multi-objectives tabu search based algorithm for progressive resource allocation

European Journal of Operational Research 177 (2007) 1779–1799

www.elsevier.com/locate/ejor

Multi-objectives Tabu Search based algorithmfor progressive resource allocation

Lamia Belfares a, Walid Klibi a, Nassirou Lo a, Adel Guitouni a,b,*

a Faculte des Sciences de l’Administration, Universite Laval, Que., Canada G1K 7P4b Defense R&D Canada—Valcartier, 2459 Pie-XI Nord, Val-Belair, Que., Canada G3J 1X5

Available online 10 January 2006

Abstract

Military course of action planning involves time and space synchronization as well as resource and asset allocation.A mission could be seen as a defined set of logical ordered tasks with time and space constraints. The resources to taskrules require that available assets should be allocated to each task. A combination of assets might be required to executea given task. The couple (task, resources) constitutes an action. This problem is formulated as a multi-objectives sched-uling and resource allocation problem. Any solution is assessed based on a number of conflicting and heterogeneousobjectives. In fact, military planning officers should keep perfecting the plan based on the Commander’s criteria forsuccess. The scheduling problem and resource allocation problem are considered as NP-Hard Problems [A. Guitouni,B. Urli, J.-M. Martel, Course of action planning: A project based modelling, Working Paper, Faculte des sciences de l’administration, Universite Laval, Quebec, 2005]. This paper is concerned with the multi-objectives resource allocationproblem. Our objective is to find adequate resource allocation for given courses of action schedule. To optimize thisproblem, this paper investigates non-exact solution methods, like meta-heuristic methods for finding potential efficientsolutions. A progressive resource allocation methodology is proposed based on Tabu Search and multi-objectives con-cepts. This technique explores the search space so as to find a set of potential efficient solutions without aggregating theobjectives into a single objective function. It is guided by the principle of maximizing the usage of any resource beforeconsidering a replacement resource. Thus, a given resource is allocated to the maximum number of tasks for a givencourses of action schedule. A good allocation is a potential efficient solution. These solutions are retained by applyinga combination of a dominance rule and a multi-criteria filtering method. The performance of the proposed Pareto-basedapproach is compared to two aggregation approaches: weighted-sum and the lexicographic techniques. The resultshows that a Pareto-based approach is providing better solutions and allowing more flexibility to the decision-maker.� 2005 Published by Elsevier B.V.

Keywords: Tabu Search; Courses of action; Multi-criteria; Resources allocation; Pareto optimality

0377-2217/$ - see front matter � 2005 Published by Elsevier B.V.doi:10.1016/j.ejor.2005.10.012

* Corresponding author. Address: Defense R&D Canada—Valcartier, 2459 Pie-XI Nord, Val-Belair, Que., Canada G3J 1X5.E-mail addresses: [email protected] (L. Belfares), [email protected] (W. Klibi), [email protected] (N. Lo),

[email protected], [email protected] (A. Guitouni).

mailto:[email protected]





1780 L. Belfares et al. / European Journal of Operational Research 177 (2007) 1779–1799

1. Introduction

Most real-world optimization problems are multi-objectives in nature: several objectives should be con-sidered at the same time. However, multi-objectives optimization problems (MOP) are very complex andthe complexity besides the combinatorial aspect, comes from the fact that there is no single optimal solutionfor these problems, but rather a set of trade-offs called efficient solutions or Pareto-optimal solutions. Inmany multi-objectives planning situations, the most important challenge is to find efficient feasible solu-tions. Space and Time interdependent tasks are subject to resource constraints.

For instance, the course of action (COA) development stage of the military operational planning process(OPP) involves many Staff members. There is no doubt about the importance of the OPP as a fundamentalproblem solving process to guide a military staff in their thinking process with guidance from the Com-mander. The OPP is composed of five stages: initiation, orientation, COA development, plan developmentand plan review. The initiation stage starts with the reception of a mission statement or simply in anticipat-ing of a new mission. In the orientation stage, the Staff begins the analysis and definition of the mission,prepares the planning guidance and describes Commander’s intent and the desired end state of the opera-tion. The Commander’s guidance and intent help the staff to focus on the development of comprehensiveand flexible plans within the allocated time. These COAs ‘‘should answer the fundamental questions ofwhen, who, what, where, why and how’’. Each COA should be suitable, feasible, acceptable, exclusiveand complete. A good COA prepares the force for future operations and provides flexibility to meet unfore-seen events during its execution. The ‘‘who’’ in a COA does not specify individual units, but rather usesgeneric assets and capabilities. During the COA development stage, staff analyzes the relative combatpower of friendly and enemy forces, and generates comprehensive COAs. The decision stage is based onthe analysis and comparison of the proposed COAs, and the primary approach used in this analysis iswar-gaming. Plan approval and review consists of a choice of the best COA according to the Commander’sbeliefs and estimates.

The COA development stage involves the entire staff. They should identify the assigned and impliedtasks to perform a given mission. These tasks can be decomposed into sub-tasks. Then, planners shouldallocate available resources and capabilities to the tasks. Synchronizing COA requires scheduling startingand ending times of all tasks according to resource availability, deployment constraints and task relation-ships. Any resource or capability has an availability calendar, in-use costing, required preparations,required staffing, etc. Guitouni et al. [15,16] proposed to model a COA planning as a multiple moderesource-constrained project-scheduling (RCPS) problem since, from a methodological point of view, plan-ning and scheduling are not much different. Multi-objectives RCPS problem has a complex combinatorialaspect. The RCPS problem is a generalization of the job-shop problem. It is NP-Hard and can be solved byusing a heuristic procedure [23].

Scheduling and resource allocation problems are ones of the most studied problems in combinatorialoptimization theory. In this work, we propose a progressive resource allocation methodology to solvemulti-objectives RCPS for a given task schedule. This methodology should address the following question:Given a set of limited resources/capabilities, what is the ‘‘best’’ allocation for a given task schedule accord-ing to several objectives? The ‘‘best’’ or potential efficient solutions should be determined considering a setof heterogeneous and conflicting criteria (objectives).

Most studies reported in the literature have focused on the scheduling rather than on the resource allo-cation optimization. Moreover, in most cases, an aggregative approach is used to reduce the multi-objec-tives problem to a single objective optimization problem. The aggregation, despite its simplicity and itsefficiency from a computational point of view, presents a serious difficulty when the objectives are non-com-mensurable or simply heterogeneous. Normalization of the objective functions is then unavoidable; all theobjectives are transformed to the same scale. This transformation is sometimes the source of errors and dif-

L. Belfares et al. / European Journal of Operational Research 177 (2007) 1779–1799 1781

ficulties [8]. Any scale transformation should respect the measurement theory principles. Moreover, it issometimes impossible to make such transformation without asking for additional information. The aggre-gative approach does not generate proper Pareto optimal solutions when the solution set is a non-convexspace [10].

The notion of ‘‘optimum’’ when having several objective functions was originally proposed by Edge-worth [11] and generalized later by Vilfredo Pareto [30]. In a maximization optimization for example, a vec-tor solution x* is called Pareto optimal if there exists no feasible vector x which would increase some criteriawithout causing a simultaneous decrease in at least one of the remaining criteria [8]. Generally speaking,Pareto optimum is not a single solution but it is rather a set of solutions called efficient or non-dominatedsolutions.

Let us consider a decision problem with Z objectives to be maximized. Dominance relationships betweentwo solutions u and x are defined as follows:

–x absolutely dominates u denoted (xDau)() fi(x) > fi(u) for i = 1, 2, . . . , Z,–x strictly dominates u denoted (xDsu)() fi(x) P fi(u) for i = 1, 2, . . . , Z, and $j where fj(x) > fj(u),–x weakly dominates u denoted (xDwu) for i = 1, 2, . . . , Z, fi(x) P fi(u),–x and u are incomparable if neither x (Da [ Ds [ Dw) u nor u (Da [ Ds [ Dw) x.

We have Da � Ds � Dw. The set of efficient solutions is also denoted Pareto set (Ptrue) in the decisionspace and Pareto front (PFtrue) in the objective space. A priori characterization of PFtrue in NP-Hard prob-lems is very difficult. Therefore, the solutions obtained from a multi-objectives optimization method arereferred to as the known Pareto front PFknown which can be a subset of PFtrue. This is the notation adoptedin this work. In some multi-objectives combinatorial problems, weak dominance is applied during thesearch process because it allows more solutions to be reached from the existing ones with regard to the con-nectedness of the search space [26]. The progressive resource allocation (PRA) procedure, proposed in thiswork, uses this kind of dominance. This procedure starts with an initial feasible solution. Local potentialefficient solutions are then generated by using a multi-objectives Pareto-based optimization techniqueinspired from the Tabu Search meta-heuristic. The optimization method is based on a posteriori preferencearticulation approach [9]: ‘‘search before making decision’’. The goal is to find the largest number of welldiversified efficient solutions. An interactive filtering or choice process might then be carried out with thedecision makers (DMs) to select the most adequate solutions. In order to assess the value added of Pareto-optimality, we compare the proposed optimization method to two other methods: an aggregative basedoptimization technique, the weighted-sum and a non-compensatory based optimization technique, thelexicographic.

This paper is organized as follows. In Section 2, a brief overview of Tabu Search (TS) and its appli-cation for RCPS problems is presented. In Section 3, the COA representation is discussed and the modelformulation is introduced. Section 4 presents the PRA procedure that has been developed. In Section 5,empirical computational results are analyzed. The last section summarizes the work and concludes thispaper.

2. Recent studies on RCPS problems and the multi-objectives optimization

Project planning and activity scheduling under resource constraints are combinatorial NP-Hard prob-lems [35]. RCPS problems reported in the literature have multiple features like: multi-mode, single mode,with non-pre-emptive or pre-emptive resources, renewable or non-renewable resources. Diverse methods


have been reported to solve each category of RCPS problems. For instance, TS was used extensively forscheduling the activities. This meta-heuristic is known for its simplicity and its efficiency when adequatelyimplemented. According to Glover [13,14], TS borrows some of its concepts from the artificial intelligencefield such as the notion of memory. Moves, neighborhood and descent procedures are the basic concepts ofthis local search method. Recently, Klein [23] has generalized this method to RCPS problems by mention-ing among many operational disadvantages, the definition of the neighborhood that he estimated as beingvery wide. This could be partially overcame by limiting the investigation using for example a candidate liststrategy. The selection of interesting moves from current solutions is difficult and sometimes the deteriora-tion of solutions is considered. Generally, every feasible schedule represents a solution and every movetransforming a solution into another one defines a neighbor. The three well-known types of moves arethe forward change, backward change and the exchange of randomly chosen activities. Among TS appli-cations to RCPS problems found in the literature, one can cite Lee and Kim [27], Baar et al. [1], Bruckerand Knust [7], Boctor et al. [5]. In the work of Barros and de Souza [2], four types of task exchanges andtask insertions have been used as moves in their TS method. These authors defined three types of Tabuattributes for the moves. Eight strategies, combining different parameters were tested and the best movestrategy identified was that of randomly selected task insertion. In their approach, solutions were evaluatedusing a single cost function to be minimized. Thomas and Salhi [31] proposed a TS algorithm where a sub-set of neighbors is selected when the move candidate list is too large. These authors proposed various strat-egies combining intensification, diversification, and strategic oscillations, used according to the frequencyof the attributes generating the best solutions. More recently, Haouari and Al-Fawzan [21] proposed abi-objective model which introduces an evaluation function to maximize the scheduling quality. This func-tion was aggregated with the makespan of the project. The results obtained did not improve the solutionson the project duration, but the function of robustness was maximized.

The multi-objectives optimization using TS meta-heuristic was developed by Gandibleux et al. [12] andHansen [18–20] at the end of 1990s. Gandibleux et al. [12] use a scalarization function to browse, in a bal-anced way, the non-dominated frontier. This function corresponds to a direction of search, starting from areference point and updated at each iteration by the best neighbor solution found in that direction. Hansenhandled the multi-objectives problem by aggregating the objectives into a scalar linear function and byusing normalized weights. These weights may vary from iteration to another to find efficient solutions.Ben Abdelaziz et al. [4] proposed a hybrid heuristic to solve the multi-objectives knapsack problem, whichis not directly related to the RCPS. The proposed heuristic is based on a double application of the TabuSearch algorithm. Other multi-objectives TS (MOTS) meta-heuristics were developed for academic andreal-world applications, but it is beyond the scope of this paper to discuss all those methods. We referthe interested reader to the review by Ehrgott and Gandibleux [29].

All the studies on RCPS problem focus on optimizing activity scheduling rather than the resource allo-cation. This later aspect of the RCPS problem is tackled as a multiple execution mode as reported by Kolishand Drexl [25]. In this paper, we suppose that the task schedule is already given, for example achievedthrough another optimization algorithm. The problem is to allocate limited resources to tasks while opti-mizing the overall plan (COA) performances (objectives). These resources are limited in quantities and con-strained by their availability in time and space. Therefore, we consider that the existing proposals areincomplete and do not address the problem at hand. A multi-objectives meta-heuristic based on TS is devel-oped to solve this resource allocation problem. Mechanisms used in this meta-heuristic are inspired by thesuccessive stages of search in the neighborhood, the type of moves as well as the mechanisms of intensifi-cation and diversification. The Pareto optimality principle is considered as no subjective preference mod-eling parameters (e.g. weights, trade-offs . . .) are required to explore the solution space. In our opinion,the flexibility of TS enables to perform a search guided in a multi-objectives space where an interestingsearch direction can be determined by more than one function. That is what we try to point out in thisstudy.


3. Course of action representation

3.1. Notation

Let M be a mission with a planning period H defined by its earliest date and its latest date. A mission isrepresented by a set of tasks T and sets of spatial and temporal relationships between these tasks. Let usconsider the following notation:

Indexes

• j: task,• k: generic resource,• i: specific resource,

Problem parameters

• T: the set of all tasks to cover during a mission T = {t1, . . ., tJ},• Rk: the generic resource of type k,• Gr: the set of all generic resources Gr = {R1, . . . , Rk},• Ik: the set of specific resources corresponding to a generic resource of type k,• EF: the set of objective functions,• Lk: the set of locations (depots) where Rk 2 Gr is stored,• Pj: the set of the predecessor of task tj,• tj: the task j,• J: the number of tasks = jTj,• K: the number of generic resources = jGrj,• rik: a specific resource i of a generic resource of type k,• ½sj;�sj�: the time window on task tj,

• ½hik; �hik�: the availability time window of the resource rik,• dj: the duration time of task j,• ts

j: the scheduled starting time of task tj,

• tfj : the scheduled ending time of task tjðtf

j ¼ tsj þ djÞ,

• tikll0 : the displacement time of rik between two locations l and l 0,

• qkj : the quantity of generic resource Rk required to accomplish task tj,

• cikj: the cost of assigning the resource rik to tj,• CFik: the fixed cost of assigning the resource rik,• CUik: the in-use cost of assigning the resource rik,• Prik

j : the probability that resource rik successfully completes its stage of operation if allocated to tj,• kik: the expected failure rate,• aj: is an elementary scheduled task j with the combination of the required specific resources,• sj: is the location where task j should take place.

Variables

• xikj: is a binary variable which takes value 1 if the resource rik is used by tj and 0 otherwise,• Tikj: the updated starting availability time of the resource rik after accomplishing tj.

A COA can be represented by a set of elementary tasks to be accomplished, taking into account con-straints like precedence relationships between tasks and resource availability [15]. Tasks and sub-taskscan be represented by means of a hierarchical structure called work break-down structure. Leafs of thishierarchical structure are called elementary tasks. Synchronization analyses lead to identify temporal


and spatial relationships between elementary tasks (e.g. End–Start, Start–Start, End–End, Time Laps,Same Spatial Zone . . .). Planners should then consider all available resources and capabilities and assignthem to the tasks. The model consists of representing generic activities (tasks with specific combina-tions of resources) into elementary (or primitive) actions interrelated to accomplish the objectives of themission. This process implies the identification of the tasks (when and where) as well as their prece-dence relationships, the pool of available resources with their localization, and finally the objective func-tions. A COA is then represented as an oriented time–space graph (see Fig. 1). Different COA networkscould be obtained. They constitute variants (or alternatives) of a mission with different evaluations onobjectives.

Each task tj has a time window characterized by its earliest starting and latest ending time: ½sj;�sj�. Eachtask might have an operation area where it should take place: a spatial location coordinate constraints. Wesuppose that we know the quantities of each generic renewable and non-renewable resources requiredto accomplish the task. Each task tj might have a set of logical scheduling relationship within the mis-sion referred to as a set of predecessors, Pj, characterized by the tasks that temporally and/or spatiallyprecede tj.

We distinguish between three types of tasks according to the temporal and precedence constraints:(i) Strongly constrained tasks: those to be accomplished according to a time window and precedenceconstraints, (ii) Weakly constrained tasks: are those subject to either precedence constraints or a timewindow constraint, and (iii) Floating Tasks: those subject neither to precedence nor to time windowconstraints.

A pool of resources dispatched at different depots is considered with the following attributes:

– Availability time windows ½hik; �hik] (resource’s calendar),– Quantities available during this interval of time,– Location of the resources (depot) (x, y, z),– Type of resource (generic resource),– Other specific characteristics such as in-use cost, mean speed (for mobile resources), reliability, etc.

Fig. 1. COA graph representation.


We distinguish two levels of representation of the resources, generic and specific resources in each gen-eric set. They can be either renewable or non-renewable. Complementary relationships determine whichgeneric resources could be combined to accomplish which tasks. Every generic resource is available in agiven quantity represented by specific resources as illustrated in Fig. 2. A specific resource can be allocatedseparately and independently and has its own characteristics that affect the objectives of the COA. Theavailability is represented by a calendar in the form of time interval during which the specific resourcecan be used once or several times before coming back to its depot. This information is known beforehandand not subject to modification during the mission.

For a given activity schedule generated by a heuristic based on a network approach and CPlex [32], thequantity of generic resources required by each task is provided by assigning the corresponding number ofspecific resources (Fig. 3). The COA is completed and feasible when all resource requirements are met andthe precedence and resource constraints are fulfilled.

3.2. COA evaluation

For any generic resource, the specific resources have some common functional characteristics and othersparticular to each one. Resources do have different technical characteristics such as, speed, mass, auton-omy, capacity, performance, location, etc. Therefore, they do not have the same performances when achiev-ing the same task; e.g. same reliability, in-use cost or the same probability of success to accomplish themission.

For every specific resource assigned to a task j, we have defined:

Fig. 2. Example of resource calendar.

r2

T1 T3T2

R1 R2

r1

r1

r1

r1

r2

r2

r2

r2r2

T1 T3T2

R1 R2

r1

r1

r1

r1

r2

r2

r2

T1 T3T2

R1 R2

r1

r1

r1

r1

r2

r2

r2

Fig. 3. Multiple specific resource allocations for three tasks.


A cost: cikj ¼CF ik þ CU ik � ðtik

j0j þ djÞ if tj is the first task served by rik;

CU ik � ðtikj0j þ djÞ otherwise.

8<: ð1Þ

A reliability: e�kik�dj ð2Þ

A probability of success: Prikj ð3Þ

Consequently the COA evaluation on mission objectives varies according to the specific resources used.In this work, we define a cost objective (Eq. (4)) as a linear function of the fixed and in-use costs (CFik andCUik, resp.) of the specific resources required by all the tasks including the transition cost (e.g., travel,setup) (Eq. (1)). The COA reliability (Eq. (5)) is the second objective computed as a non-linear functionof the resource reliability (Eq. (2)) (Poisson’s law with time-based failure frequency). The computationof the evaluation of a COA according to this objective calls upon the concepts of the reliability theory[15]. The third function is the probability of success of the COA (Eq. (6)), which is computed based on par-tial probabilities that a given resource successfully achieves its part of the task to which it is allocated inspecific conditions. An a priori probability of success is given (Eq. (3)). For example, a helicopter is ableto transport troops from a location A to B under specific conditions. Let us just consider weather condi-tions that could be assessed as green, yellow or red. For the sake of simplicity, one could consider thatthe partial probabilities of success of assigning a helicopter to transport the troops under green, yellowor red weather conditions are respectively represented by the following intervals: [80%, 100%],[40%, 80%] and [0%, 40%]. The overall probability of success for a given COA is computed by using a bias-ing network representation. In this paper, we simplified the probability of success computation formula asshown by Eq. (6). These three COA evaluations will guide the optimization process for the resourceallocation.

3.3. Problem formulation

The problem is to coordinate the availability of the resources with the task schedule. In this case study,we try to satisfy simultaneously various criteria under resource constraints, with the followingcharacteristics:

– For every task, a combination of different generic resources is required,– The initial availability calendar of these resources is not subject to modification,– The required quantity for each task is invariant during the project,– A task cannot be interrupted and its time window is fixed for a given schedule.

The problem can be formulated as follows:COA

MinimizeXk2Gr

Xi2Lk

Xj2T

cikjxikj ð4Þ

Maximize Hðe�kik�djÞ ð5Þ

Maximize1

J

Xj2T

Xi2Lk

Prikjxikj

,Xk2Gr

qkj

!ð6Þ


Subject to:Xk2Gr

Xi2Lk

xikj ¼ qkj ðj 2 T Þ; ð7Þ

xikj T ikj0 þ tikj0j � ts

j

� �6 0 ðk 2 Gr; i 2 Lk; j 2 T ; j0 is the last location of rikÞ; ð8Þ

xikjðT ikj � tfj Þ ¼ 0 ðk 2 Gr; i 2 Lk; j 2 T Þ; ð9Þ

T ikj þ tikjl 6

�hik ðk 2 Gr; i 2 Lk; j 2 T ; l depot of rikÞ; ð10Þ

xikj 2 f0; 1g ðk 2 Gr; i 2 Lk; j 2 T Þ; ð11Þ

T ikj; integer ðk 2 Gr; i 2 Lk; j 2 T Þ. ð12Þ

Eqs. (4)–(6) are the objective functions. The COA cost (Eq. (4)) is a linear function of fixed and in-usecosts (CFik and CUik, resp.) of the specific resources required by all the tasks in the mission. The COA reli-ability (Eq. (5)), computed heuristically, is a non-linear function of the reliability of the resources (see Eq.(2)) and depends on the position of the actions in the COA network (serial or in parallel) [15]. The prob-ability of success of the mission (Eq. (6)) is calculated according to the probability of success of the com-binations task–resources involved in the COA.

Constraint (7) ensure that each task i is covered exactly by the exact quantity qki of resources of type k

(generic resource). The time availability of a specific resource is updated by Eqs. (8,9). Constraint (10) guar-antees that each resource is released and returned back to its depot within its availability time window.Constraint (11) and (12) are variables of the problem.

4. The progressive resource allocation algorithm

4.1. The approach

In this section, we propose an approach based on TS to perform a progressive resource allocation. Tofind the best solution, this technique makes use of the Pareto optimality concept. The goal is to find a set ofnon-dominated and sufficiently diversified solutions to cover the entire Pareto frontier. Five key aspects areinvolved in the PRA method:

– Only renewable non-pre-emptive resources are considered. They are available in limited quantity for agiven period.

– Maximal re-use of the resources: this means that the resources are allocated to the maximum number oftasks in the schedule.

– Allocation of specific resources according to their evaluation on every objective separately: that meansthe resources presenting the best performance on each objective selected alternately, are privileged forallocation. This strategy aims to diversify the search by generating solutions where the resource combi-nations in the COA have good evaluations on all objectives. The specific resources are sorted out accord-ing to the selected objective before the allocation begins.

– The use of non-dominance concept and a multi-criteria filtering method to eliminate the dominatedsolutions. Every neighbour candidate generated from the current solution is evaluated using the domi-nance rule. When several non-dominated neighbours are generated, a multi-criteria filtering procedure(MFP) [17] is used to filter these solutions. This procedure, based on the disjunctive and conjunctivemethods, consists of selecting a sub-set of non-dominated solutions characterized by one best-scored


objective or by all objectives achieving minimal threshold values. A disjunctive method consists ofselecting solutions based on their exceptional performances. Thresholds on specific criteria could befixed and each solution achieving or exceeding at least one of these thresholds will then be selected. Con-versely, the conjunctive method is different since it selects solutions that achieve a minimal performanceaccording to each one of the selected decision criteria. Let A be a subset of non-dominated solutions andCard(A) the user-defined cardinality of this subset. In the first step of MFP, the disjunctive methodretains all the solutions that score a maximal value on at least one objective Fi. If the number of retainedsolutions is lower than Card(A), the conjunctive method is then applied to reach this number. In thisstep, the solutions characterized by objective values higher than thresholds are retained. Let e1 be thisthreshold vector. e1 is computed dynamically by a dichotomic method between S1 and S2,S1 ¼ fF �1 ; F �2 ; . . . ; F �z g. being the anti-ideal point of the set A and S2:

� ¼ fF �1; F �2; . . . ; F �zg the set of idealpoints of A, and Z the number of objectives. It is worth noting that in MFP, a maximization problem isconsidered; a transformation procedure is applied when a minimization is required.

– Time windows of the tasks are unchanged for a given COA solution. This means that the algorithm doesnot modify the structure of the schedule and does not change the tasks starting and ending times. Thisdecreases locally the complexity of the problem and concentrates the search on the possibilities of assign-ing differently, for any feasible schedule, the resources.

4.2. Adaptation of the Tabu Search approach

Several strategies of exploitation are used to apply the PRA method. The first principle of the approachdeals with the management of Tabu lists. We define a first list, Tabu Resources, in which we forbid the selec-tion of a generic resource twice in the same schedule. This insures to generate different COA solutions. Thislist is static and its length depends on the number of generic resources detected in the COA. Resourcesremain Tabu as long as we work on the same COA schedule. We define a second list, Tabu Tasks, in whichthe assignment of rik to tj is forbidden if this combination is present in a dominated COA solution. Suchrestriction avoids the cycling phenomena if a dominated solution is found. Forbidding these resourceassignments will change all the combinations resource–task in the COA and consequently its performance.This list is dynamic, because according to the generic resource considered, the number of combinations(resource assignments) changes. The Tabu status of these combinations is applied until a non-dominatedCOA is found or until the local stopping criterion is reached. According to this local stopping crite-rion, if after a given number of resource allocation attempts, no efficient solution is found from thegenerative (current) COA, this later is retained as the best one for the generic resource and the objectiveselected.

The second and more important principle of the PRA method is the definition of the neighborhoodstructure. A neighborhood of a COA solution is defined as all the possibilities of allocating specificresources of a generic set. This set can be very large, that is why we use a local search strategy based onbest-scored allocation for reducing the number of iterations to reach a non-dominated neighbor. Thisnew solution remains candidate until the filtering procedure stage. The global stopping criterion is reachedwhen all the generic resources become Tabu.

Links between our strategy and the TS approach can be described through three points. The first point isthat the original method is generally based on statistical information; the neighborhood is described by thesuccessive shape modifications of the solution. Our strategy does proceed to modifications, from a neighborto another one, without loosing the structure of the current solution.

The second point is that TS is a method of local improvement and accepts sometimes a deterioration ofthe objective in order to escape local optima. In our strategy, the improvement is done based on each objec-tive in turn to escape local optima, but only non-dominated solutions are retained at each step.

COA COA COA

COA COA

D1

D1

D1

D2

D2 D3

D3

SC SC SCSC: solution of best compromise

Fi*: extreme solution

3

COA0

COAij COAij COAij

Non dominance

Filtering

COAij COAij COAij

Lev

eli

R1

R2

R3

D2 D3

D2 D3

F1* F2* F3*SC SC SC SC

1

2

0In

tens

ific

atio

n

Diversification

D3

Fig. 4. Tree of solution generation using 3 generic resources.


The third point is that the local search in the neighborhood consists of a succession of intensification anddiversification phases as shown in Fig. 4. The intensification phase towards a single objective improvement(direction Dj) is done by generating extreme solutions on every objective function. The diversification isdone at each level i. It comprises the investigation of the multi-objectives search space by generating bestcompromise solutions. This is done by improving along with a direction Dj, a solution already optimized onDi (i 5 j) at the precedent level. The advantage of this approach is to generate, at each level, both extremesolutions (noted F �i in Fig. 4), by selecting resources that improve always the same objective (direction Dj),and solutions of best compromise (noted SC in Fig. 4), by selecting resources that improve the objectives,alternately (on different directions). The algorithm starts from an initial feasible COA and generates a set ofrich and diversified solutions by local improvements.

4.3. Description of the algorithm

We distinguish five stages in the PRA algorithm as illustrated in Fig. 5. The first stage consists of a ran-dom choice of a generic resource from EGR. Once the search is performed with this generic resource(through step 2 to step 4), it is added to the Tabu Resources list. The second stage consists of the withdrawalof corresponding specific resources from the current solution. These specific resources are put back to theirinitial depot to be available for a new allocation. Their calendar is updated consequently. In the third stage,allocation of the best scored resources is done, according to the objective function chosen from EF. Steps ofthis key stage are detailed in Fig. 6. They consist of ranking the resources from the best to the worst accord-ing to the objective Fi (step b), selecting the first resource in the list (step c) and the first task in the schedule(step d), checking the Tabu status of the combination (step e). If the combination is Tabu, the next task isselected. The availability of the resource is then checked (step f). If it is available, the allocation is done (stepg) (Eq. (11)) and the resource calendar Eqs. (8)–(10) as well as the quantity required by the task are updated(step h). At this stage, if all the required resources are supplied to this task, it is no more considered (steph1). If the resource is not yet available for another allocation it is no more considered (step h2). If all thetasks are completed (the required resources are provided) according to this generic resource or no more

Fig. 5. PRA algorithm.

Fig. 6. Steps in the stage 3 of PRA algorithm.



specific resource is available (step h3), the constructed solution is considered as a new one. Then this newsolution is checked for its completeness (or feasibility) (step i) (Eq. (7)); if the test fails, all the resource–taskcombinations are stated Tabu for this objective function. If this new solution is feasible then it is evaluatedEqs. (4)–(6) and compared to the current solution (step j). If it is not dominated then it is stored in the solu-tion list (step k). Otherwise, the Tabu status is applied to all the task–resource combinations performed.

The fourth stage checks if all objectives are considered for the current solution. It consists of an internalloop where the stage 3 is repeated for all the objective functions to generate diversified solutions along withall directions (level i in Fig. 4). In the last stage of the algorithm, we check if all generic resources are con-sidered. This external loop assures the intensification of the search in all directions.

5. Computational results

5.1. Cases studies

The efficiency of the PRA algorithm proposed in this work is investigated using instances of severalproblems. A set of complex COA schedules, summarized in Table 1, are generated using a heuristic of con-struction based on a network approach and CPLEX [32]. It is a network problem characterized by addi-tional resource constraints and solved as a mixed integer program. As the Branch and Bound techniqueis used to solve it, its complexity may be exponential. Thus the computational time to generate a solutionvaries between 7 seconds for the cases study C1 to 57 minutes for the case study C16. The tests are carriedout on a Xeon 2 GHZ processor with 2 GB of RAM. These feasible COAs are used as initial solutions forthe TS meta-heuristics. All the algorithms have been implemented in C++.

5.2. Metrics for performance assessment

Different investigations to evaluate the performance of multi-objectives meta-heuristics have been pro-posed. A critical survey on this topic can be found in [34,28,24]. Comparing the performance of multi-objectives optimization techniques is not an easy task. Most current methods do not outperform each otherbut work better regarding different performance indicators showing better proximity (nearest to the truePareto frontier) or better diversity (spread all over the approximation Pareto frontier, PFknown) [6]. Com-parison is done by evaluating the quality of their outcomes (called approximation sets). In this work, weexamine the performance of the different approaches as proposed by Guitouni and Belfares [3]. First thebinary indicator Ie [34] is used. If no conclusion could be drawn from the performance of an algorithm com-pared to another we evaluate the diversity and the proximity characteristics of their outcome (approxima-tion sets) using three metrics: the cardinality of the approximation set (number of efficient solutionsobtained by the method) noted Ns, the proximity noted C*, and the diversity in the objective space [33]noted COV.

Table 1Case studies

Classes C1 C2 C3 C4 C5 C6 C7 C8 C9 C11 C13 C14 C16

Number of tasks 6 6 6 6 50 50 50 50 50 100 100 100 100Number of generic resources 3 5 3 5 3 5 3 5 3 3 3 5 5Avg. number of predecessors (1–2) (1–2) (2–3) (2–3) (1–2) (1–2) (2–3) (2–3) (3–4) (1–2) (2–3) (2–3) (3–4)


Ie binary indicator: The performance comparison between two algorithms ‘‘a’’ and ‘‘b’’ is done, as pro-posed by Zitzler et al. [34], by examining their outcomes. Algorithm ‘‘a’’ outperforms algorithm ‘‘b’’ impliesthat the approximation set A generated by algorithm ‘‘a’’ is better than the approximation set B generatedby the algorithm ‘‘b’’. A x B (A is better than B) means that every solution xB 2 B is weakly dominated byat least one solution xA 2 A and A 5 B. Interpretation of the comparison method using this indicator is asfollows:

The set A strictly dominated set B: ðI eðA;BÞ < 1Þ () A �� B; ð13Þ

The set A is better than the set B: ðI eðB;AÞ > 1 ^ I eðA;BÞ 6 1Þ () A . B; ð13aÞ

The sets A and B are incomparable : ðI eðA;BÞ > 1 ^ I eðB;AÞ > 1Þ () AkB; ð13bÞ

where I eðA;BÞ ¼ max8xB min8xA max16i6ZF iðxAÞF iðxBÞ

n ofor a minimization problem. In this work, Ie is calculated

after the objectives to be maximized (success and reliability) are transformed.Proximity indicator: For the proximity of the algorithm’s outcome, we have chosen the similarity to ideal

solution index, C*, proposed by Hwang and Yoon [22] in their multiple attribute decision making proce-dure called Technique of Order Preference by Similarity to Ideal Solution (TOPSIS). In this work, the prox-imity index C* is calculated as follows. The closeness of the algorithm’s outcome in the objective space,PFknown, to the Pareto frontier, PFtrue, could be represented by the similarity of the centroid of PFknown

to the ideal solution:

C� ¼ 1

Ts

Xi

C�i ; ð14Þ

where

C�i ¼ S�i =ðS�i þ S�i Þ ð14aÞ

and

S�i ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXZ

j¼1

mij � F �j� �2

vuut ð14bÞ

and

S�i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXZ

j¼1

mij � F �j� �2

vuut ; ð14cÞ

where F �j and F �j are respectively the ideal and the anti-ideal points of the jth objective function

F jðxiÞ; mij ¼ F jðxiÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPiF 2

j ðxiÞp is the normalized objective Fj (xi), with 1 6 j 6 Z and 1 6 i 6 Ts, Ts being the total

number of efficient solutions to be compared. Note that, unlike in the original work, no weight is associatedto the objectives Fj. The ideal solution is represented by A� ¼ fF �1; F �2; . . . ; F �j ; . . . ; F �zg and the anti-idealsolution A� ¼ fF �1 ; F �2 ; . . . ; F �j ; . . . ; F �z g. 0 6 C* 6 1 where C* = 0 for A� and C* = 1 for A*.

Diversity indicator: When designing a multi-objectives optimization technique, the diversification of thegenerated solutions is one of the goals aimed at by the exploration strategy. The diversity of the solutions, itis to be noted, represents a desirable result as it gives to the DM a larger range of options.


We propose to express the diversification of the solutions spread all over PFknown by the extent of cov-erage in each objective’s dimension separately. This indicates the size of the objective space covered by thealgorithm’s outcome. The extent of coverage relative to each objective fi is defined as follows:

Covi ¼ maxx;y2P knownjF iðxÞ � F iðyÞj

jF �i � F �i j� 100; ð15Þ

where the numerator expresses the maximum distance among solutions, in the dimension i, and F �j andF �j are, respectively, the anti-ideal and ideal values of Fi. When these two values are unknown, onecan take reference points or the best values generated by the simulations as explained in Section4.3.

For a better appreciation of the dispersion, it is more appropriate to consider the coverage array on allthe objectives:

COV ¼ ðCov1;Cov2; . . . ;CovMÞ. ð16Þ

Comparing these coverage vectors could be achieved using dominance analysis, multiple criteria decisionanalysis or statistical techniques. Since the objectives, in this work are the cost, the reliability, and the suc-cess, the diversity COV of the algorithm’s outcome is defined as

COV ¼ ðcovcost; covreliabilty; covsuccessÞ. ð17Þ

5.3. Comparison with single objective optimization-based techniques

The performance of the proposed progressive resource allocation method, denoted by strategy S1, iscompared to two classical single objective optimization techniques: the lexicographic method denoted bystrategy S2 and the weighted sum method denoted by strategy S3. This comparison is carried out forthe different RCPS problem sizes as presented in Table 1.

Note that the evaluations on all objectives have been normalized. To simplify the comparison, wedecided to transform all function into maximization objectives (e.g. cost function has become T-cost tobe maximized). We define the ‘‘relative ideal’’ and ‘‘relative anti-ideal’’ scores for each objective. Thesescores represent, respectively, the best and worst values obtained on all runs for the same problem class,regardless of the optimization strategy applied.

The strategy (S1) implementation is shown by Fig. 5. The filtering procedure MFP is used at the end ofthe process to retain the Ns best solutions among all efficient COAs generated by the algorithm. We haverun trials and we have set Ns equal to 40.

The lexicographic strategy (S2) consists of investigating the search space based on dictatorial rankingamong the objectives. Thus, one objective is considered at a time. The algorithm is shown in Fig. 7. Severalruns are performed for all ranking possibilities of the objectives in order to discover the greatest number ofnon-dominated solutions.

The third strategy (S3) is based on sorting specific resources according to a single weighted sum aggre-gative function. This function is a weighted sum of the objectives with random normalized weights. Thecandidate solution is also evaluated using an aggregated function with uniform weights. Several runs areperformed with several set of normalized weights to approximate the Pareto frontier.

To compare the performance of the three strategies, the Ie indicator metric is first used as shown byTables 2–4. For large size problems C7, C13, and C16, the strategy S1 outperforms both strategies S2and S3 as shown by the following empirical results (see Eq. (13)):

Fig. 7. Strategy S2 algorithm.

Table 2Comparison between the 3 strategies performance for 6 tasks problems

Ie C1 C2 C3 C4

S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3

S1 1.000 14223 1E + 05 1.000 1.358 2E + 05 1.000 14223 1E + 05 1.000 1.000 1E + 05S2 1.333 1.000 1E + 05 28450 1.000 2E + 05 1.333 1.000 1E + 05 20571 1.000 1E + 05S3 1.000 8.072 1.000 2.129 0.993 1.000 1.000 8.072 1.000 1.349 0.966 1.000


– C7: Ie(S1, S2) = 0.982 and Ie(S1, S3) = 0.983;– C13: Ie(S1, S2) = 0.619 and Ie(S1, S3) = 0.589;– C16: Ie(S1, S2) = 0.9 and Ie(S1, S3) = 0.8.

However, S1 outperforms S3 is case studies C6 (Ie(S1, S3) = 0.9) and C14 (Ie(S1, S3) = 1.0 ^ Ie(S3, S1) =4E + 05). S1 outperforms S2 in case studies C8 (Ie(S1, S2) = 1 ^ Ie(S2, S1) = 261.3) and C11(Ie(S1, S2) = 0.614).

When no outranking conclusion is obvious because (Ie(S1, S2) > 1 ^ Ie(S2, S1) > 1) and (Ie (S1, S3) >1 ^ Ie(S3, S1) > 1), then we examine the quality of solutions. Table 5 reports the average CPU time for solv-ing each problem class by each one of the three strategies. When only three generic resources are consid-ered, S1 outperforms S2 and S3. However, when five generic resources are considered, S1 is slower onaverage due to the high dimensionality of the search space.

Tables 6 and 7 show simulations results for the case studies where the three strategies are incomparable.The strategy S1 generates often larger approximation sets than S2 and S3 (e.g. for case study C1:Ns(S1) = 40, Ns(S2) = 9, Ns(S3) = 4). S1 diversity COV is higher than those of S2 and S3. However,for cases C1, C2, C4 and C9, the proximity C* metric of strategy S1 is slightly lower than these of S2and S3. This can be explained by the high number of solutions generated by the former that are spread overthe objective space contrary to the strategy S3 which generates compact approximation sets. Although


Ie C5 C6 C7 C8 C9

S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3

S1 1.000 3E + 05 3E + 05 1.000 1E + 05 2E + 05 1.000 3E + 05 3E + 05 1.000 261.3 430.8 1.000 1E + 05 3E + 05S2 1.062 1.000 1.66 1.097 1.000 1.31 0.982 1.000 1.534 1.000 1.000 1.648 1.175 1.000 1.963S3 1.076 1.094 1.000 0.96 1.003 1.000 0.983 1.095 1.000 2.281 2.522 1.000 1.233 1.095 1.000

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17

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20

07

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77

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179

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Ie C11 C13 C14 C16

S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3

S1 1.000 127.3 127.3 1.000 78.34 78.34 1.000 2E + 05 4E + 05 1.000 2210 2210S2 0.614 1.000 1.003 0.619 1.000 1.192 14398 1.000 14398 0.9 1.000 1.124S3 6.686 1.000 1.000 0.589 1.000 1.000 1.000 1.000 1.000 0.8 1.114 1.000

Table 5CPU running time

CPU (seconds) C1 C2 C3 C4 C5 C6 C7 C8 C9 C11 C13 C14 C16

S1 1 5 1 5 26 285 11 160 19 180 104 409 1700S2 6 6 6 6 11 114 10 108 11 426 432 504 570S3 5 8 5 8 75 114 76 109 79 516 528 598 671

Table 6Proximity and diversity of incomparable sets of the 3 strategies in 6 tasks problems

C1 C2 C3 C4

Ns Cov C* Ns Cov C* Ns Cov C* Ns Cov C*

S1 40 (99.6; 100; 89) 0.43 40 (90.3; 56.9; 93.1) 0.48 40 (100; 100; 100) 0.46 40 (100; 67.8; 100) 0.34S2 9 (97.5; 84.1; 81.6) 0.46 18 (94.3; 95.7; 86.25) 0.52 6 (97.5; 84.1; 81.6) 0.44 13 (96.2; 100; 77.9) 0.48S3 4 (67.8; 69.3; 44.5) 0.50 3 (45.8; 43.8; 7.5) 0.61 4 (33.1; 65.5; 37.4) 0.47 10 (73.3; 60.5; 41.3) 0.45

Table 7Proximity and diversity of incomparable sets of the 3 strategies in 50 tasks problems

C5 C9

Ns Cov C* Ns Cov C*

S1 40 (99.9; 100; 98.9) 0.22 40 (76.9; 36.4; 100) 0.37S2 20 (71.8; 18.8; 31.5) 0.18 22 (67.2; 17.9; 31.4) 0.40S3 9 (19.1; 11.1; 41.9) 0.22 10 (19.4; 11.9; 31.3) 0.47


several runs (40) are processed with different weighted functions to approximate the Pareto frontier, thissearch procedure is absorbed in local optima. Moreover, S3 suffers from the compensatory effects, whichlead to discard extreme solutions. Other empirical results shows that strategy S1 better performs whenNs = 30 (higher C*). This could be explained by the fact that the filtered solutions are closer to the idealpoint (i.e. solutions with objectives characterized with higher thresholds) so their centroid is also closer.This is well illustrated by Fig. 8 for case study C7.

In cases C6 and C14 strategy, S1 and strategy S2 generate incomparable sets. However, one can remarkthat S1 provides a larger number of non-dominated solutions with a higher diversity as reported in Tables 8and 9. In conclusion, the empirical results obtained in this study show that the Pareto based approach out-performs lexicographic as well as the aggregative methods when considering large size problems. When the

Fig. 8. Empirical comparison between S1 (filled points), S2 (cross points) and S3 (empty points) for case study C7.

Table 8Proximity and diversity of incomparable sets of strategy S1 and strategy S3

C6 C14

Ns Cov C* Ns Cov C*

S1 40 (92; 96; 87) 0.3 40 (69; 58; 100) 0.31S2 2 (35; 16; 34) 0.3 10 (35; 80; 38) 0.34

Table 9Proximity and diversity of incomparable sets of strategy S1 and strategy S2

C8 C11

Ns Cov C* Ns Cov C*

S1 40 (99; 42; 100) 0.22 40 (97; 55; 99) 0.31S3 7 (26; 15; 32) 0.57 1 (0; 0; 0) 0.6


three strategies are incomparable (no solution provided by the SOO approach is dominated by solutionsgenerated by the Pareto approach), the quality of the Pareto-based approach generate better set of potentialefficient solutions (cardinality and the diversity).


6. Conclusions

In this paper, we proposed a progressive resource allocation (PRA) heuristic to solve multi-objectivesRCSP resources allocation with time window constraints. The PRA is based on Pareto optimization andTabu Search heuristic. The proposed meta-heuristic considers conflicting and heterogeneous objectives(ordinal and cardinal) to explore the feasible space to finding a set of efficient solutions. The proposedmeta-heuristic improves the resource to task allocation and preserves a given plan schedule (e.g. time syn-chronization of the COA). PRA strategy is based on the idea of maximizing the re-use of the best specificresources. This idea is motivated by the fact that in practice changing resources during the execution mightresult in performance decrease even if the new resource is better suited for the new task. In fact, transition,learned curve, set-up time and adaptation are key factors to consider when using multiple heterogeneousresources in the same plan. Moreover, this multi-objective optimization leads to accept compromise, effi-cient, solutions. The proposed PRA meta-heuristic applies the dominance concept and a multi-criteria fil-tering procedure to generate the set of potential efficient solutions.

The problem described in this paper has unique features. Thus, we decided to compare the proposedmeta-heuristic to single objectives strategies. Empirical results seem to show that Pareto based approachis superior to the lexicographic and to the aggregative optimization techniques for different problems size.In fact, the Pareto-based approach generates potential efficient solution set with higher cardinality and bet-ter diversity along the Pareto frontier. When combined with task scheduling algorithm, the proposed PRAmethod allows to generate the best potentially efficient set of solutions for each tasks schedule. An inter-esting perspective would be to combine the principles of the local search heuristic proposed in this workto other meta-heuristics such as evolutionary algorithms. The proposed heuristic could be generalized toother variants of RCPS problems.

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