0-1 Parametric Mixed Integer Programming with Multiple Parameters for Objective Function Coefficients

Abstract

While continuous parametric analysis is of some interest in integer linear programming, it is desirable to deal with parametric integer programming problems which include multiple parameters for the right-hand sides or objective function coefficients in the model. Because, by the introduction of multiple parameters into the integer linear model, the interdependences among the entire system, its environment and the subsystems can be varied. Thus, multipara-metric programming can become an important tool for solving various problems arising in design and/or control of real life engineering systems.
Although several literature on parametric integer programming (PIP) with single parameter for the right-hand sides or objective function coefficients have been published, no work has been done on PIP with multiple parameters in the model.
The purpose of this paper is to present an effective method for finding optimal solution of parametric mixed integer linear programming with multiple parameters for the objective function coefficients. The present method is an extension of the branch-and-bound algo-rithm for 0-1 parametric mixed integer programming with single parameter. In order to solve a subproblem defined on each node of the branch-and-bound tree, a relaxation problem is defined, which is formulated as a multiparametric linear programming. For each node, a range set of multiple parameters is defined and it must be convex. Otherwise, thealgorithm for solving the multiparametric linear programming cannot be applied. To construct the convex set of parameters for each node, parameter regions obtained by solving the multiparametric linear programming are effectively used. To show the effectiveness of the present method, a factility location problem is considered, where the objective function coefficients are varied continuously by the ntroduction of multiple parameters.