On Stability in Multiobjective Integer Linear Programming: A Stochastic Approach

In this paper we consider a multiobjective integer linear stochastic programming problem individual chance constraints. We assume that there is randomness in the right-hand sides of the constraints only and that the random variables are normally distributed. Some stability notions for such problem are characterized. An auxiliary problem is discussed and an algorithm as well as an illustrative example are presented Introduction Decision problems of stocastic or probabilistic optimization arise when certain coefficients of an optimization model are not tixed or known but are instead, to some extent, stochastic (or random or probabilistic) quantities. In recent years methods of mu1tiobjective stochastic optimization have become increasingly important in scientifically based decision-making involved in practical problems arising in economic, industry, health care, transportation, agriculture, military purposes and technology. We refer the Stochastic programming Web Site (2002)[1] for links to software as well as test problem collections for stochastic programming. In addition, we should point the reader to an extensive list of papers maintained by Maarten van der Vlerk at the Web Site: http:// mally.eco.rug.nl/biblio/ SP list.html.


Introduction
Decision problems of stocastic or probabilistic optimization arise when certain coefficients of an optimization model are not tixed or known but are instead, to some extent, stochastic (or random or probabilistic) quantities.
In recent years methods of mu1tiobjective stochastic optimization have become increasingly important in scientifically based decision-making involved in practical problems arising in economic, industry, health care, transportation, agriculture, military purposes and technology. We refer the Stochastic programming Web Site (2002) [1] for links to software as well as test problem collections for stochastic programming. In addition, we should point the reader to an extensive list of papers maintained by Maarten van der Vlerk at the Web Site: http:// mally.eco.rug.nl/biblio/ SP list.html.
In literature there are many papers that deal with stability of solutions for stochastic multiobjective optimization problems. Among the many suggested approaches for treating stability for these problems, for example, are those due to [2,3,4,5,6].
This paper is organized as follows: we start in Section 2 by formulating the model of chance-constrained multiobjective integer linear programming problem (CHMOILP) and the solution concept is introduced. In Section 3, a parametric study is carried out on the problem of concern, where some basic stability notions are characterized for the formulated model.
These notions are the set of feasible parameters; the solvability set, and the stability set of the first kind (SSK 1 ). Moreover, an algorithm is describled to determine the (SSK 1) for the (CHMOILP). In Section 4, we provide an example to illustrate our results. Finally, in Section 5, we state some open problems for future work in the area of stochastic multiobjective integer optimization problems.

Problem Formulation And Solution Concept
The chance-constrained multiobjective integer linear programming problem with random parameters in the rihgt-hand side of the constraints can be stated as follows: Here x is the vector of integer decision variables and F(x) is a vector of k-linear realvalued objective functions to be maximized. Furthermore, P means probability and a; is a specified probability value. This means that the linear constraints may be violated some of the time and at most 100(1a; ) % of the time. For the sake of simplicity, we assume that the random parametrs b1, (i =l, 2, ... m) are distributed normally with known means E{bi} and variances V ar { bi} and independently of each other.

Definition 1.
A The basic idea in treating problem (CHMOILP) is to convert the probabilistic nature of this problem into a deterministic form. In this section, the idea of employing deterministic version will be illustrated by using the interesting technique of chance-constrained programming [7]. In this case, the set of constarints X of problem (CHMOILP) can be rewritten in the deterministic form as: ( I ,2, .. n }• n X' =t x E R" I ~ a;ix J ~E{b;} + K,y, ~Var{b;}, i =1, 2, .... , m, x j 20 and integer, j where Kcr, is the standard normal value such that <D(Ka,) = 1 a; ; and <D(a) represents the ''cumulative distribution function" of the standrad normal distribution evaluted at a. Thus, problem (CHMOILP) can be understood as the following deterministic version of a multiiobjective integer linear programming problem: subject to xeX'. Now it can be obsereved, from the nature of problem (MOILP) above, that a suitable scalarization:technique for treating such problems is to use the -constraint method [8]. For this purpose, we consider the following integer linear programming problem with a single-'· objective function as: Ps(E): where s E K = { 1, 2, ... ,k} which can be taken arbitrary.
It is clear from [8] that a systematic variation of ~::i's will yield a set of efficient solutions. On the other hand, the resulting scalar problem Ps(E) can be solved easily at a certain parameter c= ~::* using the branch-and bound method [9]. If x* E X(~::*) is a unique optimal integer solution of problem Ps(E*), then x* becomes an efficient solution to problem (CHMOILP) with a probability level a~, (i = 1, 2, ... m).

A parametric Study on Problem (Chmoilp)
In this section, before we go further, we can rewrite problem l\(E) in the following scalar relaxed subproblem which may occur in the branch-and-bound process as: Ps'(~::): where the constraint y 1 ~ x J ~f3J ,j E 1 c { 1 ,2, .. n } is an additional constraint on the decision variable Xj and that has been added to the set of constraints of problem Ps( E) for obtaining its optimal integer solution x* by the branch-and-bound algorithm [9]. In addition, it is supposed that: In what follows, definitions of some basic stabiltiy notions are given for the relaxed problem P 5 ,(E) above. We shall be essentially concerned with three basic notions: the set of feasible parameters; the solvability set, and the stability set of the first kind (SSKI ). The qualitative and quantitative analysis of these notions have been introduced in details by Osman [1 0,11] for different classes of parametric optimization problems. Moreover, stability results for such problems have been derived. The feasibility condition for problem P 5 ,(E) is given in the following.
The Set of Feasible Parameters Definition 2.
The set of feasible parameters of problem P 5 ,(E), which is denoted by A, is defined by: The Solvability Set Definition 3. The solvability set of problem P 5 ,(E), which is denoted by B, is defined by: has an optimal integer solution}.
The Stability Set of the First Kind Definition 4.
Suppose that E * E B with a corresponding optimal integer solution x*, then the stability set of the first kind of problem P 5 .(E) corresponding to x*, which is denoted by S(x*), is defined by: where IvJc{1,2, ... n}, InJ <I> and all the relations of system(*) above are evaluated at the optimal integer solution x*. The variables ~r. 8i, u_i> V:i are the langrangian multi piers.
The first and last four relations of the system (*) above represent a Polytope in J..tD u vspace for which its vertices can be determined using any algorithm based upon the simplex method, for example, Balinski [12]. According to whether any of the variables ~r. rEK-{sL Di, (i=1,2, ... m), U.irUEI) and Vj, (jEJ) is zero or positive, then the set of parameters e's for which the Kuhn-Tucker necessary optimality conditions are utilized will be determined. This set is denoted by T(x*).

Determination of the Set T(x*)
In this section we propose an algorithm in series of steps to find the set of possible e which will be denoted by T(x*). For the set T(x*) , the point x* remains efficient for all values of the vector e. Clearly, T(x*)c S(x*) The suggested algorithm can be summarized in the following mannar. Step 1. Determine the means E{b;} and Var{b;} (i =1, 2, ... m).
Step 2. Convert the original set of constraints X of problem ( CHMOILP) into the equivalent set of constraints x'.

Step 4.
Formulate the integer linear problem with a single-objective function P 5 (e).
Step 6. Construct the payoff table and determine n,, Mr (the smallest and the largest numbers in the r 111 column in the payoff table).
Determine the E/s from the formula: Step 8.
Step 11. I r T2(x*) is a singleton, go to step 12. Otherwise, go to step 13.
Choose another E, =ErE 3-T 2 (x*) and go to step 9. The above algorithm terminates when the range of 3 is fully exhausted. Then, the stability set of the first kind S(x*) is given as:

Conclusions
The general purpose of this paper was to investigate stability of the efficient solution for chance-constrainted multiobjective integer linear programming problem. A parametric study has been carried out on the problem under consideration, where some basic stability notions have been defined and characterized for the formulated problem.
Many aspects and general questions remain to be studied and explored in the field of multiobjective integer optimization problems under randomness. This paper is an attempt to establish underlying results which hopefully will help others to answer some or all of these questions.
There are however several unsolved problems, in our opinion, to be studied in futuer. Some of these problems are: (i) An algorithm is rquired for solving multiobjective integer linear programming problems involving random parameters in the left-hand side of the constraints, (ii) An algorithm is needed for treating large-scale multiobjective integer linear nonlinear programming problems under randomness, (iii) An algorithm should be handled for solving integer linear and integer nonlinear goal programs involving random parameters.