Tamás Rapcsák

Singular value decomposition in AHP

Abstract The analytic hierarchy process (AHP) has been accepted as a leading multiattribute decision-aiding model both by practitioners and academics. The foundation of the AHP is the Saaty’s eigenvector method (EM) and associated inconsistency index that are based on the largest eigenvalue and associated eigenvector of an ( n × n ) positive reciprocal matrix. The elements of the matrix are the decision maker’s (DM) numerical estimates of the preference of n alternatives with respect to a criterion when they are compared pairwise using the 1–9 AHP fundamental comparison scale. The components of the normalized eigenvector provide approximations of the unknown weights of the criteria (alternatives), and the deviation of the largest eigenvector from n yields a measure of how inconsistent the DM is with respect to the pairwise comparisons. Singular value decomposition (SVD) is an important tool of matrix algebra that has been applied to a number of areas, e.g., principal component analysis, canonical correlation in statistics, the determination of the Moore–Penrose generalized inverse, and low rank approximation of matrices. In this paper, using the SVD and the theory of low rank approximation of a (pairwise comparison) matrix, we offer a new approach for determining the associated weights. We prove that the rank one left and right singular vectors, that is the vectors associated with the largest singular value, yield theoretically justified weights. We suggest that an inconsistency measure for these weights is the Frobenius norm of the difference between the original pairwise comparison matrix and one formed by the SVD determined weights. How this measure can be applied in practice as a means of measuring the confidence the DM should place in the SVD weights is still an open question. We illustrate the SVD approach and compare it to the EM for some numerical examples.

Geodesic convexity in nonlinear optimization

The properties of geodesic convex functions defined on a connected RiemannianC2k-manifold are investigated in order to extend some results of convex optimization problems to nonlinear ones, whose feasible region is given by equalities and by inequalities and is a subset of a nonlinear space.

On sensitivity analysis for a class of decision systems

The sensitivity analysis of general decision systems originated from the Bridgman model is dealt with. It is shown that these problems are equivalent to the optimization of linear fractional functions over rectangles. By using the specialities of decision problems, an O(n log n) algorithm is elaborated fro solving such problems. Computational experience is also given.

A note on synthesizing group decisions

Abstract An approach is discussed to expert group aggregation based on the Singular Value Decomposition (SVD) of Analytic Hierarchy Process (AHP) pairwise comparison matrices. The group decision phase should consist of the aggregation of the individual expert weight vectors determined by SVD, taking the voting powers of the experts and sensitivity analysis into account based on Bridgmans principle.

On minimization on Stiefel manifolds

Abstract The minimization of a smooth function f : R kn →R under the constraint that vectors x 1 , x 2 ,…, x k ∈R n , k ⩽ n , form an orthonormal system seems to be a new and interesting global optimization problem with important theoretical and practical applications. The set of feasible points determines a differentiable manifold introduced by Stiefel in 1935. Based on the nice geometric structure, the optimality conditions are obtained by the global Lagrange multiplier rule, and global optimality conditions based on local information, which make the advantages of using the Riemannian geometry in difficult smooth optimization problems clear.

On pseudolinear functions

Abstract In this paper the structure of a smooth pseudolinear function is investigated and the general form of the gradient is given explicitly.

Evaluation of tenders in information technology

Abstract Two case studies are described for evaluating tenders in information technology (IT) in public procurement process based on multiattribute group decision models and the software wingdss . The given decision methodology is further developed by discussing a method to aggregate scores measured on a scale and price. In 1997, 17 parallel tenders were handled together, 468 offers were evaluated by eight decision-makers, in two rounds, with respect to 18 and 5–8 criteria, respectively, in 5 days. In 1998, the same problem resulted in 168 offers, which were evaluated in one round by nine decision-makers with respect to 100–150 criteria, in 3 days. Since 1997, goods worth of 4–5 billion HUF (approximately 200–300 million US

R and D for group decision aid in Hungary by WINGDSS, a Microsoft Windows based group decision support system

) per year have been purchased within the frame of the contracts yielded by the tenders.

On convex generalized systems

Abstract A flexible and complex Group Decision Support System for PC-s in Microsoft Windows environment is presented. Several real-life applications have been carried out with WINGDSS. The Multi-Attribute Decision Aid type system reflects the logical structure of the problem as well as the preferences and the expertise of decision makers. We assume a heterogeneous group in a soft negotiation situation. Both factual data and subjective factors will be taken into account when qualifying the alternatives. Arrival at a group ranking satisfactory to all members is supported by a series of possibilities to use WINGDSS interactively, and by integrating feedbacks from individuals. The interface to relational databases, the tree handling ‘AROMA’ modul for problem structuring, the dialog box editor, the built-in interpreter for evaluating the alternatives ensure the applicability of WINGDSS in various decision situations. Experiences with real life decision problems are reported.

Some Global Optimization Problems on Stiefel Manifolds

In this paper, the set-convexity and mapping-convexity properties of the extended images of generalized systems are considered. By using these image properties and tools of topological linear spaces, separation schemes ensuring the impossibility of generalized systems are developed. Then, special problem classes are investigated.