A new model for the asymmetric vehicle routing problem with simultaneous pickup and deliveries

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Abstract

The asymmetric vehicle routing problem with simultaneous pickup and deliveries is considered. This paper develops four new classes of valid inequalities for the problem. We generalize the idea of a no-good cut. Together, these help us solve 45-node randomly generated problem instances more efficiently. We report results on a set of benchmark instances in literature. In this set, we are able to show an order of magnitude improvement in computational times over currently published results in literature.

Introduction

In this paper, the asymmetric vehicle routing problem with simultaneous pickup and deliveries (AVRPSPD) is considered. Given are a depot, denoted by 0, and N customers. We define N as the set of all customers, and N=N{0} as the set of all nodes, including the depot. Customers and the depot are indexed by i or j. Each customer i is characterized by a pickup amount, ai0, of a commodity and a delivery amount, bi0, of a potentially different and independent commodity. Identical vehicles, each with capacity C, are initially stationed at the depot. The pickup and delivery amounts individually do not exceed the vehicle capacity. Traveling from i to j incurs a cost cij. A vehicle begins its route with the amount of cargo equal to the sum of the delivery amounts of all customers included in its route. It is required that this total amount of starting cargo not exceed C. At the ith customer location, the delivery quantity, bi, is first unloaded from the vehicle. The pickup amount, ai, is then loaded onto the vehicle. Throughout its route, the total amount of cargo in the vehicle (which will include as yet undelivered cargo and the cargo picked up thus far) cannot exceed the vehicle capacity C. The pickup and delivery amounts and type of commodity are independent and only coupled via the finite capacity C of the vehicles. The objective is to satisfy all customer demand (pickups and deliveries) at least cost, while visiting each customer exactly once.

Problems with simultaneous pickups and deliveries occur in the context of reverse logistics. In these settings, in addition to delivering to customer locations from stocking points, the units delivered in one time period become available for pickup in the subsequent periods. Examples include distribution of beer and soft drinks in bottles, blood vials to and from hospitals (see [1]), product returns, amongst others. See [7] for a comprehensive review of various models and problems in reverse logistics.

The literature on vehicle routing problems (VRPs) is enormous. The traditional capacitated vehicle routing problem (CVRP) deals only with either pickups or deliveries. Monographs dealing with the CVRP include [12], [20], [21]. In the problem considered in this work, the same vehicle needs to handle both pickups and deliveries simultaneously.

A related problem in literature is the VRP with backhauls. This problem deals with cases where a vehicle has to complete delivery to all customers before starting pickups. Refer to [11], [14], [18], [19], [22] for exact and heuristic methods to solve this problem.

The vehicle routing problem with simultaneous pickup and deliveries (VRPSPD) has been studied in [3], [4], [10]. Models and benchmark solutions exist in the literature for the symmetric version of this problem (see [8], [16]), abbreviated as SVRPSPD. It is pertinent to note that the AVRPSPD has been considered previously in literature. To the best of our knowledge, [15] is the only previous work to study the AVRPSPD. In Section 2, we compare the model developed in this paper with the model presented in [15]. In Section 6.3, we compare the performance of our model with the performance of the model in [15] on benchmark instances reported in [15]. Asymmetric problems arise frequently in urban contexts with one-way streets, or cities with a sloping road terrain, amongst others. In addition to being able to solve such asymmetric problems, the methodology developed in our work that uses a directed model can also be used to solve versions of the VRPSPDs with time windows. (See [2] for one such application.) Such problems require a directed model even though the underlying cost matrix may be symmetric (see [6] and [9], for instance.) In the remainder of this paper, the abbreviation VRPSPD denotes the generic problem regardless of whether the costs are symmetric or asymmetric.

Section snippets

Model

For SN, we define A(S)=ΣiSai, and B(S)=ΣiSbi. The following is a non-compact formulation of AVRPSPD using directed binary variables xij, where xij=1 if the vehicle travels directly from node i to node j, and 0 otherwise. Min.iNjN{i}cijxijs.t.iN{j}xij=1jNiN{j}xji=1jNiSjS{i}xij|S|1SNiSjN{S}xijA(S)CSNiSjN{S}xijB(S)CSN(i,j)Kxij|K|1KKxij0, integeri,jN

The objective function (1) is to minimize the total cost of all vehicles. Constraints (2),

Solution approach

We note that the formulation given in the previous section is a non-compact formulation in that the constraint sets (4)–(7) each have an exponential number of constraints in them. The authors in [16] used a lazy-cut approach for solving the undirected version of the problem, and we use the same approach. In this approach, these constraints are omitted initially, and are added on an as-needed basis. The well-known CVRPSEP package (see [13]), which has been developed for traditional CVRP,

Valid inequalities for AVRPSPD

In this section, four new classes of valid inequalities for the AVRPSPD are stated and their validity proven.

Generalizing no-good cuts

The no-good cuts described in Section 2 were deployed in [16] as lazy-cuts in the undirected version of the problem. We call these cuts simple no-good cuts. Each such cut prohibits only a single specific subsequence of customer visitation in a route. (Note, there could be more than one route with a specific subsequence of customer visits.) In this section, we show how to generalize these cuts for the directed version of the problem to make them much stronger, so that each cut prohibits not one,

Computational experiments

In this section, a variety of computational tests are performed to evaluate the proposed method and to compare its performance relative to the other methods presented in literature.

All computational tests were performed on an Intel i5 processor with clock speed 3 GHz, 4 cores and 8 GB system RAM memory. Different parameter settings were tried on a few initial instances, and we report the settings that gave the best computational times on these instances. Similar to the settings proposed in [16]

Conclusion

The contribution of this work is threefold. First, four new valid classes of inequalities were developed for the AVRPSPD. Second, it is shown that for the AVRPSPD, the no-good cuts can be generalized and made tighter than currently extant methods. Most of the VRPSPD benchmark instances in the literature are characterized by very low 2CVRP gap, and are therefore relatively easy to solve. As a third contribution, a general method was outlined that helps generate difficult problem instances of the

Acknowledgments

The authors are thankful to Prof. Subramanian for sharing problem instances reported in his work. The authors are also thankful to Prof. Rieck and Zimmermann for maintaining the website that provides benchmark instances. The authors are thankful to Prof. Lysgaard for discussions about the correct usage of CVRPSEP package in the context of the VRPSPD. All problem instances on which results have been reported in this paper are available as an online supplementary or from the authors. Funding from

References (22)

  • Reverse Logistics – Quantitative Models for Closed-Loop Supply Chains

    (2004)
  • Cited by (6)

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