Interactive stability cutting-plane algorithm for multiobjective nonlinear programming problems

https://doi.org/10.1016/j.amc.2007.03.037Get rights and content

Abstract

In this paper, we introduce an interactive stability cutting-plane algorithm for determining a best-compromise solution to a multiobjective nonlinear programming problems in situations with an implicitly defined utility function. The method is called “interactive stability cutting-plane compromise programming (ISCPCP)”. The cutting-planes which I am going to derive, are based on suitable pairwise trade-offs between the objective functions, as prescribed by the decision maker (DM) at each iterate generated by the algorithm. This algorithm requires no line searches, and generates iterates that are all contained in the efficient frontier. This feature facilitates the preference judgment of the decision maker, and permits an analyst to terminate short of optimality with an efficient near-optimal solution. Also, we investigate the stability of its efficient solutions which are obtained by using this algorithm. An illustrating example is presented to clarify this algorithm.

Introduction

In an earlier work, Dyer [3] and Wehrung [16] made several suggestions for obtaining the marginal rates of substitution from the DM. Other variants were proposed by, for example, Oppenheimer [12], who presented an algorithm that used nonlinear preference functions as local proxies for the utility function, in lieu of the linearized approximation employed by the Geoffrion, Dyer and Feinberg (GDF) [5] procedure. Musselman and Talavage [11] developed a particularly significant variant, this procedure is called MT method. Loganathan and Sherali [10] presented an interactive cutting-plane algorithm for determining a best-compromise solution to a multiobjective optimization problem in situations with an implicitly defined utility function. Also, they introduced a convergence analysis which establishes that any accumulation point generated by the algorithm is a best-compromise solution and conducts an error analysis to point out the effect of inconsistencies in trade-off information provided by the decision maker. Ramazan [13] introduced a solution method for multiobjective linear programming problems, called interactive compromise programming, which is realised by using the method of compromise programming and the method of a two-person zero-sum game in an iterative way. Kassem [7], [9] introduced an interactive stability compromise programming by using the compromise weights from the concept of fuzzy set for determining the degree of closeness. Kassem [8] introduced the decomposition of the fuzzy parametric space in multiobjective nonlinear programming problems.

Recently, ElShafei [4] introduced an interactive stability compromise programming method for solving fuzzy multiobjective integer nonlinear programming problems by using the compromise weights from the pay-off table of membership function for each objective function. Abdelaziz and Krichen [2] introduced other method based on an aggregation of the individual expected utilities of the two DMs whenever a conflict occurs. A conflicting situation arises when the two DMs do not agree to choose the same decision. Kaisa and Marko [6] introduced an approach in creating interactive methods. This approach is based on the fact that classification and reference point based on scalarizing functions are closely related. In other words, the preference information obtained from the DM can be used to from different subproblems and to generate several Pareo optimal solutions. Abbas and Bellahcene [1] introduced a cutting plane technique for multiobjective stochastic integer linear programming that may be used to compute all the efficient solutions of the last model leaving the DM to choose a solution according to his preferences. Xu and Chen [17] introduced an interactive method for multiple attribute group decision making under fuzzy environment. The method can be used in exact numerical values or triangular fuzzy numbers, and the attribute values are triangular fuzzy numbers. The normalized expected decision matrices can preserve all the information that the original fuzzy decision matrices contain.

This paper presents the so-called trade-off compromise weights to obtain a new interactive cutting-plane algorithm for determining a best-compromise solution. This algorithm requires no line search; and generates iterates that are all included in the efficient solution. The solution to our problem is itself the next iterate which requires the DM to provide only the trade-off compromise weights in order to generate the augmented trade-off weighted cut. The augmented trade-off weighted cutting plane problems are attractive for generating efficient solutions and finally we determine the stability of these solutions.

Section snippets

Problem formulation

In general, a multiobjective nonlinear programming (MONLP) problem can be formulated as a vector maximum problem in the following form:maximizef1(x)=z1,maximizef2(x)=z2,maximizefm(x)=zm,subject toxX={xRn|gj(x)0,j=1,2,...,r},where f1,f2,,fm:RnR are m specified real-valued objective functions, X is a nonempty, compact subset of Rn; and each gj:RnR is a continuously differentiable and convex function for j=1,2,,r.

For some (unknown) implicit utility function we have the following problem:MOP

Eigenweight vectors for additive value functions

Here, we shall be interested in the following upper triangular matrix [18]:V=[vij]m×m,vij=cijifij0otherwise.The upper triangular matrix V enjoys a number of nice properties, including producing an interesting eigenweight vector.

Define the upper triangular matrixU=[uij]m×mwithuij=wijifij0otherwise.Let B be the diagonal matrixB=[bij]m×m,withbij=m-i+1i=j0otherwiseNote that B-1 is diagonal with component b˜ii=1m-i+1. From (7), (8), (9), (10), we see thatUw=Bw.Thus(B-1U-I)w=0,and the weight vector

Augmented trade-off weighted cutting plane

Musselman and Talavage [11] developed a particularly significant variant. This procedure (call it the MT method) operates as follows. Suppose that the objective functions are concave and continuously differentiable, and that X is nonempty interior, convex and compact and is given by {xRn|gj(x)0,j=1,2,,r}, where each gj:RnR is a continuously differentiable, convex function for j=1,2,,r. Corresponding to each iterate zl(f1(xl),,fm(xl)) generated in the objective function space, where xlX,

Stability set of the first kind

Here, we assume the problem (2) is stable, therefore the problem (19) is also stable [15].

Definition 1

Given a certain (w¯,ρ¯) with a corresponding optimal solution (x¯,η¯); then the stability set of the first kind of problem (19) corresponding to this optimal solution is defined byS(x¯,η¯)={(w,ρ)Rm+1:(x¯,η¯)is an optimal solution of problem(19)}.From the stability of problem (19), then there exists (w,ρ)Rm+1,wo,ρ>0,uRm,u0 and νRr,ν0 such that the Kuhn–Tucker conditions of problem (19) take the form:

Interactive algorithm (ISCPCP)

In this method, the solution process starts by solving the problem (2) to find the efficient solution under the given constraints.

The compromise trade-off weights of the objective functions are determined from the relation (3) and the matrix (14) and therefore employed in the system (**).

The steps of the algorithm can be summarized as follows:

  • Step 1.

    First, the DM selects one solution, x¯l for l = 1, from the efficient solution of problem (2).

  • Step 2.

    The trade-off compromise weights wij for ij can be found from

Illustrative numerical example

To illustrate the operation of the algorithm. The following simple numerical example is considered for the sake of illustration, we have assumed an a priori knowledge of the utility function. Let z1=f1(x1,x2)=-(x1-1),z2=f2(x1,x2)=-(x2-2). The problem is:maximizeU-z12-z22subject toxX={(x1,x2):x1+x28,x1+x220,x12,x23}.

  • Step 1.

    Let DM selects the efficient point x1=(3,5) as the starting solution. Hence, the corresponding point in the objective space is Z1=(-2,-3).

  • Step 2.

    From the relation (3), the trade-off

A practical application

The Practical application which can deal with an underground water problems is an open problem. The water resources problems are those related to underground water aquifers. Water, in general, is a vital resource to man, especially when we are talking about the problems of agriculture and irrigation, and the crucial need to develop new communities in developing countries.

Once the geological studies revealed the presence of underground water aquifer at a certain region, efficient management of

Conclusion and future work

We have presented in this paper an interactive stability cutting-plane algorithm to solve multiobjective nonlinear programming problems by using the trade-off compromise weights to generate the augmented trade-off weighted cutting-plane. This algorithm requires no line search and hence eliminates any associated subjective interactions with the DM, when η = 0. Also, we have studied the stability of solutions to the augmented trade-off weighted cutting-plane problems for obtaining the set of all

References (19)

There are more references available in the full text version of this article.

Cited by (0)

View full text