Ignacy Kaliszewski

Quantitative Pareto Analysis by Cone Separation Technique

Preface. Notation. 1. Introduction. 2. Basic Elements. 3. Cones, Efficiency and Proper Efficiency -- a General Setting. 4. Proper Efficiency with Respect to k+. 5. Quantitative Pareto Analysis. 6. Substantial Efficiency. 7. Computational Tractability of the Quantitative Pareto Analysis. References. Index.

Out of the mist--towards decision-maker-friendly multiple criteria decision making support

Abstract The number of real applications of interactive multiple criteria decision making methods is still modest despite the existence of computer realizations which operationalize them. We attribute this fact to high complexity of the methods as perceived by real decision makers. Much of this complexity can be attributed to third phase of decision selection processes which comprise intelligence, design, choice, and review. The paper concentrates on issues related to the choice phase. A way to abate this complexity is to provide descriptions of the methods on two separate levels: the methodological level and the technical level. In general the decision maker is interested only in issues which directly pertain to the choice process, and not in technical details. The effect of separating methodological issues from technical ones is a better focus on each of the two aspects. Developments on each level can take place independently, provided that there is a common standard of communication between these levels. The aim of this paper is twofold. First, we show the majority of MCDM methods fall into one of two prototypical classes. We also show that these two prototypical classes support a common standard of communication between the methodological and technical levels of method implementations. With a communication standard established, one backbone technical level computing engine can serve either MCDM method class. Second, we show how in the weight method class the need of solving optimization problems can be eliminated from interactive choice processes. We present a technical development by which such an elimination is possible.

A modified weighted Tchebycheff metric for multiple objective programming

Abstract We show how to determine nondominated criterion vectors (and then nondominated solutions, if necessary) by a modified weighted Tchebycheff metric in cases when sets of all criterion vectors are finite or polyhedral.

Interactive Multiple Criteria Decision Making based on preference driven Evolutionary Multiobjective Optimization with controllable accuracy

We present an approach to interactive Multiple Criteria Decision Making based on preference driven Evolutionary Multiobjective Optimization with controllable accuracy.

Efficient solutions and bounds on tradeoffs

The modified Tchebycheff method, widely used to generate efficient solutions to a vector optimization problem, provides means to identify properly efficient solutions with a preimposed common bound on all tradeoffs. In this paper, we show how to generate weakly efficient solutions when different bounds are preimposed on the subsets of tradeoffs.

Searching for psychologically stable solutions of multiple criteria decision problems

In interactive decision making, a choice behavior of the decision maker may differ depending on proximity of a current solution to satisfactory values of objectives. An interactive approach proposed in this paper allows the decision maker to use different search principles depending on his/her perception of the achieved values of objectives and trade-offs. While an analysis of the values of objectives may guide the initial search for a final solution, it can be replaced by trade-off evaluations at some later stages. Such an approach allows the decision maker to change search principles, and to identify a psychologically stable solution to the multiple criteria decision problem.

Using trade-off information in decision-making algorithms

Abstract The aim of the paper is to propose a general algorithm for interactive multiple criteria decision-making generalising the classical Zionts–Wallenius and Dell–Karwan algorithms. Such a generalisation has been made possible by theoretical results proved in this paper, which allow operationalising the concept of interactive multiple criteria decision-making with varying decision paradigms. The result extends earlier results on this subject. Scope and purpose Decision making is still a challenge for various branches of theoretical and applied science. Though the real nature of the process by which a decision maker eventually arrives at his/her final choice is still unknown, there is a wide selection of tools to assist and (to some extent) to ease the process. In cases where the decision problem is solved once or infrequently, and therefore there is no statistical data from which decision maker preferences can be inferred, interactive man-machine decision support schemes are recommended. Models and tools for the machine to work with should provide (and they do) in-depth insight into mutal relations between decisions considered during the interactive process. In this work we have attempted to make one step beyond the current state-of-the-art to provide new tools for interactive decision-making. To this aim we have revisited the notion of trade-off. Our formal findings show that using this notion proves to be conceptually simple and computationally inexpensive.

Generating Nested Subsets of Efficient Solutions

Given a vector optimization problem, the solution set is composed of nondominated elements of a set. All nondominated elements are equivalent in the sense that each of them is a solution as good as any other one. However, if the vector optimization problem is a formalization of a practical multiple objective decision problem, one may expect that some solutions are more preferred than others. To assist a decision maker, most of methods for multiple objective decision making rely upon additional information, to be obtained from the decision maker, to that contained in the definition of the vector optimization problem. In contrast to that, in this paper we show how it is possible to impose a certain order over nondominated elements of a set, based only on the information contained in the formulation of the corresponding vector optimization problem. We also present a numerically attractive method to determine nested subsets of properly efficient elements.

A theorem on nonconvex functions and its application to vector optimization

Abstract A basic result of convex analysis is the Fundamental Theorem on Convex Functions. We present a counterpart of this theorem for the case the convexity assumption does not hold and we shall show that some well known necessary conditions for weak efficiency and proper efficiency in nonconvex vector optimization are implicit in our result. The significance of this result in vector optimization and in supporting decision making is briefly discussed.

Dynamic parametric bounds on efficient outcomes in interactive multiple criteria decision making problems

Abstract Bounds on efficient outcomes in interactive multiple criteria decision making problems are derived. Bounds are dynamic, i.e., they become stronger with the growing number of explicitly identified outcomes. They are also parametric with respect to weighting coefficients. Computational cost to calculate bounds is negligible. Bounds of the sort offer a breakthrough for prohibitive size and/or solution time bottlenecks by allowing a decision maker to interact with an approximation of the underlying mathematical model rather the model itself. Possible applications of bounds to existing interactive decision making algorithms are discussed. Illustrative numerical examples are given.