Tomas Gal

On efficient sets in vector maximum problems -- A brief survey

The notion of efficient (nondominated, noninferior. Pareto-optimal, functional efficient) set to a Vectormaximum Problem (VMP) has been analysed and developed in several directions during the last 30 years. Starting with the basic notion of efficiency given by Pareto [1896], formal descriptions of the efficient, properly efficient, locally (proper-) efficient, weak or strong efficient set have been developed beginning in the 50th of this century. Based on these notions various characteristics and properties of the efficient set have been studied, whereas the feasible set X and the functions zk(x), k = 1,.., K, constituting the vector-valued criterion z(x) of VMP have various properties (e.g., convexity of X, concavity of zk(x) for all k, differentiability etc.). Based on such properties, the structure of the efficient set and the existence of efficient solutions have been analysed. A part of the corresponding publications are rather of a pure theoretical character, others try to develope theories serving as the basis for working out methods for determining the efficient set or for interactive methods determining some compromise solutions. Duality theories for more or less general cases have been developed and various aspects of stability of VMP have been investigated. A brief survey is given.

Redistribution of funds for teaching and research among universities: The case of North Rhine-Westphalia

Abstract This paper reports on a real application of a performance and success based system for redistribution of funds for teaching and research among universities in North Rhine-Westphalia. After a precise description of the decision situation, we show how goal programming and distance minimization were applied in order to find a solution on the basis of real data. Some comments on the results and the quality of the redistribution process conclude the paper.

Survey of solved and open problems in the degeneracy phenomenon

Degeneracy is a phenomenon that may arise, e.g., in linear programming (LP for short), bottleneck LP, multiparametric LP, linear vectormaximization, etc. If it does arise then it certainly influences any vertex-searching method for mathematical models based on a system of linear inequalities and in some cases it leads to misinterpretation of optimal solutions.

Consequences of dropping nonessential objectives for the application of MCDM methods

Abstract In this article we consider the problem of nonessential objectives for multiobjective optimization problems (MOP) with linear objective functions. In 1977 an approach based on the reduction of size of the matrix of objective functions has been worked out by one of the present authors (Gal, T., Leberling, H., 1977. European Journal of Operations Research 1, 176–184). Although this method for dropping nonessential objectives leads to a mathematically equivalent MOP, problems concerning the application of MOP methods may arise. For instance, dropping some (or all) of the nonessential objectives the question is, how to ensure obtaining the same solution as with all objectives involved. We consider the problem of adapting the parameters of multiobjective optimization methods. For the case of weighting methods a simple procedure for adapting the weights is analyzed. For other methods, e.g. reference point approaches, such a simple possibility for adapting the parameters is not given.

Nonessential objectives within network approaches for MCDM

In Gal and Hanne [Eur. J. Oper. Res. 119 (1999) 373] the problem of using several methods to solve a multiple criteria decision making (MCDM) problem with linear objective functions after dropping nonessential objectives is analyzed. It turned out that the solution does not need be the same when using various methods for solving the system containing the nonessential objectives or not. In this paper we consider the application of network approaches for multicriteria decision making such as neural networks and an approach for combining MCDM methods (called MCDM networks). We discuss questions of comparing the results obtained with several methods as applied to the problem with or without nonessential objectives. Especially, we argue for considering redundancies such as nonessential objectives as a native feature in complex information processing. In contrast to previous results on nonessential objectives, the current paper focuses on discrete MCDM problems which are also denoted as multiple attribute decision making (MADM).

A new pivoting rule for solving various degeneracy problems

Using some of the results of the recently developed theory of degeneracy graphs, an efficient procedure to solve problems connected with degeneracy is presented. Such problems are, for example, the determination of all neighboring vertices of a degenerate vertex, all vertices of a convex polyhedron and the avoidance of cycling and stalling in simplex-like methods. Rules for the efficient performance of sensitivity analyses and the determination of shadow prices under degeneracy are indicated.

Stability in vector maximization—A survey☆

Abstract Stability and sensitivity analysis becomes more and more attractive also in the area of multicriteria decision making (MCDM). Influences of parameter changes concerning the right-hand-side or the objective functions respectively or changes of the domination structure on the solution of various mathematical models of vectormaximization problems — as developed in several publications — are examined in this paper and a survey on the corresponding results is given. The survey is subdivided with respect to the various model structures of the decision problem — i.e. linear or nonlinear, deterministic or stochastic, static or dynamic — and with respect to the above mentioned investigated changes.

Classification of Real World Trim Loss Problems

Trim loss problems, also called cutting stock or depletion problems, are essential for production planning in some industries (e.g. in the paper, wood, metal, glass, plastic, and textile industries). In these industries the minimization of production costs is often approached in the following way: The raw material used is first produced in large standard sizes, possibly stored, and only later reduced to smaller sizes for in-plant processing or to meet customers’ orders. This production pattern, on the one hand, involves a temporal uncoupling of raw material production and manufacturing of final products, and on the other hand it avoids a frequent resetting of production facilities which would be necessary if the product size required were produced from the input material in one step only. There are drawbacks, however, since an additional stage of production (the cutting operation) is required which generates useless remainder (trim loss). “Trim loss planning” is concerned with minimizing the negative effects thus created on the production costs.

Relaxation analysis in linear vectorvalued maximization

Abstract As an analogy to the postoptimal sensitivity analysis in linear programming a theory of postefficient sensitivity analysis in linear vectorvalued maximization problems (LVMP), called relaxation analysis, is developed. Introducing parameters in the most coefficients of the given objective functions, the goal is to define an admissible region of the parameters such that the set E of all the efficient solutions of the initial LVMP does not change. Doing the same with the right hand side, the set of all admissible parameters is defined via an undirected graph which is generated by the initial LVMP.

A new interactive algorithm for multi-objective linear programming using maximally changeable dominance cone

Abstract The purpose of this research is to develop a computer applicable interactive methodology for the resolution of multi-objective linear decision problems within a decision makers (DMs) capability to supply necessary information for problem solution. Enlarging the (Pareto) dominance cone is subject to the condition that the set of all efficient solutions does not change. This leads to the recently developed concept of the Maximally changeable dominance cone which is proved to be strictly negative polar cone generated by the positive outer normal vectors of all maximally efficient facets. In this study, it is shown that by enlarging the dominance cone, the number of efficient solutions in the objective space becomes smaller. Therefore, our method reduces interactively the subset of efficient solutions in the objective space by enlarging the (Pareto) dominance cone step by step based on the DMs preference information until only a few efficient solutions remain. From these, then, the DM can select his/her most preferred one. In the beginning of the procedure, the concept of the intervals of pairwise tradeoffs between objectives, called the Marginal Rate of Substitution (MRS), is applied as a candidate for screening the efficient solutions which are less preferable. Furthermore, appropriate questions founded on the notion of the MRS are posed to the DM in order to eliminate the less preferable solutions from consideration in the solution process, which results in enlargement of the dominance cone. A numerical example is provided to illustrate the procedure described.