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Shortest Path Problems with Resource Constraints

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In most vehicle routing and crew scheduling applications solved by column generation, the subproblem corresponds to a shortest path problem with resource constraints (SPPRC) or one of its variants. This chapter proposes a classification and a generic formulation for the SPPRCs, briefly discusses complex modeling issues involving resources, and presents the most commonly used SPPRC solution methods. First and foremost, it provides a comprehensive survey on the subject.
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... They consider set partitioning formulations where columns are possible pairings. These methods hide the non-linearity in the pricing subproblem, which can be efficiently solved using resource constrained shortest path approaches [24]. As a large part of the working rules apply to duties, i.e., subsequences of a pairing formed by the flight legs operated on a ...
... We now describe an enumeration algorithm for the Monoid Resource Constrained Shortest Path Problem. It follows the standard labeling scheme [24] for resource constrained shortest paths. The specificity of our algorithm is that it uses, for each v in V , a set B v of bounds such that, ...
... Not only these bounds are used to discard more paths, but they are also used to improve the order in which the paths are considered by the algorithm. The two main resource constrained shortest path algorithms in the literature [24] differ by the order in which they consider paths. The label correcting algorithm is obtained from our one by using c(q P ) as key(P ) and removing the test of Step 14. ...
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Aircraft routing and crew pairing problems aims at building the sequences of flight legs operated respectively by airplanes and by crews of an airline. Given their impact on airlines operating costs, both have been extensively studied for decades. Our goal is to provide reliable and easy to maintain frameworks for both problems at Air France. We propose simple approaches to deal with Air France current setting. For routing, we introduce a compact MIP formulation that can be solved by current MIP solvers in at most a few minutes even on Air France largest instances. Regarding crew pairing, we provide a standard methodology to model the column generation pricing subproblem within a new resource constrained shortest path framework recently introduced by the first author. This new framework, which can be used as a black-box, leverages on bounds on paths' resource to discard partial solutions and speed-up the resolution. The resulting approach enables to solve to optimality Air France largest instances. Recent literature has focused on integrating aircraft routing and crew pairing problems. As a side result, we have been able to solve to near optimality large industrial instances of the integrated problem by combining the aforementioned algorithms within a simple cut generating method.
... The authors developed three heuristics for the problem that were tested on randomly generated and real data. The first exact approach for the m-CTP was proposed by Lopes et al. (2013), that is a branch-and-price (BP) algorithm, in which the pricing subproblem reduced to the elementary shortest path problem with resource constraints (ESPPRC), Irnich and Desaulniers (2006), and solved by a dynamic programming algorithm. The authors also proposed a column generation heuristic based on a set partitioning formulation. ...
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... Our proposed heuristic ARASP reduces the problem of SFC placement to the Resource Constrained Shortest Path (RCSP) problem [38]. When placing an SFC the basic idea is to search for the paths with the highest availability between s i and d i which also adhere to the end-to-end latency requirement and try to map VNFs on them recursively. ...
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