**1. Introduction**

Petersen and Madsen [1] introduced an optimization problem referred to as the double traveling salesman problem with multiple stacks, based on a prospective customer of a company producing computer software systems for transportation companies. The problem is set in two different regions. A vehicle carrying a container must visit customers to make pickups in one region. The container is then transported to a different region, whereupon a different vehicle carries the container while visiting customers to make deliveries. It is not possible to repack the container en route, and an opening in one end of the container provides the only access to its contents.

The items transported are standardized pallets and each container can fit a given number of *R* rows with a limited number of *L* pallets each. This information is crucial, due to the inability to repack the container. Each row of pallets forms a stack that must be loaded and unloaded based on a first-in-last-out principle. This provides a set of difficult linking constraints that must be taken into consideration when routing the vehicle in both the pickup region and the delivery region. For a given total capacity of *R* ∗ *L*, the loading constraints are most severe when *R* is low and *L* is high, whereas having a high value of *R* provides more flexibility.

Since its introduction, the double traveling salesman problem with multiple stacks has received significant attention from researchers, examining both exact and heuristic solution methods, as well as performing studies on computational complexity and polyhedral analysis. However, it seems that the initial assumptions of the underlying problem structure have not been examined. In this work, one of the hidden assumptions of the problem is questioned: the type of container available to carry out the transportation. By replacing the type of container used in the transportation, the defining characteristics of the problem seem to be lost. This suggests that a technical solution of modifying the container technology used would have cancelled out the need for advancing the frontier of operations research.

Petersen and Madsen [1] provided test instances where the container could contain three rows with 11 pallets each. Standardized Euro Pallets (EPAL, or EUR-1) have dimensions of 1200 by 800 millimetres. This provides a good match with the internal dimensions of a 45-foot pallet wide container, which has an internal width of 2400 millimetres and an internal length of about 13,600 millimetres, with the exact dimensions varying between different manufacturers. Additional test instances were presented with three rows of 22 pallets each, based on situations where the height of each pallet is less than half of the height of the container, and where a pallet can be put on top of another pallet. However, the same loading constraints could also arise from the transportation of another standardized pallet, the EUR-6 pallet, which has dimensions of 600 by 800 millimetres.

The basic original instances have *R* = 3 rows of pallets, each row of length *L* = 11, providing an overall capacity of *R* ∗ *L* = 3 ∗ 11 = 33 pallets. However, there is an alternative configuration given the dimensions of the container. Placing the EUR-1 pallets with their long side towards the short side of the container, only two rows of pallets will fit. However, there is enough capacity for either 16 or 17 pallets in each row, depending on the exact length of the container. Therefore, the longest editions of the 45-foot containers have enough space for 2 ∗ 17 = 34 pallets in total. When pallets are vertically stackable, the pattern of three rows provides a capacity of 3 ∗ 22 = 66 pallets, whereas the pattern of two rows provides a maximum capacity of 2 ∗ 34 = 68. Transporting EUR-6 pallets provides another possibility. Instead of using *R* = 3 rows of length *L* = 22 and a total capacity of 3 ∗ 22 = 66, it is possible to use *R* = 4 rows of length *L* = 17 for a total capacity of 4 ∗ 17 = 68.

The above is true for containers loaded from the back, that is, using one of the short sides. Open side versions also exist for many container sizes, where goods can be loaded from one of the long sides of the container. The existence of such capabilities would open up a wide range of options in the context of the double traveling salesman problem with multiple stacks. For any of the previously mentioned loading options, the number of rows *R* and their length *L* can be swapped, producing instances with many rows of relatively low length. Table 1 provides examples of values for *R* and *L* that may appear, depending on the type of pallet and container used. Figure 1 illustrates two of the combinations.


**Table 1.** Selected values for the number of stacks *R* and their capacity *L* based on different pallets and container technologies. The first three rows correspond to values that are covered by existing instances.

**Figure 1.** Top: a 45-foot container loaded from the back with three rows of 11 pallets each, resulting in a problem with *R* = 3 and *L* = 11. Bottom: a 45-foot container loaded from the side with two rows of 17 pallets each, resulting in a problem with *R* = 17 and *L* = 2.

The rest of this paper is structured, as follows. Section 2 reviews the related literature, while Section 3 presents the mathematical formulation of the double traveling salesman problem with multiple stacks and the solution methods used in our analysis. Section 4 contains a computational study, with the aim of determining the importance of considering different container types in the considered problem. The paper is concluded with Section 5, where some conclusions and additional thoughts about underlying assumptions of the problem are presented.

## **2. Literature**

The double traveling salesman problem with multiple stacks was first introduced by Petersen and Madsen [1], who presented several heuristic algorithms for the problem, based on iterated local search, tabu search, simulated annealing, and large neighborhood search. Variable neighborhood search for the problem was examined by Felipe et al. [2], with improvements in [3], whereas Urrutia et al. [4] presented a dynamic programming based local search method.

Several mathematical formulations of the double traveling salesman problem with multiple stacks with corresponding solution strategies were presented by Petersen et al. [5]. A specialized algorithm was given by [6] and improved by Lusby and Larsen [6], based on combining separate traveling salesman problems for the pickup region and the delivery region. Branch-and-cut was used by Alba Martínez et al. [7], whereas Carrabs et al. [8] presented a branch-and-bound algorithm for the problem with only two stacks. Two stacks, with infinite capacity each, was also solved by Barbato et al. [9] using a set covering approach, and in [10] using a branch-and-cut algorithm.

Some of the theoretical properties of the problem were investigated by Casazza et al. [11], leading to a simple heuristic method. Given a route for the pickup region and a route for the delivery region, Toulouse and Calvo [12] showed that it can be decided in polynomial time whether or not a feasible stacking exists. Furthermore, Toulouse and Calvo [12] also showed that, given a stacking, optimal routes conditional on the stacking can be determined in polynomial time. Bonomo et al. [13] discussed similar complexity results.

After the introduction of the double traveling salesman problem with multiple stacks, many variants have also been considered in the literature: Iori and Riera-Ledesma [14] presented a generalization with multiple vehicles, and gave three mathematical formulations with corresponding exact solution methods. Heuristics based on iterated local search, simulated annealing, and variable neighborhood descent were proposed for this problem by da Silveira et al. [15]. Another simulated annealing implementation was provided by Chagas et al. [16] and a variable neighborhood search by Chagas et al. [17].

Another variation arises when considering a pickup and delivery traveling salesman problem with multiple stacks, where the nodes to visit are not necessarily split into two separate regions. A large neighborhood search was proposed for this generalization by Côté et al. [18], whereas both Pereira and Urrutia [19] and Sampaio and Urrutia [20] presented branch-and-cut algorithms.

The pickup and delivery problem with time windows and multiple stacks was investigated by [21]. Other types of loading constraints have been considered in the literature as well. Doerner et al. [22] considered the transportation of wood products, while using both a tabu search and an ant colony optimization method. Gendreau et al. [23] used tabu search and Iori et al. [24] a branch-and-cut algorithm for vehicle routing problems with two-dimensional loading constraints. Fuellerer et al. [25] used ant colony optimization for a vehicle routing problem with three-dimensional loading constraints, and Chagas et al. [26] considered partial last-in-first-out loading constraints. Iori [27] presented a survey on combined routing and loading problems and is recommended for further details about related problems.

Even though a wide variety of problems with loading and capacity constraints leading to stacks of items have been approached in the literature through different methodologies, as far as the authors are aware, none of them performed an analysis of the practical consequences that the choice of container types and packing patterns used for transportation may have on the final costs. This is the research gap that this paper tries to fill, focusing on the case of the original double traveling salesman problem with multiple stacks.
