A Preference Order Dynamic Program for a Knapsack Problem with Stochastic Rewards

Steinberg, Earle; Parks, Michael S.

doi:10.1057/jors.1979.27

Cited by 58 publications

(10 citation statements)

References 19 publications

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“…Another type of stochastic knapsack problem with reversed roles of constants and random values was discussed by a number of authors starting (to the best of our knowledge) with the paper by Steinberg and Parks [445] in 1979 and continued in the work of Sniedovich [438], [439], Carraway, Schmidt and Weatherford [72], Morita,Ishii and Nishida [351], Henig [222], Morton and Wood [352] and others. The general model arises in the context of a financial decision problem where every item corresponds to an investment project with a fixed investment requirement W j but stochastic future profits Pj which implicitly include the involved risk.…”

Section: Related Models and Algorithmsmentioning

confidence: 99%

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: Related Models and Algorithmsmentioning

confidence: 99%

Knapsack Problems

2004

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“…Example 6. We consider a stochastic integer knapsack problem taken from Steinberg and Parks (1979) with the size of the knapsack to be 30. Table 4 shows the weight and mean and variance information of the utilities.…”

Section: Discrete Choice Under Budget Constraintmentioning

confidence: 99%

Persistency Model and Its Applications in Choice Modeling

2009

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Given a discrete maximization problem with a linear objective function where the coefficients are chosen randomly from a distribution, we would like to evaluate the expected optimal value and the marginal distribution of the optimal solution. We call this the persistency problem for a discrete optimization problem under uncertain objective, and the marginal probability mass function of the optimal solution is named the persistence value. In general, this is a difficult problem to solve, even if the distribution of the objective coefficient is well specified. In this paper, we solve a subclass of this problem when the distribution is assumed to belong to the class of distributions defined by given marginal distributions, or given marginal moment conditions. Under this model, we show that the persistency problem maximizing the expected objective value over the set of distributions can be solved via a concave maximization model. The persistency model solved using this formulation can be used to obtain important qualitative insights to the behavior of stochastic discrete optimization problems. We demonstrate how the approach can be used to obtain insights to problems in discrete choice modeling. Using a set of survey data from a transport choice modeling study, we calibrate the random utility model with choice probabilities obtained from the persistency model. Numerical results suggest that our persistency model is capable of obtaining estimates that perform as well, if not better, than classical methods, such as logit and cross-nested logit models. We can also use the persistency model to obtain choice probability estimates for more complex choice problems. We illustrate this on a stochastic knapsack problem, which is essentially a discrete choice problem under budget constraint.probability distribution, integer programming, utility preference, choice functions

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“…Static versions of the stochastic knapsack problem have been studied. In these, the set of objects is known and the rewards and/or weights are random (Carraway et al 1993;Henig 1990;Sniedovich 1980Sniedovich , 1981Steinberg and Parks 1979). The objective typically is to maximize the probability of attaining some prespeci ed level of utility or reward.…”

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The Dynamic and Stochastic Knapsack Problem with Deadlines

1996

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In this paper a dynamic and stochastic model of the well-known knapsack problem is developed and analyzed. The problem is motivated by a wide variety of real-world applications. Objects of random weight and reward arrive according to a stochastic process in time. The weights and rewards associated with the objects are distributed according to a known probability distribution. Each object can either be accepted to be loaded into the knapsack, of known weight capacity, or be rejected. The objective is to determine the optimal policy for loading the knapsack within a fixed time horizon so as to maximize the expected accumulated reward. The optimal decision rules are derived and are shown to exhibit surprising behavior in some cases. It is also shown that if the distribution of the weights is concave, then the decision rules behave according to intuition.dynamic programming, sequential stochastic resource allocation

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A Preference Order Dynamic Program for a Knapsack Problem with Stochastic Rewards

Cited by 58 publications

References 19 publications

Knapsack Problems

Knapsack Problems

Persistency Model and Its Applications in Choice Modeling

The Dynamic and Stochastic Knapsack Problem with Deadlines

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