text: This paper presents a parallel memetic algorithm for solving the vehicle routing problem with time windows (VRPTW). The VRPTW is a well-known NP-hard discrete optimization problem with two objectives. The main objective is to minimize the number of vehicles serving customers scattered on the map, and the second one is to minimize the total distance traveled by the vehicles. Here, the fleet size is minimized in the first phase of the proposed method using the parallel heuristic algorithm (PHA), and the traveled distance is minimized in the second phase by the parallel memetic algorithm (PMA). In both parallel algorithms, the parallel components co-operate periodically in order to exchange the best solutions found so far. An extensive experimental study performed on the Gehring and Homberger's benchmark proves the high convergence capabilities and robustness of both PHA and PMA. Also, we present the speedup analysis of the PMA.
  @cite_22: The usual column generation model for a Vehicle Routing Problem involves an elementary shortest-path subproblem. The worst-case complexity of the known algorithms for this problem being too high, the elementary-path constraint is usually relaxed. Indeed, as each customer must be visited exactly once, the two problems with and without the elementary-path constraint have the same optimal integer solutions. In this article, we propose one theoretical and several practical improvements to the algorithm for elementary paths. We obtain better lower bounds and pruning of the search tree, and these improvements allowed us to find an exact solution to 17 instances of the Solomon benchmark suite which were previously open
  @cite_7: This paper addresses the problem of finding the minimum number of vehicles required to visit a set of nodes subject to time window and capacity constraints. The fleet is homogeneous and is located at a common depot. Each node requires the same type of service. An exact method is introduced based on branch and cut. In the computations, ever increasing lower bounds on the optimal solution are obtained by solving a series of relaxed problems that incorporate newly found valid inequalities. Feasible solutions or upper bounds are obtained with the help of greedy randomized adaptive search procedure (GRASP). A wide variety of cuts is introduced to tighten the linear programming (LP) relaxation of the original mixed-integer program. To find violated cuts, it is necessary to solve a separation problem. A substantial portion of the paper is aimed at describing the heuristics developed for this purpose. A new approach for obtaining feasible solutions from the LP relaxation is also discussed. Numerical results for standard 50- and 100-node benchmark problems are reported.
  @cite_33: 
  @cite_6: In this paper we review the exact algorithms proposed in the last three decades for the solution of the vehicle routing problem with time windows (VRPTW). The exact algorithms for the VRPTW are in many aspects inherited from work on the traveling salesman problem (TSP). In recognition of this fact this paper is structured relative to four seminal papers concerning the formulation and exact solution of the TSP, i.e. the arc formulation, the arc-node formulation, the spanning tree formulation, and the path formulation. We give a detailed analysis of the formulations of the VRPTW and a review of the literature related to the different formulations. There are two main lines of development in relation to the exact algorithms for the VRPTW. One is concerned with the general decomposition approach and the solution to certain dual problems associated with the VRPTW. Another more recent direction is concerned with the analysis of the polyhedral structure of the VRPTW. We conclude by examining possible future lines of research in the area of the VRPTW.
  @cite_19: This paper considers the vehicle routing problem with time windows, where the service of each customer must start within a specified time interval. We consider the Lagrangian relaxation of the constraint set requiring that each customer must be served by exactly one vehicle yielding a constrained shortest path subproblem. We present a stabilized cutting-plane algorithm within the framework of linear programming for solving the associated Lagrangian dual problem. This algorithm creates easier constrained shortest path subproblems because less negative cycles are introduced and it leads to faster multiplier convergence due to a stabilization of the dual variables. We have embedded the stabilized cutting-plane algorithm in a branch-and-bound search and introduce strong valid inequalities at the master problem level by Lagrangian relaxation. The result is a Lagrangian branch-and-cut-and-price (LBCP) algorithm for the VRPTW. Making use of this acceleration strategy at the master problem level gives a significant speed-up compared to algorithms in the literature based on traditional column generation. We have solved two test problems introduced in 2001 by Gehring and Homberger with 400 and 1000 customers respectively, which to date are the largest problems ever solved to optimality. We have implemented the LBCP algorithm using the ABACUS open-source framework for solving mixed-integer linear-programs by branch, cut, and price.
  @cite_2: In this paper, we present an effective memetic algorithm for the vehicle routing problem with time windows (VRPTW). The paper builds upon an existing edge assembly crossover (EAX) developed for the capacitated VRP. The adjustments of the EAX operator and the introduction of a novel penalty function to eliminate violations of the time window constraint as well as the capacity constraint from offspring solutions generated by the EAX operator have proven essential to the heuristic's performance. Experimental results on Solomon's and Gehring and Homberger benchmarks demonstrate that our algorithm outperforms previous approaches and is able to improve 184 best-known solutions out of 356 instances.
  @cite_25: 
  @cite_12: The elementary shortest-path problem with resource constraints (ESPPRC) is a widely used modeling tool in formulating vehicle-routing and crew-scheduling applications. The ESPPRC often occurs as a subproblem of an enclosing problem, where it is used to generate implicitly the set of all feasible routes or schedules, as in the column-generation formulation of the vehicle-routing problem with time windows (VRPTW). As the ESPPRC problem is NP-hard in the strong sense, classical solution approaches are based on the corresponding nonelementary shortest-path problem with resource constraints (SPPRC), which can be solved using a pseudo-polynomial labeling algorithm. While solving the enclosing problem by branch and price, this subproblem relaxation leads to weak lower bounds and sometimes impractically large branch-and-bound trees. A compromise between solving ESPPRC and SPPRC is to forbid cycles of small length. In the SPPRC with k-cycle elimination (SPPRC-k-cyc), paths with cycles are allowed only if cycles have length at least k + 1. The case k = 2 forbids sequences of the form i - j - i and has been successfully used to reduce integrality gaps. We propose a new definition of the dominance rule among labels for dealing with arbitrary values of k ≥ 2. The numerical experiments on the linear relaxation of some hard VRPTW instances from Solomon's benchmark show that k-cycle elimination with k ≥ 3 can substantially improve the lower bounds of vehicle-routing problems with side constraints. The new algorithm has proven to be a key ingredient for getting exact integer solutions for well-known hard problems from the literature.
summary: Exact algorithms incorporating, among others, dynamic programming, branch-and-bound algorithms and greedy approaches, were proposed by @cite_7 , Irnich and Villeneuve @cite_12 , @cite_25 , @cite_19 and Chabrier @cite_22 . Noteworthy, only 9 instances out of 300 belonging to the Gehring and Homberger's benchmark @cite_33 have been solved to optimality @cite_2 . An extensive review of the exact methods can be found in Kallehauge @cite_6 .