tical Constraints
Practical constraints in the container loading problem turn up in different forms. Some constraints are related to the enclosing container, while some are related to the items being packed, and in some instances where the items being moved are heavy, some constraints are related to the forklift trucks used to move the items. These constraints may occur in practice as ‘hard’ or ‘soft’ constraints. Hard constraints must be satisfied, while soft constraints can have violations tolerated up to certain limits, if not completely satisfied.
A quick review of the literature turns up three basic constraints that seem to be a fundamental part of the canonical problem definition for the CLP, and were not explicitly observed, are assumed to be implicitly observed by most. These include: (i) the packing of boxes orthogonally to the walls of the enclosing container, (ii) the placing of all boxes completely within the walls of the container, and (iii) making sure boxes do not intersect each other. Due to the ubiquity of these constraints, some might not refer to them as ‘practical’ constraints; indeed these particular three have been referred to as ‘geometric’ constraints by some. In addition to these constraints, some solutions also insist that the entire base of a box must be fully supported by either the container floor or another box, while some allow a little overhang when boxes are placed on other boxes.
Apart from these basic constraints, the rest of the practical constraints typ- ically encountered are related to orientation, stacking, stability, weight distri- bution, weight capacity, multidrop prioritisation, and complete shipment of item groups. These are covered in detail in the review byBortfeldt and W¨ascher[2013], who divides them into categories in relation to the container, the individual items being loaded, the entire cargo being loaded, and the positioning of the cargo. I will briefly describe a number of these here.
In practice, a typical container has weight limits that must not be exceeded. In most container loading problems, the volume of the container is the objective
function that is maximised. When heavy goods are to be loaded, the weight limits are often met before the volume limits are encountered. In cases like these, the weight of the container becomes the objective function that is maximised as the weight limits become more restrictive than the volume limits. Weight limit constraints are typically treated as hard constraints. Another constraint related to weight is the weight distribution constraint. It requires the distribution of weight to be spread out almost evenly across the container floor. This is important in order to satisfy axle weight guidelines for the trucks that transport the containers, as well as to provide a stable load that reduces the movement of cargo when the container is in transit. When considered, weight distribution constraints are often treated as soft constraints.
The orientation constraint is the most common constraint considered in the CLP literature. In theory, there can be up to six different orientations possible i.e., three vertical orientations each having two horizontal orientations. In practice, however, the vertical orientation is often fixed e.g. goods that have stickers with directions to load ‘This way up ↑’, or items that have to be stood up a particular way. This results in there being only two possible orientations in which to pack items. Additional limits can also be placed on the horizontal orientations allowed. This can be seen for example in the problem considered in this thesis (described in Chapter 3) where different types of pallets are used to pack goods, and some of the pallets allow loading using forklift trucks from only two sides i.e., the front and the back, while others allow loading from all four sides. When we consider that we load the palletised goods orthogonally in the container, and account for symmetry, then the pallets that allow loading from all four sides give us two possible horizontal orientations while the pallets that allow loading from only two sides give us only one possible orientation in which the pallet can be packed. Orientation constraints are treated as hard constraints in the literature.
Stacking constraints are usually introduced as hard constraints to help prevent damage to the items being packed. They are concerned with restrictions on how boxes are placed on top of each other and are also referred to as ‘load- bearing’ constraints in the literature (see Junqueira et al. [2012b]) because they are concerned with the load-bearing strength of boxes and how much weight they can sustain before they get damaged. Several methods exist for dealing with
stacking constraints in practice. One such method is to always require heavier items to be placed below lighter ones; another separates items into groups of ‘stackable’ and ‘non-stackable’ items where stackable items are those that can have other items placed on them and non-stackable items can not. The choice of which items are stackable or not could be determined by the shape of the top surface of the item to be packed, for example, if the top surface is uneven. It could also be determined by a ‘Do not double-stack’ directive. Items might also be marked as ‘fragile’ with a meaning attached that directs loaders not to place the items on any other items and not to have other items placed on them. Another observed method considers the item density and places items of higher density below items of lower density.
The complete shipment constraint refers to the case where if an item that belongs to a subset of items is loaded into a container, all other items belonging to the same subset must be loaded as well. Examples of this might include the loading of different furniture parts, or as with the problem dealt with in this thesis, items belonging to the same customer order. In both examples, if a single part of the subset is loaded, the rest must be loaded as well. Another case for this constraint is mentioned in literature, the difference lying in the number of containers the items are loaded into. In this case, if an item that belongs to a subset of items is loaded into one of many containers used to load a given shipment, it is sufficient that the other items in the same subset are loaded as part of that shipment and not necessarily in the same container. When dealt with in literature (e.g. see Eley [2003]), complete shipment constraints are treated as hard constraints.
Loading priority constraints refer to a situation where a subset of items must be loaded from a given set of items. For example, this might be because of a deadline placed on the delivery of the particular subset of items, or because of the items having a higher delivery priority than others e.g. first class post items will be given more priority than second class post items. This constraint is usually treated as a hard constraint where we find that all high priority items must be loaded first before any low priority item is loaded.
Positioning constraints deal with the restriction on the position of items within a container. The literature distinguishes between ‘absolute’ and ‘relative’ posi-
tioning. With absolute positioning, items are restricted to (or restricted from) very specific locations, i.e. absolute positions, within a container. Relative posi- tioning, on the other hand, restricts the placement of items relative to each other e.g. requiring that items belonging to the same customer order be placed next to each other within the container. In practice, situations that require the delivery of packed items to multiple locations exhibit both absolute and relative position- ing constraints. The items that will be delivered to the same location are kept close together relative to each other, while groups of items are kept in absolute positions in the container such that each group can be unloaded according to the order of the locations being delivered to. Here, they are mostly treated as hard constraints.
The stability constraint deals with how stable the items are when being packed or unpacked, and how stable an entire packed load is when being moved. They are of significant importance in the literature and are often presented in the form of requiring that the bottom surface area of an item to be packed must be completely supported by either the top surface area of another item or the container floor. In some cases, where an item is placed on another item, partial support that results in a little overhang may be allowed (see Gehring and Bortfeldt [1997]; Tarantilis et al. [2009]). Interlocking arrangements in the load may also be used to reduce motion during transit. The use of ‘filler’ materials is also introduced in practice to plug any gaps left after loading to keep entire loads stable.
Pattern complexity constraints deal with how easy it is for generated loading patterns to be understood and implemented by human loaders or robots. It is of importance because complex patterns that achieve a very compact and high container fill may not be implementable by human loaders or loading robots without considerable extra effort. This results in loading patterns that are easy to describe and pack being more desirable in practice than complex patterns that might obtain a higher fill. An example of the complexity constraint in practice is the use of the ‘guillotine’ pattern. It is a pattern that can be obtained by a series of cuts made parallel to the walls of the container. It is frequently considered in the literature as it can easily be described and packed.