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Vehicle Routing Problems with Loading Constraints: An Overview of Variants and Solution Methods

Ines SBAI1 and Saoussen KRICHEN1

1Université de Tunis, Institut Supérieur de Gestion de Tunis, LARODEC Laboratory, Tunisia

This chapter combines two of the most studied combinatorial optimization problems, namely, the capacitated vehicle routing problem (CVRP) and the two/three-dimensional bin packing problem (2/3D-BPP). It focuses heavily on real-life transportation problems such as the transportation of furniture or industrial machinery. An extensive overview of the CVRP with two/three-dimensional loading constraints is presented by surveying over 76 existing contributions. We provide an updated review of the variants of the L-CVRP studied in the literature and analyze some of the most popular optimization methods presented in the existing literature. Alongside this, we discuss their variants and constraints, their applications for solving real-world problems, as well as their impact on the current literature.

1.1. Introduction

Although the vehicle routing problem (VRP) is the most studied combinatorial optimization problem, the challenge still remains to achieve the most optimal and effective results (Sbai et al. 2020a). The VRP aims to minimize total traveling cost in cases where a fleet of identical vehicles is used to visit a set of customers. The VRP is used in many real-world applications, for example: pharmaceutical distribution, food distribution, the urban bus problem and garbage collection. The basic version of the VRP is known as the capacitated VRP (CVRP); each vehicle has a fixed capacity which must be respected and must not be exceeded when loading items. It is aimed at minimizing the total cost of serving all the customers. The CVRP can be extended to the VRP with time windows (VRPTW) by adding time windows to define the overall traveling time for a vehicle. It can also be extended to the VRP with pickups and deliveries (VRPPD) where orders may be picked up and delivered. Another variant of the basic CVRP is the VRP with backhauls (VRPB). Here, pickups and deliveries may be combined in a single route; all delivery requests therefore need to be performed before the empty vehicle can collect goods from customer locations. Two surveys, conducted by Cordeau et al. (2002) and Laporte (2009), provide further details.

Loading and transporting items from the depot to different customers are practical problems that are regularly encountered within the logistics industry. The loading problem can be extended to the BBP. When taking into account the number of dimensions that are relevant to the problem, packing problems are classified into 2D and 3D problems. The first related problem is the 2D-BPP (Zang et al. 2017; Wei et al. 2018; Sbai and Krichen 2019) where both items and bins are rectangular and the aim is to pack all items, without overlap, into the minimum number of bins. The second one is the 3D-BPP (Araujo et al. 2019; Pugliese et al. 2019); this consists of finding an efficient and accurate way to place 3D rectangular goods into the minimum number of 3D containers (bins), while ensuring goods are housed completely within the containers.

In recent years, some researchers have focused on the combined routing and loading problem. The combinatorial problem includes the 2D loading VRP, denoted as 2L-CVRP and the 3D loading VRP, denoted as 3L-CVRP. The purpose of addressing these problems is to minimize the overall travel costs associated with all the routes that serve each of the customers, as well as to satisfy all the constraints of the loading dimensions. The two problems are solved by exact and metaheuristic algorithms which are reviewed in detail in the sections that follow. For further information, we refer the reader to Pollaris et al. (2015) and Iori and Martello (2010), wherein detailed surveys are presented in relation to vehicle routing with packing problems.

This chapter is organized as follows: section 1.2 provides an overview of the literature concerning VRPs in combination with 2D loading problems and the existing variants and constraints. Section 1.3 focuses on VRPs with 3D loading problems and the existing variants and constraints. Finally, in section 1.4, we close with conclusions and opportunities for further research.

1.2. The capacitated vehicle routing problem with two-dimensional loading constraints

The 2L-CVRP is a variant of the classical CVRP characterized by the two-dimensionality of customer demand. The problem aims to serve a set of customers using a homogeneous fleet of vehicles with minimum total cost. The 2D loading constraints must be respected.

The 2L-CVRP is available in a set of real-life problems (Sbai et al. 2020b), for example household appliances and professional cleaning equipment. Table 1.1 presents a comparative study of the existing literature for the 2L-CVRP, which includes solution methods, variants and constraints.

Table 1.1. Comparative study of the 2L-CVRP

1.2.1. Solution methods

The 2L-CVRP is an NP-hard problem, it is solved by exact, heuristic and metaheuristic algorithms:

Iori et al. (2007) use the first exact algorithm for solving small-scale instances of the 2L-CVRP and only for the sequential variant. They proposed a branch-and-cut approach for the routing problem and branch-and-bound for the packing problem.

Gendreau et al. (2008) use a Tabu search (TS) metaheuristic algorithm. They considered two loading heuristics for the sequential and unrestricted case, known as the LH2S L and the LH2U L.

Zachariadis et al. (2009) propose another metaheuristic algorithm which integrates TS and guided local search, referred to as GTS. For the loading problem, they used five packing heuristics and three neighborhood searches to generate the initial solution, namely: customer relocation, route exchange and route interchange.

Fuellerer et al. (2009) present an algorithm based on ant colony optimization (ACO) while bottom-left-fill and touching perimeter algorithms are proposed for solving the packing problem.

Leung et al. (2011) propose an extended guided Tabu search (EGTS) algorithm for the routing problem and a lowest line best-fit heuristic (LBFH) to solve 2D-BPP.

Duhamel et al. (2011) use the greedy randomized adaptive search procedure and the evolutionary local search algorithm, denoted GRASP-ELS.

Leung et al. (2013) study the heterogeneous fleet vehicle routing problem (2L-HFVRP). They propose six packing heuristics to check the feasibility of loading (presented in Table 1.1) and simulated annealing with a...