مدل های ریاضی برای مسائل بارگذاری چندین کانکتور  Mathematical models for multicontainer loading problems

1. Introduction Everyday a distribution company has to decide how to put goods onto pallets to serve the customers’ orders and then how to efficiently distribute these pallets among the trucks, in order to get the right goods to the right place, and in the desired conditions, while minimizing the number of trucks used. In this paper we will take as a reference a large company in Europe, but the problem is common to many other distribution companies around the world. Solving the problem in an optimal way that produces a reduction in the transportation costs for the companies, thus increasing the profits and also decreasing the greenhouse emissions. The loading problem consists of two, interrelated, phases. All items have to be placed on pallets; we call this phase pallet building, and all pallets have to be placed onto trucks; we call this phase truck loading. In the pallet building phase the items are grouped in layers and the layers are stacked on the pallet base. A layer is an arrangement of units of the same product, forming a rectangle whose dimensions and number of items are known. A layer completely covers the pallet base in horizontal directions without gaps, ensuring sufficient support for other layers that may be placed on top of it to make up the pallet. The layer composition of each product is decided in advance by the company, so the layer composition is given and we are left with the problem of stacking layers to build pallets. Once the pallets are built, they have to be placed onto the trucks. In the following, we suppose that there is an infinite supply of identical trucks. The loading problem we deal with is a Multi Container Loading Problem (MCLP). The means of transport, in this case the use of trucks, introduces some constraints that have to be respected for safety reasons. One important feature is the weight limit. There is a maximum weight that can be loaded and this limit cannot be exceeded. There are also limits on the maximum weight each axle can bear. Indeed, there is not only a limit on the total weight of the cargo, but there are also limits on the weight that each axle can bear. Excesses over these weight limits represent a risk for traffic safety and can cause damage to the road. Therefore, they are strictly controlled and the violations severely punished. Some roads have Weight-In-Motion systems, that monitor axle weight violations while driving (see, e.g., Jacob and Feypell-de La Beaumelle [1]). Moreover, for safety reasons the load has to be well spread into the truck to avoid movements during the journey. This means that the center of gravity has to be placed in between the axles and as near as possible to the geometric center of the truck.