Optimization algorithms can be classified in two main categories: either exact or heuristic methods. For most N P -hard combinatorial optimization problems, exact algorithms have very large run times or even become intractable for instances that are not very small (assuming that P 6= N P ). Despite all the recent advances both in mathematical programming theory and in the computer technology, in a real-world context, it may be unrealistic to try to solve these problems with an exact method. Heuristic algorithms, on the other hand, have typically much smaller computational times, although their solution is not (proven to be) optimal. This may also be undesirable in situations where small improvements in the solution are translated into considerable savings.
The trade between the quality of the solution and computational time must be contextualized in each particular situation, so that, depending on the reality to consider, the best method can be chosen.
The heuristic methods [196] may be classified as constructive heuristics, local search heuristics or metaheuristics. Constructive heuristics build solutions step by step, incrementing partial solutions iteratively, until a global one is reached. When
2.9. Heuristic Solution Methods 27
the criterion to choose the next part to increment is based on the immediate max-imization of profit, the heuristic is said to be greedy. Local search heuristics [1]
iteratively try to improve a given solution by searching another one in its neigh-borhood. This can be achieved by provoking perturbations on the initial solution.
The best solution of a given neighborhood is called a local optimum. Metaheuris-tics are heuristic solution schemes for optimization problems, whose concept was introduced in [103]. They represent a heuristic algorithmic framework where the solution space of a combinatorial optimization problem is explored in a more guided and efficient way than in the more basic heuristics. They even admit intermedi-ate non improving or even infeasible solutions in order to escape local optima.
Interestingly, some of the most known metaheuristics are inspired in natural pro-cesses, like animal behavior, genetic evolution or physical processes. Descriptions of some of the most important metaheuristics in the literature are provided in [104, 35]. The following methods are among some of these: tabu search [103, 105], simulated annealing [127], genetic algorithms [112], ant colony optimization [71], variable neighborhood search [159]. These methods have shown significant success in solving difficult combinatorial optimization problems by finding near-optimal, and sometimes even optimal solutions.
In the last decades, a new approach has been considered, the combination of metaheuristics and exact algorithms for combinatorial optimization problems. This combination may be collaborative or integrative [170, 36]. In the former method, the different algorithms run separately, in a sequential or parallel way, while exchanging information. In the latter one, there is a main method that embeds the other in some subroutine. In [121], the authors propose a taxonomy to methods that hybridize exact and metaheuristic algorithms.
Chapter 3
Cutting & Packing Problems
“To understand is to perceive patterns.”
Isaiah Berlin
Contents
3.1 Problem Definition . . . 30 3.2 One-dimensional Cutting Stock Problems . . . 34 3.3 Two-dimensional Cutting Stock Problems . . . 39
29
30 3. Cutting & Packing Problems
3.1 Problem Definition
Cutting and Packing (C&P) problems represent a class of combinatorial optimiza-tion problems, belonging to the geometric combinatorics field, due to their geomet-ric structure. Generally speaking, in a C&P problem there is a set of small items that must be placed, without overlapping, in a set of large objects. These problems are greatly connected to a wide variety of commercial and industrial real-world sit-uations, including all those that consider the cutting of raw materials, packing of objects or loading vehicles but also, in a broader sense, other ones concerning the cutting or packing of abstract objects, like packing time in scheduling problems.
Most extensions and variants of Cutting Stock Problems (CSP) are well known to be NP-hard. This means, as discussed in chapter 2, that all algorithms cur-rently known for finding optimal solutions may require a number of computational steps that grows exponentially with the problem size rather than according to a polynomial function. Over the years, many different optimization formulations and solution approaches have arisen in the literature, all including different dimen-sions, application fields and special constraints and requirements. This originated a variety of different definitions and notations for problems with similar logical struc-ture. Therefore, many researchers provided surveys and categorized bibliographies on this subject (Dowsland and Dowsland [72], Dyckhoff and Finke [75],Dyckhoff et al. [76], Hopper and Turton [115] Lodi et al. [142, 144]). Moreover, Dyckoff et al. [74] developed a typology of these problems. In this typology, there is a classi-fication scheme that combines four main characteristics in the form of four-tuples α/β/γ/δ, in order to classify any C&P problem as one of 96 possible subproblems.
These characteristics are:
Dimensionality (α): the minimum number of spatial dimensions necessary to describe a cutting pattern of a feasible solution, that can be (1), (2), (3) or (N > 3).
3.1. Problem Definition 31
Kind of assignment (β): describes whether all objects and only a selection of small items must be used (B), or if all small items and only a selection of objects are considered (V).
Assortment of large objects (γ): describes whether there is one or many, identical or different large objects to assign. (O) for one large object, (I) for many identical large objects and (D) for many different large objects.
Assortment of small items (δ): describes whether there are few/many identical/different small items to assign. (F) if there are few small items of different dimensions, (M) or (R) if there are many small items of, respectively, many or relatively few different dimensions or (C) if there are many identical small items.
In [200], W¨ascher et al. developed an improved typology of C&P problems, based on the previous one [74], with new categorization criteria, in order to over-come some deficiencies detected in the previous work. They considered five main criteria to classify problems in standard problem types. The first criterion, kind of assignment, is similar to the one in [74] and distinguishes the problems according to their objective functions (input minimization, for the cases where all small items can be assigned to the large objects, or output maximization, for the cases where only a subset of small items can be assigned to the large objects). In what concerns the assortment of small items, there can be three different situations: identical small items or a weakly or strongly heterogeneous assortment of small items. For the assortment of large objects, there can be one large object or several large objects.
Whenever there is only one large object, it can have fixed or variable dimensions.
If there are several large objects, they can all be identical or have a weakly or strongly assortment. With respect to the problem’s dimensionality, the authors only consider one, two or three-dimensional problems. Finally, they introduce a new criterion for the two or three-dimensional problems that further distinguishes
32 3. Cutting & Packing Problems
Multiple Identical Large Object PP MILOPP
Multiple Heterogeneous Large Object PP MHLOPP
Figure 3.1: Cutting and packing typology [200]
the shape of small items, which can be regular or irregular. By combining the first two criteria, kind of assignment and assortment of small items, the authors define six basic types of C&P problems. These types are then structured further into four-teen intermediate problem types by the additional application of the third criterion assortment of large objects. Finally, the application of the criterion dimensional-ity and, for the two and three-dimensional problems, the criterion shape of small items, provides the refined problem types. The identification of any C&P prob-lem should be stated in the form {dimensionality}-{shape}-{intermediate probprob-lem type}. Whenever a problem presents all the properties of its refined problem type, but it also has some additional characteristics or constraints, it is called a second-level standard problem. Problems of second-second-level standard types whose additional constraints are related with aspects that do not belong to the cutting or packing
3.1. Problem Definition 33
field are said to belong to an extended problem type. Finally, when the assump-tions of a given problem are different from those of the standard problems, it is considered as a variant.
Some approaches in the literature integrate the standard versions of the problem with additional aspects. This is the case of the pattern minimization problem, where there are setup costs to consider, and thus the objective is to minimize the number of different cutting patterns, within an optimal solution for the standard CSP [193, 186, 15]. Some versions also include constraints related with due dates [139, 120, 173], multiple lengths for the large objects [30, 57, 16], minimization of open stacks [203, 18, 204], which imply constraints with the sequencing of cutting patterns, or even the reuse of generated waste in a multiperiod context [183], among others.
The interest in the field of C&P over the last decades, by practitioners and researchers in this area, has been so significant that the European Journal of Op-erations Research (EJOR) has published four special issues on C&P problems over the last twenty years [77, 34, 198, 168]. These special issues are composed by papers that have covered both exact and heuristic methods, nesting problems, real-world application problems, surveys and typologies of C&P problems.
A list of classified articles from 1980 to 2005 (according to [200]) concerning C&P problems can be found in the web page of ESICUP [78]. ESICUP (EURO Special Group on Cutting and Packing) is a working group of the association of European Operations Research Societies ( EURO [79]), composed of researchers with an interest on this particular field.
In chapters 4 and 5, we propose new lower bounds for the one-dimensional pat-tern minimization problem and an exact solution algorithm for the two-dimensional cutting stock problem. These two C&P problems are classified according to this typology as 1D-SSSCSP and 2D-regular-SSSCSP, respectively.
In this thesis, we are, therefore, specially concerned with cutting stock problems
34 3. Cutting & Packing Problems
(CSP), in particular the one-dimensional and two-dimensional cases. The knapsack problem (KP) is also approached, as a pricing subproblem of a column generation model framework proposed in chapter 4. A thorough presentation of this last problem and several of its variants and extensions, as well as a comprehensive description of its the most relevant algorithmic methods is presented in [152, 126].
The CSP is equivalent to the Bin Packing Problem (BPP), in the sense that they both have essentially the same logic structure. According to [200], and as shown in figure 3.1, the only difference between the two problems relies on the fact that in the CSP there is a weakly heterogeneous assortment of small items, whereas in the BPP this assortment is strongly heterogeneous. This basically means that the main difference between these two problems is that in the CSP the demands are high (there are several copies of each dimension of small items), whereas in the BPP the demands are very low (there is typically a single copy of each dimension of small items). Thus, most solution methods devised for one of the problems can easily be adapted to solve the other, although it can happen that a given algorithm is more efficient for one of the problems than it is for the other. That is the example of algorithms that rely on the construction of patterns and the determination of the number of times those patterns should be used. These algorithms are clearly more appropriate for the CSP, where there are identical items, and thus patterns to repeat. A review of LP-based algorithms, sequential heuristics and hybrid solution procedures for the 1D- and 2D-CSP is provided in [109].