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

Abstract

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.

Criticality analysis of activity networks under interval uncertainty

“…In particular, the algorithms for evaluating the necessary criticality and for computing the maximal float of an activity (arc) are useful in a preprocessing of an instance of the problem, since a necessarily critical activity (arc) can be automatically added to a constructed min-max regret path. In consequence, one can decompose an instance with a necessarily critical arc into two separate instances in smaller subgraphs [29]. An activity (arc) with a small maximal float can be considered as a candidate for belonging to a min-max regret path.…”

“…Hence, an optimal min-max regret path can be viewed as a possibly critical one, which minimizes a "distance" to the necessary criticality. Furthermore, necessarily non-critical tasks can be removed from an instance of the minmax regret longest path without violating optimal min-max regret paths and necessarily critical tasks can be automatically added to a constructed optimal min-max regret path (see [29] for a deep discussion concerning the above relations).…”

Multidimensional Knapsack Problem Based on Uncertain Measure

“…For this condition, the usual way is to obtain the uncertain data by means of experience evaluation or expert advice. It should be noted that there are several researchers who characterize the relevant parameters as fuzzy variables, such as Okada and Gen [13], Kasperski and Kulej [18], and Kasperski and Zieli nski [32]. However, as pointed out in Section 2, the fuzzy set theory is not a suitable tool to model the imprecise weights, imprecise values, and imprecise capacities in the multidimensional knapsack problem when the relevant parameters are given by domain experts.…”

“…In addition, the current work can be extended to the uncertain random environment [34,35] where uncertainty and randomness coexist in a system. What's more, the situation in which the decision-makers adopt the minimax regret criterion [32,36] to make the decisions in the uncertain multidimensional knapsack problem might be considered. While these issues beyond the scope of the present study, we believe they can be potential avenues for future studies.…”

Mixed uncertainty sets for robust combinatorial optimization

“…In this paper we consider combinatorial problems of the form min x x x∈X c c cx x x (P) with X ⊆ {0, 1} n and uncertain cost vector c c c. To find a solution x x x that still performs well under the possible cost realizations, different approaches have been proposed. These include fuzzy optimization [KZ10], stochastic programming [BL11], or robust optimization [GS16,GMT14].…”