A two-commodity flow formulation for the capacitated truck-and-trailer routing problem
Introduction
Truck-and-trailer routing problems (TTRPs) form a class of vehicle-routing problems (VRPs) that are characterized by the availability of non-autonomous trailers, which must be pulled by a truck. A composite vehicle consisting of a truck pulling a trailer permits to increase the load capacity but is subject to accessibility restrictions for composite vehicles at some of the customers, e.g., because the customers are located in a mountain area or in a city center with bans on large vehicles. To reflect such restrictions, TTRPs assume that a subset of the customers can only be served by a truck alone (so-called truck customers) whereas the remaining ones can be served by both a composite vehicle or by a truck alone (so-called vehicle customers). A number of real-world applications have been reported for TTRPs, like, e.g., milk collection [[10], [29]], fuel oil delivery to private households [15], distribution of goods [[18], [34]], postal mail delivery [[7], [8]], and container movement [35]. Cuda et al. [13] provide an up-to-date survey of TTRPs and review the solution methods proposed in the literature. TTRPs have also been discussed in surveys on location-routing problems (see, e.g., [[27], [30]]).
In this paper, we consider the capacitated TTRP (CTTRP) introduced by Chao [11], in which a limited number of capacitated trucks and a limited number of capacitated trailers are available at a central depot. The composite vehicles are allowed to park their trailer at any vehicle customer and to transfer load between the truck and the trailer. From there, it is possible to perform a truck subtour serving a subset of customers, possibly inaccessible for trailers, and then return to the trailer and continue the trip. The objective is to minimize the transportation cost for serving all customers with a compatible vehicle of the given fleet while respecting the vehicle capacities.
The CTTRP is the most studied problem in the class of TTRPs, and several heuristics are available in the literature (see [[9], [11], [14], [23], [33], [36], [37]]), of which the paper by Villegas et al. [37] currently defines the state-of-the-art. To the best of our knowledge, no exact solution method for the CTTRP has yet been proposed in the literature, however, a few exact methods have been introduced for variants of the CTTRP and are described in the following.
Drexl [15] studies the generalized TTRP as a unified modeling framework for TTRPs with a fixed truck–trailer assignment. The problem considers additional transshipment locations at which trailers can be parked and load transfers take place, time windows at both customer and transshipment locations, and a heterogeneous fleet of vehicles with different fixed and distance-dependent costs. The author devises an exact branch-and-price algorithm and several heuristic variants. The exact method is able to consistently solve instances with up to 10 truck customers, 10 vehicle customers and 10 transshipment locations. Drexl [16] studies the VRP with trailers and transshipments, in which the fixed truck–trailer assignment of the generalized TTRP is abandoned, i.e., a trailer can now be pulled by any compatible truck on the parts of its route, and any truck can perform a load transfer to any trailer. In addition, load dependent transfer times between trucks and trailers are considered. For this problem, the author proposes a branch-and-cut algorithm based on two compact formulations which build on a network representation of the problem involving vertices, with being the number of customers. Both formulations require the use of three-index variables to model the vehicle tours. The largest instances that are solved to optimality contain 8 customers, 8 transshipment locations, and 8 vehicles.
Parragh and Cordeau [28] and Rothenbächer et al. [32] both consider the CTTRP with time windows and develop column-generation-based solution methods based on a set-partitioning formulation. They report optimal solutions to instances with up to 100 customers if time windows are tight. None of the two papers reports results on CTTRP instances. Rothenbächer et al. [32] also consider two real-world extensions: (i) load-dependent transfer times, and (ii) the option to collect double the amount at a customer every second day. The proposed method was also used to address the generalized TTRP instances of Drexl [15]. Here, the number of instances that can be solved increases significantly. Finally, Belenguer et al. [5] propose a branch-and-cut for the single TTRP with satellite depots (STTRPSD), in which a set of truck customers have to be served, and the trailer can be parked and load transfer can take place at a set of satellite depots. The largest instance that the authors can solve features 100 customers and 10 satellite depots.
The contributions of this paper are:
We propose the first exact method for the CTTRP based on a new formulation that extends the two-commodity flow formulation introduced by Baldacci et al. [2] for the capacitated VRP (CVRP). Our new CTTRP model requires a polynomial number of variables and is defined over a network with vertices, where denotes the number of vehicle customer locations, and is the total number of customers.
We computationally assess the effectiveness of the new formulation by developing a branch-and-cut algorithm for solving it. We report results on a set of 132 instances with different characteristics featuring up to 40 customers, which are derived from the CTTRP benchmark sets of Chao [11] and Lin et al. [24]. In the numerical experiments, we also consider a special case of the CTTRP that we call single-vehicle CTTRP, in which only one vehicle consisting of a capacitated truck pulling an uncapacitated trailer is available.
We carry out an empirical investigation of the quality of the lower bounds obtained by our two-commodity flow formulation in comparison to a single-commodity formulation. We find that the two-commodity model provides stronger lower bounds.
We adapt our CTTRP formulation to the STTRPSD. We report computational results on STRRPSD instances with up to 50 customers, which suggest that our model is general enough to successfully handle different TTRP variants.
We also consider the variant of the CTTRP in which load transfers between vehicles are forbidden and investigate the effect of this assumption in numerical experiments: several of the instances become infeasible because of the limited number of available vehicles, while the cost increase on those that remain feasible is small.
The remainder of this paper is organized as follows. In Section 2, we describe the CTTRP and introduce the main notation used in the paper. Section 3 presents the CTTRP formulation, and Section 4 introduces some classes of valid inequalities for strengthening its Linear Programming (LP) relaxation. In Section 5, we describe a branch-and-cut algorithm based on the new formulation and detail the separation procedures that it uses to detect violated inequalities. Section 6 provides a computational evaluation of our branch-and-cut algorithm on newly generated CTTRP instances and on the STTRPSD benchmark set of Belenguer et al. [5]. In addition, we study the effect of forbidding load transfers between trucks and trailers, and we compare our two-commodity formulation with its one-commodity counterpart. Conclusions are drawn in Section 7.
Section snippets
Problem description and notation
The CTTRP can be defined on a complete undirected graph with vertex set and edge set . The customer set is partitioned into the two subsets and : vertices in correspond to vehicle customers, vertices in to truck customers. We also use the notation and . Each customer has a nonnegative demand . Vertex represents the depot at which a fleet of trucks with capacity and trailers with capacity
Two-commodity flow formulation
This section describes a mathematical formulation of the CTTRP (Section 3.2). The formulation uses an extended graph (Section 3.1).
Valid inequalities
In this section, we describe some classes of valid inequalities that we use to improve the LP relaxation of the formulation presented in Section 3. The separation procedures for solving the corresponding separation problem and a branch-and-cut algorithm based on the formulation strengthened by the described inequalities are detailed in Section 5.
Branch-and-cut algorithm and separation procedures
To assess the practical usefulness of the formulation presented in Section 3, we have implemented a branch-and-cut algorithm for solving it. We first solve the LP relaxation of formulation (1)–(22) strengthened by the inequalities described in Section 4 by means of a cutting plane algorithm. At this stage, the inequalities are separated in the order (17), (27), (26), (29), and (30), and (26) are lifted to become (28) whenever possible (see Section 4.1). At most 20 violated cuts are added in
Computational experiments
This section describes the computational evaluation of the branch-and-cut algorithm described above. The algorithm was implemented in C and compiled with Visual Studio 2012 64-bit. All computational experiments were run on an Intel Core i7-3770 CPU at 3.40 GHz with 16 GB of RAM. Cplex 12.7 was used as the MILP solver. The description of the benchmark instances is given in Section 6.1, the results on the CTTRP instances (with and without load transfers) are discussed in Section 6.2, the result
Conclusions
We introduce a new mathematical formulation for the capacitated truck-and-trailer routing problem (CTTRP), which is an extension of the two-commodity flow formulation for the capacitated vehicle-routing problem, and we describe some valid inequalities for strengthening it. We develop a branch-and-cut algorithm based on the new formulation and evaluate it computationally on a set of CTTRP instances with up to 40 customers and diverse characteristics. Our computational results suggest that the
Acknowledgments
We thank three anonymous referees for several comments and suggestions that allowed us to improve the paper.
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