Vector Maximization Problem Research Articles

The Vector Maximization Problem: Proper Efficiency and Stability

Vector maximization problems arise when more than one objective function is to be maximized over a given feasibility region. While the concept of efficiency has played a useful role in the analysis of such problems, a slightly more restricted concept of efficiency, that of proper efficiency, has been proposed in order to eliminate efficient solutions of a certain anomalous type. In this paper, necessary and sufficient conditions for an efficient solution to be properly efficient are developed. These conditions relate the proper efficiency of a given solution to the stability of certain single-objective maximization problems. The conditions are useful both in verifying that certain efficient solutions are properly efficient and in identifying efficient solutions that are not proper. An immediate corollary of the theory is that all efficient solutions in linear vector maximization problems are properly efficient. Examples are given to illustrate our results.

Multiple Objective Linear Programming with Interval Criterion Weights

The purpose of this research is to develop a computer applicable methodology for analyzing multiple objective linear programming problems when interval, rather than fixed, weights are assigned to each of the objectives. Rather than obtaining a single efficient extreme point as one would normally expect with fixed weights, a cluster of efficient extreme points is typically generated when interval weights are specified. Then, from the subset of solutions generated, it is contemplated, that the decision-maker will be able to qualitatively identify his efficient extreme point of greatest utility (which should either be the decision-maker's optimal point or be close enough to it to terminate the decision process). The procedure presented in this paper involves converting the multiple objective linear programming problem with interval criterion weights into an equivalent vector-maximum problem. Once in such form, an algorithm for the vector-maximum problem can then be used to determine the subset of “efficient extreme” points corresponding to the interval weights specified. General computational experience regarding the operation of the equivalent vector-maximum problem is reported. Also, a numerical example is provided to illustrate the total procedure described.

The theorem of the alternative, the key-theorem, and the vector-maximum problem

The theorem of the alternative, the key-theorem, and the vector-maximum problem

Technical Note—Proper Efficiency and the Linear Vector Maximum Problem

A vector maximum problem arises when more than one scalar-valued objective function is to be maximized simultaneously over a given range of definition. This note considers linear vector maximum problems and shows that each efficient solution of a linear vector maximum problem also satisfies the requirements of a restricted concept of efficiency, the notion of proper efficiency.

Algorithms for the vector maximization problem

We consider a convex setB inR n described as the intersection of halfspacesa i T x ≦b i (i ∈ I) and a set of linear objective functionsf j =c j T x (j ∈ J). The index setsI andJ are allowed to be infinite in one of the algorithms. We give the definition of theefficient points ofB (also called functionally efficient or Pareto optimal points) and present the mathematical theory which is needed in the algorithms. In the last section of the paper, we present algorithms that solve the following problems:

Some duality theorems for the non-linear vector maximum problem

In this paper the well-knownLegendre transform-type of non-linear duality is extended to the general vector maximum problem. Duality is studied in terms of the primal and dual objective sets rather than in terms of the underlying feasible solutions. The structure of the objective sets is fully explored. The main result in duality is that under reasonable regularity assumptions there are no duality gaps between the primal and dual objective sets.

Letter to the Editor—Improper Solutions of the Vector Maximum Problem

Many problems in operations research, engineering, and economics require that several objectives be maximized simultaneously. It is well known that ‘maximizing’ the vector whose components are these objectives can yield more solutions than maximizing one linear combination of the objectives. Kuhn and Tucker (Kuhn, H. W., A. W. Tucker. 1951. Nonlinear programming. Proc. Second Berkeley Symp. on Math. Stat. and Prob. University of California Press, Berkeley, Calif. 1951.) discuss the vector maximum problem and derive necessary and sufficient conditions for a vector x0 to be a proper solution. However, their basic paper presents only a definition of ‘proper’ and a particular vector maximum problem for which an ‘improper’ solution has an undesirable property. The purpose of this note is to show that all ‘improper’ solutions have this undesirable property, thus justifying the calculation of only the ‘proper’ solutions.

Strictly Concave Parametric Programming, Part I: Basic Theory

This paper, which is presented in two parts, develops a computational approach to strictly concave parametric programs of the form: Maximize α f1(x) + (1 − α)f2(x) subject to concave inequality constraints for each fixed value of α in the unit interval, where f1 and f2 are strictly concave and certain additional regularity conditions are satisfied. This class of problems subsumes a corresponding class of vector maximum problems and also, by means of a simple device, provides a deformation method for ordinary concave programming. The approach is based on exploiting the continuity properties of the parametric program so as to efficiently maintain a solution to the associated Kuhn-Tucker conditions as a traverses the unit interval. The same approach can be adapted to much more general parametric programs than the one above. In Part I, a Basic Parametric Procedure is derived and shown to be finite in a certain sense. It forms the basis of various parametric programming algorithms, depending on what special assumptions are made on the functions. In Part II, additional theory is developed that facilitates computational implementation, and one possible general-purpose algorithm for a digital computer is given. An illustrative graphical example is presented and several extensions are indicated.

Optimum Allocation of Sampling Units to Strata When There areRResponses of Interest

Abstract A common problem in many areas is to achieve a combination of control variables which will simultaneously maximize all of the responses which are functions of the control variables. The fact that in general we can not simultaneously maximize all of the responses has led to the formulation of compromise solutions to this vector maximum problem. A point in the control variable space is better than a second point if each response at the first point is greater than or equal to the corresponding response at the second point. If a point is such that there is no better point, it is said to be admissible, or efficient. A set of points is complete if given any point not in the set there is a point in the set which is better. In this paper this formulation is used to consider the allocation of sample size to several strata when there are several characteristics of interest.