An algorithm for the multiparametric 0–1-integer linear programming problem relative to the constraint matrix

Abstract

The multiparametric 0–1-Integer Linear Programming (0–1-ILP) problem relative to the constraint matrix is a family of 0–1-ILP problems in which the problems are related by having identical objective and right-hand-side vectors. In this paper we present an algorithm to perform a complete multiparametric analysis.

Introduction

The need for parametric analysis in Mathematical Programming arises from the uncertainty in the data. The most important surveys of parametric methods in integer linear programming (ILP) have been published by Geoffrion and Nauss [4], Holm and Klein [5] and Jenkins [8]. Most work has been done on changes in the right-hand side or in the objective vector with a scalar parameter guiding the perturbation of the data. Changes in elements of the constraint matrix have been considered by Jenkins [7] with a scalar parameter guiding the perturbation in such a manner that for the maximization case the feasible region increases continuously . The multiparametric ILP problem relative to the right-hand-side vector has been considered by Crema [2] and some of the ideas presented here may be considered as generalizations of those presented in that work. In the case of multiple changes on the constraint matrix, in such a manner that there is no scalar guiding the perturbation, no deep results are known up to now [8], [1].

Let L and U be matrices such that $\text{L∈}\text{Z}{}^{\mathrm{m\times n}}$ and $\text{U∈}\text{Z}{}^{\mathrm{m\times n}}$ with L_ij⩽U_ij for all (i,j)∈I×J={1,…,m}×{1,…,n}. Let $\text{H={A}\text{:}\text{A∈}\text{Z}{}^{\mathrm{m\times n}}\text{,}\text{L}{}_{\mathrm{ij}}\text{⩽A}{}_{\mathrm{ij}}\text{⩽U}{}_{\mathrm{ij}}\text{∀(i,j)∈I×J}}$. Let $\text{Ω={(i,j)∈I×J}\text{:}\text{L}{}_{\mathrm{ij}}\text{<U}{}_{\mathrm{ij}}\text{}}$. In this paper $\text{Ω≠∅}$. Let S⊆H defined by linear constraints in A_ij where (i,j)∈I×J.

The multiparametric 0–1-ILP problem relative to S is a family of 0–1-ILP problems in which the problems are related by having identical objective and right-hand-side vectors. A member of the family is defined as

$$\text{(P(A))}\text{max}\text{c}{}^{\mathrm{<mtext>t</mtext>}}\text{x}\text{s.t.}\text{Ax⩽b,}\text{x∈{0,1}}{}^{\mathrm{n}}$$

where $\text{c∈}\text{R}{}^{\mathrm{n}}\text{,}\text{b∈}\text{Z}{}^{\mathrm{m}}$ and A∈S.

Because all the constraints can be put in the form ⩽ and for the sake of standardization the members of the family were written in a maximization form with all the constraints in the form ⩽.

We use the following standard notation: if T is an optimization problem then F(T) denotes its set of feasible solutions, if D is a matrix $\text{D}{}_{\mathrm{i\ast }}$ denotes its ith row vector.

Constraints in the form ⩾ can be treated as follows: let us consider the constraint a^tx⩾a₀ where L_j′⩽a_j⩽U_j′, then we use with −U_j′=L_i1j⩽A_i1j⩽U_i1j=−L_j′.

Equality constraints can be treated as follows: let us consider the constraint a^tx=a₀ where L_j′⩽a_j⩽U_j′. In this case we use two constraints,

where

and

$$\text{a}{}^{\mathrm{<mtext>t</mtext>}}\text{x⩾a}{}_0\text{where}\text{L}{}_{\mathrm{j}}\text{′⩽a}{}_{\mathrm{j}}\text{⩽U}{}_{\mathrm{j}}\text{′}$$

and now the second constraint can be put in the form such as we explain above. Note that in this case the parameters of constraints i₁ and i₂ are not independent and their relations (A_i2j=−A_i1j for all j) must be included to define S.

We say that a multiparametric analysis is complete after finding an optimal solution for P(A) (if it exists) for all A∈S. In this paper we present an algorithm to perform a complete multiparametric analysis. Our algorithm works by choosing an appropiate finite sequence of non-parametric ILP problems in such a manner that the solutions of the problems in the sequence provide us with a complete multiparametric solution. This kind of approach was introduced by Jenkins [6], [7], [8] to the single parametric case.

In Section 2 we present the theoretical results and the algorithm. Computational experience is presented in Section 3.