An algorithm for the biobjective integer minimum cost flow problem

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Abstract

In this paper, we study the single commodity flow problems, optimizing two objectives simultaneously, where the flow values must be integer values. We propose a method that finds all the 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. In addition, we carry out a computational experiment showing that the number of efficient integer solutions that do not lie on the efficient boundary is greater than the number of integer solutions on the efficient boundary.

Scope and purpose

In many combinatorial optimization problems, the selection of the optimum solution takes into account more than one criterion. For example, in transportation problems or in network flows problems, the criteria that can be considered are the minimization of the cost for selected routes, the minimization of arrival times at the destinations, the minimization of the deterioration of goods, the minimization of the load capacity that would not be used in the selected vehicles, the maximization of safety, reliability, etc. Often, these criteria are in conflict and for this reason, a multiobjective network flow formulation of the problem is necessary. The solution to this problem is searched for among the set of efficient points. Although multiobjective network flow problems can be solved using the techniques available for the multiobjective linear programming problem, network-based methods are computationally better. The multicriteria minimum cost flow problem has already merited the attention of several authors and the case which has been considered in literature is that which has two objectives, where the continuous flow values are permissible. However, the integer case of the biobjective minimum cost flow problem has scarcely been studied. Whereas, in many real network flow problems, integer values on flow values are required. In this paper, we propose an approach to solve the biobjective integer minimum cost flow problem. An algorithm to obtain all efficient integer solutions of this problem is introduced. This method is characterized by the use of the classic resolution tools of network flow problems, such as the network simplex method. It does not utilize the biobjective integer linear programming methodology. Furthermore, the method does not calculate dominated solutions, so it is not necessary to incorporate tools to eliminate dominated solutions.

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 iV, let the integer bi be the supply/demand of the node i and for each arc (i,j)∈A let uij and lij be the upper bound and the lower bound on flow through arc (i,j), respectively. Let cijk be the cost per unit of flow on arc (i,j) in the kth objective function, k=1,2.

If xij denotes the amount of flow on an arc (i,j), Suc(i) the set {j∈V/(i,j)∈A} and Pred(i) the set {j∈V/(j,i)∈A}, the

Obtaining all integer solutions on the efficient boundary

The algorithm to obtain Eex[f(X)]∪Enex[f(X)] is based on the algorithm of Sedeño-Noda and González-Martı́n [13], which only generates Eex[f(X)] 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 f(x0) 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 Eex[f(X)]∪Enex[f(X)] on the efficient boundary of f(X) is generated, the next step is to find the set E[f(XI)]−(Eex[f(X)]∪Enex[f(X)]), 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, f1,f2 and f3 correspond to efficient extreme points, and, f11,f12,f22 and f32 belong to the set Enex[f(X)].

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 [−1000,1000]. 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.

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