Waldemar Kołodziejczyk

A fuzzy approach to the transportation problem

Abstract The transportation problem with fuzzy supply values of the deliverers and with fuzzy demand values of the receivers is analysed. For the solution of the problem the technique of parametric programming is used. This makes it possible to obtain not only the maximizing solution (according to the Bellman-Zadeh criterion) but also other alternatives close to the optimal solution.

Maximum flow in a network with fuzzy arc capacities

Abstract Our main concern is the maximum flow in a network in which an excess over the beforehand fixed quota of arc capacity is admissible. The problem is represented as a partially fuzzy linear programming task. A theorem equivalent to the Ford and Fulkerson one concerning the classic task of maximum flow is proved in the paper. An algorithm for searching maximum flow assuming integer values of flows on network arcs is presented.

Real-valued flows in a network with fuzzy arc capacities

Abstract The problem of the maximal (minimal) flow in a network with fuzzy capacity constraints is considered. A theorem being a natural direct generalization of the Ford-Fulkerson theorem relating the flow value with cut capacities in the network is proved. A simple algorithm, based on the mentioned theorem, for the determination of optimal real-valued flows is developed.

Integer flows in network with fuzzy capacity constraints

A generalization of the maximum flow problem in a network with one-sided (upper) and two sided constraints on arc capacities is presented. It allows for violating the capacity constraints in some ranges of tolerance. To solve the generalized problem the apparatus of the fuzzy set theory is used. The efficient algorithms are developed for integer flows.

Convergence of powers of s-transitive fuzzy matrices

Abstract Natural powers, via max-min multiplication, of s-transitive fuzzy matrices are considered. It is shown that they oscillate with period equal to 2. This is an extension of the convergence property concerning max-min transitive fuzzy matrices.

Canonical form of a strongly transitive fuzzy matrix

Abstract Several properties of so called strongly transitive fuzzy matrices are developed. The canonical form of the strongly transitive fuzzy matrix is formulated and constructed. The results presented generalize analogous results concerning max-min transitive fuzzy matrices known in the literature.

Convergence of powers of reciprocal fuzzy matrices

Abstract Convergence properties for reciprocal fuzzy matrices are established. Exemplary applications are also discussed.

Transitive Solutions of Relational Equations on Finite Sets and Linear Lattices

The set of solutions of relational equations over a finite referential space and with values from a linear lattice is considered. We determine in this set the greatest max-min transitive solution and the related minimal ones. Further, we investigate for the determination of particular max-min transitive solutions, namely those having Schein rank equal to 1. Related properties of convergence of fuzzy systems represented by the involved relations are also given.

DECOMPOSITION PROBLEM OF FUZZY RELATIONS—FURTHER RESULTS

Abstract In the paper three classes of decomposable fuzzy relations defined in the Cartesian product of a Finite space are considered. It is shown that powers of decomposable fuzzy relations are convergent. Relationships between decomposable and transitive fuzzy relations are analysed and decomposition problem of transitive fuzzy relations is considered.

On equivalence of two optimization methods for fuzzy discrete programming problems

Abstract A class of fuzzy discrete programming problems, problems with alternatives (feasible solutions) being integer linear combinations of fuzzy parameters of the problem, is considered. Two methods for solving such problems are taken into account. The first method consists in the use of classical algorithms for a respective classical problem with parameters being values of a given ranking function described on the set of fuzzy parameters of a considered fuzzy problem. According to the second approach nondominated solutions in the sense of a given fuzzy relation are determined. Conditions for the equivalence of the two mentioned above approaches are developed. They are next examined with respect to several known fuzzy preference relations.