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...osed in the literature for the PDPTW. Exact algorithms for the PDPTW have been proposed by Dumas et al. [54], Savelsbergh and Sol [112], Lu and Dessouky [84], Ropke et al. [107] and Ropke and Cordeau =-=[106]-=-. The method of Dumas et al. [54] is the first branch-and-price algorithm proposed in the literature for the PDPTW. This algorithm is based on a set partitioning formulation of the problem where each ...

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...3 of the optimal solution without transfers. If z(PDPT ) denotes the optimal solution of the problem with transfers and |R| the number of requests, then z(PDPT ) > z(PDP ) 6d √ |R|e+1 . Masson et al. =-=[26]-=- develop a Branch-and-Cut-and-Price for a special case of the PDPT, called the Pickup and Delivery Problem with Shuttles routes, where the set of distinct delivery locations is limited. In this proble...

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... problem as an integer program, and solved instances with up to 40 requests, also by dynamic programming. More recently, Cordeau [5] presented some valid inequalities and a branch-and-cut algorithm for the multi-vehicle DARP. The author solved instances involving up to 32 requests. Ropke et al. [22] later presented stronger formulations and new valid inequalities for the DARP and the Pickup and Delivery Problem with Time Windows (PDPTW), which can be viewed as a DARP without ride time constraints. They solved instances with up to 96 requests using a branch-and-cut algorithm. Ropke and Cordeau [21] then proposed a branchand-cut-and-price method for the PDPTW. This algorithm uses some of the inequalities introduced by Ropke et al. [22] within a column generation framework and it could solve some tightly constrained instances with up to 500 requests. Very recently, Bartolini [1] also formulated the PDPTW as a set partitioning problem with additional cuts. He proposed an exact algorithm for the problem, using both relaxations of the formulation and a branchand-cut-and-price algorithm. His method provided better results than that of Ropke and Cordeau [21] in terms of lower bound quality and...

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...ted annealing 15 (SA) [93, 154], tabu search [73, 70, 71], variable neighbourhood search (VNS) [113, 78], large neighbourhood search (LNS) [142] and finally adaptive large neighbourhood search (ALNS) =-=[134, 135, 121]-=-. In simulated annealing, the heuristic considers some neighbouring state xt+1 of the current state xt at each iteration, and probabilistically decides between moving the system to state xt+1 or stayi...