Heuristics for vehicle routing problems with backhauls, time windows, and 3D loading constraints

Highlights

• Vehicle routing problems with backhauls and time windows are considered.

• Three dimensional loading constraints are taken into account.

• The packing first, routing second principle is applied.

• Unloading and reloading efforts are considered if appropriate.

• Results for benchmark instances and new problem instances are presented.

Abstract

In this paper, we consider vehicle routing problems with backhauls and time windows (VRPBTW). Different backhaul variants are studied, namely clustered backhauls (CB), mixed linehauls and backhauls, and variants with simultaneous delivery and pickup and with divisible delivery and pickup. Three dimensional loading constraints are assumed. A two-phase approach following the principle packing first, routing second is proposed. In the first phase, the packing of goods is carried out by solving a 3D strip packing problem for each customer using tabu search. The resulting VRPTW instance is solved in the second phase using first a multi-start evolutionary strategy to minimize the number of vehicles while again tabu search is applied to minimize the total travel distance. We show that the various backhaul types can be incorporated into this framework. For the backhaul variants different from CB, unloading and reloading efforts are taken into account. Moreover, side loading and a separation of the loading space into separate compartments for goods of linehaul and backhaul customers are proposed. Computational results for benchmark instances and new randomly generated problem instances are presented that demonstrate that the heuristics determine high-quality solutions in a short amount of computing time. The unloading and reloading strategies outperform the strategies based on two separate compartments.

Introduction

Vehicle routing problems (VRPs) are important in many real-world applications. Recently, there is a trend to consider more real-world constraints in problem formulations (cf. Toth and Vigo 2014). This leads to the notation of rich VRPs (cf. Lahyani, Khemakhem, & Semet, 2015; Caceres-Cruz, Arias, Guimarans, Riera, & Juan, 2015). Important attributes of rich VRPs are backhaul customers, i.e., goods have to be picked up at the customer locations and have to be transported to the depot (cf. Caceres-Cruz et al., 2015). Another important class of attributes is given by loading constraints, i.e., more complex loading strategies for the boxes with the customer goods are considered in addition to the fairly simple weight or volume constraints of the vehicles in conventional VRPs (cf. Lahyani et al., 2015; Pollaris, Braekers, Caris, Janssens, & Limbourg, 2015). Time windows are important in many real-world applications. This class of constraints is extensively considered in the literature (cf., for instance, Bräysy and Gendreau, 2005a, Bräysy and Gendreau, 2005b).

VRPs with backhaul and time window constraints are discussed to some extent in the literature (cf., for instance, Küçükoğlu & Öztürk, 2015; Parragh, Doerner, & Hartl, 2008; Reimann & Ulrich, 2006 amongst others). However, this is not true for the combination of backhauls and loading constraints. We are only aware of the paper by Bortfeldt, Hahn, Männel, and Mönch (2015) where a VRP with clustered backhauls and three dimensional loading constraints is studied and the paper by Pinto, Alves, de Carvalho, and Moura (2015) where a VRP with mixed linehauls and backhauls and two dimensional loading constraints is investigated. Time windows are not considered in these two papers. More generally, the combination of 3D loading constraints and time windows is only rarely discussed in the literature. Therefore, there is a need to consider VRPs where backhauls, loading constraints, and time windows are combined. For instance, global moving companies have to deal with such problems because they have to deliver house moving items to the customers. At the same time they might collect packaging material and bring it back to the depot. Time windows are also important in this application scenario.

In the present paper, we extend the approach proposed by Bortfeldt and Homberger (2013) for the VRPTW with 3D loading constraints (3L-VRPTW) by considering various types of backhauls. It turns out that the packing first, routing second (P1R2) heuristic from Bortfeldt and Homberger (2013) can be applied to situations with backhauls too. The P1R2 heuristic is based on the idea that in the packing stage the boxes of each customer are packed in a separate segment of the loading space by solving a 3D strip packing problem (3D-SPP) for each customer. A loading length arises for the boxes of each customer. In the second stage, the corresponding routing problem is solved where the sum of the loading lengths of the customers that are assigned to a single vehicle cannot exceed the loading space length of the vehicle. In the present paper, we consider variants where we take into account the unloading and reloading effort caused by mixed sequences of linehaul and backhaul customers. We study also situations where the loading space of each vehicle is either divided into two vertical or horizontal compartments of the same size, namely one for boxes of linehaul and one for boxes of backhaul customers. Moreover, we investigate side loading in addition to rear-loading strategies.

The paper is organized as follows. The problem is described in Section 2. This includes a discussion of related work. The proposed heuristics are presented in Section 3. The results of computational experiments are described in Section 4. Finally, future research directions are discussed in Section 5, while conclusions are presented in Section 6.