Tetsuzo Tanino

Fuzzy preference orderings in group decision making

Abstract In this paper, some use of fuzzy preference orderings in group decision making is discussed. First, fuzzy preference orderings are defined as fuzzy binary relations satisfying reciprocity and max-min transitivity. Then, particularly in the case where individual preferences are represented by utility functions (utility values), group fuzzy preference orderings of which fuzziness is caused by differences or diversity of individual opinions are defined. Those orderings might be useful for proceeding the group decision making process smoothly, in the same manner as the extended contributive rule method.

Fuzzy Preference Relations in Group Decision Making

This paper surveys how fuzzy preference relations are utilized in group decision making. First, several fundamental properties of fuzzy preference relations are reviewed. Secondly, some procedures for deriving a solution (group choice) from individual fuzzy preference relations are discussed. They can be divided into two classes: direct approach and indirect approach. Finally, a systems analytic approach which aims to generate a practically meaningful group fuzzy preference relation which reflects differences or diversity of individual preferences is discussed.

Stability and sensitivity analysis in convex vector optimization

In this paper we provide some theoretical results on stability and sensitivity analysis in convex vector optimization. Given a family of parametrized vector optimization problems, the perturbation maps are defined as the set-valued map which associates to each parameter value the set of minimal points (properly minimal points, weakly minimal points) of the perturbed feasible set with respect to an ordering convex cone. Sufficient conditions for the upper and lower semicontinuity of the perturbations map are obtained. We also provide quantitative properties of the perturbation maps under some convexity assumptions.

An algorithm for solving two-level convex optimization problems

This paper provides an algorithm for solving two-level convex optimization problems. The algorithm is based on the subgradient formula for the upper level objective function which is not generally ...

Saddle points and duality in multi-objective programming

This paper deals with saddle points of vector-valued functions and duality in multi-objective programming. First, saddle points for vector-valued functions are defined as a generalization of those for scalar-valued functions. Some properties of them are derived. Secondly, it. is shown that multi-objective programming problems are closely connected with vector-valued lagrangians. Solutions to multi-objective programming problems and corresponding multiplier vectors are saddle points of vector-valued lagrangians. Multiplier vectors are also sub-gradients for perturbation maps. Finally, there exists a corresponding multiplier vector, which is a solution to the dual problem, for each solution to some class of primal problems (stable problems).

Maintenance policies in markovian deterioration models of parallel unit systems

Abstract Markovian deterioration models of parallel unit systems are dealt with. Units of a system are supposed to be of different types, and a state of the system is represented by a vector whose component corresponds to the state of each unit. Therefore all the states of the system are only partially ordered, while they are totally ordered in ordinary markovian deterioration models. It is proved under some conditions that the optimal maintenance policy is a partial control limit policy, which is a generalization of the well-known control limit policy in the ordinary case. An algorithm improving the successive approximation method is proposed for obtaining the optimal policy. A typical case of the model is also discussed.

CONJUGATE MAPS AND CONJUGATE DUALITY

This paper considers conjugate duality in multiobjective optimization, in which minimality (efficiency, noninferiority, or Pareto optimality) is a natural solution concept. First, conjugate maps and subgradients are defined for vector-valued functions and point-to-set maps. Embedding the primal problem into a family of perturbed problems enables us to define a dual problem in terms of the conjugate map of the perturbed objective function. Every solution to the stable primal problem is associated with a solution to the dual problem, which is characterized as a subgradient of the perturbation map. This pair of solutions is also a saddle point of the Lagrangian.

On Optimization Methods for Two-Level Hierarchical Systems

In this paper, optimization methods for hierarchical systems are dealt with. We consider a problem in which the upper level objective function includes the lower level optimal value function.First, we review several useful results in sensitivity analysis which are already obtained by many resarchers. Moreover we point out that information on sensitivity analysis for optimal value function (i.e. gradient, generalized gradient or directional derivative) depends on the uniqueness of the optimal solution and of the Lagrange multiplier vector.Next, under the uniqueness of optimal solution and Cottle constraint qualification, we consider a problem for finding the upper level descent direction. This problem is formulated as a quadratic programming problem. Therefore, we may apply a proper solution method to the problem to obtain the upper level descent direction.Finally, we construct an optimization algorithm for the two-level hierarchical system based on the above results.

Optimization of Hierarchical Systems with Multiple Objectives

Optimization problems in the real world often have hierarchical structures and multiple objectives. For example, in a regional investment allocation problem, cities of subregions are lower level decision makers (DMs) and a prefecture consisting of them is an upper level DM. Moreover, each DM must take into account several objectives (criteria) such as economy, welfare, environment and so on. (Kobayashi et al., 1987).

Some Basic Theoretical Results in Multiobjective Optimization

This paper overview some basic theoretical results in multiobjective optimization (vector optimization). The topics dealt with are nonemptiness, external stability and perturbation stability of the efficient set, and proper efficiency. First, some sufficient conditions are obtained for the existence of efficient solutions and for the external stability of the efficient set. Secondly, several definitions of proper efficiency are overviewed along with their relationships. Finally, the stability of the efficient set is studied with respect to the perturbation of the feasible set and the domination cone.