Vehicle Routing with Time Windows and Split Deliveries Dominique Feillet Laboratoire d’Informatique d’Avignon Universit´e d’Avignon, Agroparc BP 1228, 84911 Avignon Cedex 9, France fax.: (33) 4 90 84 35 01,
e.mail:
[email protected]
Pierre Dejax IRCCyN, Ecole des Mines de Nantes, France Michel Gendreau Centre de Recherche sur les Transports, Universit´e de Montr´eal, Canada Cyrille Gueguen Department of Operations Research, Air France, France October 30, 2002
Abstract In this presentation, we consider the Split Delivery Vehicle Routing Problem with Time Windows (SDVRPTW) in which the delivery to a customer can be split between any number of vehicles. This problem has not been extensively studied in the literature. The main contribution of this paper is a Branch and Price approach for solving the SDVRPTW without imposing restrictions on the split delivery options. The computational results show that this approach works well and that split delivery can lead to significant improvements in terms of cost and number of vehicles used.
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The Split Delivery Vehicle Routing Problem with Time Windows
Let G = (V, A) be a directed graph with vertex set V = {v0 , ..., vn } and arc set A. Vertices v1 to vn correspond to customers. The set of these customers is noted V ? . Customers require a delivery of some commodity. The expected quantity is assumed to be positive and is noted di . Vertex v0 1
represents a depot at which a fleet of vehicles is based. An unlimited number of identical vehicles of capacity Q are assumed to be available. To every arc (vi , vj ) ∈ A is associated a cost cij and a travel time tij . The cost and time matrices are assumed to be nonnegative and to satisfy the triangle inequality. A time window [ai , bi ] and a service time si are defined for every customer vi ∈ V . We assume that the service time of a customer does not depend upon the quantity delivered. The Split Delivery Vehicle Routing Problem with Time Windows (SDVRPTW) consists in determining a least cost set of vehicle routes such that every route starts and ends at the depot and that every customer is served by one or several vehicles, the vehicle capacity being respected and the service of customers beginning inside their time windows. Contrary to the classical Vehicle Routing Problem with Time windows, the demand of customers can be split between several vehicles, possibly leading to significant decreases of the costs. Even so, this problem (with or without time windows) has not received a lot of attention in the literature. No efficient exact solution algorithm is proposed in the literature, even if several relaxations are assessed (Dror, Laporte and Trudeau 1994, Belenguer et al. 2000). Sierksma and Tijssen (1998) and Mullaseril and Dror (1996) present optimal solution algorithms based on covering models and column generation methods, respectively for the SDVRP and the SDVRPTW. However, in both cases, only some predefined proportions of split are allowed. The purpose of this presentation is to propose a new set covering formulation for the SDVRPTW and to describe a complete algorithm that solves problems of moderate size to optimality without making any assumption on the fashion way in which demands are split . We base our approach on the following model : X minimize ck xk ,
(1)
rk ∈Ω
subject to X
aik yik ≥ di
(vi ∈ V ? ),
(2)
X
aik yik
(rk ∈ Ω),
(3)
xk ≥ 0 and integer
(rk ∈ Ω),
(4)
(vi ∈ V ? , rk ∈ Ω).
(5)
rk ∈Ω
Qxk ≥
vi ∈V ?
yik ≥ 0
where Ω is the set of feasible tours for the SDVRPTW, aik is a constant value equal to 1 if and only if route rk ∈ Ω visits customer vi , ck represents the cost of route rk , xk is an integer variable equal to the number of times route rk is used in the solution and yik is the quantity delivered by route rk to customer vi . We first study the linear programming bound of this model. We bring out an analytic formula for its computation, that demonstrates its weakness. We then introduce some valid inequalities 2
to reinforce the quality of the bound. We derive two column generations approaches to compute the new relaxations, depending on the valid inequalities considered. These two column generation approaches induce two original (and surprisingly very different) shortest path problems as subproblems. Computations on (modified) Solomon’s instances are used to evaluate the effectiveness and efficiency of these approaches. Finally, we adapt classical branching schemes to fit the specificities of the problem and to obtain optimal integer solutions. We especially propose to branch on the number of times the demand is split for a customer (i.e., the number of times this customer is visited). Results demonstrate the capabilities of this approach.
References [1] M. Dror and G. Laporte and P. Trudeau, ”Vehicle Routing with Split Deliveries”, Discrete Applied Mathematics 50, 239-254 (1994). [2] G. Sierksma and G.A. Tijssen, ”Routing Helicopters for Crew Exchanges on Off-Shore Locations”, Annals of Operations Research 76, 261-286 (1998). [3] J.M. Belenguer and M.C. Martinez and E. Mota, ”A Lower Bound for the Split Delivery Vehicle Routing Problem”, Operations Research 48(5), 801-810 (2000). [4] P.A. Mullaseril and M. Dror, ”A set-covering approach for directed node and arc routing problems with split delivery and time windows”, Working Paper, MIS department, University of Arizona, Tucson, Arizona, 1996.
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