Dynamic window reduction for the multiple depot vehicle scheduling problem with time windows

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Abstract

We consider a widespread branch-and-price approach to solve the multiple depot vehicle scheduling problem with time windows. We describe a dynamic time window reduction technique to speed up this approach. The time windows are transferred from nodes to arcs in order to take advantage of dual information and to tighten as much as possible the time variable domains. The performance of the proposed technique is evaluated through computational experiments on randomly generated instances involving several depots and up to 900 tasks.

Introduction

The multiple depot vehicle scheduling problem with time windows (MDVSPTW) is an important optimization problem that arises in freight transport systems and in urban bus scheduling. It can be described as follows. We are given a fleet of vehicles housed in several depots indexed by a set K and denoted by D1,D2,,D|K|, where each depot Dk, kK, can house at most vk vehicles, along with a set of n trips (or tasks) {T1,T2,,Tn}. Each trip Ti, 1in, starts within its respective prescribed time window [li,ui]. Let tij denote the duration of trip Ti plus the travel time between the ending point of trip Ti and the starting point of trip Tj, tn+k,i denote the travel time between depot Dk and the starting point of trip Ti and ti,n+|K|+k the duration of trip Ti plus the travel time between the ending point of trip Ti and depot Dk. Note that waiting at starting points of trips before specified time windows is permitted. Let also cij be the cost incurred if a vehicle performs the trip Ti immediately before the trip Tj and cn+k,i (resp. ci,n+|K|+k) be the cost incurred if Ti is the first (resp. last) trip undertaken by a vehicle housed at depot Dk, kK. A route, performed by a vehicle of depot Dk, is an ordered sequence of q trips Ti1,,Tiq such that:

  • (i)

    the vehicle returns to the starting depot;

  • (ii)

    the starting time sih of trip Tih, 1hq, lies within [lih,uih];

  • (iii)

    any two consecutive trips (Tih,Tih+1) form a compatible pair (i.e., sih+tih,ih+1sih+1);

  • (iv)

    the cost of the route is cn+k,i1+h=1q-1cih,ih+1+ciq,n+|K|+k.

A feasible assignment of trips to vehicles is a set of routes satisfying the following conditions:
  • (1)

    no more than vk routes originate at depot Dk, k=1,,|K|;

  • (2)

    each trip is covered by exactly one route.

Finally, the MDVSPTW is the problem of finding a feasible assignment of minimum cost, where the cost of an assignment is the sum of the costs of its routes. When each time window reduces to a single time point, the MDVSPTW is actually the classical multiple depot vehicle scheduling problem (MDVSP). Note that, in this no time window case, a route performed by a vehicle of depot Dk must satisfy conditions (i), (iii) and (iv) given above.

The MDVSPTW is NP-hard since it includes, as special cases, the MDVSP that is known to be NP-hard when |K|2 (Bertossi et al. [1]) and the Traveling Salesman Problem with Time Windows (exactly one vehicle housed in one depot) for which even finding a feasible solution is an NP-complete problem (Savelsbergh [2]).

The MDVSP has been studied for more than 20 years (see Bodin et al. [3]) and several heuristic algorithms have been proposed (see the survey paper of Desrosiers et al. [4]). As well, exact algorithms for solving the MDVSP have been developed by Carpaneto et al. [5], Ribeiro et al. [6], Bianco et al. [7], Forbes et al. [8], Löbel [9] and more recently by Fischetti et al. [10] and Hadjar et al. [11]. Relatively few MDVSPTW investigations have been reported in the literature. To the best of our knowledge, the papers of Mingozzi et al. [12] and Desaulniers et al. [13] are the only published papers that deal with the MDVSPTW.

In Mingozzi et al. [12], the authors generalize the method they proposed for the MDVSP. They formulate the MDVSPTW as a set partitioning problem and they describe a branch-and-bound algorithm for solving it. They succeeded in solving randomly generated instances having up to 120 tasks and five depots. Desaulniers et al. [13] developed a branch-and-price algorithm for the MDVSPTW and solved randomly generated instances involving up to 300 tasks and five depots.

Generally, the number of variables and the domain (the set of all possible values) of each one of them are liable, to a considerable degree, for the computational difficulties encountered in solving integer programming problems. When the variable number and the variable domains are small the solution is usually carried out in reasonable time even for NP-hard problems. Thus any technique that reduces the number and the domains of the variables should play an important role in the improvement of solution methodologies.

Several authors proposed arc elimination procedures in order to reduce the number of variables while solving routing and scheduling problems. In Bianco et al. [7] and Mingozzi et al. [12], [14], the authors formulate, respectively, the MDVSP, the MDVSPTW and the crew scheduling problem as set partitioning problems. By combining a number of different procedures, they compute a heuristic solution to the dual of the linear relaxation of the set partitioning problem. Then they consider the reduced costs of the routes relative to this dual solution in order to remove arcs from the underlying networks and to reduce the number of feasible routes taken into account in the set partitioning formulation. As well, Hadjar et al. [11] present a branch-and-price-and-cut algorithm for the MDVSP where, at each node of the branch-and-price tree, a variable fixing procedure computes the reduced costs of the arc variables and reduces the size of the subproblem networks. This dynamic variable fixing procedure allowed them to save up to more than 85% of the total computing time. Recently, in order to speed up the pricing process in column generation algorithms, Irnich et al. [15] describe an arc elimination procedure that consists of removing an arc when the lower bound on the reduced cost of all feasible routes that use this arc is at least a given bound. The reduced cost lower bound is evaluated by solving each subproblem twice, i.e., in the (forward) corresponding network and in the backward one obtained by reversing the direction of all arcs. The authors tested their approach on the vehicle routing problem with time windows and showed that it can yield to a significant overall speedup of factor going up to 18.9 with an average of 2.8.

For time window constrained routing and scheduling problems, the complexity of the proposed algorithms depends on the size of the network and on the width of the time windows. Several researchers suggest static time window reduction and static arc elimination procedures (see the survey paper of Desrosiers et al. [4]). These procedures are principally based on the investigation of the earliest (resp. latest) departure and arrival times for each node and they are performed prior to starting the solution process.

In the present paper, we design a time window reduction technique to speed up the classical exact branch-and-price approach for the MDVSPTW. The reduction technique is applied at every node of the branch-and-price tree and tries, once the lower bound is computed by column generation, to tighten the time windows. By transferring time windows from nodes to arcs, this technique takes advantage of dual information and offers the possibility to tighten dynamically and as much as possible the time variable domains and to fix to zero, as well, arc variables whose corresponding reduced windows contain no time point. The results of a computational study conducted on randomly generated instances involving several depots, up to 900 tasks and up to 30 min time windows, show the significant improvement brought by the proposed technique.

The paper is organized as follows. In Section 2 we review the mathematical formulations of the MDVSPTW and we describe a column generation scheme for computing lower bounds. Section 3 gives a detailed presentation of the time window reduction technique. In Section 4 we present the branch-and-price algorithm and we report extensive computational results in Section 5. Finally, Section 6 draws some conclusions and outlines avenues for future research.

Section snippets

Mathematical formulation and column generation

First, let us associate with our problem the directed multigraph G=(V,A) where each trip Ti is represented by a trip-vertex i and each depot Dk, kK, by a source-vertex n+k and a sink-vertex n+|K|+k (every depot is duplicated into a source-depot and a sink-depot). The vertex set V of G is then the union of sets N={1,2,,n}, {n+1,n+2,,n+|K|} and {n+|K|+1,n+|K|+2,,n+2|K|}; the arc set A consists of |K| sets A1,,A|K| such that each Ak is the union of the following three arc sets: {(n+k,i)|iN}{(

Time window reduction

The mathematical formulation presented in the previous section involves a huge number of binary flow variables and continuous time variables. The domains of the time variables depend on the width of the time windows. Hence the complexity of the solution approach is also a function of the time window width; when the windows are too wide, much time and memory is consumed by the dynamic programming algorithm for solving the subproblems (this is due to the huge number of generated labels). It would

The branch-and-price algorithm for the MDVSPTW

The branch-and-price algorithm we present in this section is similar to those ones described in Desaulniers et al. [13] and in Hadjar et al. [11]. This algorithm uses column generation to compute lower bounds. At each node of the branch-and-price tree, and just after the lower bound is evaluated, the time window reduction procedure is solicited. This procedure consists of applying Proposition 3 to every arc of the current network. The reduced time windows are substituted for the original ones,

Computational results

The branch-and-price algorithm, described in the previous section, has been implemented using the GENCOL software package (see Desaulniers et al. [18]) and experimentally evaluated on test problems of different sizes. Numerical results are reported in this section in order to assess the effectiveness of the time window reduction procedure. We compare two versions of the branch-and-price algorithm: a first version (referred to as the std-algorithm) where the time window reduction procedure is

Conclusion

In this paper, we have considered a classical branch-and-price method to solve the MDVSPTW. We have presented a dynamic time window reduction procedure to speed up this standard algorithm. This procedure is a non-trivial generalization of the variable fixing procedure of Hadjar et al. [11] developed for the MDVSP since the formulation of MDVSPTW involves nonlinear and non-convex constraints. The time windows were transferred on arcs to obtain the tightest domains for the time variables. Through

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