János Fülöp

On optimal completion of incomplete pairwise comparison matrices

An important variant of a key problem for multi-attribute decision making is considered. We study the extension of the pairwise comparison matrix to the case when only partial information is available: for some pairs no comparison is given. It is natural to define the inconsistency of a partially filled matrix as the inconsistency of its best, completely filled completion. Here we study the uniqueness problem of the best completion for two weighting methods, the Eigenvector Method and the Logarithmic Least Squares Method. In both settings we obtain the same simple graph theoretic characterization of the uniqueness. The optimal completion will be unique if and only if the graph associated with the partially defined matrix is connected. Some numerical examples are discussed at the end of the paper.

Game theoretic models for climate change negotiations

Abstract Negotiations to reduce greenhouse gas accumulation in the atmosphere are modeled as extensive games of perfect information. Various solution concepts, such as Nash equilibrium, reaction function equilibrium, correlated equilibrium and bargaining solutions are applied, analyzed and computed. Special reduction techniques are used when the size of the game tree becomes excessive. A new solution concept, the tree-correlated equilibrium is also introduced. Main features of an Excel add-in designed to compute various solutions are briefly described and a sample policy analysis for a special negotiating scenario is discussed.

A cutting plane algorithm for linear optimization over the efficient set

In this paper, we consider the problem of optimizing a linear function over the efficient set of a multiple objective linear programming problem. The problem is formulated as a linear program with a special reverse convex constraint. We propose a finite method using simplex steps on adjacent efficient extreme points, and convexity and disjunctive cuts.

On pairwise comparison matrices that can be made consistent by the modification of a few elements

Pairwise comparison matrices are often used in Multi-attribute Decision Making for weighting the attributes or for the evaluation of the alternatives with respect to a criteria. Matrices provided by the decision makers are rarely consistent and it is important to index the degree of inconsistency. In the paper, the minimal number of matrix elements by the modification of which the pairwise comparison matrix can be made consistent is examined. From practical point of view, the modification of 1, 2, or, for larger matrices, 3 elements seems to be relevant. These cases are characterized by using the graph representation of the matrices. Empirical examples illustrate that pairwise comparison matrices that can be made consistent by the modification of a few elements are present in the applications.

A method for approximating pairwise comparison matrices by consistent matrices

In several methods of multiattribute decision making, pairwise comparison matrices are applied to derive implicit weights for a given set of decision alternatives. A class of the approaches is based on the approximation of the pairwise comparison matrix by a consistent matrix. In the paper this approximation problem is considered in the least-squares sense. In general, the problem is nonconvex and difficult to solve, since it may have several local optima. In the paper the classic logarithmic transformation is applied and the problem is transcribed into the form of a separable programming problem based on a univariate function with special properties. We give sufficient conditions of the convexity of the objective function over the feasible set. If such a sufficient condition holds, the global optimum of the original problem can be obtained by local search, as well. For the general case, we propose a branch-and-bound method. Computational experiments are also presented.

A finite cutting plane method for solving linear programs with an additional reverse convex constraint

Abstract The paper deals with linear programs with an additional reverse convex constraint. If the feasible region of this problem is nonempty and the objective function to be minimized is bounded below on it, then the problem has a finite minimum obtained on an at most one-dimensional face of the polyhedron as well. This paper presents a finite cutting plane method using convexity and disjunctive cuts for solving the considered problem. The method is based upon a procedure which, for a given nonnegative integer q, either finds such an at most q-dimensional face of the original polyhedron which has a point feasible to the cuts generated previously or proves that there exists no such a face. Computational experience is also provided.

Branch-and-bound variant of an outcome-based algorithm for optimizing over the efficient set of a bicriteria linear programming problem

The paper presents a finite branch-and-bound variant of an outcome-based algorithm proposed by Benson and Lee for minimizing a lower-semicontinuous function over the efficient set of a bicriteria linear programming problem. Similarly to the Benson-Lee algorithm, we work primarily in the outcome space. Dissimilarly, instead of constructing a sequence of consecutive efficient edges in the outcome space, we use the idea of generating a refining sequence of partitions covering the at most two-dimensional efficient set in the outcome space. Computational experience is also presented.

On reducing inconsistency of pairwise comparison matrices below an acceptance threshold

A recent work of the authors on the analysis of pairwise comparison matrices that can be made consistent by the modification of a few elements is continued and extended. Inconsistency indices are defined for indicating the overall quality of a pairwise comparison matrix. It is expected that serious contradictions in the matrix imply high inconsistency and vice versa. However, in the 35-year history of the applications of pairwise comparison matrices, only one of the indices, namely

A finite procedure to generate feasible points for the extreme point mathematical programming problem

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