Karel Zimmermann

Disjunctive optimization, max-separable problems and extremal algebras

The paper was motivated by solution methods suggested in the literature for solving linear optimization problems over (max,+)- or (max, min)-algebras and certain class of so called max-separable optimization problems. General features of these optimization problems, which play a crucial role in the optimization methods were used to formulate a general class of optimization problems with disjunctive constraints and a max-separable objective function and suggest a solution procedure for solving such problems. Linear problems over (max, +)-algebras and the max-separable problems are contained in this general class of optimization problems as special cases.

A strongly polynomial algorithm for solving two-sided linear systems in max-algebra

An algorithm for solving mxn systems of (max,+)-linear equations is presented. The systems have variables on both sides of the equations. After O(m^4n^4) iterations the algorithm either finds a solution of the system or finds out that no solution exists. Each iteration needs O(mn) operations so that the complexity of the presented algorithm is O(m^5n^5).

One class of separable optimization problems: solution method, application

A solution method for solving optimization problems with a max-separable objective function and min-separable inequality constraints is suggested. The method is based on the results of 5, 6. Application in some location problems and small illustrative numerical examples are presented.

Decision Making and Optimization

Well, someone can decide by themselves what they want to do and need to do but sometimes, that kind of person will need some decision making and optimization references. People with open minded will always try to seek for the new things and information from many sources. On the contrary, people with closed mind will always think that they can do it by their principals. So, what kind of person are you?

A note on a paper by E. Khorram and A. Ghodousian

Abstract The aim of this short note is to show on a numerical example that one of the two optimization algorithms proposed in [E. Khorram, A. Ghodousian, Linear objective function optimization with fuzzy relation equation constraints regarding max-av composition, Appl. Math. Comput. 173 (2006) 872–886] may not lead to the optimal solution in some cases. Besides it will be pointed out that the other algorithm (correct, but unfortunately ineffective for larger problems) proposed in [E. Khorram, A. Ghodousian, Linear objective function optimization with fuzzy relation equation constraints regarding max-av composition, Appl. Math. Comput. 173 (2006) 872–886] can be extended to a wider class of non-convex optimization problems.

Biobjective center – balance graph location model *

In the paper, a new algorithm for finding the absolute center of the graph is proposed. Later on problem of the absolute balance point of the graph is investigated the objective function of which is defined as the difference in the distance from the located facility to the farthest and the nearest demand point. An algorithm for finding it is presented By following the balance criterion it may happen, that the balance point is located too far from the demand points. Conversely, if we are interested not only in the distance to the farthest demand point, but (what is more realistic) also in the structure of the distances, a criterion of balance is appropriate. Consequently, we investigate the center balance constrained model that simultaneously used center and balance criteria. By considering the center and the balance objective in a multiobjective framework, efficient solutions are generated by minimizing former objective such that latter satisfies an upperbound

Duality of optimization problems with generalized fuzzy relation equation and inequality constraints

Abstract A generalization of fuzzy relation equations and inequalities is introduced. An explicit method for solving optimization problems, the feasible set of which is described by a system of generalized fuzzy relation equations and inequalities, is considered. The objective function of the problems is equal to the maximum of continuous increasing functions of one variable. A duality concept for such problems is proposed. The problems are considered in a finitely dimensional space.

FUZZY SET COVERING PROBLEM

The classical set covering problem is one of the well known NP-hard problems from discrete optimization. It consists of finding the cheapest covering of a finite set with a subsystem of a finite system of its subsets and has been investigated by many authors in various formulations. Here, two formulations are considered and corresponding results are presented.

GENERALIZED FUZZY RELATIONAL INEQUALITIES AND RELATED PROBLEMS

Various authors investigated fuzzy relational equalities constructed on the basic of the classical max-min extension principle (Czogala, E.. Drewniak. J. and Pedrycz. W. (1982) Fuzzy Sets and Systems, 7,89-101; Sanchez, E. (1976) Information andComrol, 30, 38-48). In this contribution, generalized fuzzy relational inequalities constructed by making use of compensatory operators or t-norms are studied (for the introduction of these concepts (Schwcizer and Sklar, A. (I960) Pacific J. Math., 10,313-334; (Clement, E.P.. Mesiar. R. and Pap, A. (1994) Preprint). Fuzzy relations considered here have a finite support. Solvability of generalized fuzzy relational inequalities is investigated.

An optimization problem on the image set of a (max, min) fuzzy operator

Abstract The paper deals with the unsolvability of (max, min) linear equation systems with coefficients in the unit interval (for fuzzy systems), or in an arbitrary real interval (general case). If the system has no solution, then the nearest vector to the right-hand side of the system for which the system is solvable is computed. A polynomial algorithm for the problem is presented. The method is illustrated by numerical examples.