Vector Maximization Problem Research Articles

Efficiency and proper efficiency in vector maximization with respect to cones

Vector maximization problems arise when more than one objective function is to be maximized over a given feasibility region. The concepts of efficiency and proper efficiency have played a useful role in the analysis of such problems. Recently these concepts have been extended to vector maximization problems in which the underlying domination cone is a convex cone. In this paper, efficient and properly efficient solutions for the vector maximization problem in which the underlying domination cone is any nontrivial, closed convex cone are examined. Differences between properly and improperly efficient solutions are established. Characterizations of efficient and properly efficient solutions are presented, and conditions under which efficient solutions exist and fail to exist are derived.

A method using fuzzy mathematics to solve the vectormaximum problem

In this paper, a new solution of the vectormaximum problem will be defined in terms of fuzzy mathematics and may be called the fuzzy solution. It will overcome difficulties which appear if otherwise solved by nonfuzzy mathematics. In this paper, the relation between this solution and the important concepts of vecormaximum problem, the efficient and weak-efficient solutions, will be shown.

Compensatory operators in fuzzy linear programming with multiple objectives

Previously, H.J. Zimmermann [11] presented a fuzzy approach to solve the linear vectormaximum problem. In the mentioned approach, the author uses the minimum or the product operators to translate the semantic meaning of the 'and'. The aim of this paper is to reconsider this approach by using operators which allow some degree of compensation between aggregated membership functions. Firstly the @c-operator already defined in [12] is used to combine our fuzzy objectives. It is shown that solutions obtained by this way are always efficient. Unfortunately, the substitute problem is not generally computationally feasible. We suggest then the min-bounded sum operator which is a compensatory one, has some desired properties and is supported by empirical arguments. It allows a considerable decrease of the computational burden in the substitute problem and leads to a solution which is attractive from the stand-point of efficiency.

Properly efficient and efficient solutions for vector maximization problems in euclidean space

Recently Benson proposed a definition for extending Geoffrion's concept of proper efficiency to the vector maximization problem in which the domination cone K is any nontrivial, closed convex cone. We give an equivalent definition of his notion of proper efficiency. Our definition, by means of perturbation of the cone K, seems to offer another justification of Benson's choice above Borwein's extension of Geoffrion's concept. Our result enables one to prove some other theorems concerning properly efficient and efficient points. Among these is a connectedness result.

On finding compromise solutions in multicriteria problems using the fuzzy min-operator

This paper presents an application of fuzzy approaches to the linear vectormaximum problem. It shows that using the fuzzy min-operator together with linear as well as special nonlinear membership functions the obtained solutions are always compromise solutions of the original multicriteria problem.

Discussion and modification of the Fandel-method solving linear vector maximum problems

In this paper, the convergence-model of Fandel for determining an optimal compromise solution for a vector maximum problem is discussed. It will be shown that applied to linear vector maximum problems, the method will not always generate efficient compromise proposals and under certain conditions fails. A modification of the method is proposed which eliminates the demonstrated shortcomings.

The determination of a subset of efficient solutions via goal programming

A wide variety of models and methods have been proposed to solve the vectormaximum problem. Many of these approaches center their attention on linear programming with several objective functions and seek to obtain the set of efficient (Pareto optimal) solutions. Another approach to the same problem is to rank the objectives according to a priority structure and seek the lexicographic minimum of an ordered function of goal deviations. This latter approach, known as goal programming with preemptive priorities, has, in the literature, usually been treated as a separate topic. In this paper we show that the solution to the linear goal programming problem can be made to always be an efficient solution from which we may conduct a practical investigation of a subset of efficient solutions which form a useful compromise set. While perhaps lacking the elegance of the more esoteric approaches, this technique nonetheless has worked well in practice on actual problems.

Nondominance In Goal Programming

AbstractThis paper investigates the notion of nondominance in goal programming (GP). Nondominance in the standard sense of a vector maximum (VM) problem is not always desirable. A definition of “GP-nondominance” and a test for determining whether an optimal GP solution is GP-nondominated are provided. It is demonstrated that one or more GP-nondominated solutions can be found by exploring the alternative optima of the GP solution. A formulation for determining the set of GP-nondominated solutions is provided and examples are used to illustrate the concept of GP nondominance.

The dependency of automatically designed multicriterion decisions on scale transformations

A solution procedure L z for the vector-maximum problem ‘max’ { z( x) | xϵX} should relay on some rationality axioms. In this setting z( x) is considered as an element of n-dimensional Euclidean space and X as the set of relevant alternatives. Together with some more technical requirements on L z decisions made by such solution procedures depend on the chosen scale determinants.

Inverse programming and the linear vector maximization problem

This paper presents a new concept of efficient solution for the linear vector maximization problem. Briefly, these solutions are efficient with respect to the constraints, in addition to being efficient with respect to the multiple objectives. The duality theory of linear vector maximization is developed in terms of this solution concept and then is used to formulate the problem as a linear program.

Goal programming sensitivity analysis using interval penalty weights

This paper presents an application of the vector-maximum research [4–8] to the sensitivity analysis of goal programming problems as several of the criterion function penalty weights are simultaneously and independently varied. A generalized goal programming capability is presented and a six-stage analytic procedure is described. The problem is generalized in the sense that the regular goal programming penalty weights can be expanded to intervals if desired. The solution procedure is new in that it depends upon an algorithm for the vector-maximum problem, “criterion cone” contraction procedures, and “filtering” techniques. Together they are able to generate and process all extreme points on the portion of the surface of the goal programming “augmented” feasible region corresponding to the interval penalty weights specified. In effect, the procedure and adapted algorithm of this paper delivers to goal programming an operational power of sensitivity analysis not previously available to users. A numerical example is provided in order to illustrate the computerized application of the total goal programming procedure outlined.

A survey of multicriteria optimization or the vector maximum problem, part I: 1776?1960

In this survey, the history of the subject from 1776 until 1960 is presented. A brief biographical sketch of Vilfredo Pareto is given first. Then, the more or less simultaneous development of the concepts of utility, preference, and welfare theory follows, with results which go back to Hausdorff and Cantor. A brief discussion of the work of Borel and von Neumann as initiators of game theory is included. Each of these areas has developed enough to warrant its own survey; hence, they are reviewed here only insofar as they provide necessary foundations. Thereafter, the concepts of efficiency, vector maximum problem, and Pareto optimality are reviewed in connection with production theory, programming, and economics. The survey is presented within a unified mathematical framework, and the emphasis is on mathematical results, rather than psychological or socio-economic discussion. To enable the reader to draw conclusions without having to obtain each article himself, the results have been presented in somewhat more detail than usual.

An improved definition of proper efficiency for vector maximization with respect to cones

Recently Borwein has proposed a definition for extending Geoffrion's concept of proper efficiency to the vector maximization problem in which the domination cone S is any nontrivial, closed convex cone. However, when S is the non-negative orthant, solutions may exist which are proper according to Borwein's definition but improper by Geoffrion's definition. As a result, when S is the non-negative orthant, certain properties of proper efficiency as defined by Geoffrion do not hold under Borwein's definition. To rectify this situation, we propose a definition of proper efficiency for the case when S is a nontrivial, closed convex cone which coincides with Geoffrion's definition when S is the non-negative orthant. The proposed definition seems preferable to Borwein's for developing a theory of proper efficiency for the case when S is a nontrivial, closed convex cone.

Vector maximization with two objective functions

This note gives a characterization of an efficient solution for the vector maximization problem with two objective functions. This characterization yields a parametric procedure for generating the set of all efficient solutions for this problem. The parametric procedure can also be used to solve certain bicriterion mathematical programs.

Existence of efficient solutions for vector maximization problems

The vector maximization problem arises when more than one objective function is to be maximized over a given feasibility region. The concept of efficiency has played a useful role in analyzing this problem. In order to exclude efficient solutions of a certain anomalous type, the concept of proper efficiency has also been utilized. In this paper, an examination of the existence of efficient and properly efficient solutions for the vector maximization problem is undertaken. Given a feasible solution for the vector maximization problem, a related single-objective mathematical programming problem is investigated. Any optimal solution to this program, if one exists, yields an efficient solution for the vector maximization problem. In many cases, the unboundedness of this problem shows that no properly efficient solutions exist. Conditions are pointed out under which the latter conclusion implies that the set of efficient solutions is null. As a byproduct of our results, conditions are derived which guarantee that the outcome of any improperly efficient point is the limit of the outcomes of some sequence of properly efficient points. Examples are provided to illustrate these results.

Fuzzy programming and linear programming with several objective functions

In the recent past numerous models and methods have been suggested to solve the vectormaximum problem. Most of these approaches center their attention on linear programming problems with several objective functions. Apart from these approaches the theory of fuzzy sets has been employed to formulate and solve fuzzy linear programming problems. This paper presents the application of fuzzy linear programming approaches to the linear vectormaximum problem. It shows that solutions obtained by fuzzy linear programming are always efficient solutions. It also shows the consequences of using different ways of combining individual objective functions in order to determine an “optimal” compromise solution.

Testing for complete efficiency in a vector maximization problem

Testing for complete efficiency in a vector maximization problem

A general method for determining the set of all efficient solutions to a linear vectormaximum problem

To solve a linear vectormaximum problem means, in general, to determine the set E of all efficient solutions. A multiparametric method based on earlier works of the author is presented. In the procedure efficient vertices and efficient edges are generated via one subprogram, which works as a simple linear programming problem, and just by inspection of these results higher dimensional efficient faces are determined. The procedure does not depend on special properties of the restriction set and/or of the system of given objective functions. Illustrative examples are presented. Two appendixes provide a survey on a multiparametric algorithm and on a solution procedure for the auxiliary problem, both of which are the background for the method.

Lexicographic quasiconcave multiobjective programming

Letf i :A → R ben real-valued objective functions on a convex setA ⊂-K m ,K:=R orC, n, m∈N. Letg: A → R n be defined by $$g(x): = (g_1 (x),...,g_n (x)): = (f_{i_1 (x)} (x),...,f_{i_n (x)} (x))$$ , where for eachx∈A, (i 1 (x), ..., i n (x)) is a permutation of (1, ...,n) such that $$(f_{i_1 (x)} (x) \leqslant ... \leqslant f_{i_n (x)} (x))$$ . In this paper we treat the problem of findingx *∈A such that $$g(x^ * ) = \mathop {l--\max }\limits_{x \in A} g(x)$$ , wherel-max denotes the lexicographic maximum. If the fi's are strongly quasiconcave we can reduce the problem stepwise until finally it is in the form of a scalar programming problem. Further, we consider conditions for the existence and uniqueness of a solution and discuss the relationship of the problem to the vector maximum (i.e. Pareto) and maxmin (i.e. Chebychev) problems.

Redundant objective functions in linear vector maximum problems and their determination

Suppose that in a multicriteria linear programming problem among the given objective functions there are some which can be deleted without influencing the set E of all efficient solutions. Such objectives are said to be redundant. Introducing systems of objective functions which realize their individual optimum in a single vertex of the polyhedron generated by the restriction set, the notion of relative or absolute redundant objectives is defined. A theory which describes properties of absolute and relative redundant objectives is developed. A method for determining all the relative and absolute redundant objectives, based on this theory, is given. Illustrative examples demonstrate the procedure.