A solution procedure for general knapsack problems with a few constraints

Ozden, Mufit

doi:10.1016/0305-0548(88)90007-x

Cited by 6 publications

(5 citation statements)

References 15 publications

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“…For the unbounded version of (d-KP) there is an advanced algorithm by Ozden [364] for problems with a small number of constraints (d :S 5) based on the concept of DPwith-Lists. It applies a straightforward generalization of the simple dominance from Section 8.2.2 as a dominance test in a preprocessing phase to reduce the number of variables.…”

Section: Dynamic Programmingmentioning

confidence: 99%

“…This deadlock situation is broken by Ozden [364] through a partitioning of the given problem instance. On the other hand, we would like to have even tighter bounds and thus more effective reductions when the number of states increases.…”

Section: Dynamic Programmingmentioning

confidence: 99%

“…The implementation of [364] limits the number of subproblems, i.e. different classes in the multiple-choice problem, to at most 5.…”

Section: Dynamic Programmingmentioning

confidence: 99%

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Knapsack Problems

2004

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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...

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Section: Dynamic Programmingmentioning

confidence: 99%

Section: Dynamic Programmingmentioning

confidence: 99%

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Knapsack Problems

2004

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“…, w hd ) for each h ∈ I. The literature reports different exact solution algorithms for DKP (see Ozden, 1988;Fréville, 2004). It seems that none of these approaches allows the direct retrieval of the solution to the WSI separation problem in a straightforward way.…”

Section: Dynamic Separation Of Violated Dual Inequalitiesmentioning

confidence: 99%

Dual Inequalities for Stabilized Column Generation Revisited

Gschwind

Irnich²

2016

INFORMS Journal on Computing

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Column generation (CG) models have several advantages over compact formulations, namely, they provide better LP bounds, may eliminate symmetry, and can hide non-linearities in their subproblems. However, users also encounter drawbacks in the form of slow convergency a.k.a. the tailing-off effect and the oscillation of the dual variables. Among different alternatives for stabilizing the CG process, Ben Amor et al. [Ben Amor, H., Desrosiers, J., and Valério de Carvalho, J. M. (2006). Dual-optimal inequalities for stabilized column generation. Operations Research, 54(3), [454][455][456][457][458][459][460][461][462][463] suggest the use of dual-optimal inequalities (DOIs) in the context of cutting stock and bin packing problems. We generalize their results, provide new classes of (deep) DOIs, and show the applicability to other problems (vector packing, vertex coloring, bin packing with conflicts). We also suggest the dynamic addition of violated dual inequalities in a cutting-plane fashion and the use of dual inequalities that are not necessarily (deep) DOIs. In the latter case, a recovery procedure is needed to restore primal feasibility. Computational results proving the usefulness of the methods are presented.

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“…It is also a general formulation for problems in which resources are limited and decisions have to be made in order to satisfy the capacity constraints while maximizing benefits. Some examples include resource allocation problems (Ozden 1988) and sensor fusions for signal detection in limited bandwidth (Pao 1995). The wide applicability of this formulation and its variants have attracted much attention using, for example, neural networks (Ohlsson, Peterson and Soderberg 1992) and simulated annealing (Drexl 1988).…”

Section: The Knapsack Problemmentioning

confidence: 99%

Super‐heuristics and Their Applications to Combinatorial Problems

Lau

1999

Asian Journal of Control

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Combinatorial problems are known to be difficult because of the shear size of the solution space and the lack of polynomial time algorithms to “solve” them. Heuristics are often devised to produce acceptable solutions in an affordable time. In this paper, we propose a method called super‐heuristic that expands the capabilities of heuristics using randomization and sampling techniques. We submit that heuristics are in general strategies that map from available information of a problem instance to decisions in solution constructions/improvement. We show that it is important to utilize the information effectively in the randomization process. More importantly, the possibility of randomization around a heuristic spells out the demarcation between the roles of human and machines in complex optimization problems.

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A solution procedure for general knapsack problems with a few constraints

Cited by 6 publications

References 15 publications

Knapsack Problems

Knapsack Problems

Dual Inequalities for Stabilized Column Generation Revisited

Super‐heuristics and Their Applications to Combinatorial Problems

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