Milan Vlach

Satisficing solutions and duality in interval and fuzzy linear programming

In this paper, we introduce a class of fuzzy linear programming problems and define the concepts of feasible and satisficing solutions--the necessary tools for dealing with such problems. In this way, we show that the class of crisp (classical) LP problems can be embedded into the class of FLP ones. Moreover, for FLP problems we define the concept of duality and prove the weak and strong duality theorems. Further, we define a class of interval linear programming problems as a special subclass of FLP problems and apply the previous results to this special case.

Conditions for the existence of solutions of the three-dimensional planar transportation problem

Abstract A survey of conditions for the nonemptiness of the three planar sums transportation polytype is given together with several open problems.

Scheduling of parallel identical machines to maximize the weighted number of just-in-time jobs

We study the problem of nonpreemptively scheduling n jobs on m identical machines in parallel to maximize the weighted number of jobs that are completed exactly at their due dates. We show that this problem is solvable in polynomial time even if positive set-up times are allowed. Moreover, we show that if due date tolerances are permitted, then already the single-machine case is NP-hard even if all set-up times are zero and all weights are the same.

Generalized Concavity in Fuzzy Optimization and Decision Analysis

Preface. Acknowledgments. Part I: Theory. 1. Preliminaries. 2. Generalized Convex Sets. 3. Generalized Concave Functions. 4. Triangular Norms and T-Quasiconcave Functions. 5. Aggregation Operators. 6. Fuzzy Sets. Part II: Applications. 7. Fuzzy Multi-Criteria Decision Making. 8. Fuzzy Mathematical Programming. 9. Fuzzy Linear Programming. 10. Fuzzy Sequencing and Scheduling.

Linear coalitional games and their fuzzy extensions

The paper deals with fuzzifications of the coalitional game models in which the expected pay-offs of players and coalitions are known only vaguely. There exists a class of games, here we call them linear coalitional games, which can be modelled as games with, as well as without, side payments. In the deterministic case both types of models of a linear coalitional game lead to the same results. As the fuzzification of games with side-payments is based on other principles than the fuzzification of more general games without side-payments, the linear coalitional games offer situations in which both approaches to the fuzzification of coalitional game models can be compared and their differences can be clearly illustrated. In this paper the attention is focused to the analysis of concepts of superadditivity and core of coalitional games.

Fuzzy Mathematical Programming: A Unified Approach Based On Fuzzy Relations

Fuzzy mathematical programming problems (FMP) form a subclass of decision - making problems where preferences between alternatives are described by means of objective function(s) defined on the set of alternatives. The formulation a FMP problem associated with the classical MP problem is presented. Then the concept of a feasible solution and optimal solution of FMP problem are defined. These concepts are based on generalized equality and inequality fuzzy relations. Among others we show that the class of all MP problems with (crisp) parameters can be naturally embedded into the class of FMP problems with fuzzy parameters. We also show that the feasible and optimal solutions being fuzzy sets are convex under some mild assumptions.

Maximizing Weighted number of Just-in-Time Jobs on Unrelated Parallel Machines

We are concerned with problems of scheduling jobs non-preemptively with the objective to maximize the weighted number of jobs that are completed exactly at their due dates. It has been shown that the problems for single machine and identical parallel machines are polynomial time solvable. The purpose of this paper is to establish the complexity status of the problem for unrelated parallel machine, which was left open. First, we present a polynomial time algorithm for solving the problem when the number of machine is fixed. Second, we show that when the number of machine is a part of input, the problem becomes NP-hard in the strong sense.

Oblique fuzzy vectors and their use in possibilistic linear programming

In this paper, we propose oblique fuzzy vectors to treat the interactivity among fuzzy numbers. Oblique fuzzy vectors are extensions of fuzzy numbers and vectors of non-interactive fuzzy numbers. The interactivity among fuzzy numbers can be treated by a non-singular matrix in an oblique fuzzy vector. We discuss characterization of an oblique fuzzy vector and the tractability of manipulation of oblique fuzzy vectors in fuzzy linear functions. Moreover, we discuss possibilistic linear programming problems with oblique fuzzy vectors. It is shown that the possibilistic linear programming problems are reduced to linear programming problems with a special structure.

Pareto-optimality of compromise decisions

The questions of the existence of Pareto-optimal decisions and the existence of compromise decisions for decision problems in finite dimensional spaces are investigated. Several conditions for a compromise decision to be Pareto-optimal are derived. The concept of the compromise decision and some of the results are then generalized by allowing in the definition of the compromise solution arbitrary triangular norms instead of the triangular norm defined by the minimum operation only.

Nonpreemptive flowshop scheduling with machine dominance

Abstract Flowshop scheduling deals with processing a set of jobs through a set of machines, where all jobs have to pass among machines in the same order. With the exception of minimizing a makespan on two machines, almost all other flowshop problems in a general setup are known to be computationally intractable. In this paper we study special cases of flowshop defined by additional machine dominance constraints. These constraints impose certain relations among the job processing times on different machines and make the studied problems tractable.