María del Mar Muñoz

Efficient Solution Concepts and Their Relations in Stochastic Multiobjective Programming

In this work, different concepts of efficient solutions to problems of stochastic multiple-objective programming are analyzed. We center our interest on problems in which some of the objective functions depend on random parameters. The existence of different concepts of efficiency for one single stochastic problem, such as expected-value efficiency, minimum-risk efficiency, etc., raises the question of their quality. Starting from this idea, we establish some relationships between the different concepts. Our study enables us to determine what type of efficient solutions are obtained by each of these concepts.

Stochastic approach versus multiobjective approach for obtaining efficient solutions in stochastic multiobjective programming problems

In this work, we deal with obtaining efficient solutions for stochastic multiobjective programming problems. In general, these solutions are obtained in two stages: in one of them, the stochastic problem is transformed into its equivalent deterministic problem, and in the other one, some of the existing generating techniques in multiobjective programming are applied to obtain efficient solutions, which involves transforming the multiobjective problem into a problem with only one objective function. Our aim is to determine whether the order in which these two transformations are carried out influences, in any way, the efficient solution obtained. Our results show that depending on the type of stochastic criterion followed and the statistical characteristics of the initial problem, the order can have an influence on the final set of efficient solutions obtained for a given problem.

ISTMO: An interval reference point-based method for stochastic multiobjective programming problems

In this paper, we present an interactive algorithm (ISTMO) for stochastic multiobjective problems with continuous random variables. This method combines the concept of probability efficiency for stochastic problems with the reference point philosophy for deterministic multiobjective problems. The decision maker expresses her/his references by dividing the variation range of each objective into intervals, and by setting the desired probability for each objective to achieve values belonging to each interval. These intervals may also be redefined during the process. This interactive procedure helps the decision maker to understand the stochastic nature of the problem, to discover the risk level (s)he is willing to assume for each objective, and to learn about the trade-offs among the objectives.

An additive achievement scalarizing function for multiobjective programming problems

The success of the reference point scheme within interactive techniques for multiobjective programming problems is unquestionable. However, so far, the different achievement scalarizing functions are, more or less, extensions of the Tchebychev distance. The reason for this is the ability of this function to determine efficient solutions and to support every efficient solution of the problem. For the same reasons, no additive scheme has yet been used in reference point-based interactive methods. In this paper, an additive achievement scalarizing function is proposed. Theoretical results prove that this function supports every efficient solution, and conditions are given under which the efficiency of each solution is guaranteed. Some examples and computational tests show the different behaviours of the Tchebychev and additive approaches, and an additive reference point interactive algorithm is proposed.

Relations among Several Efficiency Concepts in Stochastic Multiple Objective Programming

In this paper, the resolution of stochastic multiple objective programming problems is studied. The existence of random parameters in the objective functions has resulted in the definition of several efficient solution concepts for such problems in the literature. We will focus our attention in the study of some of these concepts, namely, minimum risk and s probability. Once these concepts are defined, the relations among the sets of efficient solutions obtained are studied.

Satisfactory solution concepts and their relations for Stochastic Multiobjective Programming problems

This work deals with the concept of satisfactory solution for Stochastic Multiobjective Programming (SMP) problems. Based on previous literature, we will introduce different concepts of satisfactory solutions for SMP problems, define a new concept of solution (where the decision maker (DM) sets his/her preferences in terms of two aspiration levels for the stochastic objective and two probabilities to reach those levels), and establish some relationship between these concepts. The results will aim at featuring these concepts and determine the differences between them. Moreover, the paper proposes a new step by step procedure to exchange information between the analyst and DM prior to solving the problem. Thus, the DM will be able to choose the transformation criterion for each stochastic objective and the aspiration level.

Analysis and comparisons of some solution concepts for stochastic programming problems

The aim of this study is to analyse the resolution of Stochastic Programming Problems in which the objective function depends on parameters which are continuous random variables with a known distribution probability. In the literature on these questions different solution concepts have been defined for problems of these characteristics. These concepts are obtained by applying a transformation criterion to the stochastic objective which contains a statistical feature of the objective, implying that for the same stochastic problem there are different optimal solutions available which, in principle, are not comparable. Our study analyses and establishes some relations between these solution concepts.

A graphical characterization of the efficient set for convex multiobjective problems

In this paper, a graphical characterization, in the decision space, of the properly efficient solutions of a convex multiobjective problem is derived. This characterization takes into account the relative position of the gradients of the objective functions and the active constraints at the given feasible solution. The unconstrained case with two objective functions and with any number of functions and the general constrained case are studied separately. In some cases, these results can provide a visualization of the efficient set, for problems with two or three variables. Besides, a proper efficiency test for general convex multiobjective problems is derived, which consists of solving a single linear optimization problem.

The stochastic bottleneck linear programming problem

In this paper we consider some stochastic bottleneck linear programming problems. We overview the solution methods in the literature. In the case when the coefficients of the objective functions are simple randomized, the minimum-risk approach will be used for solving these problems. We prove that, under some positivity conditions, these stochastic problems are reduced to certain deterministic bottleneck linear problems. An application of these problems to bottleneck spanning tree problems is given. Two simple numerical examples are presented.