The double travelling salesman problem with multiple stacks – Formulation and heuristic solution approaches

Abstract

This paper introduces the double travelling salesman problem with multiple stacks and presents four different metaheuristic approaches to its solution. The double TSP with multiple stacks is concerned with determining the shortest route performing pickups and deliveries in two separated networks (one for pickups and one for deliveries) using only one container. Repacking is not allowed, instead each item can be positioned in one of several rows in the container, such that each row can be considered a LIFO (last in, first out) stack, but no mutual constraints exist between the rows. Two different neighbourhood structures are developed for the problem and used with each of three local search metaheuristics. Additionally some simpler removal and reinsertion operators are used in a Large neighbourhood search framework. Finally some computational results are given along with lower bounds on the objective value.

Introduction

As congestion is an ever-growing problem on the roads all over the world, intermodality is playing an increasingly important role in the transportation of goods. Furthermore, the complexity of the resulting planning problems presents additional requirements to the tools available to planners.

The project that forms the basis of this paper was initiated in cooperation with a company producing computer software systems for operation and fleet management in small and medium-sized transportation companies. The software company encountered this problem at one of its prospective customers, and the problem is intriguing in that it does not seem to have been treated previously in the literature, at the same time as it is conceptually simple.

The double travelling salesman problem with multiple stacks (DTSPMS) is concerned with finding the shortest routes performing pickups and deliveries in two separated networks/regions. The problem permits neither repacking nor vertical stacking, instead the items can be packed in several rows (horizontal stacks) in the container, such that each row must obey the LIFO (last in, first out) principle, while there are no mutual constraints between the rows.

In the DTSPMS a set of orders is given, each one requiring transportation of one item from a customer in the pickup region to a customer in the delivery region, i.e. each order contains a pickup customer and a delivery customer for one item. The items are required to be boxes/pallets of identical dimensions and each region has a depot. The two regions are far apart, and thus some long-haul transportation is required between the depots. This long-haul transportation is not part of the problem considered here. All pickups and deliveries must be carried out using the same container, which cannot be repacked along the way, and items in the container can only be accessed from the opening in one end of the container. Hence the problem to be solved consists of determining the shortest Hamiltonian tour through each of the networks, in such a way that a feasible loading plan exists. No time windows are considered in this problem.

In practice this situation can occur when the container is loaded onto a truck to perform the pickup operations, then returned by that truck to a local depot/terminal where it is transferred onto a train, ship or another truck, which then performs the long-haul transportation. Upon arrival at the depot/terminal in the delivery region, the container is again transferred to a truck, which carries out the actual deliveries. The terminals only have facilities to perform container movements, and do not offer any opportunities for opening or repacking the container.

It is assumed that each order consists of exactly one item, thus if one pickup or delivery location is shared by several orders the corresponding node in the graph will be duplicated.

To state the problem more formally two weighted complete graphs $(V^G\text{,}{E}^G)\text{,}G\in \{\mathrm{P}\text{,}\mathrm{D}\}$ are given for pickup (P) and delivery (D), respectively, and the purpose is to find a Hamiltonian tour through each graph, such that the sum of the weights of the edges used is minimised. Each graph G has a depot node $v_0^G$, $G\in \{\mathrm{P}\text{,}\mathrm{D}\}$, customer nodes $v_1^G\text{,}\dots \text{,}{v}_n^G$, and symmetric edge costs $c_{\mathit{ij}}^G$. A set of n orders $\{1\text{,}\dots \text{,}n\}$ is given, where order i must be picked up at $v_i^{\mathrm{P}}\in V^{\mathrm{P}}$ and delivered at $v_i^{\mathrm{D}}\in V^{\mathrm{D}}$. Finally, $V_C^G=V^G⧹\{v_0^G\}\text{,}G\in \{\mathrm{P}\text{,}\mathrm{D}\}$ denotes the set of customer nodes in graph G. Whenever superscript G is used in the following it will indicate a distinction between pickup and delivery, $G\in \{\mathrm{P}\text{,}\mathrm{D}\}$.

The edge costs are assumed to be symmetric throughout this paper, but the presented algorithms do not rely on this property, and can also handle problems with asymmetric edge costs.

Throughout the paper it will be assumed that the number of orders n equals the number of loading positions. The loading container will have R rows, each of length L, and thus $n=R·L$. The total number of nodes in the problem is $2n+2$.

The connection between the two tours to be found is given by the loading of the container. Since no repacking is allowed, the only items that can be delivered “next” at any time during delivery are the ones that can be accessed from the opening of the container. This implies that the loading is subject to LIFO constraints.

However, in the DTSPMS there is no LIFO ordering for the container as a whole. Rather, it contains several loading rows, each of which can be considered a LIFO stack, but all rows are independently accessible.

In real life the items to be transported would typically be standardised Euro Pallets, which fit 3 × 11 on the floor area of a 40-foot pallet container, providing three independent loading rows.

A solution to a given problem consists of a pickup route, a delivery route, and a row assignment, which for each item indicates which loading row it must be placed in. A row assignment only gives the row that each item should be placed in, and does not indicate which position the item will occupy in that row. Given a route (pickup or delivery) and a row assignment, one can construct the loading plan, which gives the exact position of each item inside the loaded container.

The problem may at first glance seem purely theoretical, since the extra mileage incurred by not being able to repack may seem prohibitive. However the problem has been encountered in real-life applications, where this extra mileage is justified by the wages stemming from handling and requirements to comply with union restrictions (the driver is not allowed to handle the goods).

Special cases of the DTSPMS occur when the number of loading rows is equal to 1 or to the number of orders n. In both cases the problem of finding a row assignment for the solution becomes irrelevant.

In the single row case the pickup route will strictly dictate the delivery route (or vice versa), and the two routes will be exact opposites. In this case the problem can be solved by adding the transposed distance matrix of the delivery graph to the distance matrix of the pickup graph, and solving a regular TSP for the resulting distance matrix.

Conversely, when the number of loading rows equals the number of orders n, the two routes do not impose any restrictions on each other and the optimal solution to the problem consists of the optimal solutions to the two independent TSPs.

The DTSPMS as described here is a combination of the travelling salesman problem (TSP) and pickup and delivery problems (PDPs) and does not seem to have been treated previously in the literature, however early presentations of the work presented in this paper have inspired additional work presented in [9], which uses several new operators, along with the ones presented here, in a variable neighbourhood search (VNS) framework.

Although being concerned with pickups and deliveries, the DTSPMS differs significantly from the “regular” PDP as described in e.g. [6], [7], and earlier in [15], [21]. A number of variations of the PDP have been described in the more recent survey [18].

The main additional complication is the availability of multiple LIFO loading rows and thus the need to present a loading plan as part of the solution. The regular PDP with LIFO ordering (one stack only) has been treated using both heuristics [3] and exact methods [5], [2].

In the regular PDP it is necessary to make sure that each pickup is performed before the corresponding delivery. This is automatically ensured in the DTSPMS, since all pickups are performed before all deliveries. Furthermore when capacity constraints are present in a PDP these must be checked for every node that is visited. In the DTSPMS all items will need to be kept in the container at the same time, and therefore it would suffice to check the capacity of the full vehicle if any kind of capacity constraints were present (which would not happen in the plain DTSPMS, but could occur with an extension to multiple vehicles, or if not all orders need to be served).

Apart from the regular PDP, another class of problems that show similarities with the DTSPMS is the TSP with Backhauls (TSPB) (cf. e.g. [23]). Here the property that “all pickups lie before all deliveries” is preserved, however there is no longer any constraints tying a pickup to its corresponding delivery.

The DTSPMS is a special case of the generalised pickup and delivery problem with loading constraints in 2 dimensions. Routing problems with more general loading constraint are described in [13], [11], [10].

The Multi-Pile VRP (MPVRP), is a problem somewhat similar to the DTSPMS, combining routing and loading, using several available stacks/piles. The paper [8] solves the MPVRP using tabu search and ant colony optimisation. The MPVRP is a generalisation of the DTSPMS, with varying dimensions of the transported items (leading to overlap between the stacks/piles).

The paper is organised as follows: First a mathematical formulation of the problem is presented in Section 2 and some comments are made on its implementation in GAMS/CPLEX. Next, four different heuristic solution approaches are presented in Section 3, with emphasis on the developed neighbourhood structure that is based on the structure of the problem, and is used for the first three approaches. Finally, Section 4 describes the implementations and gives some computational results, and Section 5 concludes on the described heuristic solution approaches and gives some suggestions for future work on the DTSPMS.