Particle Swarm Optimization for the Vehicle Routing ...routing problem, where the only constraint that was taken into account was the capacity of the vehicles (and later the maximum

Particle Swarm Optimization for the VehicleRouting Problem: A Survey and a ComparativeAnalysis

Yannis Marinakis, Magdalene Marinaki , and Athanasios Migdalas

Abstract

In the last few years, a number of books and survey papers devoted to the vehiclerouting problem (VRP) or to its variants or to the methods used for the solutionof one or more variants of the VRP have been published. Also, in these years,the field of swarm intelligence algorithms has a significant growth. One of themost important swarm intelligence algorithms is the particle swarm optimization(PSO). Although the particle swarm optimization was first published in 1995,it took around 10 years in order researchers to publish papers using a PSOalgorithm for the solution of variants of the VRP. However, in the last 10 years, alot of journal papers, conference papers, and book chapters have been publishedwhere a variant of VRP is solved using a PSO algorithm. Thus, it is significant topresent a survey paper where a review and brief analysis of the most importantof these papers will be given. This is the main focus of this chapter.

KeywordsVehicle routing problem • Particle swarm optimization

Y. Marinakis (�) • M. MarinakiSchool of Production Engineering and Management, Technical University of Crete, Chania,Greecee-mail: [email protected]; [email protected]

A. MigdalasIndustrial Logistics, Luleå Technical University, Luleå, Sweden

Department of Civil Engineering, Aristotle University of Thessalonike, Thessalonike, Greecee-mail: [email protected]; [email protected]

© Springer International Publishing AG 2017R. Martí et al. (eds.), Handbook of Heuristics,https://doi.org/10.1007/978-3-319-07153-4_42-1

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mailto:[email protected]




https://doi.org/10.1007/978-3-319-07153-4_42-1

2 Y. Marinakis et al.

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Application of Particle Swarm Optimization Algorithm in Vehicle Routing Problems . . . . . . 8

Main Variants of VRP Solved by PSO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Different PSO Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Analysis of the Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Introduction

The vehicle routing problem is one of the most important problems in the field ofsupply chain management, of logistics, of combinatorial optimization, of transporta-tion, and, in general, of operational research. The interest in this problem has beenrecently increased both from theoretical and practical aspect. There are a numberof reasons for this growth. From the practical point of view, the problem is one ofthe most important problems in the supply chain management, and, thus, the findingof the optimal set of routes will help the decision makers to reduce the cost of thesupply chain and to increase the profit. Also, in the formulation of the problem,the managers could simulate the complete network and add any of the constraintsconcerning the customers, the vehicles, the routes, and, also, the traffic conditionsof the network and the energy consumption of the vehicles. Thus, someone couldsolve a realistic problem and find a near optimal set of solutions.

The reason that usually a near optimal set of solutions is found is that the problemfrom its origin (it was first introduced by Dantzig and Ramser in 1959 [46]) wasproved to be NP-hard problem even in its simpler version, the capacitated vehiclerouting problem, where the only constraint that was taken into account was thecapacity of the vehicles (and later the maximum tour length constraint was added).Thus, it is impossible in real-life applications to find an optimal solution. For thisreason, a number of heuristic, metaheuristic (mainly), evolutionary, and nature-inspired approaches have been proposed for the solution of the VRP and its variants.Also, exact algorithms have been proposed in order to solve the problem. They are,mainly, used for the solution of the simplest vehicle routing problems (the problemswith as few as possible constraints) and for a small number of nodes.

From the theoretical point of view, there are a huge number of researchers thatdeal with the solution of a variant (or more variants) of the problem. These variantsof the problem focus on a specific constraint, and the researchers are trying to findan algorithm that gives new best solutions in a specific set of benchmark instancesin short computational time with as less as possible parameters in order to give amore general algorithm. Thus, a new researcher that he/she would like to focus toa specific variant of a vehicle routing problem, he/she could find a large numberof very good papers focusing to VRP or to its variants and papers focusing to themethods that he/she would like to implement for the solution of the problem. Themost important and well-studied variants of the vehicle routing problem are the

Particle Swarm Optimization for the Vehicle Routing Problem: A Survey and. . . 3

capacitated vehicle routing problem [39,40,55,98,100], the vehicle routing problemwith time windows [43, 49, 50, 159, 160], the open vehicle routing problem [152],the vehicle routing problem with simultaneous pickup and deliveries [123,170], thevehicle routing problem with backhauls [29,30,171], the multidepot vehicle routingproblem [32, 99, 122], the stochastic vehicle routing problem [62, 77, 142, 162], thedynamic vehicle routing problem [61, 142, 145, 146], the periodic vehicle routingproblem [5], the split delivery vehicle routing problem [6], the heterogeneous fleetvehicle routing problem [64], the asymmetric vehicle routing problem [178], thevehicle routing problem with two- or three-dimensional loading constraints [21,66],etc. Nowadays, more complicated problems have been published where more thanone from the previously mentioned problems are combined in order to create a newmore challenging to be solved problem. These problems are denoted from someresearchers as rich vehicle routing problems [96]. However, a number of otherproblems have been introduced in the last years in order to cope with new needsthat arise from new realistic situations of the life, like the cumulative capacitatedvehicle routing problem [127], the evacuation vehicle routing problem [189], thegreen vehicle routing problem [103], the pollution routing problem [15], etc. Finally,there are a number of combined problems that need a solution of a vehicle routingproblem, like the location routing problem [97, 126, 144], the inventory routingproblem [26, 27, 54], the location inventory routing problem [78], the productionrouting problem [1], the production inventory distribution problem [13], and theship routing problem [37, 38, 150, 151]. Also, a number of publications with a realcase application have been realized where the authors simulate the realistic scenario,formulate the problem (or use an existing formulation) using a combination of theexisting variants or introducing a new one if it is possible, and use an existingalgorithm (or propose a new one or a suitable modified variant of an existing one)to solve it.

Thus, in the last years, the EURO Working Group on Vehicle Routing and Logis-tics Optimization, the VeRoLog, was created (www.verolog.eu); two different seriesof conferences devoted to the vehicle routing problem’s variants and applicationshave been introduced, the one is an annual conference which is organized fromthe VeRoLog working group since 2012 and the second is a triennial workshopon freight transportation and logistics denoted as Odysseus started on Crete 2000.A number of books devoted to the vehicle routing problem have been published.The first one was published in 1988, and it was a very inspiring book for anynew researcher that would like to study the vehicle routing problem or just toknow about the vehicle routing problem [70]. Twenty years later, one of the twoauthors (Bruce Golden) of the first published book [70] published a new book[72] that summarizes the work performed in these 20 years and gives some newdirections for future research. Before the publication of this book, another verysuccessful and inspiring book was published by Paolo Toth and Daniele Vigo [170]which is the most widely read and most cited book in the field. The success ofthis book led the same team of authors to publish in the end of 2014 the secondedition of the book [172] which covers all the field of the vehicle routing problemwith a more complete rewriting of some of the chapters of the previous edition

www.verolog.eu


of the book and with the new directions that have been created based on the newchallenges in the field. A volume of the Handbooks in Operations Research andManagement Science was devoted to vehicle routing problems and to more generalapplications, like the arc routing problem [9], and it was published in [10]. Finally,two other books devoted to the vehicle routing problem have been published in[138, 192]. Also, a large number of surveys have been published, initially, devotedto general vehicle routing problems and, nowadays, to variants of the vehiclerouting problem or to methods that are applied for the solution of the vehiclerouting problem or of its variants. Some of these survey papers can be found in[7,8,17,19,20,23,24,28,52,55,60,63,65,71,81,96,98,100,103,113,140,165,177].

Thus, in this chapter, as a general review for the vehicle routing problem cannotbe restricted in one chapter, we decided to focus in a specific algorithm, the particleswarm optimization (PSO) algorithm, and in the application of this algorithm for thesolution of the vehicle routing problem and its main variants. This is the first surveychapter, at least to our knowledge, that is devoted to this method for the VRP. In thefollowing sections, initially, a brief presentation of the particle swarm optimizationalgorithm will be given, and, then, we will present the variants of the vehicle routingproblem in which a particle swarm optimization algorithm has been applied for theirsolution.

Particle Swarm Optimization

Particle Swarm Optimization (PSO) is a very popular global optimizationmethod that was originally proposed by Kennedy and Eberhart as a simulation ofthe social behavior of social organisms such as bird flocking and fish schooling[86]. PSO uses the physical movements of the individuals in the swarm. Completesurveys, in the time that they are published, for the particle swarm optimizationcan be found in [11, 12, 41, 141]. Nowadays, a complete review for the particleswarm optimization is very difficult to be performed as the range of variants and ofapplications of PSO covers the whole field of optimization, and, thus, only surveysin a specific subject like the one presented in this chapter (application of PSOalgorithm in vehicle routing problems) can be presented without the length of thepaper to increase dramatically.

In general, in a PSO algorithm, a set of solutions are used where each solution isdenoted as a particle. These solutions create the swarm. In many algorithms, morethan one swarms exist. Initially, the solutions are randomly initialized in the solutionspace. Two are the main vectors that describe a particle, the position vector (xij ,where i denotes the particle (i D 1; � � � ; N , N is the swarm size) and j denotesthe corresponding dimension of the particle (j D 1; � � � ; d , d is the dimension ofthe problem)) and the velocities vector (vij ). The performance of each particle isevaluated on the predefined fitness function (f .x/).

Thus, each particle is randomly placed in the d -dimensional space as a candidatesolution. One very simple and effective way to initialize the particles was given in


[53] where the minimum value of the solution space is denoted as xmin;j and themaximum value is denoted as xmax;j . Then, the values of the solution vector foreach particle are calculated from the following equation:

xij .0/ D xmin;j C rj .xmax;j � xmin;j / (1)

where rj is a random number between .0; 1/.The velocity of the i -th particle vij is defined as the change of its position. The

algorithm completes the optimization through following the personal best solutionof each particle and the global best value of the whole swarm. Thus, in each iteration,except of the current position, another vector is used which is the personal bestposition of each particle through the iterations. The best member of the personal bestposition vector is the global best position of the whole swarm. Each particle adjustsits trajectory toward its own previous best position and the global best position,namely, pbestij and gbestj , respectively. The velocities and positions of particlesare updated using the following equations [86]:

vij .t C 1/ D vij .t/C c1rand1.pbestij � xij .t//C c2rand2.gbestj � xij .t//

(2)

xij .t C 1/ D xij .t/C vij .t C 1/ (3)

where t is the iterations’ counter, c1 and c2 are the acceleration coefficients, andrand1 and rand2 are two random variables in the interval (0, 1). The values of c1and c2 could be constant, or they could be adapted during the iterations. Usuallyin the most published papers concerning an application of PSO in a continuous ordiscrete problem, these values were set equal to 2. However, there is a possibilityto take different values or to adjust their values during the iterations, for example,using the following equations [53]:

c1 D c1;min Cc1;max � c1;min

i termax� t (4)

c2 D c2;min Cc2;max � c2;min

i termax� t (5)

where itermax is the maximum number of iterations and c1;min, c1;max, c2;min, andc2;max are the minimum and maximum values that c1 and c2 can take, respectively.In the beginning of the procedure, the values of c1 and c2 are small and, then,are increasing until they reach their maximum values. By doing this, on the firstiterations, there is a great freedom of movement in the particles’ solution space inorder to find the optimum quickly.

A number of different velocities equations have been proposed during the lastyears. Nowadays, other researchers use one of the velocities equations proposed inthe past that have been proved to perform well (mainly, the inertia equation or theconstriction equation), or, less frequently, they propose a new one which either is a


variant of a previously published equation or simulates something from the naturein which the specific researcher focuses in his/her study. The most important andknown velocities equations are the following:

• Inertia particle swarm optimization (IPSO) [157]:

vij .t C 1/ D wvij .t/C c1 rand1.pbestij � xij .t//C c2rand2.gbestj � xij .t// (6)

The difference between this variant and the one presented on Eq. (2) is the useof the inertia weight w. The inertia weight w is adapted during the iterations andis given by the following equation:

w D wmax �wmax � wmin

itermax� t (7)

where wmax and wmin are the maximum and minimum values of the inertia weight.• Constriction particle swarm optimization [42]:

vij .t C 1/ D �.vij .t/C c1rand1.pbestij � xij .t//

C c2rand2.gbestj � xij .t/// (8)

where the constriction factor, �, is used:

� D2

j2 � c �pc2 � 4cj

and c D c1 C c2; c > 4 (9)

• Another version of the constriction particle swarm optimization [53]:

vij .t C 1/ D �.vij .t/C c1rand1.pbestij � xij .t//

C c2rand2.gbestj � xij .t/// (10)

The only difference from the previous version is the use of the parameter k inthe constriction factor [53]:

� D2k

j2 � c �pc2 � 4cj

and c D c1 C c2; c > 4 (11)

• Clerc and Kennedy [42] proposed a simpler form of the constriction factor, thecondensed form, for the calculation of the velocities of the particles:

vij .t C 1/ D �.vij .t/C c.pmj � xij .t/// (12)


where

� D2

j2 � c �pc2 � 4cj

and c D c1 C c2; c > 4 (13)

and

pmj Dc1pbestij C c2gbestj

c: (14)

• Cognition-only particle swarm optimization [85]In this variant, the velocities equation is given by:

vij .t C 1/ D vij .t/C c1rand1.pbestij � xij .t// (15)

The social factor of velocities equation is not used.• Social-only particle swarm optimization [85].

In this variant, the velocities equation is given by:

vij .t C 1/ D vij .t/C c2rand2.gbestj � xij .t// (16)

The cognition factor of velocities equation is not used.• Local neighborhood topology particle swarm optimization [53]:

vij .t C 1/ D vij .t/C c1rand1.pbestij � xij .t//

C c2rand2.lbestij � xij .t// (17)

where the term of gbest in the previous algorithms has been replaced with theterm lbest , which means that instead of a global best population, a local bestpopulation is used.

lbestij 2 Ni jf .lbestij / D minf .xij /;8x 2 Ni (18)

The neighbor Ni is defined by [53]:

Ni D pbesti�nNi .t/; : : : ; pbesti�1.t/; pbesti .t/;

pbestiC1.t/; : : : ; pbestiCnNi .t/ (19)

A particle’s best position (pbestij ) in a swarm is calculated from the equation:

pbestij D

�xij .t C 1/; if f .xij .t C 1// < f .xij .t//pbestij ; otherwise

(20)


The optimal position of the whole swarm is calculated by the equation:

gbestj 2 fpbest1j ; pbest2j ; � � � ; pbestNj gjf .gbestj / D

minff .pbest1j /; f .pbest2j /; � � � ; f .pbestNj /g (21)

Most applications of PSO have concentrated on the optimization in continuousspace, while a lot of work has been done to the discrete optimization problemsbeginning from the next two papers [87,157]. In the following, a pseudocode of theparticle swarm optimization algorithm is presented.

algorithm Particle Swarm OptimizationInitializationSelect the number of swarmsSelect the number of particles for each swarmInitialization of the position and velocity of each particleCalculation of the initial cost function (fitness function) value of each particleKeep global best particle (solution) of the whole swarmKeep personal best of each particleMain PhaseDo until the maximum number of iterations has not been reached:

Calculate the velocity of each particleCalculate the new position of each particleEvaluate the new fitness function of each particleUpdate the best solution of each particleUpdate the best particle of the whole swarm

EnddoReturn the best particle (the global best solution)

Application of Particle Swarm Optimization Algorithm in VehicleRouting Problems

In this chapter, the papers in which a variant of the particle swarm optimizationalgorithm is applied for the solution of a variant of the vehicle routing problemare presented. In this section, an analysis is given on how these papers have beenselected. As there are a large number of papers in the literature, we will in the mostimportant of them. A very difficult task is to define which of the papers are the“most important” ones. We use a number of criteria in order to show the impactof each paper in the research community. The first criterion is the impact factorof the journal or the importance of the conference in which the paper has beenpublished. The second criterion is the number of citations (based on Google Scholar)that each paper has received through the years. Papers that have been published inthe current year are not expected to have a large number of citations. However, someof them based on their computational results are included in the analysis. Finally, the


third criterion is the computational results given in each paper. There are a numberof papers that have demonstrated their PSO variant giving results in one or in asmall set of instances. These papers are not analyzed as the effectiveness of theirvariant is not sufficiently proved. Also, there are a number of papers that, althoughthey have been published in very good journals, they present a PSO variant forthe solution of a real-life application and they do not give a comparative analysiswith other algorithms from the literature in benchmark instances; thus, it is verydifficult to show the effectiveness of the PSO variant. These kinds of papers are notincluded in the analysis. Finally, there are a number of papers that they present anovel PSO procedure, they apply the method in classic sets of benchmark instances,and their computational results are competitive with the results of the most effectivealgorithms from the literature. This last category of papers is those that are analyzedin this chapter.

Main Variants of VRP Solved by PSO

The main variants of the vehicle routing problem that a particle swarm optimizationalgorithm has been applied are the following.

Capacitated Vehicle Routing ProblemAbbreviation: CVRP Definition: Each vehicle must start and finish its tour atthe depot. Capacity constraints of the vehicles. Maximum tour duration of eachroute. Not split deliveries allowed. Customers have only demands and servicetime [19, 20, 72, 170].

Open Vehicle Routing ProblemAbbreviation: OVRP Definition: Same constraints as in the CVRP except that thevehicles do not return in the depot after the service of the customers [152].

Vehicle Routing Problem with Time WindowsAbbreviation: VRPTW Definition: Same constraints as in the CVRP and inaddition each customer must be serviced within a specific time window. Vehiclesand depots may, also, have a time window. Minimization, initially, of the numberof routes and, then, minimization of the total traveled distance [70, 72, 138, 158,159, 170].

Vehicle Routing Problem with Simultaneously Pickup and DeliveryAbbreviation: VRPSPD Definition: Same constraints as in the CVRP. Thecustomers require not only delivery of products but, also, a simultaneous pickup of products from them. It is assumed that the delivery is performed beforethe pickup and that the vehicle load should never be negative or larger than thevehicle capacity [70, 72, 138, 170].

Vehicle Routing Problem with Stochastic DemandsAbbreviation: VRPSD Definition: A vehicle with finite capacity leaves from thedepot with full load and has to serve a set of customers whose demands are knownonly when the vehicle arrives to them. A route begins from the depot and visitseach customer exactly once and returns to the depot. This is called an a priori


tour, and it can be seen as a template for the visiting sequence of all customers[18, 62, 162].

Dynamic Vehicle Routing ProblemAbbreviation: DVRP Definition: In dynamic problems, part or all of the inputis unknown and is revealed dynamically during the design or execution of theroutes [140].

Multidepot Vehicle Routing ProblemAbbreviation: MDVRP Definition: More than one depots are used for thecustomers’ service. There is a possibility for each customer to be clustered andserved from only one depot, or the customers may be served from any of thedepots using the available fleet of vehicles [122, 149].

Vehicle Routing Problem with Stochastic Travel TimesAbbreviation: VRPSTT Definition: In this variant, the travel time between everypair of nodes is a random variable related to traffic jam, road maintenance, orweather conditions. The stochasticity appears in the arcs (road) of the networkdue to unexpected conditions [196].

Vehicle Routing Problem with Fuzzy DemandsAbbreviation: VRPFD Definition: In this variant, the demands could berepresented as fuzzy variables.

Periodic Vehicle Routing ProblemAbbreviation: PVRP Definition: In this problem, vehicle routes must be con-structed over multiple days where during each day within the planning period,a fleet of capacitated vehicles travels along routes that begin and end at a singledepot [56]. The objective of the PVRP is to find a set of tours for each vehicle thatminimizes total travel cost while satisfying the constraints of the problem [56].

Location Routing ProblemAbbreviation: LRP Definition: In this variant, the optimal location to be used forthe storage facilities have to be decided. From these locations, the vehicles willbegin their routes, in a way that the total cost of the routing (distance, fuel, time,etc.) and facility location (running costs, rent or property cost, etc.) will be theminimum. At the same time, the optimal routes for the vehicles have to be foundin order to satisfy the demand of the customers [47, 119, 126, 144].

Location Routing Problem with Stochastic DemandsAbbreviation: LRPSD Definition: In this variant, the same constraints as inthe case of the location routing problem hold except that the demands of thecustomers have stochastic and not deterministic values.

Team Orienteering ProblemAbbreviation: TOP Definition: This is a variant of the vehicle routing problemwith profits or pricing. In this problem, a set of locations is given, each one witha score. The goal is to determine a fixed number of routes, limited in length,that visit some locations and maximize the sum of the collected scores [174].The objective of the TOP is to construct a certain number of paths starting at anorigin and ending at a destination that maximize the total profit without violatingpredefined limits [154].


Production Routing ProblemAbbreviation: PRP Definition: This variant is a hybridization of VRP withproduction problems. These problems are very complicated problems as theyinclude, among others, decisions concerning the number of workers to be hired,the quantities of products to be produced using various strategies, the amountof inventories to be maintained, the assignment of the customers in differentmanufacturing plans or depots, and, finally, the design of the routes in order tosatisfy the customers’ demands.

Ship Routing ProblemAbbreviation: SRP Definition: In this variant, the optimum routing and schedul-ing of different types of ships are calculated [37, 38, 48, 150, 151].

Vehicle Routing Problem with heterogeneous fleetAbbreviation: HVRP Definition: In this variant, instead of an homogeneous fleetof vehicles, the company uses a set of vehicles with different capacities, and,thus, the fleet of vehicles is heterogeneous.

Inventory Routing ProblemAbbreviation: IRP Definition: In this variant of the vehicle routing problem,a simultaneous allocation of inventories and decision of routing schedules isrealized. The objective of the IRP is to minimize the total cost (the distributionand inventory costs of retailers) [26, 27].

Vehicle Routing Problem with Uncertain DemandsAbbreviation: VRPUD Definition: In this variant, the demand is uncertain withunknown distribution.

Vehicle Routing and Scheduling ProblemAbbreviation: VRSP Definition: In general, the goal of VRSP is to determinethe set of trips that a vehicle will make during the day in order to reduce thetransportation costs.

Other variants of VRP, combinations of VRP with other problems, or real-life applications of VRP that have been solved using a variant of a particleswarm optimization algorithm are the distance constraint vehicle routing problem[169], the multiple destination routing problem (MVRP) [193], the multivehicleassignment problem (MVAP) [51], the production routing problem [1], the berthallocation problem [168], the disaster relief logistics [22], the garbage collectionsystem [95], the integrated production and distribution [175], the urban transitrouting problem [84], the production and pollution routing problem [94], thereducing vehicle emissions and fuel consumption [133], the pedestrian-vehiclemixed evacuation (P-VMEP) [201], the time-dependent vehicle routing problem(TDVRP) [130], the emergency vehicle scheduling problem [57], the emergencylogistics (EL) [189, 197], the evacuation vehicle routing problem (EvVRP) [191],the environmental vehicle routing problem (EnVRP) [59], the integrated productionscheduling and vehicle routing [35], and the reverse logistics (RL) [134].

Table 1 presents the number of papers (journal papers, international conferencepapers, and book chapters, denoted as Journal, Conf, and BC in the table, respec-tively) in which a variant of the particle swarm optimization algorithm has been used


Table 1 Number of papers using particle swarm optimization in each variant of the vehiclerouting problem

Variant Journal Paper Conf Paper

and BC

CVRP 8 [4, 33, 92, 109, 116, 147, 167, 176] 7 [79, 93, 101, 111, 156, 163, 183]

VRPSPD 3 [3, 36, 69] 4 [68, 164, 184, 195]

LRP 2 [104, 108] 2 [14, 135]

HVRP 2 [16, 188]

VRPTW 2 [2, 73] 9 [25, 31, 58, 80, 105, 118, 124, 182, 200]

VRPUD 2 [34, 186]

TOP 2 [125, 154] 3 [44, 45, 153]

MVAP 1 [51]

DVRP 2 [91, 132] 3 [89, 90, 131]

OVRP 2 [120, 128] 4 [110, 139, 180, 199]

VRPSD 1 [117] 3 [112, 137, 198]

PVRP 3 [129, 148, 166]

RL 1 [134]

VRPFD 1 [185] 1 [136]

VRPSTT 1 [155]

MDVRP 1 [82] 4 [161, 179, 181, 194]

DCVRP 1 [169]

VRSP 1 [173]

EvVRP 1 [191] 2 [189, 190]

MVRP 1 [193]

EL 1 [197]

LRPSD 1 [107]

EnVRP 1 [59]

IRP 1 [106]

TDVRP 1 [130]

PRP 1 [1]

SRP 1 [48]

for the solution of a variant of the vehicle routing problem. As it was expected, mostof the papers concern an application of a PSO variant to the capacitated vehiclerouting problem with eight journal papers published. There is no other variant ofVRP with more than three journal papers. However, in the vehicle routing problemwith time windows, there are a large number of conference papers and book chapters(nine in total). As we can see, a variant of PSO algorithm has been applied in 27different variants of vehicle routing problem. In total, there are 37 journal papersand 52 international conference papers and book chapters with a PSO variant in avehicle routing problem.

Although when the particle swarm optimization algorithm was initially proposed,there were no any applications of the algorithm for the solution of the vehicle routing


Table 2 Journals that have published at least one PSO algorithm for the solution of a vehiclerouting problem variant

Journal Abbreviation I.F. Nr TNrCit

Transportation Science TS 3:295 1 60

Networks and Spatial Economics NETS 3:250 1 23

Expert Systems with Application ESWA 2:981 3 295

Applied Soft Computing ASOC 2:857 6 201

Neurocomputing NC 2:392 1 2

Engineering Applications of Artificial Intelligence EAAI 2:368 1 130

Transportation Research Part E TRE 2:279 2 57

IEEE Transactions on Systems, Man, andCybernetics-Part C

IEEE SMC 2:171 1 74

Computers and Industrial Engineering CIE 2:086 6 363

Journal of Intelligent Manufacturing JIM 1:995 3 94

Computers and Operations Research COR 1:988 2 297

Entropy Entropy 1:743 1 20

Measurement Measurement 1:742 1 12

International Journal of Advanced ManufacturingTechnology

IJAMT 1:568 2 57

Annals of Operations Research ANOR 1:406 1 33

Applied Mathematics and Computation AMC 1:345 1 38

Applied Intelligence APIN 1:215 1 8

Optimization Letters OptL 1:019 1 5

Journal of Zhejiang University SCIENCE A SCIENCE A 0:941 1 198

Memetic Computing MEME 0:900 1 13

International Journal of Operational Research IJOR � 1 49

Procedia Engineering PrE � 1 8

Journal of Mathematical Modelling and Algorithms JMMA � 1 67

American Journal of Applied Sciences AJAS � 1 13

Transportation Research Journal TRJ � 1 4

problem; in the last years, a number of researchers have solved a variant of VRPusing a PSO algorithm. In Table 2 journals that have published at least one paperwith this topic are presented. Journals are sorted based on their impact factor for2015 (denoted as I.F. in the table) that it is presented in column three. Finally, incolumns 4 and 5, the number of papers (denoted as Nr in the table) that has beenpublished in each journal and the total number of citations of all papers based onGoogle Scholar on 30 May 2017 (denoted as TNrCit in the table) are presented.

Different PSO Variants

As it is mentioned earlier, the main problem of the application of a PSO algorithmfor the solution of a VRP was that the PSO algorithm is suitable for continuous


optimization problems and the VRP is a combinatorial optimization problem witha specific structure in the solution which needs in most cases a path representationin order to have an effective algorithm. Thus, there are a number of algorithms thatuse a 0/1 representation of the solution. However, these algorithms are not the mosteffective algorithms for the solution of this kind of problems. We mentioned thatfrom the first years that PSO was proposed, there were implementations of PSO[157] suitable for discrete optimization problems but not for routing type problems.Thus, the first effective application of a PSO algorithm in VRP was in 2006 [33].Since then a number of different algorithms have been proposed that are suitablefor the solution of a VRP variant. Most of them are hybridized with a local searchalgorithm, as almost every evolutionary algorithm if it is applied for the solution of aVRP problem. In the following, we present and analyze the most important variantsof PSO that have been applied for the solution of a VRP variant. The choice wasperformed using the same criteria that were mentioned earlier, especially the oneconcerning the effectiveness of the algorithms.

SR-PSO. One of the most used representations of PSO for the solution of a VRPvariant denoted as SR-PSO (SR-1 PSO or SR-2 PSO) was proposed in [4] wheretwo representations of the particles were used for the solution of the CVRP. Inthe first one, the particle consists of n C 2m dimensions. The first n dimensionsare related to customers (each customer is represented by one dimension). Thelast 2m dimensions are related to vehicles (each vehicle is represented by twodimensions as the reference point in the Cartesian map). The priority matrix ofvehicles is constructed based on the relative distance between these points andcustomers’ location [4]. A customer is prioritized to be served by the vehicle whichhas the closer distance. In the second representation, the particle consists of a 3m-dimensional particle where it is decoded as a real number. All dimensions are relatedto vehicles; each vehicle is represented by three dimensions: two dimensions forthe reference point and one dimension for the vehicle coverage radius [4]. Thealgorithms use a number of local search algorithms. A number of papers either fromthe same authors or from other research groups have used the same representationsfor solving a VRP variant.

The research group that proposed this representations have applied them forthe solution of the vehicle routing problem with time windows [2], of the vehiclerouting problem with simultaneously pickups and deliveries [3] (the most importantpublication of an application of PSO-based algorithm for solving a VRP based onthe number of citations), of the multidepot VRP with pickup and deliveries [161]and [82], and of the location routing problem [104].

Other researchers that used the same encoding and decoding scheme for solvinga VRP variant are the one from Hu et al. [76] that proposed a hybrid chaos-particle swarm optimization algorithm (HPSO) for solving VRPTW. In [121], avariant of the algorithm was proposed for the solution of a vehicle routing problemwith uncertain demands. In [16], a variant of the algorithm was presented for arich vehicle routing problem, the vehicle routing problem with heterogeneous fleet,mixed backhauls, and time windows (VRPHFMBTW).


Quantum PSO – QPSO. One of the first PSO implementations for the solution ofthe CVRP was the one published in [33]. In this PSO implementation, a quantumdiscrete PSO algorithm is used [187]. The algorithm is hybridized with a simulatedannealing algorithm [88] for the improvement of the particles. They use a zeroone encoding scheme where value equal to one means that the vehicle servesthe customer and value equal to zero means that another vehicle will serve thecustomers. The dimension of the vector is equal to N �K where N is the numberof nodes and K is the number of vehicles.

HybGenPSO. In [109], a hybridization of a genetic algorithm with a particleswarm optimization, algorithm for the solution of the CVRP was presented. Inthis memetic algorithm, the role of the genetic algorithm was to create the newgenerations, and the role of the PSO algorithm was to improve the populationbetween two different generations. Thus, it was a memetic algorithm with thedifference that usually in a memetic algorithm, each member of the populationis evolved between two generations using a local search algorithm, while in thismemetic algorithm, the population was evolved using a global search algorithm.This idea was inspired by the fact that in real life, each member of a population isevolved by reacting with other members of the population and not by itself. Thealgorithm uses the inertia equation for the velocities (Eq. 6) and was tested in thetwo classic sets of benchmark instances, and a number of best-known solutions werefound.

HybPSO. In [116], a hybrid particle swarm optimization algorithm was usedfor the solution of the capacitated vehicle routing problem. The algorithm is animproved version of the algorithm used in [108] for the solution of the locationrouting problem, and it uses the same three algorithms in the hybridization phase,the multiple phase neighborhood search – greedy randomized adaptive searchprocedure (MPNS-GRASP), the expanding neighborhood search (ENS) [114], andthe path relinking (PR) [67]. The first method is used for the creation of the initialsolutions, the second one is used as a local search phase, and the third one is usedfor the movement of the particles. The difference of the algorithm proposed in[116] from the algorithm denoted as combinatorial neighborhood topology PSO(proposed in [111]) is that the first one uses a transformation between continuousand discrete values and vice versa based on the relative position indexing [102]and it, also, uses a procedure that does not affect the solution of each particle. Inthe combinatorial neighborhood topology PSO, no transformation is needed at all.The algorithm proposed in [116] uses the inertia equation for the velocities (Eq. 6).The algorithm was tested in a classic set of benchmark instances, and a number ofbest-known solutions were found.

In [110], a new version of the hybrid particle swarm optimization algorithmthat was presented in [116] for the solution of the CVRP was presented whereinstead of using the expanding neighborhood search algorithm for the improvementof each particle separately, the variable neighborhood search algorithm was used


[74]. Another difference is that the inertia velocities equation has been replaced bythe constriction velocities equation (Eq. 8).

In [117], an improved version of the hybrid particle swarm optimization algo-rithm that was presented in [110] for the solution of the OVRP was presented whereinstead of using the variable neighborhood search algorithm for the improvement ofeach particle separately, a simpler local search algorithm based on 2-opt and 3-optalgorithms was used. Also, in the initialization phase of the algorithm, the particlesstart from random values, thus avoiding the use of a more sophisticated procedure.The reason that this simpler local search algorithm and the random initial valueswere used was that the calculation of the objective function of the VRPSDs is muchmore difficult and time-consuming than the calculation of the objective functionof a simpler variant of VRP and, thus, the use of a simpler local search algorithmavoids the increase of the computational time of the whole procedure. Eight differentversions of velocities equation were used in order to test which one of them is themost appropriate for the selected problem.

CNTPSO. In [108], a hybrid particle swarm optimization algorithm was used forthe solution of the location routing problem. The PSO algorithm was hybridizedwith three other effective algorithms, the multiple phase neighborhood search –greedy randomized adaptive search procedure (MPNS-GRASP) [115], the expand-ing neighborhood search (ENS) [114], and the path relinking [67].

The novelty of the paper was that a procedure was used where there is notransformation from continuous to discrete values and the movement of the particleswas performed using a path relinking procedure [67], where the current solutionof the particle was used as starting solution and the target solution was either thelocal best particle or the global best particle. This was the first step of the verysuccessful and effective topology that was denoted as combinatorial neighborhoodtopology and was used for the solution of the capacitated vehicle routing problem[111]. The velocities equation used was the inertia one (Eq. 6). In [111], one ofthe most successful versions of the PSO for the CVRP was published, denotedas combinatorial neighborhood topology PSO. This algorithm does not need atransformation from continuous to discrete values, and, thus, there is no loss ofany information of good solutions that have been created in an iteration. In thisalgorithm, the position equation (Eq. 3) of the particles has been replaced by apath relinking strategy. The role of the velocities equation is limited to show whichparticle (or combination of particles) will be followed by the selected particle basedon some conditions. This version of PSO is a very efficient version as it wasproved by its application in the classic set of benchmark instances where bettersolutions were found compared to the ones found by other versions of PSO in lesscomputational time.

In [112], an improvement of the CNTPSO was presented where a combinationof this topology with an expanding neighborhood topology (ENT) is used. In thistopology, there are not a standard number of local neighbors, but the number of localneighbors begins from a small number and increases using some conditions. Whenthe number of neighbors becomes equal with the number of particles, then a global


best PSO algorithm is used. The algorithm uses the same local search algorithm andthe same initialization procedure as the algorithm presented in [117].

An improved version of CNTPSO with expanding neighborhood topology waspresented in [107] for the solution of the location routing problem and of the locationrouting problem with stochastic demands. The difference of this algorithm from theinitial CNTPSO is that it uses simultaneously a global neighborhood topology andthe expanding neighborhood topology as it was described previously.

An adaptive version of CNTPSO is presented in [118] for the solution ofthe vehicle routing problem with time windows. In this version, all parameters(acceleration coefficients, iterations, local search iterations, upper and lower boundsof the velocities and of the positions, and number of particles in each swarm) areadapted during the procedure, and, thus, the algorithm works independently andwithout any interference from the user.

h_PSO. In [68, 69], two different algorithms for the solution of VRPSPD problemusing a PSO algorithm were presented. Initially, it was presented as a conferencepaper [68], and its improved version was published as a journal paper in [69]. Theauthors presented a hybridization of PSO algorithm with a variable neighborhoodsearch algorithm and used an annealing-based mechanism in order to maintain thediversity of particles. The inertia velocities equation was used (Eq. 6). They useda representation that began with a giant tour without taking into account the routerestrictions, and, then, they used a decoding scheme to partition the giant tour intofeasible routes. The procedure that they used is denoted as split procedure, and it wasfirst presented by Prins in the frame of an evolutionary algorithm for the solution ofthe CVRP [143]. The algorithms were tested in the usually used benchmark set ofinstances for the problem, and a number of new best solutions were found.

DAPSO. The most important application of a particle swarm optimization algo-rithm for the solution of a dynamic vehicle routing problem was presented in [91]. Inthis paper, the authors solved the DVRP with dynamic requests, and a procedure wasused where a partial static VRP was solved each time a new request was received.The authors hybridized the particle swarm optimization algorithm with a variableneighborhood search algorithm for the solution of the problem (DAPSO). Therepresentation used in this work is a simple discrete representation which expressesthe route of m vehicles over the n customers to serve. The representation allows theinsertion of dynamic customers in the already planned route, and it is a permutationof the customers where as a DVRP is solved, the authors keep the coordinates ofthe customers, the time in which each customer is served, and if a customer hasbeen served or not. Another very interesting application of PSO algorithm in thisproblem was presented in [89] where a multiswarm PSO algorithm was applied, anumber of benchmark instances were solved, and comparisons with algorithms fromthe literature were given. Finally, a third publication of the same group of authorsusing a PSO algorithm was presented in [90].


SPSO. One of the most important applications of a PSO algorithm in the solutionof the vehicle routing problem with time windows is the one presented in [73]. Inthis paper, the proposed SPSO-VRPTW treated the discrete search space as an arcset of the complete graph that was defined by the nodes in the VRPTW and regardedthe candidate solution as a subset of arcs. A set-based representation method wasproposed to characterize the discrete search space. The search space is representedby a universal set S. The elements in S can be divided into D dimensions.

2MPSO. A two-phase particle swarm optimization algorithm for the solution of theDVRP was presented in [131], and later it was improved in [132]. They presented analgorithm where a new equation of velocities for the particles was presented wherelocal and global search topologies were combined.

PVPSO. In [120], a PSO algorithm for the OVRP is presented. The authors used astandard PSO for the encoding and the decoding procedure where all the elementsof the positions vector are sorting in descending order and, then, the first is addedin the first route, the second in the route with the least residual capacity, and soon until all elements are inserted in a route. The routes, then, were improved usingone-move local search. They used the inertia equation of velocities (Eq. 6), and theytested their algorithm in a small set of instances and found very good solutions.

GLNPSO. A very interesting application of a PSO in the VRPFDs was the onepresented in [185]. In this problem, the variant of VRP used included, also, softtime windows constraints, and the authors considered two objective functions, theminimization of the total travel cost and the maximization of the average satisfactionlevel of all customers in a fuzzy environment. The authors used a global-local-neighbor particle swarm optimization with exchangeable particles. An illustrativeexample that explains how the PSO algorithm had been applied in this problem waspresented and analyzed in details.

PMPSO. In [92], the authors presented an approach that uses a probability matrixas the main device for particle encoding and decoding. In the decoding phase, notonly the assignment of the customers in vehicles was realized, but, also, the routingof the customers was calculated. They used a number of local search algorithms (1-1exchange, 2-opt, Or-opt) in order to improve the solution of the particles.

MOPSO. A very interesting formulation of a multiobjective competitive openvehicle routing problem with time windows was presented in [128]. In this problem,the reaching time to customers affects the sales amount. Therefore, distributorsintend to service customers earlier than rivals to obtain the maximum sales.Moreover, a part of a driver’s benefit is related to the amount of sales. Thus,the balance of goods carried in each vehicle is important in view of the limitedvehicle capacities. They gave a complete analysis of the new problem and of theformulation, and they used a multiobjective PSO algorithm for the solution of theproblem. In this multiobjective algorithm, the nondominated solutions are stored in


a repository, and, then, if a new solution dominates a solution from the repository ora personal best solution of a particle, then the new solution replaces the dominatedsolution.

StPSO. A very interesting paper was published in [173]. In this paper, the authorssolved a very complicated vehicle routing and scheduling problem with cross-docking. They use an encoding scheme to represent the scheduling in a string. Thus,all are encoded as genes in a 1-by-n string where the length is equal to the numberof vehicles. Thus, the digit in the ith chromosome indicates the route number towhich the ith vehicle is assigned [173]. A similar variant of the algorithm has beenproposed in [166] for the solution of the periodic vehicle routing problem.

NPSO. In [83], a nested PSO was given for the solution of the CVRP. Initially,feasible solutions are created in clusters using sweep algorithm, and, then, a routeoptimization was performed inside the clusters. The PSO was used in both phases,in the first phase for reorganizing the routes and in the second phase for leading tothe optimization of the routes.

DPSO. A discrete PSO algorithm is presented in [125] for the solution of theteam orienteering problem. The authors hybridized their algorithm with a variableneighborhood search algorithm. Another application of DPSO for the solution ofthe CVRP was proposed in [147]. The authors hybridized their algorithm with aniterated local search algorithm.

DHPD. In [75], a hybridization of a PSO algorithm with a differential evolutionalgorithm was presented. In the paper, an indirect representation was proposed.For N customers, each individual was encoded as a real number vector with Ndimensions. The integer part of each dimension or element in the vector representsthe vehicle. Thus, the same integer part represents the customer in the samevehicle. The fractional part represents the sequence of the customer in the vehicle[75]. Another similar formulation of the algorithm was proposed in [180] for thesolution of the OVRP. The authors presented a very interesting representation ofthe particles’ solution where in the encoding procedure, the customers took realvalues and in the decoding procedure, the customers with the same integer part areassigned in the same team (vehicle or group of vehicles if the capacity of the vehiclewas violated). The authors tested their algorithm using the classic set of benchmarkinstances, and they obtained very good results.

MODPSO. In [195], a PSO algorithm was used for the solution of the VRPSPD.The very interesting part of the algorithm was the solution representation where them customers were divided inmC1-dimensional particles. In the decoding process ofthe algorithm, the particles were transformed to vehicle allocation s with the sweepalgorithm, and, then, the priority matrices of customers served by the same vehiclewere evaluated. Based on the two matrices, the vehicle routes were constructed.


Analysis of the Algorithms

In Table 3, the most important papers that use a variant of PSO for a solution of avariant of VRP are presented. The selection of the papers has been performed basedon the criteria mentioned previously. The ranking in Table 3 is based on the numberof citations that each of these papers has on 30 May 2017 on Google Scholar.Also, in Table 3, papers with a small number of citations are presented, but thesepapers either present very effective algorithms based on their computational resultsor have great potential for fast increase on the number of citations due to the year ofpublication. In Table 3, in addition to the number of citations, it is presented whichvariant of the VRP is solved (column 2), with which variant of PSO (column 3),in which journal, book, and conference it is published (column 4), and the yearpublished (column 5). The number of citations is given in the last column.

As we can see, there is only one paper with more than 200 citations [3] with289 citations and one that it is near to 200 (paper [33] with 198 citations). Twoother papers have more than 150 citations (paper [109] with 167 and paper [4] with162 citations). Finally, there are two papers with more than 100 citations, paper[116] with 130 citations, and paper [73] with 117 citations. In total, there are sixpapers with more than 100 citations. Four of them present algorithms for solvingthe capacitated vehicle routing problem, and the other two present algorithms forsolving the vehicle routing problem with simultaneous pickup and delivery. Also,two of them ([3] and [4]) are works from a specific research group that have madea significant contribution in the field. In Table 3, there are, also, other four papersfrom the same research group (paper [2] for the solution of the VRPTW with 49citations, paper [161] for the solution of MDVRP with 23 citations, paper [82]for the solution of MDVRP with 8 citations, and paper [104] for the solution ofLRP with 3 citations). The other research group with significant contribution in thefield studied in this chapter is the one that has two publications in the six moresignificant publications in the field ([116] and [109]) and other six publicationsthat are presented in Table 3 (paper [117] for the solution of the VRPSD with 76citations, paper [108] for the solution of LRP with 67 citations, paper [111] forthe solution of CVRP with 9 citations, paper [110] for the solution of OVRP with8 citations, paper [112] for the solution of VRPSD with 7 citations, paper [107]for the solution of LRP and LRPSD with 5 citations). Of course, there are paperswith significant impact that are presented in this table as papers with 78 citations[91], papers with 74 citations [73], or papers with 70 citations [120] which haveinfluenced a large number of researchers.

One of the most important tables in this survey is the last one (Table 4). In thistable, the computational results of the most important papers are summarized andpresented analytically. In this table, they are not presented papers that the proposedalgorithm was tested in one or in a small set of instances, especially when theseinstances are not available in the Internet. Also, they are not presented papers thatthe proposed PSO algorithm solves a real-life problem as we cannot compare theresults and we cannot see their effectiveness. The only papers that are presented are


Table 3 Total number of citations in the most important papers that use a PSO variant for solvinga VRP variant

Paper VRP variant PSO variant Published in Year Citations

Ai and Kachitvichyanukul [3] VRPSPD SR-PSO COR 2009 289

Chen et al. [33] CVRP QPSO SCIENCE A 2006 198

Marinakis and Marinaki [109] CVRP HybGenPSO ESWA 2010 167

Ai and Kachitvichyanukul [4] CVRP SR-PSO CIE 2009 162

Marinakis et al. [116] CVRP HybPSO EAAI 2010 130

Goksal et al. [69] VRPSPD h_PSO CIE 2013 117

Khouadjia et al. [91] DVRP DAPSO ASOC 2012 78

Marinakis et al. [117] VRPSD HybPSO ASOC 2013 76

Gong et al. [73] VRPTW SPSO IEEE SMC 2012 74

MirHassani and Abolghasemi [120] OVRP PVPSO ESWA 2011 70

Marinakis and Marinaki [108] LRP CNTPSO JMMA 2008 67

Moghaddam et al. [121] VRPUD SR-PSO CIE 2012 57

Xu et al. [185] VRPTW GLNPSO TRE 2011 55

Ai and Kachitvichyanukul [2] VRPTW SR-PSO IJOR 2009 49

Belmecheri et al. [16] HVRP SR-PSO JIM 2013 42

Yao et al. [188] HVRP IPSO ANOR 2016 33

Kim and Son [92] CVRP PMPSO JIM 2012 31

Norouzi et al. [128] OVRP MOPSO NETS 2012 23

Sombuntham andKachitvichayanukul [161]

MDVRP SR-PSO IMECS 2010 2010 23

Vahdani et al. [173] VRSP StPSO JIM 2012 21

Hu et al. [76] VRPTW SR-PSO Entropy 2013 20

Okulewicz and Ma Kndziuk [131] DVRP 2MPSO ICAISC 2013 2013 16

Kanthavel and Prasad [83] CVRP NPSO AJAS 2011 13

Muthuswamy and Lam [125] TOP DPSO MEME 2011 13

Norouzi et al. [129] PVRP IPSO Measurement 2015 12

Marinakis and Marinaki [111] CVRP CNTPSO EvoCOP 2013 9

Qi [147] CVRP DPSO PrE 2011 8

Marinakis and Marinaki [110] OVRP HybPSO ANTS 2012 8

Kachitvichyanukul et al. [82] MDVRP SR-PSO CIE 2015 8

Hu and Wu [75] OVRP DHPD WCICA 2010 7

Marinakis and Marinaki [112] VRPSD CNTPSO GECCO 2013 7

Marinakis [107] LRPSD CNTPSO ASOC 2015 5

Tavakkoli Moghaddam et al. [166] PVRP StPSO TRJ 2012 4

Liu and Kachitvichyanukul [104] LRP SR-PSO IEAS 2013 3

Okulewicz and Ma Kndziuk [132] DVRP 2MPSO ASOC 2017 0

the ones that the proposed algorithm was tested in one or more well-known set(s)of benchmark instances, and the authors presented their results in such a way thatthe results are comparable with the computational results of other algorithms fromthe literature. In Table 4 in the first column, a reference of the paper is given, in


Table 4 Computational results of papers that used a variant of a PSO algorithm for the solutionof a vehicle routing problem variant

Paper VRP Sets NBI Nodes NBS EBS Quality Quality range

Ai and VRPSPD 3 84 20–40 Average results are given

Kachitvichyanukul [3] 40 50 Average results are given

14 50–199 Average results are given

Chen et al. [33] CVRP 1 16 29–134 0 7 0:969 0.00–4.56

Marinakis and CVRP 2 14 51–200 0 10 0:046 0.00–0.23

Marinaki [109] 20 200–483 0 1 0:60 0.00–0.91

Ai and [4] CVRP 2 16 29–134 0 10 0:065 0.00–0.29

Kachitvichyanukul 14 51–200 0 4 0:874 0.00–2.51

Marinakis et al. [116] CVRP 1 14 51–200 0 7 0:084 0.00–0.29

Goksal et al. [69] VRPSPD 2 40 50 0 40 0:00 0.00

14 50–199 0 6 0:00 0.00–3.16

Khouadjia et al. [91] DVRP 1 21 50–199 5 0 NM| NM

Marinakis et al. [117] VRPSD 2 21 51–200 21 0 �1:105 �3.13 to �0.001

40 16–60 New set of benchmark instances

Gong et al. [73] VRPTW 3 56 25 0 0 8:72 0.18–60.56

29 best solutions based on number of vehicles

56 50 0 0 6:99 0.00–41.33

32 best solutions based on number of vehicles

56 100 6 9 2:96 �12.69 to 15.70

MirHassani andAbolghasemi [120]

OVRP 1 15 32–50 0 12 0:196 0.00–2.605

Marinakis andMarinaki [108]

LRP 1 19 12–318 6 13 �0:08 �0.68 to 0.00

Moghaddam et al.[121]

VRPUD 1 (CVRP) 60 32–101 0 53 0:04 0.00–1.027

Ai and [2] VRPTW 2 56 25 0 0 0:323 0.2–1.2

Kachitvichyanukul 56 50 0 0 1:148 0.2–7.2

Kim and Son [92] CVRP 2 16 29–134 0 10 0:142 0.00–0.733

14 51–200 0 6 0:712 0.00–2.95

Sombuntham andKachitvichayanukul[161]

MDVRP 1 29 100 0 15 0:96 0–13.94

Kanthavel and Prasad[83]

CVRP 1 16 29–134 0 16 0:00 0.00–0.00

Qi [147] CVRP 1 12 32–80 0 4 0:577 0.00–2.40

Marinakis and OVRP 3 14 51–200 0 7 0:11 0.00–0.32

Marinaki [110] 8 51–200 0 4 0:13 0.00–0.38

8 200–480 0 0 0:07 0.01–0.21

Marinakis and CVRP 2 14 51–200 0 11 0:019 0.00–0.14

Marinaki [111] 20 200–483 0 1 0:37 0.00–0.81

(continued)


Table 4 continued

Paper VRP Sets NBI Nodes NBS EBS Quality Quality range

Marinakis andMarinaki [112]

VRPSD 1 40 16–60 27 13 �0:64 �3.18 to 0.00

Liu andKachitvichyanukul[104]

LRP 1 30 50–200 1 0 3:29 �0.02 to 16.12

Marinakis [107] LRP 6 16 12–150 0 8 0:24 0.00–2.1230 20–200 0 13 0:42 0.00–2.5436 100–200 0 6 0:86 0.00–3.10

LRPSD 16 12–150 15 1 �0:11 0.00–0.4130 20–200 10 20 0:00 �0.02 to 0.0036 100–200 32 4 �0:04 �0.08 to 0.00

the second column the problem(s) that is (are) solved is mentioned, and in the third,fourth, and fifth columns, the number of sets of benchmark instances, the number ofinstances in each set (NBI), and the range in which the number of nodes of each setfluctuates are given for each paper, respectively. Finally, in the last four columns, thenumber of new best solutions found in the paper in the year of its publication (NBS),the number of instances in which the proposed algorithm found a solution equal tothe best-known published solution (EBS), the average quality of the solutions, andthe range of the qualities of the solutions are given, respectively. The quality of asolution is calculated using the following equation Quality D .cPSO�cBKS/

cBKS%, where

cPSO denotes the cost of the solution found by the mentioned PSO algorithm in thecorresponding paper and cBKS is the cost of the best-known solution.

It was a very difficult task to summarize all the results and to create this tableas every researcher presents the produced results in a different way. For example,in the most important and cited paper [3], the authors present the average results inthe well-known benchmark instances, and, then, they give an extensive analysis invariants of the studied problem, and it was difficult to present analytically the resultsof the paper. In other papers, a new set of benchmark instances is created [117], and,thus, it was difficult to give comparisons for this set of instances. However, in mostof the papers that are presented in this table, we could extract the information that wewould like to have in order to see the effectiveness of each one of the algorithms.In most of the algorithms presented in the table, at least a number of EBS werefound, and in few of them, new best solutions were given. It is very difficult toanalyze each paper separately due to space limitation, and this analysis can be foundeasily in each one of the papers. However, we could say that the application ofPSO in VRP variants gives very interesting computational results. In most cases,the computational results are competitive with the results of the most effectivealgorithms from the literature. Thus, alternative propositions exist for someone thathe/she would like to apply a PSO algorithm for the solution of a VRP variant.


Conclusions

In this chapter, an analytical review of the papers proposing and applying a PSOalgorithm for the solution of VRP variants was presented. In total, around 100 paperswere found. In some of them, new best solutions in VRP variants were presented,in other papers the authors presented the solutions that have been found that arevery good and competitive solutions and in some instances equal to the best-knownsolutions but not new best solutions in the VRP variant solved, in other papers theauthors presented and solved a new variant of the VRP with a PSO algorithm, and,finally, there are some papers that gave only illustrative examples of how the authorsworked in specific instances of a VRP variant. In general, this chapter gives to theresearch community a basis that someone could use if he/she would like to apply aPSO algorithm for solving a specific variant of the VRP as he/she could find whathas already been published and with what results and he/she could see how theresearchers cope with the problems that may arise from the application of a PSOalgorithm to the VRP or to its variants.

Cross-References

�Hybrid Heuristics� Particle Swarm Methods� Path Relinking

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Particle Swarm Optimization for the Vehicle Routing ...routing problem, where the only constraint that was taken into account was the capacity of the vehicles (and later the maximum