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Chapter 3 Satisficing Solutions for Multiobjective Stochastic Linear Programming Problems
Preamble
In this chapter, we introduce new solution concepts for MSLP problems. The solution concepts are based on the chance constrained approach for tackling randomness (see e.g., Kall [77], Liu [89]).
We also briefly discuss their advantages over other optimality/efficiency notions discussed in Chapter 3.
These solution concepts are then characterized under the assumptions that involved ran-dom variables have normal, chi-squared, exponential and gamma distributions.
Recall that the optimization problem at hand is of the form
Definitions of satisficing solutions for (P 1)
We say that x∗ ∈ Rn is an (α, β)−satisficing solution of type 1 for (P 1) if (x∗, s∗) is optimal for the following optimization problem Where cx is an aggregation of c1x, . . . , cK x, α = (α1, . . . , αm) with αi ∈ (0, 1]; i = 1, . . . , m and β ∈ (0, 1]. αi(i = 1, . . . , m) and β are probability levels a priori fixed by the decision maker.
Definition. ((α, β)−satisficing solution of type 2 )
We say that x∗ ∈ Rn is an (α, β)−satisficing solution of type 2 for (P 1) if (x∗, s∗), where s∗ ∈ RK , is efficient for the following multiobjective optimization problem:
Here δk, νk and tk are respectively the upper deviation of the kth goal from the kth target, the weight of the kth goal and the target associated with the kth objective function.
We note that αi ∈ (0, 1]; i = 1, . . . , m and βk ∈ (0, 1]; k = 1, . . . , K are as in Definition (3.2.2).
Interpretation of satisficing solutions for (P 1)
Solution concepts discussed here are based on the individual chance-constrained approach [4]. One may equally well, consider the joint chance-constrained approach [112].
The Chance constrained approach allows partial violation of the constraints, and it may be viewed as a technique for providing an appropriate safety margin [122].
The Chance constrained technique in general has been applied to various industrial and economic problems [135].
Intuitive ways of thinking about the above solution concepts are given below.
An (α, β)−satisficing solution of type 1 for (P 1) is an alternative that keeps the probability of the aggregate cx of the objective functions c1x, . . . , cK x of this prob-lem, lower than a variable s to a desired extent β, while minimizing this variable.
Moreover the chance of meeting the ith constraint of the problem is requested to be higher than a fixed thresholds αi. It is clear that for αi and β close to 1, such an alternative is a an interesting one.
As a matter of fact, it maintains the probability of keeping the aggregation of ob-jective functions low, while keeping the probabilities of realization of constraints to desired extents.
An (α, β)−satisficing solution of type 2 for (P 1) is an action that keeps the proba-bility of the kth objective ckx, for all k less than some variable sk to a level βk while the vector (s1, . . . , sK ) is minimized in the Pareto sense.
In the meantime, the chance of having the ith constraint, for all i, be satisfied is greater than αi.
We recall that a random vector is said to have the multivariate normal distribution if any linear combination of its com-ponents is normally distributed. The following lemma due to Kataoka [80] will be needed in the sequel.
are independent and normally distributed. Suppose also that for any i = 1, . . . , m, the random variables aij; j = 1, . . . , n and bi are normally distributed and independent. Then x∗ is an (α, β)-satisficing solution of type 1 for (P 1) if and only if x∗ is optimal for the following optimization problem:
If the Decision maker is interested in a satisficing solution of type 1, he should use the weighted sum method described in Subsection (1.3.7). For the following weights: λ1 = 0.4 and λ2 = 0.6 and for αi = 0.99 for all i, and βk = 0.99; k = 1, 2.
We have by virtue of Proposition (3.3.2) to solve the following problem, in a way to obtain an (α, β)−satisficing solution of type 1; with α = β = (0.99, 0.99).
We use LINGO 13.1 to solve this optimization problem. Local optimal solution was found at iteration: 38, and the optimal solution: z∗ = −5383.792 at the points x∗1 =14.96307, x∗2 = 48.13617.
As far as (α, β)−satisficing solution of type 2 is concerned we have the following result.
Satisficing solution of type 2
Proposition. Consider (P 1) and assume that the random variables
are independent and normally distributed. Assume further that for any i = 1, . . . , m, the random variables aij; j = 1, . . . , n, and bi are also independent and normally distributed. Then x∗ is an (α, β)-satisficing solution of type 2 for (P 1) if and only if x∗ is efficient for
the following mathematical program where α1, . . . , αm, β1, . . . , βK ∈ (0, 1] are as in Definition (3.2.2), φ(x) is as in Proposition (3.3.2).
And clearly, this problem is equivalent to (P 5)iv in the sense that they have the same efficient frontie
Example. Consider Example (3.3.1) and assume now, that, a satisficing solution of type 2 is sought with αi = 0.99 ∀ i, and βk = 0.99 ∀ k.
Then by Proposition (3.3.3), one has to solve the following problem:
1 Preliminaries
1.1 Elements of Probability Theory
1.2 Mathematical Programming
1.3 Multiobjective Programming
2 Multiobjective Stochastic Linear Programming: An overview
2.1 Preamble
2.2 Transformation of the feasible set of (P1)
2.3 Solution concepts
2.4 Related mathematical results
2.5 Methodological approaches for solving Multiobjective Stochastic Linear
2.6 Applications
3 Satisficing Solutions for Multiobjective Stochastic Linear Programming Problems
3.1 Preamble
3.2 Satisficing solutions for (P1)
3.3 Characterization of satisficing solutions for the Normal case
3.4 Characterization of satisficing solutions for the Chi-squared case
3.5 Characterization of satisficing solutions for the Exponential case
3.6 Characterization of satisficing solutions for the Gamma case
4 Extension
4.1 Multiobjective Programming problems with fuzzy random coefficients
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SATISFICING SOLUTIONS FOR MULTIOBJECTIVE STOCHASTIC LINEAR PROGRAMMING PROBLEMS
