PROMISE/scenarios: An interactive method for multiobjective stochastic linear programming under partial uncertainty

Abstract

In many real contexts where the multiobjective stochastic linear programming can be used as a modelling approach, the decision maker is in general placed in a situation of incomplete information concerning the uncertain parameters of the problem. A good way to express that incomplete information consists in resorting to the idea of scenarios relatively to the objectives and constraints of the stochastic program. While the authors who have suggested methods based on scenarios suppose that the probabilities of those scenarios are known, in this paper we propose a scenarios approach where the probabilities of scenarios is only incompletely specified according to a ranking. That interactive method, called PROMISE/scenarios, is presented and is illustrated by a didactic example.

Introduction

About 20 years ago, researchers began to develop the multiobjective stochastic linear programming as an appropriate tool to model some real complicated problems in context of uncertainty. The way of modelling those problems through that tool and subsequently obtaining a solution depends in large part on the nature of available information about uncertain parameters of the problem. So, in the framework of the PROTRADE method, Goicoechea et al. (1982) assume that the decision maker (DM) is placed in a situation of risk, that is to say that he is able to specify practically the probability distribution of the random parameters of the problem; after the DM has to build his partial utility function relatively to each objective and then, on the basis of the preceding probability distribution, he makes his choice by means of the expected utility criterion. However, in the most of real situations, the DM is rather placed in a situation of a partial uncertainty (or incomplete information): even if he has some information about uncertain parameters of the problem, he is practically unable to specify a probability distribution for those parameters. Thus, in Urli and Nadeau, 1990a, Urli and Nadeau, 1990b, Urli and Nadeau, 1992, we suppose that the DM can specify only variation limits of uncertain parameters and possibly their central values and we propose methods adapted to these situations. Another way to deal with situations of incomplete information (that can be appropriate to many real contexts), consists in resorting to the idea of scenarios: those scenarios represent few representative and well contrasted values for uncertain parameters and are used to approximate them. The scenarios are called partial or global depending that states of the nature (that are supposed to exist) affect only a part or globally the whole of objectives and constraints; moreover, the probabilities of those scenarios can be supposed known or not. So, for contexts of partial scenarios on objectives and constraints with know probabilities, Teghem et al. (1986) have proposed the STRANGE method where the uncertainty in constraints is taken into account by an approach inspired by a recourse approach. Moreover, in the context of global scenarios on the whole of objectives and constraints with know probabilities for those scenarios, Klein et al. (1990) and Teghem (1993) have proposed interactive approaches with recourse.

In this paper, we deal with situations which are modelled by a global scenarios approach, but where the DM is unable to assign specific probabilities to those scenarios; he is just able to rank the probabilities of scenarios according to a total order, from the most probable to the less probable with possibility of ex-aequo. In some strategic decision problems when the use of scenarios seems to be suitable, those scenarios can cover a relativity long period of time; for example, in some production of hydro-electricity planning problems, given the usable life of a hydro-dam can be until one hundred years, an important implied parameter expresses the price of a kilowatt/hour of electric energy for example in the year 2030. The possible value of that random variable can be resumed by a small number of representative scenarios; but, in such a context, it is very difficult to assign a specific probability to each of those scenarios: then it appears more realistic to try only to rank the scenarios from the most probable to the less probable (with possibility of ex-aequo).

For such multiobjective stochastic linear programming problems where the uncertain parameters are modelled by scenarios and where the probabilities of those scenarios are only partially specified (according to a ranking), that partial information cannot be used in a resolution method in order to calculate expected utilities as, for example, in the method PROTRADE from Goicoechea et al. (1982). A way of taking into account that partial information consists in using it in the resolution phases of an interactive method: that is what we propose in a new pragmatic method which deals with such an uncertain context and that we called PROMISE/scenarios (in French, PROgrammation Multiobjectif Interactive StochastiquE). At first, on the basis of the scenarios, the multiobjective stochastic program (in which we introduce violation variables for the constraints) is transformed into a multiobjective deterministic program. However, in the resolution of that last multiobjective program, instead of using more or less a typical recourse approach (as in the aforementioned methods), we propose to the DM to take advantage of the available information about the violation of the constraints and the partially specified probabilities on scenarios, in order to progress interactively towards a satisfactory compromise solution; that proposed interactive method is derived from the STEM method (Benayoun et al., 1971). In what follows, after that our method will presented, it will be briefly illustrated by means of a didactic example.