Continuous Optimization
The compromise hypersphere for multiobjective linear programming

https://doi.org/10.1016/S0377-2217(01)00388-5Get rights and content

Abstract

For a linear-programming problem with q(⩾2) objective functions (that is, a multiobjective linear-programming problem), we propose a method for ranking the full set or a subset of efficient extreme-point solutions. The idea is to enclose the given efficient solutions, as represented by q-dimensional points in objective space, within an annulus of minimum width, where the width is determined by a hypersphere that minimizes the maximum deviation of the points from the surface of the hypersphere. We argue that the hypersphere represents a surface of compromise and that the point closest to its surface should be considered as the “best” compromise efficient solution. Also, given a ranked (sub)set of efficient solutions, a procedure is given that associates to each efficient solution a set of q positive weights that causes the efficient solution to be optimal with respect to the given set of efficient solutions.

Introduction

The general multiobjective linear-programming problem (MOLP) can be stated as:(P1)Maximizec1Xc2XcqXsubjecttoAX=b,X0,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, X=(x1,…,xn) is an n-dimensional column vector, b=(b1,…,bm) is an m-dimensional column vector, ck=(ck1,…,ckn) are the n-dimensional objective function row vectors, and the matrix A=(aij) is (m×n).

If we letC=c1=(c11,…,c1n)c2=(c21,…,c2n)cq=(cq1,…,cqn),we can interpret this problem equivalently as the vector maximization problem:(P2)“Maximize”CXsubjecttoAX=b,X0.

The problem is of interest as, in general, there is no one point in the solution space S={X:AX=b,X0} that simultaneously maximizes the q objective functions. Thus, a decision process or rule must be used to select an XS 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:

To find an X0 that maximizes the vector function GX constrained by FX⩾0, X⩾0; that is, to find an X0 satisfying the constraints and such that GXGX0 for no X satisfying the constraints. (F and G involve differentiable functions.)

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:

Definition 1

X0 is said to be an efficient solution to problem (P2) if X0S and there exists no other XS such that CXCX0 and CXCX0. We let E be the set of efficient solutions {X0S} and let N(E) be the number of efficient extreme point solutions in S.

Equivalent definitions for efficient solutions are the following:

Definition 2

X0S is an efficient solution to problem (P2) if and only if there exists no other XS such that ckXckX0 for all k=1,…,q and ckX>ckX0 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 X0 to another XS and increase one ckX without having to decrease some other ckX.)

From these definitions, we conclude that a rational decision maker (DM) should be concerned only with efficient solutions, as the objective function values ckX for any nonefficient solution XS will, for some efficient X0S, always be less than or equal to the corresponding ckX0, with at least one ckX0>ckX. The difficulty in applying this decision rule is that N(E) 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 X0 as the solution to problem (P2)?

Definition 3

An ideal solution Xk is the (finite) optimal solution to the single-objective function linear-programming problem:(Pk)MaximizeckX(k=1,…,q)subjecttoAX=b,X0.If Xk is a unique solution to problem (Pk), then Xk is an efficient solution to (P2). For notational convenience, we let zk(X)=ckX for any XS and define the associated criterion vector z(X)=[z1(X),…,zq(X)], or simply, z=[z1,…,zq]. For the set of ideal solutions {Xk}, we have the ideal or utopian criterion vector z=(c1X1,…,cqXq)=(z1,…,zq).

The DM would really like to find a utopian solution X∗∗S such that all ckX∗∗=ckXk. For problems of interest, we assume that such an X∗∗ 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:(P3)MinimizekyksubjecttockX−yk=ckXk(k=1,…,q),AX=b,X0,yk⩾0.

A key theorem that supports the search for efficient solutions to problem (P2) is the following:

Theorem 1

Consider the multiparameter problem(P4)Maximizez=∑kλk(ckX)=λ(CX)subjecttoAX=b,X0for λ=(λ1,…,λq) and all λk>0. A vector X0 that solves problem (P4) for fixed λk>0(k=1,…,q) 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 S(E) 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 S(E) is the set of q ideal solutions (assuming all are efficient). We denote a generic efficient solution by XE and its corresponding q-dimensional vector of objective function values by z(XE)=[z1(XE),…,zq(XE)].

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

Maximizez1=x1z2=x2subjecttox1+x2⩽8,2x1+x2⩽12,x1+2x2⩽14,9x1+7x2⩽63,−4x1+10x2⩽61,2x1−x2⩽8,14x1+3x2⩽72,xj⩾0.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 E of efficient solutions to be considered by the DM.

  • (ii) For points XE in E, calculate the associated q-dimensional criterion space vectors z(XE)=[z1(XE),…,zq(XE)] (Table 2).

  • (iii) For the set of points z(XE), formulate the associated compromise dual problem (D0*) and determine the hypersphere and min-max annulus that contains the set z(XE) for E.

  • (iv) Rank the efficient solutions based on how

References (45)

  • Chou, S.-Y., Woo, T.C., Pollock, S.M., 1994. On characterizing circularity. Department of Industrial and Operations...
  • J.L. Cohon

    Multiobjective Programming and Planning

    (1978)
  • M. Dror et al.

    Interactive scheme for a MOLP problem given two partial orders

    Applied Mathematics and Computation, Part I

    (1987)
  • G.W. Evans

    An overview of techniques for solving multiobjective mathematical programs

    Management Science

    (1984)
  • J.P. Evans et al.

    A revised simplex method for linear multiple objective programs

    Mathematical Programming

    (1973)
  • S.I. Gass

    Linear Programming: Methods and Applications

    (1985)
  • S.I. Gass et al.

    The computational algorithm for the parametric objective function

    Naval Research Logistics Quarterly

    (1955)
  • S.I. Gass et al.

    Fitting circles and spheres to coordinate measuring machine data

    International Journal of Flexible Manufacturing

    (1998)
  • R. Hartley

    Survey of algorithms for vector optimization problems

  • D.W. Hearn et al.

    Efficient algorithms for the (weighted) minimum circle problem

    Operations Research

    (1982)
  • S.C. Huang

    Note on the mean-value strategy for vector-valued objective functions

    Journal of Optimization Theory and Applications

    (1972)
  • C.-L. Hwang et al.

    Multiple Objective Decision Making – Methods and Application: A State-of-the-Art Survey

    (1979)
  • Cited by (20)

    • A class of rough multiple objective programming and its application to solid transportation problem

      2012, Information Sciences
      Citation Excerpt :

      A lot of researchers describe the act of finding a compromise solution of a multiple objective programming problem, such as Zadeh [46], Zeleny [47], and Wierzbicki [37]. Based on the former researchers’ work, Gass and Roy [7] summarized three basic approaches to obtain a compromise solution: A priori articulation of preferences: the DM states a set of criteria weights and the corresponding parametric system is solved.

    • Applications of AHP Methodology for Decision-Making in Cleaner Production Processes

      2022, Applications of AHP Methodology for Decision-Making in Cleaner Production Processes
    • Fully Piecewise Linear Vector Optimization Problems

      2021, Journal of Optimization Theory and Applications
    View all citing articles on Scopus
    View full text