Continuous OptimizationThe compromise hypersphere for multiobjective linear programming
Introduction
The general multiobjective linear-programming problem (MOLP) can be stated as:where “Maximize” represents the simultaneous maximization of the q objective functions subject to a set of linear equations and nonnegativity conditions on the variables. Here, is an n-dimensional column vector, is an m-dimensional column vector, are the n-dimensional objective function row vectors, and the matrix is (m×n).
If we letwe can interpret this problem equivalently as the vector maximization problem:
The problem is of interest as, in general, there is no one point in the solution space that simultaneously maximizes the q objective functions. Thus, a decision process or rule must be used to select an for consideration, that is, a compromise solution must be determined. In this paper, we describe a new approach for the selection of such a compromise.
The earliest reference to the vector maximization problem (P2) by that name is, we believe, due to Kuhn and Tucker (1951). There they give a general statement of the nonlinear vector maximum problem as:
They cite Koopmans's (1951b) economic analysis of the multicommodity production model and his conceptional approach to its resolution in terms of efficient solutions, also termed Pareto optimal, nondominated or noninferior solution. Efficient solutions of problem (P2) are defined as follows:To find an that maximizes the vector function GX constrained by , ; that is, to find an satisfying the constraints and such that for no X satisfying the constraints. (F and G involve differentiable functions.)
Definition 1
is said to be an efficient solution to problem (P2) if and there exists no other such that and . We let E be the set of efficient solutions and let be the number of efficient extreme point solutions in S.
Equivalent definitions for efficient solutions are the following: Definition 2 is an efficient solution to problem (P2) if and only if there exists no other such that for all k=1,…,q and for at least one k. Definition 2a Cohon, 1978 A feasible solution to a multiobjective programming problem is noninferior if there exists no other feasible solution that will yield an improvement in one objective without causing a degradation in at least one other objective. (This states that it is not possible to move from an efficient point to another and increase one without having to decrease some other .)
From these definitions, we conclude that a rational decision maker (DM) should be concerned only with efficient solutions, as the objective function values for any nonefficient solution will, for some efficient , always be less than or equal to the corresponding , with at least one . The difficulty in applying this decision rule is that is usually equal to at least the number of objective functions q, and, as we shall see, quite often much larger. How does one choose one as the solution to problem (P2)? Definition 3 An ideal solution is the (finite) optimal solution to the single-objective function linear-programming problem:If is a unique solution to problem (Pk), then is an efficient solution to (P2). For notational convenience, we let for any and define the associated criterion vector , or simply, . For the set of ideal solutions , we have the ideal or utopian criterion vector .
The DM would really like to find a utopian solution such that all . For problems of interest, we assume that such an does not exist, as is the case for most real-world problems. That is, for problem (P2), the following linear-programming problem would be infeasible:
A key theorem that supports the search for efficient solutions to problem (P2) is the following: Theorem 1 Consider the multiparameter problemfor and all λk>0. A vector that solves problem (P4) for fixed is an efficient solution to problem (P2) (Koopmans, 1951a, Koopmans, 1951b; Kuhn and Tucker, 1951; Geoffrion, 1968; Chankong and Haimes, 1983; Gass, 1985; Steuer, 1986). The λk can be interpreted as objective function importance weights and are usually given in normalized form, i.e., ∑kλk=1, λk>0.
For problem (P2), the DM must choose an efficient solution from the set of efficient solutions. For methodological and practical reasons, we assume that a partial or full set of efficient extreme point solutions has been generated by the ADBASE program of Steuer (1986) or similar efficient point generating algorithms (Evans and Steuer, 1973; Yu and Zeleny, 1975; Chankong and Haimes, 1983; Hartley, 1983). Efficient edge solutions, if known, can also be included. If (P2) has two objective functions, then the parametric programming procedure of Gass and Saaty (1955) can be used to generate all efficient solutions.
Section snippets
Compromise solutions
For our purposes, we define a compromise efficient solution as follows: Definition 4 For problem (P2), a compromise efficient solution is an efficient solution to problem (P2) selected by the DM as being the best solution, where the selection is based on the DMs explicit or implicit selection criteria.
Zeleny (1982, p. 315), as well as most authors, describes the act of finding a compromise solution to problem (P2) as “…an effort to approach or emulate the ideal solution as closely as possible.” Zeleny (1982,
The compromise hypersphere
We assume that the DM has found (or is presented) with either the full set E of efficient solutions or a subset that is considered to be an adequate representation of E. In either case, the DM has agreed to select one of the efficient solutions as a compromise solution. Included in is the set of q ideal solutions (assuming all are efficient). We denote a generic efficient solution by and its corresponding q-dimensional vector of objective function values by .
Multicriteria problems and their solution via hyperspheres
In this section, we investigate the solution of a number of vector maximum problems and interpret their solution in terms of the hypersphere compromise approach. The problems chosen do not necessarily have any real-world motivation; the purpose of examining them is to illustrate the method. Example 4 The problem has six efficient solutions, as found by ADBASE. They are given in Table 2, along with the
Summary of the compromise solution approach
(i) Using an MOLP efficient-point solution procedure such as ADBASE (Steuer, 1986), determine the set of efficient solutions to be considered by the DM.
(ii) For points in , calculate the associated q-dimensional criterion space vectors (Table 2).
(iii) For the set of points , formulate the associated compromise dual problem (D0*) and determine the hypersphere and min-max annulus that contains the set for .
(iv) Rank the efficient solutions based on how
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