Extending the study to a new set of test instances

classes BR8-BR15 of strongly heterogeneous instances. The number of box types ranges from 30 for BR8 to 100 for BR15.

The classes BR8-BR15 have not been solved yet in any study about load-bearing constraints, but it is interesting to include them in this compu- tational study in order to assess the performance of the algorithm in these cases. In a similar way to classes BR1-BR7, we have added weights which are proportional to the box volumes and load-bearing strengths taken from a uniform distribution.

3.9. Summary 83 Table 3.12 compares the results of these new BR8-BR15 classes with those of BR1-BR7 when solving them with a limit of 5000 GRASP itera- tions. It can be seen that the average occupied volume decreases when heterogeneity increases. This effect was already observed when only full support was required and it also appears here when additional constraints for load-bearing are added.

For our GRASP algorithm, the computing times increase very sharply because in strongly heterogeneous instances there are no large blocks. Small blocks or even individual boxes are packed at each step and many more steps are required to complete a solution.

Class Average Class Average

BR1 81.4 BR8 85.5 BR2 85.7 BR9 84.8 BR3 87.3 BR10 84.0 BR4 86.9 BR11 82.8 BR5 86.6 BR12 81.3 BR6 86.3 BR13 80.2 BR7 85.7 BR14 78.9 BR15 78.0 Av. BR1-7 85.7 Av. BR8-15 83.7

Time BR1-7 (sec.) 9.8 Time BR8-15 (sec.) 64.7 Table 3.12. Comparing classes BR1-BR7 with BR8-BR15

3.9

Summary

We have developed a GRASP algorithm with a new constructive proce- dure and some new improvement movements. The main contribution is the simultaneous use of several criteria for selecting the blocks, defining different strategies for filling the container. As none of them is shown to be the best and no correlation between the objective function and the char- acteristics of the instances has been found, the procedure starts by giving each objective function the same probability of being chosen. The history

of the search is then used to adjust these probabilities to favor the best performing rules for the instance being solved. Some improvement moves have also been defined and implemented. The proposed algorithm obtains better results than other existing algorithms for all the classes in the test problems.

Chapter 4

A case study of delivering

products by trucks

4.1

Introduction

The mission of logistics is basically to get the right goods or services to the right place, at the right time, and in the desired conditions, while mak- ing the greatest contribution to the company [66]. In the previous chapter we covered the problem of loading boxes directly into containers, which forms part of the container loading problem. But, as mentioned in the in- troduction, the loading depends on the type of products. Some products have to be placed first onto pallets and then the pallets are placed into containers or trucks. The first problem is known as thepallet loading prob- lem, whereas the second problem is the container loading problem. Both issues have attracted the attention of professionals and researchers be- cause they are complex optimization problems that have to be solved as efficiently as possible. These are the problems covered in this chapter.

This study began as a collaboration between our research group and some colleagues in the Netherlands, with the idea of applying the algo- rithms we had developed to real problems. In this way, thanks to the work they do for a distribution company, we were provided with a real scenario in which to carry out our research.

Everyday a distribution company has to decide how to put goods onto pallets according to the customers’ orders and then how to efficiently dis- tribute these pallets among the number of trucks so as to reduce the trucks needed to supply the customers. In this scenario, the company described to us its working process in order to check whether our algorithms could improve their processes in any way.

We deal with the inter-depot transportation problem. The company sup- plies a group of depots spread around the country. At the company, a daily decision has to be taken about which trucks will be sent to each depot to distribute the goods.

The problem begins when a depot sends a set of orders. Anorder is a request to deliver a collection ofitemsof oneproductthat the depot wants to be supplied with before or at a certain due date. Usually there are many items of one product, all having the same (rectangular) size and weight. The loading problem consists of two, interrelated, phases:

• All items have to be placed on a pallet; we call this phasepallet build- ing.

• All pallets have to placed in one of the trucks; we call this phasetruck loading.

There is only one type of pallet, the ISO pallet of horizontal dimensions 1219.2× 1016 millimeters. The trucks are all of the same characteristics and available in a large quantity. In the next two sections we describe the constraints of the pallet building and the truck loading phases.