Solving stochastic multiobjective vehicle routing problem using probabilistic metaheuristic

MATEC Web of Conferences 105, 00001 (2017 ) DOI: 10.1051/ matecconf/201710500001 IWTSCE'16 Solving stochastic multiobjective vehicle routing problem using probabilistic metaheuristic Asmae Gannouni1,2 , , Rachid Ellaia1 , and El-Ghazali Talbi2 1 2 LERMA, Mohammadia School of Engineering, Mohammed V University Rabat, Morocco CRIStAL/Dolphin, University Lille 1 Abstract. In this paper we propose a new metaheuristic algorithm for solving stochastic multiobjective combinatorial optimization(SMOCO) problems. Indeed, we find that the various initiatives that have been launched recently on this subject, they propose the classical metaheuristics to solve a stochastic multi-objective problems, but when the stochasticity effects is not taken into account, the choice of an arbitrary value leading to a particular configuration and a high loss of information. To conserve the stochastic nature of SMOCO problems, the pareto ranking should be defined on the random objectives functions directly rather than converted deterministic objectives. From these considerations, the scope of this research should consist of: (i)Proposed novel methodology of stochastic optimality for ranking objective functions characterized by non-continuous and no closed form expression. This novel approach is based on combinatorial probability and can be incorporated in a multiobjective evolutionary algorithm. (ii)Provide probabilistic approaches to elitism and diversification in multiobjective evolutionary algorithms. Finally, The behavior of the resulting Probabilistic Multi-objective Evolutionary Algorithms (PrMOEAs) is empirically investigated on the multi-objective stochastic VRP problem. 1 Introduction Many real-life optimization problems encountered in logistics, transportation, and Markets finance have several conflicts objectives under aleatory uncertainty to be satisfied, traditionally such problems were handled by converting the stochastic multiple objectives to deterministic multiple objectives using statistical aggregation: mean value, extreme value, variance..., then applying the classical metaheuristics [1], such as pareto evolutionary algorithms for generate a set of well-distributed pareto-optimal solutions [13]. Such an approach has many problems, the including the loss of significant trade-off information and the inability to search the true objective space because it is incapable to describe the relationship among stochastic(random) objectives. Personally speaking, to represent and conserve the stochasticity of original SMOCO problems, we should proposed other methodology of optimality under aleatory uncertainty. Therefore, the aim of this study is to provide a probabilistic definition, to dominance, elitism and diversification in multi-objective evolutionary algorithms. Based on the literature, the authors have classified methods for solving stochastic multiobjective combinatorial optimization problems, into three specific categories based on the way the objective function is estimated: i)Deterministic approaches: have been focused on treating it in multiobjective design by avoiding or ignoring un e-mail: certainty[18], then the solution of the stochastic multiobjective optimization problems is not essentially different from that of a deterministic multiobjective optimization problems, and exact(lerning method, random optimization methods, stochastic approximation) [5] or metaheuristic techniques(NSGAII, SPEAII,...) can be used [4,15] easily. ii) Robust Approaches: In this categorization, researchers reduced the stochastic nature of the problem by using for example the expectation of random functions: sample average approximation [17], or pareto means, pareto variance, pareto Quantile,...[6,12], these approaches are similar to the deterministic approaches, pareto dominance is used after changing the methodologies of the evaluation. Finally, for (iii)Stochastic Approaches: only few techniques are available and are still in nascent [11,16], this work is part of these Approaches, here the stochastic optimality is the major challenge. These approaches are well the resulting of hybridization between the probabilistic programming, simulation [8, 10] and metaheuristic algorithms. The remainder of this paper is organized as follows: In section 2, we investigate the formula of stochastic multiobjective optimization problems and we discusse the main concepts in multiobjective evolutionary algorithms that can be extended in stochastic context. Section 3, describes the design of Probabilistic approaches: probabilistic dominance, probability memory and diversification for practical use. He whole methodology is then illustrated on analytical test case: Multiobjective stochastic VRP in section © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). MATEC Web of Conferences 105, 00001 (2017 ) DOI: 10.1051/ matecconf/201710500001 IWTSCE'16 denoted by x  x if fi (x) ≤ fi (x ), ∀i ∈ {1, ..., m}. Definition.2 (Strong Dominance) x strongly dominates x denoted by x  x if fi (x) < fi (x ), ∀i ∈ {1, ..., m}. Definition.3 (Incomparable) x is incomparable with x denoted by x ∼ x if fi (x) > fi (x ), ∃i ∈ {1, ..., m} and f j (x) > f j (x ) ∃ j∈ {1, ..., m}. Definition.4 (Nondominated vector) An objective vector F(x) is said to be nondominated iff there does not exist another objective vector F(x ) such that x x. Definition.5 (Efficient solution) A solution x ∈ Xis said to be efficient(or pareto optimal, nondominated) iff its mapping in the objective space results in a nondominated vector. The set of all efficient solutions is called efficient(or Pareto optimal) set, and its mapping in the objective space is called Pareto front. A possible approach in multiobjective combinatorial optimization problem solving is to find the minimal set of efficient solutions. But, generating the entire set of pareto optimal solutions is usually, due to the complexity of the underlying problem or to the large number of optima. Therefore, the overall goal is often to identify a good efficient set approximation in stochastic objectives space. These relationships can not compared objec- 4, followed by conclusions and future prospects in Section 5. 2 Stochastic Multi-objective Optimization Stochastic multi-objective optimization problems[7] have been investigated thoroughly in the operations research community. We start this section by giving a general formula of SMOCO problems, we then expose general concepts that share these problems and the challenge that it entails. Then we present pareto multi-objective evolutionary algorithms which will be be extended on stochastic context[21-23] in the next section. 2.1 Stochastic Multi-objective Optimization Problems In the stochastic multi-objective optimization problems, two or more usually conflicting stochastic(uncertain) objectives are required to be optimized s (...truncated)