W¨ ascher’s Improved Typology for C&P Problems

In 2006, W¨ascher et al. [157] attempted to improve Dyckoff’s proposed typology. Some weak-

nesses in his typology had become apparent during the fifteen years since Dyckhoff first proposed his typology for C&P problems. For example, Dyckhoff proposed that the strip-packing problem should be coded as 2/V/D/M, while other researchers preferred to code it as 2/V/O/M [157, p. 4]. Another problem, identified by Gradiˇsar et al. [64, p. 1208], was that there was no possibility in the assortment of large objects for few groups of identical objects. They proposed a fourth possibility for this characteristic, labelled G (increasing the possible number of categories of C&P problems to 128). This eliminated the ambiguous notation for the case of items being packed into many variably-sized large objects versus the case where small items are packed into many large items that can be sorted into few groups of identically-sized items. Now, instead of both these problems being labelled as 1/V/D/R problems (which may be solved by means of

2.1. Classifications of Cutting and Packing Problems 9

an item-oriented approach), the latter may be differentiated by being labelled as a 1/V/G/R problem (which may be solved by means of a pattern-oriented approach).

Example 2.2 The addition of the labelling proposed by Gradiˇsar et al. for an assortment of

large objects consisting of few groups of many identical objects to Dyckhoff ’s typology allows the case in Example 2.1 to be relabelled. In that example the large objects were few groups of many

identical boards. Thus, the cutting problem of Example 2.1 may now be labelled as a 2/V/G/R

problem.

Examples of pattern-oriented approaches to solving C&P problems are described by Eisemann [45], Gilmore and Gomory [57,58], Haessler and Talbot [66], Pandit [132] and Yanasse et al. [159]. These pattern-oriented approaches to C&P problems are typically solved by a column generation method. Lodi et al. [106] describe item-oriented algorithms for C&P problems as procedures in which each item is considered individually for packing into an object. These algorithms include, for example, the First Fit, Best Fit, Next Fit and Worst Fit algorithms, etc. and all their derivatives. Many of these algorithms are described in more detail later in this chapter. Other authors who have studied item-oriented packing include Coffman et al. [31], Lodi [101], Lodi et al. [106] and Ntene [125].

W¨ascher et al. [157] agreed with Dyckhoff’s dimensionality characterisation and left it un-

changed. However, they felt that the German notations Verladeproblem and Beladeproblem should be avoided, leading to their proposal to change the problem categories to either input minimisation (a set of small items must be assigned to a set of large objects, such that all large objects are used), or output maximisation (a set of small items must be assigned to a set of

large objects, such that all small items are used). Although W¨ascher et al. did not develop

codes for problem types in the same manner that Dyckhoff did, Ntene [125, p. 2] labelled the problem types IM (equivalent to Dyckhoff’s V) and OM (equivalent to Dyckhoff’s B), respec- tively. The assortment of small items characterisation was reduced to three options, namely identical small items (denoted by IS by Ntene, corresponding to Dyckhoff’s C labelling), a weakly heterogeneous assortment of small items (many items are identical, labelled as W by Ntene, corresponding to Dyckhoff’s R labelling) and S for a strongly heterogeneous assortment of small items (very few items are identical, labelled as S by Ntene, corresponding to Dyckhoff’s M and F labels).

Although the improved typology is still similar to the original, it is in the assortment of large

objects that the major change occurs. Here W¨ascher et al. [157] group C&P problems into two

categories, each with subcategories. The first set of problems is the class dealing with only one large object (labelled as O by Ntene), and this class may be partitioned into problems where all dimensions of the objects are fixed (subset labelled Oa by Ntene, identical to Dyckhoff’s type O), and those where one dimension is variable (labelled Oo by Ntene, for strip packing problems) or where more dimensions of the object are variable (subset labelled Om by Ntene). The second set of problems are those dealing with several large objects (labelled Sf by Ntene). This set of problems may be divided into three subsets, namely those problems where the large objects are identical (labelled Si by Ntene, identical to Dyckhoff’s type I), those problems where the objects are weakly heterogeneous (labelled Sw by Ntene) and those problems where the objects are strongly heterogeneous (labelled Ss by Ntene). The final two sets make up the grouping Dyckhoff called type D. W¨ascher et al. claim that it does not seem important to differentiate between problems that deal with variable dimensions and those that do not within the Sf group, as only problems with fixed dimensions had been considered in literature [157, p. 8].

Example 2.3 The case described in Example 2.1 may be characterised as a2/IM/Sw/W prob-

lem using Ntene’s labelling of W¨ascher et al.’s typology. W¨ascher et al. call it a Multiple Stock

Size Cutting Stock Problem.

There are some problems that W¨ascher et al. do not include in their typology. These include

• problems where large objects are non-rectangular (such as disks).

• problems where items/objects are inhomogeneous, for example, stock material with de- fects.

• problems where items have irregular shapes (such as in the clothing industry).

These are considered problem variants. The packings in all problems of the typology by W¨ascher

et al. are also all assumed to be orthogonal, i.e. the edges of the small items must be parallel

or perpendicular to the edges of the large objects into which packing occurs. Orthogonal and non-orthogonal packings of regular items are illustrated in Figure 2.1.

(a) Orthogonal Packing (b) Non-Orthogonal Packing

Figure 2.1: A comparison of orthogonal and non-orthogonal packings (shaded areas denote empty spaces). A packing is orthogonal if the edges of a rectangular item are parallel or perpendicular to the sides of the large object.

W¨ascher et al. named all possible problems in their typology. These names may be found in

Table 2.1 for output maximisation problems and in Table 2.2 for input minimisation problems.