Introduction to Fuzzy and Possibilistic Optimization

Introduction

Deterministic optimization is a normative process which extracts the best from a set of options, usually under constraints. It is arguably true that optimization is one of the most used areas of mathematical applications. It is the thesis of this book that applied mathematical programming problems should be solved predominantly by using a fuzzy and possibilistic approaches. Rommelfanger ([42], p. 295), states that the only operations research methods that is widely applied is linear programming. He goes on to state that even though this is true, of the 167 production (linear) programming systems investigated and surveyed by Fandel [18], only 13 of these were ”purely” (my interpretation) linear programming systems. Thus, Rommelfanger concludes that even with this most highly used and applied operations research method, there is a discrepancy between classical linear programming and what is applied. Deterministic and stochastic optimization models require well-defined input parameters (coefficients, right-hand side values), relationships (inequalities, equalities), and preferences (real valued functions to maximize, minimize) either as real numbers or real valued distribution functions. Any large scale model requires significant data gathering efforts. If the model has projections of future values, it is clear that real numbers and real valued distributions are inadequate representations of parameters, even assuming that the model correctly captures the underlying system. It is also known from mathematical programming theory that only a few of the variables and constraints are necessary to describe an optimal solution (basic variables and active constraints), assuming a correct deterministic normative criterion (objective function). The ratio of variables that are basic and constraints that are active compared to the total becomes smaller, in general, as the model increases in size since in general large-scale models tend to become more sparse. Thus, only a few parameters need to be obtained precisely. Of course the problem is that it is not known a priori which variables will be basic and which constraints will be active.