Adam Kasperski

An approximation algorithm for interval data minmax regret combinatorial optimization problems

The general problem of minimizing the maximal regret in combinatorial optimization problems with interval data is considered. In many cases, the minmax regret versions of the classical, polynomially solvable, combinatorial optimization problems become NP-hard and no approximation algorithms for them have been known. Our main result is a polynomial time approximation algorithm with a performance ratio of 2 for this class of problems.

Minimizing maximal regret in the single machine sequencing problem with maximum lateness criterion

In this paper, the single machine sequencing problem with maximum lateness criterion is discussed. The parameters of the problem are imprecise and they are specified as intervals. The maximal regret criterion is applied to calculate the optimal sequence. A polynomial algorithm for the studied problem is constructed.

Minimizing maximum lateness in a single machine scheduling problem with fuzzy processing times and fuzzy due dates

Abstract A single machine scheduling problem with fuzzy processing times and fuzzy due dates is considered. Two approaches to this problem are investigated. In the first one it is assumed that for each job J i , i=1,…,n , there is given a real valued function fi of its fuzzy completion time and its fuzzy due date. The objective is to minimize the maximum value among fi values. It is shown that if the cost functions fi are F-monotone with respect to fuzzy completion time (the concept of F-monotonicity is defined in the paper), the generalized Lawlers algorithm can be used for solving the problem. Four problems, being special cases of the above general problem, are also considered. In the second approach, called a goal approach, each sequence of jobs is evaluated by fuzzy maximum of weighted lateness and the objective is to maximize the degree of possibility that this value is not greater than a certain fuzzy number (fuzzy goal), provided by a decision maker. The efficient solution method for this problem is proposed, in which again the Lawlers algorithm is essentially used.

On two single machine scheduling problems with fuzzy processing times and fuzzy due dates

Abstract Two single machine scheduling problems with fuzzy processing times and fuzzy due dates are considered. In both we define the fuzzy tardiness of a job in a given sequence as a fuzzy maximum of zero and the difference between the fuzzy completion time and the fuzzy due date of this job. In the first problem we minimize the maximal expected value of a fuzzy tardiness and in the second we minimize the expected value of a maximal fuzzy tardiness. The problems look similar but we show that they have quite different computational complexity. The first problem can be solved by a polynomial algorithm if we only can calculate easily the fuzzy tardiness. We propose a such algorithm assuming that all processing times and all due dates are fuzzy numbers of the L–R type with power shape functions. We prove that the second problem is NP-hard.

The robust shortest path problem in series-parallel multidigraphs with interval data

In this paper the robust shortest path problem in edge series-parallel multidigraphs with interval costs is examined. The maximal regret criterion is applied to calculate the optimal solution. It is shown that this problem is NP-hard. A pseudopolynomial algorithm for the studied problem is constructed.

A 2-approximation algorithm for interval data minmax regret sequencing problems with the total flow time criterion

In this paper we discuss a minmax regret version of the single-machine scheduling problem with the total flow time criterion. Uncertain processing times are modeled by closed intervals. We show that if the deterministic problem is polynomially solvable, then its minmax regret version is approximable within 2.

On combinatorial optimization problems on matroids with uncertain weights

In this paper the combinatorial optimization problem on weighted matroid is considered. It is assumed that the weights in the problem are ill-known and they are modeled as fuzzy intervals. The optimality of solutions and the optimality of elements are characterized. This characterization is performed in the setting of possibility theory. A method of choosing a solution under uncertainty is also proposed.

On the approximability of minmax (regret) network optimization problems

In this paper the minmax (regret) versions of some basic polynomially solvable deterministic network problems are discussed. It is shown that if the number of scenarios is unbounded, then the problems under consideration are not approximable within log^1^-^@eK for any @e>0 unless NP@?DTIME(n^p^o^l^y^l^o^g^n), where K is the number of scenarios.

Approximating a two-machine flow shop scheduling under discrete scenario uncertainty

This paper deals with the two machine permutation flow shop problem with uncertain data, whose deterministic counterpart is known to be polynomially solvable. In this paper, it is assumed that job processing times are uncertain and they are specified as a discrete scenario set. For this uncertainty representation, the min–max and min–max regret criteria are adopted. The min–max regret version of the problem is known to be weakly NP-hard even for two processing time scenarios. In this paper, it is shown that the min–max and min–max regret versions of the problem are strongly NP-hard even for two scenarios. Furthermore, the min–max version admits a polynomial time approximation scheme if the number of scenarios is constant and it is approximable with performance ratio of 2 and not (4/3−ϵ)-approximable for any ϵ>0 unless P=NP if the number of scenarios is a part of the input. On the other hand, the min–max regret version is not at all approximable even for two scenarios.

Minmax regret approach and optimality evaluation in combinatorial optimization problems with interval and fuzzy weights

This paper deals with a general combinatorial optimization problem in which closed intervals and fuzzy intervals model uncertain element weights. The notion of a deviation interval is introduced, which allows us to characterize the optimality and the robustness of solutions and elements. The problem of computing deviation intervals is addressed and some new complexity results in this field are provided. Possibility theory is then applied to generalize a deviation interval and a solution concept to fuzzy ones.