Discrete Optimization
Cutting plane method for multiple objective stochastic integer linear programming

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Abstract

This paper grapples with the problem of incorporating integer variables in the constraints of a multiple objective stochastic linear program (MOSLP). After representing uncertain aspirations of the decision maker by converting the original problem into a deterministic multiple objective integer linear program (MOILP), a cutting plane technique may be used to compute all the efficient solutions of the last model leaving the decision maker to choose a solution according to his preferences. A numerical example is also included for illustration.

Introduction

Consider the multiple objective stochastic linear programming (MOSLP) problem defined by:minZk=Ck(ξ)x,k=1,,K,subjecttoAx=b,T(ξ)x=h(ξ),x0,where Ck, T, h are random matrices of respective dimensions (1 × n), (m0 × n) and (m0 × 1) defined on some probability space (Ω, E, P); A and b are deterministic matrices of dimensions (m × n) and (m × 1), respectively.

The (MOSLP) methodologies has been few in number, undoubtedly due to the addition of uncertainty to the difficulty of multicriteria. Some of the popular approaches concerning this problem first transform it into a deterministic multiple objective program which is then solved via an interactive method as in PROTRADE of Goicoechea et al. [4], STRANGE of Teghem et al. [11] and PROMISE of Urli and Nadeau [15]. These methods have been successfully tested in real world contexts.

In this paper, we will discuss solutions to (MOSLP) problems with the additional complication brought about by the incorporation of integer variables:minZk=Ck(ξ)x,k=1,,K,(P1)subjecttoAx=b,T(ξ)x=h(ξ),x0,xinteger.The presence of integer variables in (MOSLP) structure introduce new difficulties to characterize efficient solutions [9], [13], [14] and to develop interactive methods. Many of the difficulties inherent to the general multiple objective stochastic integer linear programming (MOSILP) problems already show up when we consider what theoretical features of multiple objective integer linear programming (MOILP) [5], [8], [10], [12], [16] contribute to the success of (MOSLP). The road to computational success for (MOSILP) problems is in general through the exploitation of special structure. To our knowledge there exist today only one interactive method for (MOSILP) problem: the STRANGE-MOMIX which is developed by Teghem [12].

We propose an algorithm that combines the cutting plane technique developed by Abbas and Moulai in [1] and the L-shaped decomposition method described in [7]. Note that the L-shaped method is known as “Benders” decomposition [2] in other areas of mathematical programming and its original goal was to solve unicriterion mixed integer programming problems.

This paper is organized as follows: in Section 2 we shall transform the problem (P1) into an equivalent deterministic (MOILP) problem. Section 3 introduces notations, definitions and some results concerning the L-shaped decomposition method. Section 4 investigates in detail the solution procedure. Every step of the method will be illustrated in Section 5 by a numerical example. Section 6 concludes the paper.

Section snippets

Associate deterministic problem

Assume that we have a joint finite discrete probability distribution (ξr, pr), r = 1, …, R, of the random data.

• In the first step, for each realisation ξr of ξ we associate a criterion Zkr = Ck(ξr)x, a matrix T(ξr) and a vector h(ξr) to take into account the different scenarios affecting the K objectives and the stochastic constraints.

• The second step is to coming back exactly to the idea of recourse used in single-criterion stochastic programming [3], [6], [7]. Of course, we assume that the

Notations and basic definitions

In this section we pay attention to some basic results which can help the reader to understand the algorithm of the next section.

Solution method

The procedure is to generate all the efficient solutions of problem (P3) by solving a sequence of progressively more constrained single objective integer linear programming problems. The additional constraints exclude previously generated efficient points and ensure that newly generated solutions will be efficient.

Step 1: As said in Section 3, the algorithm starts with θ = −∞ and without feasibility and optimality cuts. The objective Z1=E[C1(ξ)x] is minimized under the deterministic constraints.

An illustrative example

Let us consider the following example with a similar structure to that of problem (P1), K = 3, n0 = 4, m0 = m = n = 2.

Deterministic constraints:-4x1+2x2-8.x1+x25.Two scenarios (R = 2) affect the three objectives and the stochastic constraints.C1(ξ1)=(-9,4),C2(ξ1)=(3,-5),C3(ξ1)=(8,-11),C1(ξ2)=(3,-2),C2(ξ2)=(7,1),C3(ξ2)=(-4,9),T(ξ1)=12-21,T(ξ2)=1034,h(ξ1)=35,h(ξ2)=61,q(ξ1)=(1,0,6,2)T,q(ξ2)=(5,3,2,1)T,p(ξ1)=12,p(ξ2)=12,W(ξ)=W=-2-12132-5-6,Z1=E[C1(ξ)]=p1C1(ξ1)+p2C1(ξ2)=12(-9,4)+12(3,-2)=(-3,1),Z2=E[C2(ξ)]=p

Conclusion

In this paper, we have considered multiple objective stochastic linear programming problem with the additional complication brought about by the incorporation of integer variables. The stochastic data are treated by recourse approach to obtain an equivalent deterministic two-stage (MOILP) program. An algorithm that generates the set of all integer efficient solutions of this equivalent program is presented. This approach has the advantage to give the decision maker two informations: the

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