Abstract
In this paper a class of combinatorial optimization problems is discussed. It is assumed that a feasible solution can be constructed in two stages. In the first stage the objective function costs are known while in the second stage they are uncertain and belong to an interval uncertainty set. In order to choose a solution, the minmax regret criterion is used. Some general properties of the problem are established and results for two particular problems, namely the shortest path and the selection problem, are shown.
Highlights
Consider the following deterministic single-stage combinatorial optimization problem: optP (c) = min cT x, (P )x ∈X where X ⊆ {0, 1}n is a set of feasible solutions and c ∈ Rn+ is a vector of nonnegative objective function costs
In the single-stage minmax regret version of P we seek a solution minimizing the maximum regret, i.e. we study the following problem: min max(cT x − optP (c))
We show that for this problem the maximum regret of a given first-stage solution can be computed in polynomial time and the optimal first-stage solution can by determined by using a compact mixed-integer programming (MIP) formulation
Summary
In the single-stage minmax regret version of P we seek a solution minimizing the maximum regret, i.e. we study the following problem: min max(cT x − optP (c)). In these papers the robust minmax criterion has been applied, i.e. a first-stage solution is determined minimizing the largest total first and second-stage cost. We proceed with the study of two variants of this problem, which have different computational properties We show that both computing an optimal first-stage solution and the maximum regret of a given first-stage solution are NP-hard. We show that for this problem the maximum regret of a given first-stage solution can be computed in polynomial time and the optimal first-stage solution can by determined by using a compact MIP formulation.
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