Optimization and Machine Learning. Patrick Siarry

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

Author	Problem	Routing problem Solution methods	Loading problem Solution methods
lori et al. (2007) Gendreau et al. (2008) Fuellerer et al. (2009) Zachariadis et al. (2009)	2L-CVRP 2L-CVRP 2L-CVRP 2L-CVRP	Branch-and-cut TS ACO GTS	Branch-and-bound LH2SL, LH2U L LB, Branch and Bound Bottom-Left Fill (L,W axis) Max Touching Perimeter Max Touching Perimeter No Walls Min Area Bottom-Left Fill(L,W axis) Max Touching Perimeter Max Touching Perimeter No Walls Min. Area LBFH GRASP-ELS PRMP
Leung et al. (2011)	2L-CVRP	EGTS
Duhamel et al. (2011) Zachariadis et al. (2013) Wei et al. (2015) Wei et al. (2017) Sbai et al. (2020b) Leung et al. (2013)	2L-CVRP 2L-CVRP 2L-CVRP 2L-CVRP 2L-CVRP 2L-HFVRP	GRASP-ELS LS VNS SA GA SAHLS	Skyline heuristic Open space based heuristic ALWF Bottom-Left Fill (L,W axis) Max. Touching Perimeter Max. Touching Perimeter No Walls Min. Area Max. fitness value Bottom-Left Fill (L,W axis) Max Touching Perimeter Max. Touching Perimeter No Walls Min. Area Max. fitness value Lower Bound L-cuts MA ALWF ILP GVNS BLH Best-Fit LS VNS VNS LS Bottom-Left Heuristics
Sabar et al. (2020)	2L-HFVRP	MA
Cote et al. (2013) Cote et al. (2020) Khebbache-Hadji et al. (2013) Sbai et al. (2017) Attanasio et al. (2007) Song et al. (2019) Pinto et al. (2015) Dominguez et al. (2016) Zachariadis et al. (2017) Pinto et al. (2017) Pinto et al. (2020) Zachariadis et al. (2016) Malapert et al. (2008)	S2L-CVRP S2L-CVRP 2L-CVRPTW 2L-CVRPTW 2L-CVRPTW 2L-CVRPTW 2L-VRPMB 2L-VRPB 2L-VRPB 2L-SPD 2L-VRPB 2L-VRPMB 2L-SPD 2L-VRPPD	L-Cuts L-Cuts MA GA ILP GVNS Insert-heur LNS Touch-Per LS VNS VNS LS Scheduling based-model

      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.