An algorithm for the biobjective integer minimum cost flow problem
Introduction
The multicriteria minimum cost flow problems have already merited the attention of several authors. The following papers consider the case where the continuous flow values are permissible and use network-based methods to solve minimum cost flow and transportation problems: Aneja and Nair [2]; Calvete and Mateo [3]; Isermann [5]; Klingman and Mote [7]; Lee and Pulat [8]; Maholtra and Puri [10]; Pulat, Huarng and Lee [11] and Ringuest and Rinks [12]. However, the integer case of the biobjective minimum cost flow problem has scarcely been studied. In the literature, only in the paper of Lee and Pulat [9] have we found that the flow variables are imposed to take integer values for the biobjective minimum cost flow problem.
In this paper, we study the bicriteria minimum cost flow problem, where the flow variables must take integer values, proposing a method that finds all efficient integer points in the objective space. Our algorithm performs two phases. In the first phase, all integer points on the efficient boundary are found and in the second phase, the efficient integer points that do not lie on the efficient boundary are calculated. Our method is a network-based method where each efficient solution is generated only once. The method of Lee and Pulat [9] does not calculate all efficient integer solutions for the problem in consideration. The reason is that they do not consider combinations on the nonbasic variables when they perform parametric analysis on the biobjective minimum cost flow problem. In addition, our method never generates dominated solutions and so does not incorporate tools to eliminate them. However, in the method of Lee and Pulat, these tools are necessary.
After this introduction, in Section 2, the mathematical notation of the problem appears. In Section 3, we introduce the algorithm to generate all integer solutions on the efficient boundary in the objective space. In Section 4, we propose the algorithm to find all efficient integer solutions that do not lie on the efficient boundary and give an example that will help to understand both algorithms. In Section 5, we expose the computer results of our method. In Section 6, we will conclude with some comments.
Section snippets
Formalization of the problem
Given a directed network G=(V,A), let V={1,…,n} be the set of nodes and A be the set of arcs. For each node i∈V, let the integer bi be the supply/demand of the node i and for each arc let uij and lij be the upper bound and the lower bound on flow through arc , respectively. Let cijk be the cost per unit of flow on arc in the kth objective function, .
If xij denotes the amount of flow on an arc , Suc(i) the set and Pred(i) the set , the
Obtaining all integer solutions on the efficient boundary
The algorithm to obtain is based on the algorithm of Sedeño-Noda and González-Martı́n [13], which only generates for the continuous case. The method is a biobjective network simplex method in essence. It starts with the efficient extreme point in the objective space which results when only the first objective is considered in the BNF problem and finds the remaining efficient points in the same space by a finite sequence of pivots.
To compute x0 we use the
Obtaining all efficient integer solutions that do not lie on the efficient boundary
Once the set of integer points on the efficient boundary of is generated, the next step is to find the set , that is, the set of efficient integer points that do not lie on the efficient boundary. The points belonging to this set lie inside the triangles shown in Fig. 1. In Fig. 1, and correspond to efficient extreme points, and, and belong to the set .
Two adjacent efficient extreme points on the
Computer results
EIP is a PASCAL code that has been tested in HP9000/712 workstation running at 60 MHZ. The test problem was generated by using NETGEN [6]. The cost values of the first objective function are provided by NETGEN and the cost values of the second objective function were uniformly generated in the interval . These values are integer. Table 4 shows the levels of the number of nodes (n), number of arcs (m), the ratio m/n and the maximum capacity of arcs (U):
Through the combinations of
Conclusions
In this paper, we present a method for solving the biobjective minimum cost flow problem where the flow variables must take integer values. This method is characterized by the use of the classic resolution tools of network flow problems. Furthermore, the method never generates dominated solutions. Therefore, the method does not incorporate tools to eliminate dominated solutions. In addition, each efficient solution is generated only once.
In addition, in this paper we perform a computer
A. Sedeño-Noda has a B.Sc. degree in Physics from the University of Santiago de Compostela in Spain and is currently preparing his Ph.D. dissertation on network optimization. He is a lecturer and research assistant in the Departamento de Estadı́stica, Investigación Operativa y Computación at the University of La Laguna (Spain). His current research interests are the design of algorithms for network flow problems with a single objective and various objectives, multiobjective programming. He has
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A. Sedeño-Noda has a B.Sc. degree in Physics from the University of Santiago de Compostela in Spain and is currently preparing his Ph.D. dissertation on network optimization. He is a lecturer and research assistant in the Departamento de Estadı́stica, Investigación Operativa y Computación at the University of La Laguna (Spain). His current research interests are the design of algorithms for network flow problems with a single objective and various objectives, multiobjective programming. He has three publications in Spanish journals.
C. González-Martı́n has a B.Sc. degree in Mathematics and Ph.D. in Mathematics from the University of La Laguna, Spain. Currently, he is Professor of Operations Research in the Departamento de Estadı́stica, Investigación Operativa y Computación at the University of La Laguna. His current research interests are network optimization, linear and non-linear programming, multiobjective linear programming. He has many publications in Spanish journals and he has published in European Journal of Operational Research.