A hybrid meta heuristic algorithm for bi-objective minimum cost flow (BMCF) problem

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Abstract

In this paper we study the bi-objective minimum cost flow (BMCF) problem which can be categorized as multi objective minimum cost flow problems. Generally, the exact computation of the efficient frontier is intractable and there may exist an exponential number of extreme non-dominated objective vectors. Hence, it is better to employ an approximate method to compute solutions within reasonable time. Therefore, we propose a hybrid meta heuristic algorithm (memetic algorithm hybridized with simulated annealing MA/SA) to develop an efficient approach for solving this problem. In order to show the efficiency of the proposed MA/SA some problems have been generated and solved by both the MA/SA and an exact method. It is perceived from this evaluation that the proposed MA/SA outputs are very close to the exact solutions. It is shown that when the number of arcs and nodes exceed 30 (large problems) the MA/SA model will be more preferred because of its strongly shorter computational time in comparison with exact methods.

Introduction

In this paper we consider bi-objective minimum cost flow (BMCF) which is special case of multi objective minimum cost flow (MMCF). This problem can be categorized in class of network flow problems. MMFC can be employed to solve a large set of real-world problems and therefore, they can be considered as important building blocks and sub-problems in complex models, such as transportation, assignment and transshipment problems. Furthermore, they possess nice theoretical properties such as the integrality of its basic solution [9]. In addition, this problem can be categorized in class of multi objective problems. In general, in many combinatorial optimization problems, selecting the optimum solution needs evaluation of more than one criterion [16]. For example, in transportation problems, criteria such as cost for selected routes, arrival times at the point of destination, deterioration of goods, load capacity that would not be used in the selected vehicles, safety, reliability, etc. can be referred as the most important criteria.

In general, in most practical situations these objectives have conflict with each other and consequently, there is no solution which simultaneously optimizes all preferred objectives and practitioners desire to find the global solution concerning all objectives. Therefore, finding all or a suitable subset of all pareto or efficient solutions is the goal of multi objective optimization. Regarding the literature of the subject, a few authors have worked on combining these two fields to consider multi objective network flow problems [9].

There are so many real life problems in which continuous flows pass through networks from a set of beginning stations to supply the demand of the sinks. Gas or chemical pipe lines, waste water networks and in general transportation of materials or services in continuous quantities are samples of this problem. Hence, in this paper we consider the continuous case of BMCF.

This paper is organized as follows. In Section 2 we briefly discuss previous studies concerned with MMCF problem. In this section we point out some shortages in previous works in this field which has become the motivation for this study. In Section 3 we briefly describe both ‘multi objective linear programs’ and ‘minimum cost flow problem’. Then we combine them to make multi objective minimum cost flow problem and its mathematical model. In Section 4 our proposed algorithm (MA/SA) along with an example is presented to illustrate the modeling of the problem and discuss the results obtained from MA/SA. In Section 5 the performance of the algorithm is evaluated through solving several problems with differing dimensions. Therefore, computational time and quality of outputs are measured and compared with outputs of an exact method obtained by MATAB software. Finally, in Section 6 concluding remarks are provided.

Section snippets

Literature review

In this section we study and discuss some previous studies concerned with the MMCF problem. In general as illustrated in Fig. 1 we can divide these studies into two major categories (continuous multi objective minimum cost flow and integer multi objective minimum cost flow). Similar to linear programming, there are more papers on the continuous multi objective minimum cost flow problem (MMCF) than on its integer counterpart. An obvious reason is that some of the techniques of multi objective

Problem description

In this section we introduce some relevant concepts, formulas and properties of multi objective linear programming and network flow theory. For more information and complete introduction to these topics, refer to Steuer [17], Ehrgott [6] and Ahuja et al. [1].

The proposed algorithm

Ruhe [14] shows that the exact computation of the efficient frontier is, in general, intractable, since there may exist an exponential number of extreme non-dominated objective vectors. Furthermore, BMCF is known to be generalization of the bi-criteria shortest path problem which is NP-hard [6]. Hence, in order to deal with this complexity, utilizing heuristics or Meta Heuristics can be helpful to obtain suitable solutions within a reasonable computation time. In this section we introduce a

Performance evaluation of the MA/SA algorithm

In this section we discuss the performance of our proposed algorithm regarding the following factors:

  • (1)

    Quality: comparing the quality of outputs obtained from proposed MA/SA algorithm with outputs of MATLAB. This is prepared by measuring the RE of approximated efficient frontier.

  • (2)

    Computational time length: comparing the computational time of the MA/SA algorithm with MATLAB.

For this purpose,the MATLAB software and the MA/SA algorithm were run for 15 sample problems with different dimentions on a

Conclusions

In this paper we studied the bi-objective minimum cost flow BMCF problem that is the special case of multi objective minimum cost flow problem. In the literature of the subject, it is stated that the exact computation of the efficient frontier is, in general, intractable, since there may exist an exponential number of extreme non-dominated objective vectors (Ruhe [14]). Therefore, we proposed a hybrid Meta heuristic algorithm to take the BMCF problem into consideration. Because of the strength

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