An improved particle swarm optimization algorithm for the capacitated location routing problem and for the location routing problem with stochastic demands

Highlights

• An improved variant of the particle swarm optimization algorithm is presented.

• A new formulation of the location routing problem with stochastic demands is given.

• A new neighborhood topology for PSO suitable for combinatorial optimization problems is proposed.

• The proposed algorithm is tested in the CLRP and in the LRPSDs.

• Comparisons with other algorithms from the literature are performed.

Abstract

In this paper, a new version of the particle swarm optimization (PSO) algorithm suitable for discrete optimization problems is presented and applied for the solution of the capacitated location routing problem and for the solution of a new formulation of the location routing problem with stochastic demands. The proposed algorithm combines three different topologies which are incorporated in a constriction particle swarm optimization algorithm and, thus, a very effective new algorithm, the global and local combinatorial expanding neighborhood topology particle swarm optimization, was developed. The algorithm was tested, initially, in the three classic sets of benchmark instances for the capacitated location routing problem with discrete demands and, then, as there are no benchmark instances for the location routing problem with stochastic demands, these instances were transformed appropriately in order to be suitable for the problem with stochastic demands. The algorithm was tested in the problem with the stochastic demands using these transformed sets of benchmark instances. The algorithm was compared with a number of different implementations of the PSO and with metaheuristic, evolutionary and nature inspired algorithms from the literature for the location routing problem with discrete and stochastic demands.

Introduction

In this paper, a new neighborhood structure for the particle swarm optimization (PSO), the global and local combinatorial expanding neighborhood topology (GLCENT), suitable for routing problems is presented. The PSO algorithm has been observed in the past that it is not suitable for application in routing problems as either it does not give very good results or it needs a very strong local search algorithm in order to improve its results. The reason is that as a solution in a routing problem should be represented as a path (or a set of paths), because otherwise a computational inefficient algorithm could be produced as the solutions should be transformed from continuous values (suitable for PSO) to discrete values (suitable for routing problems) and vice versa. Over the last 10 years, researchers have proposed efficient ways to avoid the transformation from continuous to discrete values and vice versa. For example, variants of the particle swarm optimization algorithm for the capacitated vehicle routing problem [62], [67], [69], the vehicle routing problem with stochastic demands [58], [68], the open vehicle routing problem [63], the vehicle routing problem with time windows [71], the probabilistic traveling salesman problem [61] and the location routing problem [60]. These algorithms were competitive with the most effective algorithms from the literature for the solution of routing problems.

Two years ago we have proposed a topology, the combinatorial neighborhood topology [67], where the role of the velocities equation has been changed and the position vector was not used at all. Thus, with this topology the PSO algorithm was applied effectively in routing problems, most specifically in capacitated vehicle routing problem, and the results were competitive and in most of the instances equal to the best solutions from the literature. In a following publication we applied an expanding version of this topology in a different more difficult routing problem, the vehicle routing problem with stochastic demands [68], that resulted an algorithm that gave new best solutions in most of the instances used in the tests. The improvement in this algorithm in the part of the combinatorial neighborhood topology was that instead of a global topology in the velocities a local one was used, where, initially, the neighborhood was equal to two neighbors and it was expanding during the iterations until it became equal to a global topology and, then, it was restarted from the beginning (this topology was first applied by my research group in a solution of a flowshop scheduling problem [64] and in a solution of a feature selection problem [66]). In the present paper, the combinatorial neighborhood topology is applied in two more demanding problems, the one is the capacitated location routing problem and the other is the newly formulated location routing problem with stochastic demands. The reason that these two problems were selected was that the location routing problems in general combine two different problems the capacitated facility location problem and the vehicle routing problem where in the capacitated facility location Problem the solution can not be mapped as a path, as in the capacitated vehicle routing problem, but it must be mapped using a binary representation in which the ones mean that the facility is open and the zeros mean that the facility is closed. Thus, the challenge of this paper was to apply effectively the combinatorial neighborhood topology in a solution vector where its one part has values equal to zeros and ones and the other part has a path that represents the routes. The algorithm was tested in both problems as there are two different problems. The one problem has stochastic parameters, the demands of the customers and the other has only deterministic parameters. It is desirable to see how this algorithm can be applied in both problems. In the part of the combinatorial neighborhood topology a combined version of the previous two version, the global and the expanding one was proposed. This was achieved by the addition of a fourth parameter in the equation of velocities. Thus, in the new equation the particle, except from a movement towards a new direction, a movement towards his previous best and a movement towards the global best of the swarm, moves towards the local best of his neighborhood. The reason that this improvement was selected was that as both versions were proved to be very effective when applied separately in different problems, a combination of them would probably give an even more effective algorithm.

Particle swarm optimization (PSO) is a population-based swarm intelligence algorithm that was originally proposed by Kennedy and Eberhart [44] and simulates the social behavior of social organisms by using the physical movements of the individuals in the swarm. There is a number of review papers [6], [7], [86] that have been published for the particle swarm optimization. Initially, most of the review papers focused in all improvements and variants of PSO, in velocities and positions equations, in the topologies (local or global) etc. In recent years with the increased in the applications of PSO such a review is very difficult to be written and to present the most important publications. Thus, nowadays the survey papers are mainly focused on the application of PSO in a problem or in a group of similar problems that can be reviewed together [4], [46]. In this paper, as it was mentioned previously, the innovation is the way that a new very challenging topology, which combines a local and a global topology, is applied in two difficult routing problems, the location routing problem and the location routing problem with stochastic demands. In addition, in recent years a number of PSO implementations have been published for routing problems [1], [2], [3], [17], [33], [35], [47].

In the past, a number of algorithms have been published using local neighborhood topologies but most of them are applied in global optimization problems or combinatorial optimization problems and not in routing problems as the ones studied in this paper. The first paper that designs and investigates neighborhood topologies (circle, wheel, star and random) was published by Kennedy [43]. In a following work Kennedy and Mendes [45] proposed a number of population topologies. Mendes et al. [76] proposed another local neighborhood topology, denoted as full informed particle swarm optimization algorithm (FIPS). The first one that published an expanding neighborhood topology for the solution of global optimization problems was Suganthan [97]. Another two earlier adaptations of local neighborhood topologies are presented in [38], [83]. In recent years, the adaptations of local neighborhood topologies in PSO have been increased. More precisely, in [104] a hybrid PSO algorithm is presented, which is denoted as DNSPSO where a diversity enhancing mechanism and neighborhood search strategies are used in order to achieve a trade-off between exploration and exploitation abilities. An improved version of this strategy is presented in [100], where an enhanced particle swarm optimization with diversity and neighborhood search (EPSODNS) approach was presented. In [51] a comprehensive learning particle swarm optimizer (CLPSO) was presented. An improvement of this method was proposed in [81], denoted as dynamic neighborhood learning particle swarm optimizer (DNLPSO), which uses a learning strategy whereby all other particles’ historical best information is used to update a particle's velocity. In [105] a multi-layer PSO method (MLPSO) consisting of global MLPSO and local MLPSO by increasing the swarm layers from two to multiple layers was proposed. Another PSO algorithm using a local search topology was presented in [50] where a ring topology is used. In [52], a particle swarm optimization with increasing topology connectivity (PSO-ITC) was proposed to solve unconstrained single-objective optimization problems with continuous search space while in [53] a PSO denoted as PSO with adaptive time-varying topology connectivity (PSO-ATVTC) was proposed where the ATVTC module is used in order to balance the algorithm's exploration/exploitation searches by varying the particle's topology connectivity with time according to its searching performance. A time adaptive topology is proposed for constrained optimization problems in [11]. In [42] an age-group topology particle swarm optimization algorithm (PSOAG) is presented where the concept of age is used to measure the search ability of each particle in local area. The particles are divided in different age-groups by their age and particles in each age-group can only select the ones in younger groups or their own groups as their neighborhoods [42]. A cyclic neighborhood topology is presented in [74]. An analysis of local best topologies is presented in [30].

In [102], [103] two mutation neighborhood based PSO implementations have been proposed for the solution of fuzzy stochastic programming problems. Especially in [103], a fuzzy random facility location model (VaR-FRFLM) was formulated where the costs and demands are assumed to be fuzzy random variables and the capacity of each facility was not fixed but a decision variable assuming continuous values was used [103]. For the solution of the problem, the authors presented two PSO implementations a continuous Nbest-Gbest-based PSO and a genotype–phenotype-based binary PSO. The first one was used in order to deal with the continuous capacity decision variables and the second one was used in order to deal with the binary location decision variables. Afterwards, a mutation operator is incorporated in order to improve even more the quality of the produced solutions. These are the two most relevant publications in the literature compared the one proposed in this paper. However, a number of differences between them exist. First of all, the authors in [103] solved a facility location problem while in this paper a location routing problem is solved. Thus, while the authors in [103] have to deal with binary variables that concern the assignment of customers in depots, we, in addition, have to solve a routing problem, meaning to find the order that the selected number of vehicles, assigned in each depot, will visit the customers assigned to the depots. Thus, in this paper, in addition with the binary variables a part of the solution will be the paths that the vehicles will follow. In the paper [103] the stochastic variables are treated using a fuzzy approach while in this research the stochastic variables are treated finding the a priori routes of the customers. Also, both researches use a local neighborhood topology for the PSO algorithm. However, in the proposed algorithm in this paper, there is no need to transform any solution to continuous values as the innovation of the method is that a topology, denoted as combinatorial neighborhood topology, is used in order not to transform the solutions from continuous to discrete values and vice versa. This is very important in a routing problem as when a solution (a path) is transformed into continuous values (in order to be suitable for the velocities’ and positions’ equations of PSO), then, good parts of solutions (i.e. a sequence of customers) may be destroyed and, thus, when the solution is transformed back to a path, a completely new path will be created without having any memory of the good sequences of the nodes.

In order to give the effectiveness of each one of the features of the proposed algorithm, different versions of the algorithm that use separately each one of the features are implemented and tested and, then, their results are compared with the results of the proposed algorithm. The algorithm is compared with a number of other implementations of PSO algorithm for the solution of the capacitated location routing problem published in [60], [65]. Also, it is compared with other algorithms from the literature. The rest of the paper is organized as follows: In the next section, the description of the capacitated location routing problem, the new formulation of the location routing problem with stochastic demands and the presentation of the most important algorithms for the solution of the capacitated location routing problem are given. In Section 3, the proposed algorithm is analyzed and described in detailed. The results of the algorithm when used for the solution of these two problems are presented in Section 4 and, finally, in the last section conclusions and future directions are presented.