Solving the Pickup and Delivery Problem with Time Windows by Branch-and-Cut Jean-Fran¸cois Cordeau1, Stefan Ropke2, Gilbert Laporte1 1
Canada Research Chair in Distribution Management, HEC Montr´eal
3000, chemin de la Cˆ ote-Sainte-Catherine, Montr´eal, Canada, H3T 2A7 {cordeau,gilbert}@crt.umontreal.ca 2
DIKU, University of Copenhagen
Universitets Parken 1, 2100 Copenhagen, Denmark
[email protected]
January 26, 2005
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Introduction
In the Vehicle Routing Problem with Pickup and Delivery (VRPPD), a set of transportation requests must be satisfied by capacitated vehicles based at one or several depots. Each request is defined by an origin (or pickup point), a destination (or delivery point), and a demand to be transported from the origin to the destination. Typical applications are parcel services and dial-a-ride transportation systems for the elderly and the disabled. In the Dial-a-Ride Problem (DARP), additional constraints are usually considered to control the quality of service. In particular, narrow time windows are often imposed on pickup and delivery times, and ride time constraints limit the time spent by users in the vehicles. The VRPPD is NP-hard and, in the presence of time windows, even checking the feasibility of the problem is NP-complete. As a result, most work on the VRPPD has concentrated on heuristics (Desaulniers et al., 2002). When the problem is sufficiently constrained, exact solutions can nevertheless be obtained for small to medium size instances. Column generation has for example been used by Dumas et al. (1991) to address the VRPPD with time windows. Recently, Cordeau (2004) developed a branch-and-cut algorithm for the DARP based on a 3-index formulation of the problem with time and load variables. This algorithm relies on existing inequalities for the TSP with precedence constraints (see, e.g., Balas et al., 1995; Ruland and Rodin, 1997) and on new inequalities that take advantage of the particular structure of the problem.
In this paper, we study the slightly more general VRPPD with time windows, compare three different formulations of the problem and introduce new inequalities and separation procedures to be used within a branch-and-cut algorithm. Let n denote the number of requests to be satisfied. Assuming that all vehicles are based at a single depot, the problem may be defined on a directed graph G = (N, A) where N = P ∪ D ∪ {0, 2n + 1}, P = {1, . . . , n} and D = {n + 1, . . . , 2n}. Subsets P and D contain pickup and delivery nodes, respectively, while nodes 0 and 2n + 1 represent the origin and destination depots. With each request i are thus associated an origin node i and a destination node n + i. Let K be the set of vehicles. A 3-index formulation of the problem is obtained by defining, for each arc (i, j) ∈ A and each vehicle k ∈ K, a binary variable xkij = 1 if and only if vehicle k travels from node i to node j. For each node i ∈ N and each vehicle k ∈ K, let also Bik be the time at which vehicle k begins the service at node i, and Qki be the load of the vehicle after visiting node i. The problem can be be formulated as a mixed-integer linear programming problem with covering and pairing constraints to ensure that each request is served exactly once and that the associated pickup and delivery nodes are visited by the same vehicle. In the presence of the time variables Bik , precedence constraints can be stated k ≥ Bik + di + ti,n+i where di denotes the service time directly by an inequality of the form Bn+i at node i and tij is the travel time between nodes i and j. Similarly, capacity constraints are imposed as simple upper bounds on the load variables Qki . This formulation is very compact in terms of the number of constraints but it requires a large number of variables. A formulation with fewer variables can be obtained by removing the vehicle index k from all variables xkij , Bik and Qki . In this case, however, imposing the pairing constraints between the pickup and delivery nodes can no longer be accomplished through the vehicle indices. Instead, a set of precedence constraints (whose size is exponential in the number of requests) must be considered, and violated constraints must be generated dynamically during the branch-and-bound search. An even more compact formulation in terms of the number of variables can be obtained by removing the time and load variables, and replacing them with two sets of constraints: capacity inequalities and infeasible path inequalities. The former correspond to the well-known capacity constraints for the VRP while the latter have been used for the VRP with time windows (see, e.g., Ascheuer et al., 2000). With all formulations, several types of additional valid inequalities can be used to improve the LP lower bounds. Various strengthenings of subtour elimination constraints and capacity constraints are possible because of the pickup and delivery structure. Similarly, reachability cuts (Lysgaard, 2004) and infeasible path inequalities can be extended by combining the time window and pickup and delivery aspects together with the ride time constraints. Finally, a large number of problem reductions can be performed by detecting subsets of incompatible users or arc sequences that are infeasible because of the precedence constraints. This paper compares the computational efficiency of the different formulations and evaluates the impact of the various types of valid inequalities. Computational results are reported on instances with up to 50 requests (i.e., 100 nodes).
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