutions.

In order to construct the item chromosome, we assign

an index to every item in the cargo manifest. In general,

each item receives a unique index; however, for manifests

containing large numbers of items of the same type it is

sometimes advantageous to assign indices on the basis of

item type instead. In addition, each item type has a specified

set of up to four allowable orthogonal orientations relative

to the longitudinal axis of the aircraft cargo bay. The item

chromosome is simply an ordered list of all item indices; for

each item, one of the allowable orientations is also specified.

The aircraft utilization chromosome is generated in a similar

manner. We assign an index to each distinct airlift asset type;

given a specified number of aircraft of each type available

for use, an ordered list of aircraft labelled by their type

index is formed as the aircraft chromosome. Each “aircraft”

appearing in the aircraft chromosome is more accurately a

chalk of materiel assigned to be carried by an aircraft of the

specified type. If a specified aircraft type appears in the aircraft

chromosome six times, it does not matter whether those six

loads are transported one after the other by a single aircraft

or simultaneously by six distinct aircraft.1

To generate the aircraft chromosome, we need to specify

the maximum number of chalks to be allocated to each aircraft

type. This approach allows the user to focus the search onto

a restricted area of the aircraft utilization space based on

the availability of the various airlift asset types; however, it

can suffer from the drawback that insufficient aircraft may

be provided to allow all items to be loaded. If one wishes

to explore the entire space of efficient loading solutions,

one needs to ensure that sufficient aircraft of each type are

provided.

If we let Ij denote the subset of items that can feasibly be

loaded into an aircraft of type j, then a loose upper bound nj

on the required number nj of aircraft of type j is given by:

nj = max

⎛⎝4

∑i∈Ij

(li + s)(wi + s)(Lj − s)(Wj − s)

, 2∑i∈Ij

mi

Mj

⎞⎠ , (1)

where s is the minimum inter-item spacing, li, wi and mi

are the length, width and weight of item i, and Lj , Wj and

Mj are the cargo bay length, cargo bay width and maximum

allowable payload for an aircraft of type j, respectively. This

upper bound is constructed through use of the continuous

1The number of aircraft available to carry out the movement does affect boththe cost of the movement and the time required to complete it, but GALAHAD

is concerned only with generating the set of efficient loading solutions.

lower bounds on the number of bins required, while the factors

of 2 and 4 explicitly capture the worst-case performance ratio

for the one- and two-dimensional bounds, respectively. The

conservative nature of this bound will generate long aircraft

chromosomes and thereby adversely affect computational ef-

ficiency during genetic operations—development of tighter

upper bounds is desirable.

B. Bin packing

With our genetic representation of individuals in hand, we

now require a means of mapping each individual from its

genetic representation to a point in d-dimensional aircraft

utilization space. As noted in the discussion above, permuting

the order of indices in either or both components of the

macrochromosome will lead to different loading solutions for

any deterministic, sequential, list-based bin packing heuris-

tic. For ease of implementation during the prototyping of

GALAHAD, we have elected to use the simple BLF heuristic;

future research will examine the impact of various bin packing

heuristics on the overall performance of GALAHAD.

For each individual, we break the macrochromosome into its

two components. Using the BLF heuristic, we move sequen-

tially through the two chromosomes, loading each item into the

first available aircraft into which it will fit given the specified

orientation. Once all items are loaded, we count the number

of non-empty aircraft of each type and compose the number

of aircraft used into a vector corresponding to the individual’s

location in d-dimensional aircraft utilization space.

C. Fitness function

Given the location of each individual in aircraft utilization

space, we wish to determine the relative fitness of each

individual with respect to our objective function. Given our

representation of the populations as a set X of points in Rd,

it is of interest to compute the convex hull, denoted conv(X),of the point set. The convex hull is defined as

conv(X) =

{ |X|∑i=1

λixi

∣∣∣∣ xi ∈ X, λi ≥ 0,

|X|∑i=1

λi = 1

}; (2)

from a physical standpoint, the convex hull is the minimal

convex set containing X . The space contained within the

convex hull can be described as a convex polytope P ∈ Rd

with a halfspace or H-representation as

P =

{�x ∈ R

d

∣∣∣∣d∑

j=1

aijxj ≤ bi, i = 1, 2, . . . , m

}(3)

for d-dimensional real vectors �aTj = (a1j , a2j , ..., adj) and real

numbers bi. Each of the m halfspace relations defines a facet

Fi of the convex polytope

Fi = P⋂ {

�x ∈ Rd

∣∣∣∣d∑

j=1

aijxj = bi

}, (4)

which lies on a (d−1)-dimensional hyperplane. The complete

set of facets defines the boundary of the convex polytope Pinduced by the point set X .

The prototype implementation of GALAHAD uses the lrsconvex hull algorithm (Avis, 2000) for facet enumeration.

While efficient convex hull algorithms exist for two- and three-

dimensional cases, the scaling of the computational complexity

as the number of dimensions increases is of some concern.

lrs is a pivoting algorithm using a combination of linear

programming and reverse search techniques and can handle

problems of quite large size. However in general there is no

known algorithm for facet enumeration that is polynomial in

both the input and output size (see Avis et al., 1997). For

typical aircraft loading problems, this computational time is

unlikely to be a limiting factor, as there are usually O(1000)points in X and it is unlikely that the number of aircraft

types will exceed 8. However, the scaling behaviour of the

complexity of computing convex hulls in higher dimensions

could be of concern if the GALAHAD algorithm were to be

adapted to other, larger, convex multi-objective optimisation

problems.

Since we want to find solutions that minimise the utilization

of all aircraft types subject to the constraint that all items

are loaded, it seems reasonable that solutions on or near the

“lower” boundary of P should be assigned higher fitness

values than those far away from the lower boundary. To

formalise this concept of the lower boundary, we note that the

outward pointing normal of each facet can be thought of as

the direction in which the facet would move if the volume of

the polytope were to be increased. As we wish to minimise the

number of aircraft used, we are interested in those facets for

which at least one component of the outward pointing normal

is negative—expansion of the polytope in this direction would

allow for reduction of the usage of at least one type of aircraft,

albeit potentially at the expense of increasing the usage of

other aircraft types. We therefore rigorously define the frontierof the point set X , denoted frontier(X), as a subset of facets

of the induced convex polytope:

frontier(X) ={Fi | ∃ j � aij < 0, ∀ i

}. (5)

If we define d(xi;�aTj , bj) to be the Euclidean distance from

a point xi ∈ X to the hyperplane defined by the equation

�aTj · �x = bj , then the distance di from xi to the nearest facet

of the frontier is given by

di = minj∈frontier(X)

d(xi;�aTj , bj). (6)

If we define dmax = maxj dj to be the largest distance from

the frontier observed in a given generation, then we define the

fitness, fi, of the i-th individual to be

fi =dmax − di

dmax. (7)

By this definition, individuals lying on the frontier will have

the highest possible fitness, f = 1, while the individual

furthest away from the frontier will be assigned f = 0.

D. Genetic operators and local temperature

Once the fitness of the population has been established,

GALAHAD uses a hybrid GA-SA metaheuristic algorithm to

search for improved solutions. In the discussion that follows,

we describe in turn the genetic operators used to evolve the

population, the SA-inspired local temperature assigned to each

individual, and how the local temperature is used to influence

the application of the genetic operators on the population.

There are three broad classes of genetic operators used

in GALAHAD—immigration, mutation and reproduction. Im-

migration is simply the introduction of a small number of

new, randomly generated individuals at each generation in an

effort to prevent premature convergence to a local optimum.

Mutation refers to the “noisy” duplication of an existing

individual in the population—a set of (usually small) changes

are introduced into the genetic code of the new individual. In

the context of our specific problem, these mutations manifest

as permutations of indices or altering item orientations in one

or both of the item and aircraft chromosomes. Reproduction

refers to the creation of new individuals by combining features

of the genetic representation of two parent individuals—

GALAHAD generates offspring via partially mapped crossovers

performed on both the aircraft and the item chromosomes of

the selected pair of parents.

We also assign a fitness-dependent temperature to each

individual in the population, drawing inspiration from the

cooling process in simulated annealing. Given a cooling factor,

α ∈ (0, 1], and a reference temperature, Tref , we define the

local temperature, Ti, of the i-th individual as

Ti = αfiNTref , (8)

where N is the number of individuals in the population.

Adjusting α modulates the degree of variation of local temper-

atures across the population, with the temperature distribution

approaching uniform as α → 1. The resulting local tempera-

ture is inversely proportional to the fitness of the individual—

solutions on the frontier have low temperature while those far

away have high temperature.

At present, the local temperature of an individual is only

used in the selection of those individuals that will act as

parents during reproduction. As we wish to bias selection

towards fitter individuals, in each generation, we randomly

select a fitness threshold f ∈ (0, 1) and then determine the

probability pi that a randomly selected individual will be

accepted as a parent using the expression

pi =

{1 fi ≥ f

1− exp(

fi−fkTi

)otherwise,

(9)

where k is a proportionality constant. However, the local

temperature could also be applied to other decisions during

the evolution (e.g., the magnitude of allowable mutations)

through the use of probabilities of the form of the Boltzmann

distribution

pi = exp(−Δzi

k′Ti

), (10)

where k′ is a proportionality constant and Δzi measures the

amount of change between the genetic representations of the

original and modified individuals.

At each generation, the population is evolved using the three

genetic operators according to a user-specified ratio of the

number of new individuals to be introduced via each method.

Once the genetic evolution is completed, every individual in

the population for which fi < 1, i.e., those individuals not

on the frontier, have their age incremented by one unit. Those

individuals older than a specified maximum age are then culled

from the population.

E. Refinements

As we have seen above, GALAHAD maps each individual

from a macrochromosome in genetic space to a point in d-

dimensional aircraft utilization space denoting the number of

each of the d aircraft asset types required to transport all items.

The fitness of each individual is then determined by computing

the Euclidean distance of each individual to the nearest facet of

a subset of the d-dimensional convex hull that we have called

the frontier. Over the course of a large number of generations,

the frontier of the population grows and evolves towards the

set of Pareto-efficient loading solutions for the given manifest

of items and available aircraft types.

In the discussion that follows, we discuss possible refine-

ments of the basic algorithm laid out above that could be used

to improve the rate of convergence to near-optimal solutions

or to improve the real world fidelity of the generated solutions.

1) Genetic engineering: In early testing, we observed that

for initial populations consisting entirely of randomly gener-

ated individuals, the corresponding set of points in solution

space tended to be closely clustered in aircraft utilization

space, as these solutions were generated from individuals

whose DNA consisted of lists with randomly ordered mixtures

of the various aircraft asset types. A large number of gener-

ations are “wasted” while waiting for the frontier to expand

to include more extremal loading solutions—those solutions

which make maximal use of one aircraft type and minimal

use of all others. It is consequently likely that convergence of

the frontier to the Pareto-efficient set will be uneven as the

local neighbourhood of certain facets on the frontier will tend

to have a higher average population density over the course

of the optimisation.

From the structure of the genetic representation of our

individuals, solutions that make maximal use of a particular

aircraft type are most likely to be generated when all the

aircraft of that type are located near the beginning of the

aircraft chromosome. We can therefore ensure that extremal

load solutions are found at the very start of the process through

“genetic engineering”—we introduce artificially constructed

individuals into the population in which we have ordered the

aircraft chromosome such that all aircraft types are grouped

together. As a minimum, a single individual can be introduced

for each of the d distinct aircraft types in which all aircraft of

the specified type are located at the beginning of the aircraft

list; for small values of d, it is even feasible to add individuals

consisting of all d! permutations of the clusters of aircraft

types. Other portions of the frontier can also be explored

by introducing engineered individuals in which the list of

aircraft consists of a regular pattern in which small, contiguous

blocks of each aircraft type appear in the appropriate relative

proportions.

2) Incorporating load balance through periodic extinctions:Each load plan generated by the GALAHAD algorithm outlined

above consists of an allocation of items to aircraft such that

there is a feasible physical layout of the cargo manifest

for each individual aircraft that also satisfies aircraft-specific

item height and total payload constraints. GALAHAD however

neglects an important real-world constraint on aircraft loading

solutions—aerodynamic stability concerns require that the CG

of the payload fall within a specified envelope within the

cargo bay dictated by an aircraft’s specifications. Recently,

Kaluzny and Shaw (2008, 2009) developed a mixed integer

linear program (MILP) formulation of the problem modelling

realistic problem features such as item CG offsets from the

physical centre, free placement, the ability to rotate items

to achieve all four orthogonal orientations, fixed obstacles

or passageways, etc. Unfortunately, while the Kaluzny-Shaw

MILP is generally able to determine whether feasible layouts

of a specified set of cargo items exist for a particular air-

craft in less than a second, on typical GALAHAD instances

with populations of potentially thousands of individuals, each

containing O(100) aircraft loads, the computational overhead

associated with running the MILP on each load plan is

far too large to consider evaluating load balance feasibility

during each generation. However, drawing inspiration from

mass extinctions in nature, we incorporate the additional load-

balance constraints as part of an extinction operator on the

population of solutions. On a periodic basis after a large

number of generations, we evaluate each load plan for load-

balance feasibility and remove those individuals containing

chalks that are found to be unbalanceable. Only individuals

with maximal fitness, f = 1, are exempt from culling. In

this manner, we hope to obtain a reasonable tradeoff between

solution quality and running time.

The action of the extinction operator on an individual

progresses through four phases. First, a set of simple, fast

heuristics are employed on each chalk in an attempt to re-

position the loaded items so that the CG is acceptable. Chalks

not balanced by this filter are subsequently modelled by the

MILP formulation and solved. Infeasible chalks are stored

using an AVL tree database (Adelson-Velskii and Landis,

1962) that is queried in subsequent GALAHAD generations by

a modified bin packing routine to prevent known infeasible

cargo allocations from reoccurring.

a) Heuristics: For each chalk, the CG of the load plan

produced by the bin packing is computed and tested against

the aircraft’s CG envelope. If the layout is infeasible then a

combination of simple heuristics are performed: the fore/aft

mirror image layout is tested, the set of items are translated

together in the direction that minimises the CG displacement

from the envelope (until the layout is feasible or until wall

spacing constraints are violated), and items are sorted by

weight and re-packed in this order.

b) MILPs: Should the heuristics fail to find a feasible

load plan for a particular chalk, a MILP is formulated and an

external solver is called to solve the model. The MILP solver

is allotted a maximum of 5 seconds to solve the instance and

the program returns either “feasible”, “infeasible” or “timeout”

status.

c) AVL database: Chalks tagged as infeasible are stored

using an AVL binary search tree. AVL trees enable lookup

and insertion that takes O(log C) time in both the average and

worst cases, where C is the number of infeasible chalks stored

in the tree. To compactly store an infeasible load configuration

(aircraft type and item set), whose full representation may

exceed 64 bits of information, a surjective mapping onto a

computationally manageable 15 digit integer (storage number)

is defined. The first two digits represent the aircraft type, the

next two the number of items, the next two the number of

unique item types, the next six digits represent the sum of the

item weights in kilograms, and the next three digits the sum

of the item lengths and widths in metres. While the mapping

may be non-injective, it is designed to limit the possibility that

a feasible load configuration shares the same storage number

as an infeasible chalk.

d) Modified bin packing: To ensure that recorded in-

feasible chalks do not re-appear in population individuals, in

subsequent GALAHAD generations the basic BLF heuristic is

modified. Prior to each assignment, GALAHAD searches the

AVL tree database to check if the potential load configuration

is known to be infeasible. If it is, then the item under

consideration is duplicated and replaced by a dummy item with

the same dimensions but with zero weight. The dummy item is

loaded and the original item is moved to the end of the master

item list (to be loaded onto the last aircraft or an additional

one if required). By design this extinction operation eliminates

the chance of obtaining known infeasible load configurations

while confining the changes to the infeasible chalks within a

solution and possibly the last few chalks loaded.

III. PROOF OF CONCEPT

GALAHAD has been implemented as a C++ program with

a JAVA user-interface. Third party software packages lrs,

Zimpl (Koch, 2004) and CPLEX 11.2 (IBM ILOG, 2008)

are incorporated in order to compute high-dimensional convex

hulls, translate MILP formulations into standard MPS format,

and solve formatted MILPs, respectively. GALAHAD has been

used as an analysis tool for Defence R&D Canada – Centre

for Operational Research and Analysis studies (e.g., Ghanmi

and Shaw, 2008). As proof of concept, the practicality of

GALAHAD is exhibited on problem instances based on a

recent large-scale airlift conducted by the Canadian Forces.

The computational results are divided into two sections. First

GALAHAD is applied to a two-dimensional problem to illus-

trate the solution progress and to demonstrate the benefit of

genetically engineering chromosomes. Secondly, the efficacy

of the extinction operator implementing the load balancing

constraints is shown. An Intel R© Xeon R© 5160 processor

running at 2.99 GHz was used for all computations.

The item manifest used in our examples was based on

the early stages of Operation ATHENA, Canada’s participa-

tion in the International Security Assistance Force (ISAF)

in Afghanistan. Operation ATHENA required an airlift de-

ployment of 349 vehicles/trailers and 301 twenty foot sea-

containers similar to what is presented in Table I. For presen-

tation purposes, similar items have been aggregated; Table I

presents the range of item dimensions and weights associated

with each item type along with the number of items to

be moved. Similarly, for simplification purposes, an average

weight has been used for all sea-containers. A minimum inter-

item spacing constraint of 0.2 metres was set for all item pairs

as well as between items and the cargo bay walls.

TABLE IOPERATION ATHENA ITEM MANIFEST

ID Name Length Width Height Weight Qty(m) (m) (m) (kg)

1 20 FT ISO CONTAINER 6.10 2.44 2.59 6500 3012 ILTIS LIGHT UTILITY VEHICLE 3.98 1.52 - 1.85 1.84 2050 - 2180 983 1.5 TONNE CARGO TRUCK 5.61 - 5.85 2.01 - 2.13 2.59 - 2.73 3270 - 4631 474 HEAVY LOGISTICS VEHICLE 8.2 - 10.11 2.43 - 2.79 3.55 - 4.13 12423 - 20500 355 LIGHT ARMOURED VEHICLE 6.10 - 6.98 2.25 - 2.70 2.63 - 2.70 7286 - 13744 326 BISON ARMOURED VEHICLE 6.48 - 7.23 2.59 - 2.86 2.44 - 3.06 11300 - 12410 237 2.5 TONNE CARGO TRUCK 7.09 - 8.32 2.44 - 2.49 2.29 - 3.49 6414 - 8645 218 2 WHEEL CARGO TRAILER 3.69 - 4.04 1.88 - 1.98 2.01 670 149 1.5 TONNE CARGO TRAILER 4.14 - 4.22 2.11 1.98 - 2.52 1073 - 2254 11

10 DIESEL ENGINE GENERATOR 4.11 - 4.19 2.11 - 2.40 2.15 - 2.54 2000 - 2404 911 15 TONNE PLS TRAILER 8.29 2.59 1.44 5180 612 LOGISTIC SUPPORT VEHICLE 5.69 2.04 2.59 3472 613 105 MILLIMETER GUN 5.32 2.00 1.52 1520 614 14 FT SHELTER 4.30 2.32 2.41 860 615 1-PAX ALL TERRAIN VEHICLE 2.06 1.16 1.21 246 616 FORK LIFT TRUCK 5.08 2.54 2.54 12510 517 TRAILER MOUNTED KITCHEN 4.58 2.39 2.53 2554 418 1/4 TONNE CARGO TRAILER 2.89 1.52 1.07 257 419 15 TONNE PLS TRAILER WAGON 8.30 2.51 3.20 12000 320 PERSONNEL CARRIER 5.32 2.68 2.22 10390 321 2.5 TONNE CARGO TRAILER 4.41 2.41 2.37 1845 222 WHEELED TRACTOR 9.55 3.00 3.35 27000 123 FLAT BED TRAILER 6.85 2.43 1.21 22264 124 SNOW PLOW DUMP TRUCK 14.90 3.20 3.53 17219 125 CRANE 10.03 2.44 3.28 16785 126 15 TONNE PLS TRUCK 10.01 2.44 3.40 14520 127 MOTORIZED ROAD GRADER 8.40 2.50 3.34 12600 128 12 FT SHELTER SPECIAL 3.73 2.32 2.41 2860 129 12 FT SHELTER 3.73 2.32 2.41 770 1

A. Effectiveness of Engineered Chromosomes

The Operation ATHENA problem was solved for a fleet

composed of two aircraft types: the Antonov-124 (AN-124)

and the Ilyushin-76 (IL-76). The AN-124 has a maximum

payload of 90,000 kilograms and its cargo bay was modelled

with a rectangle measuring 41.5m long by 6.4m wide by

4.4m high. By comparison, the IL-76 is a smaller aircraft

with a maximum payload of 50,000kg and the cargo bay

is approximated with a rectangle measuring 20m long by

3.4m wide by 3.3m high. The algorithm parameters were set

as follows: the initial population was set at 100, maximum

population at 5,000, maximum age at 3, Tref at 100, and

α at 0.9. GALAHAD was run twice for 250 generations

with the load balancing extinction operator disabled. In each

generation, 80% of the population underwent reproduction,

10% were randomly mutated, and 10% of the population size

was introduced via immigration. Total computational time per

run was on the order of minutes. For the first run, the initial

population was completely random, while in the second run,

two engineered chromosomes were introduced to seek out the

extremal loading solutions. Figure 1 exhibits the solution pop-

ulation for both runs at 0, 125, and 250 generations. Figure 2

shows the final solution frontiers in each case, demonstrating

the improvement in the quality of solutions found after a fixed

number of generations by injecting genetically engineered

individuals into the initial population.

Engineered

Non-engineered

10 20 30 40 500

50

100

150

200

Number of AN-124 chalks

Num

berofIL-76chalks

Fig. 2. Solution improvement by injecting engineered chromosomes

B. Effectiveness of the Extinction Operator

To demonstrate the effectiveness of incorporating load bal-

ancing constraints using an extinction operator, two GALAHAD

runs were executed. The Operation ATHENA item manifest

listed in Table I was used as input with the CG of each item

assumed to correspond to its physical centre. For simplicity,

a single aircraft type was considered: the Canadian Forces

CC-177 Globemaster III aircraft. For item manifests similar to

that of Operation ATHENA, typical load plans for the CC-177

have between 4 and 10 items and can approach the aircraft’s

maximum payload constraint. Such chalks are difficult to

balance as the CG envelope is small and the size and number

of items limits the number of possible load configurations.

Detailed specifications of the CC-177 were obtained from the

United States Air Force Air Mobility Command Affiliation

Workbook (2002). The cargo bay was modeled as a 2D

rectangle measuring 27m long by 5.2m wide by 3.6m high.2

For this example, a conservative peacetime-planning maximum

payload of 35,000kg is used for the CC-177. As with most

aircraft, the extent of the allowed CG envelope is a function

of the weight of cargo loaded. The longitudinal extent of the

CG envelope for the CC-177 is illustrated in Figure 3. In the

figure, the shaded region represents the acceptable forward

and aft limits of the envelope; the CG constraints become

progressively tighter as the payload carried increases. In all

cases the latitudinal extent of the CG envelope is assumed to

be +/- 1m from the centreline of the aircraft.

For each of the two runs GALAHAD was executed for 100

generations with a maximum population set at 1000. The

remaining algorithm parameters were set as per Section III-A.

2Note that due to the global height restriction of 3.6m, 28 of the HEAVY

LOGISTIC VEHICLE items can not be loaded. However in the real world, thereis sufficient clearance aft of the wing box to allow for these vehicles to betransported by CC-177.

0 generations

0 10 20 30 40 500

50

100

150

200

Number of AN-124 chalks

Num

berofIL-76chalks

0 generations

0 10 20 30 40 500

50

100

150

200

Number of AN-124 chalks

Num

berofIL-76chalks

125 generations

0 10 20 30 40 500

50

100

150

200

Number of AN-124 chalks

Num

berofIL-76chalks

125 generations

0 10 20 30 40 500

50

100

150

200

Number of AN-124 chalks

Num

berofIL-76chalks

250 generations

0 10 20 30 40 500

50

100

150

200

Number of AN-124 chalks

Num

berofIL-76chalks

250 generations

0 10 20 30 40 500

50

100

150

200

Number of AN-124 chalks

Num

berofIL-76chalks

Fig. 1. GALAHAD populations: without engineered chromosomes (left) vs. with engineered chromosomes (right)

The same seed was used to initialize the pseudorandom

number generator for both runs. In the first GALAHAD run

the extinction operator was disabled. In the second run the

extinction operator was invoked midway at the 50th genera-

tion. On average the extinction operation took a tenth of a

second to evaluate a single chalk. Performing extinction on

a population of 1,000 individuals with 100–120 chalks each

amounts to a few hours of computation. For both GALAHAD

runs, statistics on the chalks and solutions were gathered after

50 and 100 generations. These are presented in Table II. The

results support the lasting effect of the extinction operator on

the solution population. After 50 generations, 18.3% of all

chalks were balanceable. Each individual was found to have

between 9.9% and 28.6% of its component chalks balance-

able. A total of 7,990 unique unbalanceable load plans were

found amongst the solutions. After invoking the extinction

operator at the 50th generation, over 97.2% of all chalks

in the final population were balanceable in comparison to

18.6% when GALAHAD continued to run without extinction.

Without extinction, between 10.8% and 30.1% of the chalks

were balanceable, which is comparable to the proportions

observed after 50 generations. In contrast, when extinction is

incorporated, a minimum of 92.7% of chalks per individual

were balanceable with 20.9% of the individuals completely

free from infeasible chalks.

It is also interesting to note that—in this example—

extinction did not detriment the final solutions, but rather

yielded an improvement in the total number of aircraft chalks

0 5 10 15 20 250

20

40

60

Distance from front of cargo bay (metres)

Payload

(1000kg)

Fig. 3. The longitudinal extent of the centre of gravity envelope as a functionof the payload carried for the CC-177.

required. This may be attributed to the extinction operator

discarding all solutions (0% were all-balanced) requiring more

than 110 chalks followed by generations starting with a very

fit population in the asset utilization space. While these results

cannot be generalized without more extensive experimentation,

they provide a proof of concept that the extinction operator

proposed is viable and effective.

TABLE IIEXTINCTION EFFECT IN GALAHAD

Generation: 50th 100thExtinction: Disabled Enabled

Percent of all chalks balanceable 18.3% 18.6% 97.2%Minimum percent balanceable per solution 9.9% 10.8% 92.7%Maximum percent balanceable per solution 28.6% 30.1% 100%Percent of all-balanced solutions 0% 0% 20.9%Number of assets per solution 110-115 110-114 108-110

IV. CONCLUSION

GALAHAD uses a combination of genetic algorithm and

simulated annealing metaheuristic optimisation techniques,

coupled with a modified BLF heuristic, a novel convex hull

based fitness function, and a novel extinction operator to

solve the problem of determining the optimal allocation of

a manifest of cargo items across a heterogeneous fleet of

airlift assets. We have implemented a proof of concept version

of the algorithm; computational results illustrate the potential

of the approach. Solutions improve over the span of many

generations, and we are able to improve solution quality and

fidelity through genetic engineering and implementation of the

extinction operator.

Work is currently underway to develop a suitable set of

benchmark problems to allow for more rigorous testing of

the algorithm in order to determine parameter values that

lead to rapid, robust convergence. We are also interested in

quantifying the optimality gap associated with the algorithm

as the number of generations increase.

ACKNOWLEDGMENT

The authors would like to thank Geoffrey Foster, Joseph

Fourny, Lani Haque, Kimberly Phillips and Antonio Sanchez

for their efforts in implementing GALAHAD.

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