Abstract
In this paper, the two-stage orienteering problem with stochastic weights is studied, where the first-stage problem is to plan a path under the uncertain environment and the second-stage problem is a recourse action to make sure that the length constraint is satisfied after the uncertainty is realized. First, we explain the recourse model proposed by Evers et al. (2014) and point out that this model is very complex. Then, we introduce a new recourse model which is much simpler with less variables and less constraints. Based on these two recourse models, we introduce two different two-stage robust models for the orienteering problem with stochastic weights. We theoretically prove that the two-stage robust models are equivalent to their corresponding static robust models under the box uncertainty set, which indicates that the two-stage robust models can be solved by using common mathematical programming solvers (e.g., IBM CPLEX optimizer). Furthermore, we prove that the two two-stage robust models are equivalent to each other even though they are based on different recourse models, which indicates that we can use a much simpler model instead of a complex model for practical use. A case study is presented by comparing the two-stage robust models with a one-stage robust model for the orienteering problem with stochastic weights. The numerical results of the comparative studies show the effectiveness and superiority of the proposed two-stage robust models for dealing with the two-stage orienteering problem with stochastic weights.
Highlights
In this paper, the two-stage orienteering problem with stochastic weights is studied, where the first-stage problem is to plan a path under the uncertain environment and the second-stage problem is a recourse action to make sure that the length constraint is satisfied after the uncertainty is realized
We theoretically prove that the two-stage robust models are equivalent to their corresponding static robust models under the box uncertainty set, which indicates that the two-stage robust models can be solved by using common mathematical programming solvers (e.g., IBM CPLEX optimizer)
With the box uncertainty set defined, the two-stage robust models are equivalent to their corresponding static robust models, and the two two-stage robust models are equivalent to each other even though they are based on different recourse models. ese conclusions we obtained indicate that the two-stage robust models for OP with stochastic weights (OPSW) can be solved to optimality by solving their corresponding static robust models
Summary
The formal definition and the mathematical model of the deterministic orienteering problem (OP) are introduced. Let xij be a binary decision variable, where xij 1 if and only if arc (i, j) is visited by the path; otherwise, xij 0. An auxiliary variable ui is used to denote the position of node i in the path. E formulation of the deterministic OP is as follows: DOP: maximize xij, i∈N j∈N+\{i}. Constraint (1b) is the path length constraint. Constraint (1d) is the flow conservation constraint ensuring that a vertex is visited at most once. Constraint (1e) ensures the connectivity of the path. Constraints (1f ) and (1g) are the boundary and integrality constraints on the auxiliary variables and decision variables, respectively
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