A computational study of LP-based heuristic algorithms for two-dimensional guillotine cutting stock problems

Álvarez-Valdés, Ramón; Parajón, Antonio; Tamarit, José Manuel

doi:10.1007/s00291-002-0093-3

Cited by 36 publications

(17 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…£ = to, which means that a deviation of 10 percent from the optimal solution value is still acceptable, we have a running time of O(n 9 ) or still O(n 8 ) in the improved implementation. However, if the desired accuracy gets higher its running time basically explodes.…”

Section: Approximation Schemesmentioning

confidence: 99%

See 1 more Smart Citation

Knapsack Problems

2004

View full text Add to dashboard Cite

PrefaceThirteen years have passed since the seminal book on knapsack problems by Martello and Toth appeared. On this occasion a former colleague exclaimed back in 1990: "How can you write 250 pages on the knapsack problem?" Indeed, the definition of the knapsack problem is easily understood even by a non-expert who will not suspect the presence of challenging research topics in this area at the first glance.However, in the last decade a large number of research publications contributed new results for the knapsack problem in all areas of interest such as exact algorithms, heuristics and approximation schemes. Moreover, the extension of the knapsack problem to higher dimensions both in the number of constraints and in the number of knapsacks, as well as the modification of the problem structure concerning the available item set and the objective function, leads to a number of interesting variations of practical relevance which were the subject of intensive research during the last few years.Hence, two years ago the idea arose to produce a new monograph covering not only the most recent developments of the standard knapsack problem, but also giving a comprehensive treatment of the whole knapsack family including the siblings such as the subset sum problem and the bounded and unbounded knapsack problem, and also more distant relatives such as multidimensional, multiple, multiple-choice and quadratic knapsack problems in dedicated chapters.Furthermore, attention is paid to a number of less frequently considered variants of the knapsack problem and to the study of stochastic aspects of the problem. To illustrate the high practical relevance of the knapsack family for many industrial and economic problems, a number of applications are described in more detail. They are selected subjectively from the innumerable occurrences of knapsack problems reported in the literature.Our above-mentioned colleague will be surprised to notice that even on the more than 500 pages of this book not all relevant topics could be treated in equal depth but decisions had to be made on where to go into details of constructions and proofs and where to concentrate on stating results and refer to the appropriate publications. Moreover, an editorial deadline had to be drawn at some point. In our case, we stopped looking for new publications at the end of June 2003. VI PrefaceThe audience we envision for this book is threefold: The first two chapters offer a very basic introduction to the knapsack problem and the main algorithmic concepts to derive optimal and approximate solution. Chapter 3 presents a number of advanced algorithmic techniques which are used throughout the later chapters of the book. The style of presentation in these three chapters is kept rather simple and assumes only minimal prerequisites. They should be accessible to students and graduates of business administration, economics and engineering as well as practitioners with little knowledge of algorithms and optimization.This first part of the book is also well suited to introd...

show abstract

Section: Approximation Schemesmentioning

confidence: 99%

“…We choose to pair the items in the order we meet them. Finally we pair (9,10) noticing that 10 is dominated and hence pairing (9,11). Next we pair (4,5) then (6,7) but since 7 is dominated we pair (6,8).…”

Section: If L~lmentioning

confidence: 99%

Knapsack Problems

2004

View full text Add to dashboard Cite

show abstract

“…From the round-down solution and the extra cutting patterns we know that the number of stock sheets needed is 61, so that this is must be the optimal solution of the master problem (1). The visualization of the selected cutting patterns and the extra cutting patterns can be seen in figures 3 and 4.…”

Section: Examplementioning

confidence: 99%

“…The column generation technique for solving the twodimensional cutting stock problem is actually has been studied in Alvarez-Valdes, et.al [1]. In this paper we propose an alternative sub problem for generating a new column, which is based on the stripe method proposed by Hifi [8].…”

Section: Introductionmentioning

confidence: 99%

Column Generation Technique for Solving Two-Dimensional Cutting Stock Problems: Method of Stripe Approach

Novianingsih

Hadianti

Uttunggadewa

2012

JIMS

View full text Add to dashboard Cite

Abstract. We consider two-dimensional cutting stock problems where single rectangular stocks have to be cut into some smaller rectangular so that the number of stocks needed to satisfy the demands is minimum. In this paper we focus our study to the problem where the stocks have to be cut with guillotine cutting type and fixed orientation of finals. We formulate the problem as an integer programming, where the relaxation problem is solved by column generation technique. New pattern generation is formulated based on method of stripe. In obtaining the integer solution, we round down the optimal solution of the relaxation problem and then we derive an extra mix integer programming for satisfying the unmet demands. The optimal solution of the original problem is the combination of the round-down solution and the optimal solution of the extra mix integer programming.A numerical example of the problem is given in the end of this paper.

show abstract

“…Among the most famous studies from that time are those of Gilmore and Gomory (Gilmore and Gomory 1961, 1963, 1965Alvarez-Valdes et al 2001;Eshghi and Javanshir 2005). Since the 1960s, many other cutting and packing problems with the same logic and structure emerged but were given different names in the literature, such as cutting stock and trim-loss problems, bin packing, strip packing, knapsack problems, loading pallets, and loading containers (Faina 1999;Eshghi and Javanshir 2005).…”

Section: Introductionmentioning

confidence: 99%

A simple approach to the two-dimensional guillotine cutting stock problem

et al. 2012

View full text Add to dashboard Cite

Cutting stock problems are within knapsack optimization problems and are considered as a non-deterministic polynomial-time (NP)-hard problem. In this paper, two-dimensional cutting stock problems were presented in which items and stocks were rectangular and cuttings were guillotine. First, a new, practical, rapid, and heuristic method was proposed for such problems. Then, the software implementation and architecture specifications were explained in order to solve guillotine cutting stock problems. This software was implemented by C++ language in a way that, while running the program, the operation report of all the functions was recorded and, at the end, the user had access to all the information related to cutting which included order, dimension and number of cutting pieces, dimension and number of waste pieces, and waste percentage. Finally, the proposed method was evaluated using examples and methods available in the literature. The results showed that the calculation speed of the proposed method was better than that of the other methods and, in some cases, it was much faster. Moreover, it was observed that increasing the size of problems did not cause a considerable increase in calculation time. In another section of the paper, the matter of selecting the appropriate size of sheets was investigated; this subject has been less considered by far. In the solved example, it was observed that incorrect selection from among the available options increased the amount of waste by more than four times. Therefore, it can be concluded that correct selection of stocks for a set of received orders plays a significant role in reducing waste.

show abstract

A computational study of LP-based heuristic algorithms for two-dimensional guillotine cutting stock problems

Cited by 36 publications

References 13 publications

Knapsack Problems

Knapsack Problems

Column Generation Technique for Solving Two-Dimensional Cutting Stock Problems: Method of Stripe Approach

A simple approach to the two-dimensional guillotine cutting stock problem

Contact Info

Product

Resources

About