variable neighborhood search for multi-objective resource allocation problems

Robotics and Computer-Integrated Manufacturing 29 (2013) 73–78

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Robotics and Computer-Integrated Manufacturing

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Variable neighborhood search for multi-objective resourceallocation problems

Yun-Chia Liang n, Chia-Yin Chuang

Department of Industrial Engineering and Management, Yuan Ze University Chungli, Taoyuan County 320, Taiwan, ROC

a r t i c l e i n f o

Article history:

Received 25 November 2011

Received in revised form

3 April 2012

Accepted 29 April 2012Available online 14 June 2012

Keywords:

Variable neighborhood search

Multi-objective resource allocation

problem

Pareto front

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x.doi.org/10.1016/j.rcim.2012.04.015

esponding author. Tel.: þ886 3 463 8800x25

ail address: [email protected] (Y.-C.

a b s t r a c t

The Resource Allocation Problem (RAP) is a classical problem in the field of operations management

that has been broadly applied to real problems such as product allocation, project budgeting, resource

distribution, and weapon-target assignment. In addition to focusing on a single objective, the RAP may

seek to simultaneously optimize several expected but conflicting goals under conditions of resources

scarcity. Thus, the single-objective RAP can be intuitively extended to become a Multi-Objective

Resource Allocation Problem (MORAP) that also falls in the category of NP-Hard. Due to the complexity

of the problem, metaheuristics have been proposed as a practical alternative in the selection of

techniques for finding a solution. This study uses Variable Neighborhood Search (VNS) algorithms, one

of the extensively used metaheuristic approaches, to solve the MORAP with two important but

conflicting objectives—minimization of cost and maximization of efficiency. VNS searches the solution

space by systematically changing the neighborhoods. Therefore, proper design of neighborhood

structures, base solution selection strategy, and perturbation operators are used to help build a well-

balanced set of non-dominated solutions. Two test instances from the literature are used to compare

the performance of the competing algorithms including a hybrid genetic algorithm and an ant colony

optimization algorithm. Moreover, two large instances are generated to further verify the performance

of the proposed VNS algorithms. The approximated Pareto front obtained from the competing

algorithms is compared with a reference Pareto front by the exhaustive search method. Three measures

are considered to evaluate algorithm performance: D1R, the Accuracy Ratio, and the number of

non-dominated solutions. The results demonstrate the practicability and promise of VNS for solving

multi-objective resource allocation problems.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Resource Allocation Problems (RAP) have been broadly appliedto different fields such as human resource allocation, medicalresource allocation, project budget allocation, raw material allo-cation, weapon targeting assignment, etc. The purpose of RAP is toallocate limited resources to optimize objective fulfillment. Deci-sion makers are frequently concerned with multiple objectives;for instance, a manager may want to simultaneously reduce totalresource costs while increasing efficiency. These two objectivesare in conflict, which increases the difficulty for finding the bestsolution, and the problem can be characterized as a so-calledmulti-objective resource allocation problem (MORAP).

In general, RAP falls in the category of NP-Hard. When thenumber of decision variables (i.e., problem dimensions) or con-straints is limited, exact methods such as dynamic programming

ll rights reserved.

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Liang).

(DP), integer programming (IP), and Branch and Bound (B&B) caneasily obtain optimal solutions. However, larger problems withmore complex instances entail exponentially greater computa-tional costs. For such problems, metaheuristic methods providemore realistic techniques for finding a solution. One of the mainadvantages of metaheuristics is that this approach can find highquality or near-optimal solutions with reasonable computationaltimes. In recent years, several metaheuristic algorithms have beendeveloped for MORAP including fuzzy dynamic approach [5],genetic algorithm (GA) [9], hybrid particle swarm optimization(PSO) [10], hybrid genetic algorithm (hGA) [7,8], and ant colonyoptimization (ACO) [1].

One of the more recently developed metaheuristics, variableneighborhood search (VNS) explores the search space by system-atically moving from one neighborhood to another and has shownexcellent performance in many combinatorial optimization pro-blems [4]. Moreover, the MORAP lends itself to a proper design ofneighborhood structures. Thus, this study proposed three variableneighborhood search algorithms to solve the MORAP. The remainderof this paper is organized as follows: Section 2 defines the MORAP,

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Y.-C. Liang, C.-Y. Chuang / Robotics and Computer-Integrated Manufacturing 29 (2013) 73–7874

while Section 3 proposes the VNS algorithms for the MORAP. Thecomputational results on benchmark instances are discussed inSection 4 and concluding remarks are provided in Section 5.

Fig. 1. Illustration of initial solution generation.

2. Problem formulation

2.1. Multi-objective resource allocation problem

The mathematical model of the MORAP can be formalized asfollows [7,8]:

MaximizeXN

i ¼ 1

XMi

j ¼ 0

eijxij, ð1Þ

MinimizeXN

i ¼ 1

XMi

j ¼ 0

cijxij, ð2Þ

Subject toXN

i ¼ 1

XMi

j ¼ 0

jxijrXN

i ¼ 1

Mi, ð3Þ

XMi

j ¼ 0

jxijrMi, 8i ð4Þ

XMi

j ¼ 0

xij ¼ 1, 8i ð5Þ

xijA 0,1f g, 8i, j: ð6Þ

where i defines the index of jobs and i¼1, 2, y, N; j denotesthe number of workers assigned to job i and j¼1, 2, y, Mi. N

represents the total number of jobs and Mi is the total number ofworkers that can be assigned to job i. cij denotes the cost when j

workers are assigned to job i, and eij represents the efficiencyassociated with job i when j workers are assigned. xij is the binarydecision variable that is equal to 1 if j workers are assigned to job i,and equal to zero if the number of assigned workers is not equal to j.

In this study, the bi-objective of MORAP consists of maximizingtotal efficiency and minimizing total cost as in Eqs. (1)–(3) indicatesthat the total number of workers assigned to all jobs should be lessthan or equal to the upper bound of workers available, while Eq. (4)limits the number of workers for each job no more than the pre-defined upper bound. Eq. (5) requires that a certain number ofworkers be assigned to each job, and Eq. (6) defines the binarydecision variable.

2.2. Pareto front

Among the approaches for dealing with multi-objective opti-mization problems, Pareto optimality is considered one of themost efficient ways to store the best solutions, i.e., the so-callednon-dominated solutions. Unlike its counterpart in single objec-tive optimization, the number of best solutions derived by Paretooptimality in multi-objective optimization is usually more than one.Therefore, an archive (called a Pareto front) is needed to updatethese elite solutions. While simultaneously evaluating all objec-tives, the members (non-dominated solutions) in this archivecannot outperform or be outperformed by any other members inthe same archive. In addition, all other solutions found in the searchprocess are dominated (i.e., are outperformed) by these non-dominated solutions. In this study, the concept of the Pareto frontis used to store the best solutions found during the search process.At the end of search process, the approximated Pareto front, i.e., the

set of the non-dominated solutions obtained by an algorithm willbe presented to decision makers.

3. Variable neighborhood search algorithm

Geiger [2] first proposed a variable neighborhood search algo-rithm (MOVNS) for multi-objective optimization problems. Thisstudy develops three variations of the VNS algorithm, VNSbasic,VNSpet, and VNSpetþbs based on different strategies on the basesolution selection and the ‘‘marking’’ of the selected non-dominatedsolutions. The procedure of the proposed VNSbasic algorithm (withthe basic structure) starts with the generation of an initial solution,and then randomly picks one of the neighborhood structures toperform local search. The set of non-dominated solutions is updatedwhenever a neighboring solution is generated. When a neighbor-hood is completely searched, a new base solution is randomlychosen from the current non-dominated set and the search processabove is then repeated. If all the non-dominated solutions in thecurrent archive have been explored, the marks denoting theselected solutions will be reset, i.e., all non-dominated solutionsin the Pareto front can again be considered as the base solution.The procedure continues until the stopping criterion is reached, i.e.,the maximum number of evaluations. The following introducesthe key steps in the proposed VNS algorithms and explains thedifferences among three variations.

3.1. Initial solution generation

The initial solution is generated randomly in all three VNSvariations. First, the encoding of the solution stores the number ofworkers assigned to each job. In addition, a dummy job is createdand attached to the end of the encoding to accommodate unas-signed workers. A random number less than or equal to the upperbound of available workers, i.e., satisfying Eq. (3), is then generated.This random number represents the total number of assignedworkers. These workers will then be consecutively and randomlyassigned to each job and the number of workers assigned to eachjob also satisfies the upper bound for each job (Eq. (4)). If therandom number is less than the upper bound of available workers,the number of remaining workers will be stored in the dummy job.An illustration of the initial solution generation process is shown inFig. 1. In this example, the total number of available workers is 10and the number of jobs is 4. If eight workers (determined by arandom number) are assigned to these four jobs, the procedurecontinues to randomly assign one worker to Job 2, and then Job 4,

Y.-C. Liang, C.-Y. Chuang / Robotics and Computer-Integrated Manufacturing 29 (2013) 73–78 75

until all eight workers are assigned. Note that the number of workersassigned to each job should satisfy the upper bound for each job aswell. The remaining two workers are then stored in the dummy job.Thus, the final sequence in Fig. 1 represents the outcome of an initialsolution generation procedure, and also a complete solution tothe MORAP.

3.2. Neighborhood search

For the MORAP, this study proposes kmax neighborhoodstructures for all three VNS variations. The kth neighborhoodðk¼ 1,. . ., kmaxÞ is performed by reducing the number of workersin a job and reassigning these workers to another job. Forexample, when k¼1, the number of workers shifted is equal toone. The principle of a feasibility check on the move is the samethat discussed in the previous section. All jobs will be consideredfor a change to their number of assigned workers, i.e., a completeneighborhood search is employed. Note that the dummy job isalso under consideration for adding or deleting workers. Thisextends the search range of each neighborhood and providesflexibility in changing the total number of assigned workers. Also,unlike traditional single-objective VNS algorithms (which imple-ments a sequence of neighborhoods for the base solution), theproposed VNS algorithms in this study employ only one neigh-borhood for the chosen base solution each time. Fig. 2 illustrates acomplete neighborhood search for a given base solution. Four jobsplus one dummy job are considered, and the number of workersassigned to each job is denoted by x1, x2, x3, x4, and V. For the kth

neighborhood, the procedure starts by moving k workers one byone from Job 1 to other jobs (as shown in Fig. 2(a)). The similarchange then applies to Job 2 (Fig. 2(b)), and the procedure goes onuntil the dummy job also changes its stored number (as illu-strated in Fig. 2(c)). All these moves still have to satisfy theconstraint defined in Eq. (4) to maintain the feasibility of theneighboring solutions.

Job 1 dummyJob 4Job 3Job 2

+k

+k

+k

+k

Job 1 dummyJob 4Job 3Job 2

+k

+k

+k

+k

x1 - k x2 x3 x4 V

x1 x2 - k x3 x4 V

Job 1 dummyJob 4Job 3Job 2x1 x2 x3 x4 V-k

+k

+k

+k

+k

Fig. 2. Example of the kth neighborhood.

3.3. Pareto front update

To make use of all search information, all neighboring solutionswill be considered to update the approximated Pareto front, i.e., theset of non-dominated solutions collected by the algorithm(s). Oncea neighborhood search generates a neighboring solution, it will beused to update the Pareto front. If the base solution of a particularneighborhood search is still sustained in the Pareto front after theneighborhood search is conducted, the base solution will be markedas ‘‘searched’’. The purpose of the marker is to avoid repeatedsearches and to reduce computational costs.

3.4. Base solution selection and perturbation mechanism

This section discusses two main strategies used to distinguish thethree proposed VNS variations—base solution selection and pertur-bation mechanism. A base solution must be determined before theneighborhood search starts. In both VNSbasic and VNSpet, the basesolution is selected randomly from the set of current non-dominated solutions. Thus all solutions in the set have an equalprobability of being chosen. However, in the VNSpetþbs version,the selection of the base solution employs the following three-step procedure: (1) sort all non-dominated solutions according toone of their objective function values, i.e., total efficiency or totalcost; (2) one of the following three strategies will be chosenuniformly—the smallest total cost, the largest total efficiency, orrandom generation; (3) a base solution is picked using the chosenstrategy. This selection strategy puts a greater emphasis on bothends of the Pareto front, i.e., the solutions with the smallest totalcost or the largest total efficiency, and hopes to help improve theconvergence of the front in a more balanced way. If the solutionchosen by the first two strategies has been marked, the secondbest solution in the same category will be selected and checked ifit is also marked. Same procedure continues until a base solu-tion is selected or all solutions are marked. Fig. 3 provides anillustration of the search direction using the strategy proposed forVNSpetþbs.

During the selection of a base solution, if all current non-dominated solutions are already ‘‘marked’’ in VNSbasic, the marks willbe reset so that all non-dominated solutions are eligible for re-selection. However, in both VNSpet and VNSpetþbs algorithms, aperturbation mechanism originally proposed in a Pareto iterated localsearch [3] is activated to generate a new neighboring solution as thebase for neighborhood search. Like the neighborhood search, theperturbation mechanism also consists of kmax structures. In the kth

neighborhood of the perturbation, a job is randomly selected and k

workers will be removed (if applicable) from this job. These k workerswill then be randomly assigned to another job. Note that the

Total Cost

Tot

al E

ffic

ienc

y

: Non-dominatedSolution

Fig. 3. Illustration of moving direction using the base selection strategy in VNSpetþbs.


neighborhood selected in the perturbation mechanism does not haveto be the same as that chosen in the neighborhood search procedure.

The VNSbasic, VNSpet, and VNSpetþbs algorithms can be sum-marized in pseudo code as follows:

Procedure (VNSbasic)

(Initialization)Randomly generate an initial solution and build the initial

Pareto front

Define a set of neighborhood structures Nk, k¼ 1,:::, kmax

(Search Procedure)Repeat the following steps until the stopping criterion is

reachedRandomly pick an ‘‘unmarked’’ non-dominated solution

x from the current Pareto frontIf all non-dominated solutions in the current Pareto

front are marked, reset all the marks and select one of thenon-dominated solutions

Randomly pick one neighborhood structure Nk

Sub-loop of Neighborhood SearchFind a neighboring solution of the chosen base solution in

the Nk neighborhood structureEvaluate the neighboring solution and update the Pareto front

Continue until all neighboring solutions have beengenerated

Mark x as ‘‘searched’’ if x is still kept in the Pareto front

Procedure (VNSpet)


Pareto front



reachedRandomly pick an ‘‘unmarked’’ non-dominated solution

x from the current Pareto frontIf all non-dominated solutions in the current Pareto

front are marked, perform Perturbation to generate a newbase solution for the following neighborhood search



the Nk neighborhood structureEvaluate the neighboring solution and update the Pareto front

Continue until all neighboring solutions have beengenerated

Mark x as ‘‘searched’’ if x remains in the Pareto front

Procedure (VNSpetþbs)


Pareto front



reachedPick an ‘‘unmarked’’ non-dominated solution x from the

current Pareto front using one of the three strategiesdepicted in Section 3.4

Else, if all non-dominated solutions in the currentPareto front are marked, perform Perturbation to generate anew base solution for the following neighborhood search



the Nk neighborhood structureEvaluate the neighboring solution and update the Pareto

frontContinue until all neighboring solutions have been

generatedMark x as ‘‘searched’’ if x is still kept in the Pareto front

4. Computational results

This section describes the performance of VNS variations bytesting on benchmark instances. All VNS variations are coded onBorland Cþþ Builder 6.0, and run on a PC with an Intel Core Quad6600 CPU 2.40 GHz and 2.0 GB of memory. We introduce fourtest instances, describe the performance measures, analyze theconvergent behavior of VNS variations, and discuss the computa-tional results.

4.1. Test instances

This study used four benchmark instances as follows:Instance 1: Based on the instance proposed by Hussein and

Abo-Sinna [5], but increasing the number of workers to 10 andkeeping the number of jobs at 4, Lin and Gen [7] proposed ahybrid genetic algorithm (hGA) to solve the problem. The numberof evaluations was not available for hGA. Later on, Chaharsooghiand Kermani [1] proposed an ACO algorithm to test the sameinstance. The stopping criterion for ACO is the number of evalua-tions equal to 200.

Instance 2: Lin and Gen [8] proposed a different type of MORAPwith 4 sales districts and 12 salespersons available for each district,making a total number of salespersons of 48. They proposed an hGAto solve the instance. The population size of hGA is 100 and thenumber of generations is 2000. The number of evaluations spent onthe local search was not specified but it was at least 200,000.

Instance 3: To further evaluate the performance of the pro-posed VNS algorithms, this study proposes two instances withlarger scales. In Instance 3, there are 6 jobs with 20 workers eachfor a total of 120 workers to be assigned.

Instance 4: The largest randomly generated instance consid-ered in this study has 6 jobs, each with a maximum of 30 workersfor a total of 180 workers.

4.2. Performance measures

To fairly compare the performance of the competing algo-rithms, several popular performance measures are employed suchas D1R [6], Accuracy Ratio (AR) [11], and the number of non-dominated solutions. The D1R value proposed by Ishibuchi et al.[6] measures the average minimum distance from each point (i.e.,the non-dominated solution) in the reference Pareto front to thenon-dominated set obtained by the competing algorithms. Thereference Pareto front in this study is constructed through anexhaustive search which enumerates all feasible solutions andthen finds the non-dominated solutions to form the true (refer-ence) Pareto front. For the D1R measure shown in Eqs. (6) and (7),the smaller the better, and this can be used to evaluate both thediversity and convergence of the competing algorithms:

D1RðYapp,iÞ ¼1

9Yref 9

XqAYref

min drq9rAYapp,i

� �ð6Þ

Y.-C. Liang, C.-Y. Chuang / Robotics and Computer-Integrated Manufacturing 29 (2013) 73–78 77

drq ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðf TEðrÞ�f TEðqÞÞ

2þðf TCðrÞ�f TCðqÞÞ

2q

ð7Þ

where 9Yref 9 denotes the number of non-dominated solutions inthe reference Pareto front ðYrefÞ, drq represents the Euclideandistance between a solution r and a reference solution q in thetwo-dimensional objective space. Yapp,i represents the approxi-mated Pareto front (i.e., the set of non-dominated solutions) of analgorithm i. f TEðUÞ and f TCðUÞ are, respectively, the objectivefunction of total efficiency and total cost.

The second measure – the accuracy ratio – evaluates thepercentage of non-dominated solutions in the reference Paretofront contributed by the approximated Pareto front (which isconstructed by the competing algorithms.) Therefore, thenumerator of the accuracy ratio is calculated by counting thenumber of non-dominated solutions obtained by the competingalgorithms and also falling on the reference Pareto front.The denominator is the number of non-dominated solutions inthe reference Pareto front. This ratio, as shown in Eq. (8), indicatesthe accuracy (i.e., convergence) of the competing algorithms.

AR¼

P9Yapp,i9

j ¼ 1

hj

9Yref 9where hj ¼

1

0

if a vector in Yapp,iAYref

otherwise

�ð8Þ

where hj¼1 denotes that the non-dominated solution j in theapproximated Pareto front ðYapp,iÞ of algorithm i can be found inthe reference Pareto front ðYref Þ, and hj¼0 represents its absence,i.e., the non-dominated solution j does not fall on the referencePareto front.

The last measure is used to compare the number of non-dominated solutions found by each algorithm. Note that this mea-sure is only used to supplement the previous two. In addition tothese three performance measures of algorithmic effectiveness,where applicable this study also compares the CPU time and numberof evaluations for the efficiency of the competing algorithms.

Table 1Performance comparison on Instance 1 (# of jobs: 4, total # of workers available: 10).

Algorithm # of non-dominated

point

AR D1R # of evaluations

hGA 7 0.1429 45.80 –

ACO 4 0.5714 49.07 200

VNSbasic 6.4 0.6857 6.25 800

VNSpet 6.6 0.9429 2.27 800

VNSpetþbs 6.6 0.9429 4.29 800

Exhaustive search 7 – – 1001

4.3. Results comparison

The stopping criterion for the proposed VNS variations is themaximum number of evaluations. To account for the complexityof test instances, different stopping criteria are employed for eachinstance. All VNS algorithms are run five times for each instanceusing different random number seeds. In the first two instances,the statistical averages of the VNS algorithms over multiple runsare compared with those from competing algorithms found in theliterature. Note that, where the number of runs was provided,most of the algorithms in the literature ran one time only.In addition, the number of neighborhood search and perturbation

-0.05

0.05

0.15

0.25

0.35

0.45

0 100 200 3

aver

age

D1R

val

ue

number of e

VNSbasic V

Fig. 4. Convergence analysis of three VNS variation

structures (kmax) (if applicable) is both set to three for all VNSvariations in all instances.

To analyze the convergence behavior of the three VNS varia-tions, the average D1R value over multiple runs are collected andplotted using instance 3, as illustrated in Fig. 4. VNSbasic convergesthe fastest among three variations but produces the most inferiorsolution. VNSpet starts with similar solution quality as VNSbasic

but improves over time to produce a significantly better finalsolution. The difference between VNSbasic and VNSpet lies mainlyin the implementation of the perturbation strategy when all non-dominated solutions have been searched. The performance ofVNSpet confirms that this strategy can help jump out of localoptima. The best performer VNSpetþbs can obtain a better solutionquality at an early stage and also continues to improve throughthe process. The selection strategy of the base solution inVNSpetþbs is more balanced, i.e., the base solution is selectedfrom both ends of the Pareto front, and also from the remainingpoints in the front. Fig. 4 shows this mixed strategy successfullyenhances the convergence speed of VNSpetþbs at an early stage ofthe search process.

Following the convergence analysis of the VNS algorithms,discussion turns to the comparison among three VNS variationsand some methods from the literature. Table 1 shows the resultsof Instance 1. Although hGA [7] is able to find more non-dominated solutions than ACO [1] and VNS, only one of thesolutions falls on the reference Pareto front. Therefore, theaccuracy ratio of hGA is lower than all three VNS variations,while the D1R is higher. In fact, the D1R values of both hGA andACO algorithms are greatly higher than the D1R values of the VNSalgorithms. This indicates that the diversity and convergence ofVNS variations are superior that of the competing algorithms,hGA and ACO. In addition, on average both VNSpet and VNSpetþbs

can find 6.6 out of 7 non-dominated solutions and almost 95%of the points on the reference Pareto front can be identified.The number of non-dominated solutions by VNSbasic is similar tothat of the other two VNS variations; however, at approximately69%, its accuracy ratio is much lower. In terms of CPU time

00 400 500 600

valuations (x103)

NSpet VNSpet+bs

s (using Instance 3 as an illustrative example).

Table 2Performance comparison on Instance 2 (# of jobs: 4, total number of workers

available: 48).


point


hGA 18 0.0000 57.8324 4200,000

VNSbasic 94.8 0.9670 0.0735 16,000

VNSpet 96.6 0.9938 0.0111 16,000

VNSpetþbs 96.6 0.9938 0.0075 16,000

Exhaustive search 97 – – 28,561

Table 3Performance comparison on Instance 3 (# of jobs: 6, total # of workers

available: 120).


point


VNSbasic 247.4 0.6695 10.0112 320,000

VNSpet 300.2 0.9666 0.0518 320,000

VNSpetþbs 300.4 0.9613 0.0572 320,000

Exhaustive search 305 – – 85,766,121

Table 4Performance comparison on Instance 4 (# of jobs: 6, total # of workers

available: 180).


point


VNSbasic 439.2 0.5217 4.0875 640,000

VNSpet 435.2 0.8664 0.3883 640,000

VNSpetþbs 444 0.9080 0.2729 640,000

Exhaustive search 452 – – 887,503,681


consumption, all VNS variations need less than 0.01 s in this smallinstance.

The results of Instance 2 over the competing algorithms aresummarized in Table 2. Once again, VNS variations dominate thehGA [8] in all categories. Both VNSpet and VNSpetþbs can find over99% of non-dominated solutions in the reference Pareto front whileVNSbasic falls short with a gap of 2.68%. Table 2 also indicates thatthe most comprehensive version of VNS, i.e., VNSpetþbs owns thelowest D1R value, while VNSpet is the runner-up and VNSbasic

performs the worst among the three VNS variations. When takinginto account the computational expense, VNS variations thatrequest 16,000 evaluations are obviously more efficient than bothhGA (at least 200,000 evaluations) and the exhaustive searchapproach (28,561 evaluations). In addition, on average VNS varia-tions need only around 0.66 s of CPU time.

To further evaluate the performance of VNS algorithms onlarger instances, two instances with over one hundred workers intotal are tested. In Instance 3, VNSpet and VNSpetþbs perform com-petitively with similar numbers in all categories (see Table 3.)On average, over 96% of non-dominated solutions can be found byboth VNSpet and VNSpetþbs, but the performance of VNSbasic isdramatically reduced with the AR value dropping to 67% whileD1R rises to 10.0112. Without the proper base solution selectionstrategy and perturbation operator, VNSbasic seems to easilybecome stuck in local optima in larger instances. Consideringthe computational efficiency, VNS variations need only 320,000evaluations while the exhaustive search method evaluates over85 millions solutions. Additionally, the average CPU time of VNSvariations are all around 50 s while the exhaustive search approachconsumes 4,780 s (approximately 80 min).

Lastly, for the largest instance (Instance 4), VNSpetþbs still per-forms very well (see Table 4), finding over 90% of non-dominatedsolutions on average, while VNSpet identifies 86.64% of the referencePareto front and VNSbasic finds only 52% of the solutions. The D1R

value of both VNSpetþbs and VNSpet are considered competitively low.Again, a low D1R value indicates excellent convergence and diversityperformance in the proposed VNS algorithms. In terms of computa-tional expense, the stopping criterion of all three VNS algorithms is

640,000 evaluations while the exhaustive search method evaluatesnearly 890 million solutions. The average CPU time for the three VNSalgorithms is all below 170 s, but the exhaustive search required69,267 s (i.e., over 19 h).

5. Conclusions

This study proposes variable neighborhood search (VNS) algo-rithms to solve multi-objective resource allocation problems(MORAP). Three VNS variations were developed based on differ-ent base solution selection strategies and the reset strategy ofnon-dominated solutions. The VNSpetþbs version which considersthe mixed base solution selection strategy and perturbationstrategy outperforms other two VNS algorithms, along with compet-ing methods (hGA and ACO) taken from the literature. While testingon large instances, VNSpetþbs and VNSpet were both highly effectiveand efficient as compared to the exhaustive search method. The VNSalgorithms in this study provide a practical alternative to theMORAP. With the flexibility and simple implementation on thealgorithmic procedure, VNS can be easily applied to other srelatedproblems.

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