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30
Algorithms for Knapsack Problems
, 1995
"... This thesis considers a family of combinatorial problems known under the name Knapsack Problems. As all the problems are A7)hard we are searching for exact solution techniques having reasonable solution times for nearly all instances encountered in practice, despite having exponential time bounds f ..."
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Cited by 66 (5 self)
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This thesis considers a family of combinatorial problems known under the name Knapsack Problems. As all the problems are A7)hard we are searching for exact solution techniques having reasonable solution times for nearly all instances encountered in practice, despite having exponential time bounds for a number of highly contrived problem instances. A similar behavior is known from the Simplex algorithm, which despite its exponential worstcase behavior has reasonable solution times for all realistic problems.
A Minimal Algorithm for the MultipleChoice Knapsack Problem.
 European Journal of Operational Research
, 1994
"... The MultipleChoice Knapsack Problem is defined as a 01 Knapsack Problem with the addition of disjoined multiplechoice constraints. As for other knapsack problems most of the computational effort in the solution of these problems is used for sorting and reduction. But although O(n) algorithms whic ..."
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Cited by 43 (4 self)
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The MultipleChoice Knapsack Problem is defined as a 01 Knapsack Problem with the addition of disjoined multiplechoice constraints. As for other knapsack problems most of the computational effort in the solution of these problems is used for sorting and reduction. But although O(n) algorithms which solves the linear MultipleChoice Knapsack Problem without sorting have been known for more than a decade, such techniques have not been used in enumerative algorithms.
New Trends in Exact Algorithms for the 01 Knapsack Problem
, 1997
"... While the 1980s were focused on the solution of large sized "easy" knapsack problems, this decade has brought several new algorithms, which are able to solve "hard" large sized instances. We will give an overview of the recent techniques for solving hard knapsack problems, with special emphasis on t ..."
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Cited by 30 (0 self)
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While the 1980s were focused on the solution of large sized "easy" knapsack problems, this decade has brought several new algorithms, which are able to solve "hard" large sized instances. We will give an overview of the recent techniques for solving hard knapsack problems, with special emphasis on the addition of cardinality constraints, dynamic programming, and rudimentary divisibility. Computational results, comparing all recent algorithms, are presented. 1 Introduction We consider the classical 01 Knapsack Problem (KP) where a subset of n given items has to be packed in a knapsack of capacity c. Each item has a profit p j and a weight w j and the problem is to select a subset of the items whose total weight does not exceed c and whose total profit is a maximum. We assume, without loss of generality, that all input data are positive integers. Introducing the binary decision variables x j , with x j = 1 if item j is selected, and x j = 0 otherwise, we get the ILPmodel: maximize z =...
An Exact Algorithm for Large Multiple Knapsack Problems
 European Journal of Operational Research
, 1999
"... The Multiple Knapsack Problem is the problem of assigning a subset of n items to m distinct knapsacks, such that the total profit sum of the selected items is maximized, without exceeding the capacity of each of the knapsacks. The problem has several applications in naval as well as financial manage ..."
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Cited by 23 (2 self)
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The Multiple Knapsack Problem is the problem of assigning a subset of n items to m distinct knapsacks, such that the total profit sum of the selected items is maximized, without exceeding the capacity of each of the knapsacks. The problem has several applications in naval as well as financial management.
Discrete Facility Location and Routing of Obnoxious Activities
, 2000
"... The problem of simultaneously locating obnoxious facilities and routing obnoxious materials between a set of builtup areas and the facilities is addressed. Obnoxious facilities are those facilities which cause nuisance to people as well as to the environment i.e. dump sites, chemical industrial pla ..."
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Cited by 16 (2 self)
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The problem of simultaneously locating obnoxious facilities and routing obnoxious materials between a set of builtup areas and the facilities is addressed. Obnoxious facilities are those facilities which cause nuisance to people as well as to the environment i.e. dump sites, chemical industrial plants, electric power supplier networks, nuclear reactors and so on. A discrete combined locationrouting model, which we refer to as Obnoxious Facility Location and Routing model (OFLR), is defined. OFLR is a NPhard problem for which a Lagrangean heuristic approach is presented. The Lagrangean relaxation proposed allows to decompose OFLR into a Location subproblem and a Routing subproblem; such subproblems are then strenghtened by adding suitable inequalities. Based on this Lagrangean relaxation two simple Lagrangean heuristics are provided. An effective Branch and Bound algorithm is then presented, which aims at reducing the gap between the above mentioned lower and upper bounds. Our Bran...
Core problems in Knapsack Algorithms.
 Operations Research
, 1994
"... Since Balas and Zemel a dozen years ago introduced the socalled core problem as an efficient way of solving the Knapsack Problem, all the most successful algorithms have been based on this idea. Balas and Zemel proved, that there is a high probability for finding an optimal solution in the core, th ..."
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Cited by 12 (1 self)
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Since Balas and Zemel a dozen years ago introduced the socalled core problem as an efficient way of solving the Knapsack Problem, all the most successful algorithms have been based on this idea. Balas and Zemel proved, that there is a high probability for finding an optimal solution in the core, thus avoiding to consider the remaining items. However this paper demonstrates, that even for randomly generated data instances, the core problem may degenerate, making it difficult to obtain a reasonable solution. This behavior has not been noticed before due to inadequate testing, since the capacity usually is chosen such that the core problem becomes as easy as possible. A model for the expected hardness of a core problem as function of the capacity is presented, and it is demonstrated that the hitherto applied test instances are among the easiest possible. As a consequence we propose a series of new randomly generated test instances, and show how recent algorithms behave when applied to these problems.
Heuristics for the container loading problem
 European Journal of Operational Research
, 2002
"... The knapsack container loading problem is the problem of loading a subset of rectangular boxes into a rectangular container of fixed dimensions such that the volume of the packed boxes is maximized. A new heuristic based on the wallbuilding approach is proposed, which decomposes the problem into a ..."
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Cited by 10 (2 self)
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The knapsack container loading problem is the problem of loading a subset of rectangular boxes into a rectangular container of fixed dimensions such that the volume of the packed boxes is maximized. A new heuristic based on the wallbuilding approach is proposed, which decomposes the problem into a number of layers which again are split into a number of strips. The packing of a strip may be formulated and solved optimally as a Knapsack Problem with capacity equal to the width or height of the container. The depth of a layer as well as the thickness of each strip is decided through a branchandbound approach where at each node only a subset of branches is explored.
Several ranking rules for the selection of the most promising layer depths and strip widths are presented and the performance of the corresponding algorithms is experimentally compared for homogeneous and heterogeneous instances. The best ranking rule is then used in a comprehensive computational study involving largesized instances. These computational results show that instances with a total box volume up to 90% easily may be solved to optimality, and that average fillings of the container volume exceeding 95% may be obtained for largesized instances.
The Core Concept for the Multidimensional Knapsack Problem
 IN EVOLUTIONARY COMPUTATION IN COMBINATORIAL OPTIMIZATION  EVOCOP 2006
, 2006
"... We present the newly developed core concept for the Multidimensional Knapsack Problem (MKP) which is an extension of the classical concept for the onedimensional case. The core for the multidimensional problem is defined in dependence of a chosen efficiency function of the items, since no singl ..."
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Cited by 10 (5 self)
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We present the newly developed core concept for the Multidimensional Knapsack Problem (MKP) which is an extension of the classical concept for the onedimensional case. The core for the multidimensional problem is defined in dependence of a chosen efficiency function of the items, since no single obvious efficiency measure is available for MKP. An empirical study on the cores of widelyused benchmark instances is presented, as well as experiments with different approximate core sizes. Furthermore we describe a memetic algorithm and a relaxation guided variable neighborhood search for the MKP, which are applied to the original and to the core problems. The experimental results show that given a fixed runtime, the di#erent metaheuristics as well as a general purpose integer linear programming solver yield better solution when applied to approximate core problems of fixed size.
Menon S.: Efficient Scheduling of Internet Banner Advertisements
 ACM Transactions on Internet Technology
, 2003
"... Despite the slowdown in the economy, advertisement revenue remains a significant source of income for many Internetbased organizations. Banner advertisements form a critical component of this income, accounting for 40 to 50 percent of the total revenue. There are considerable gains to be realized t ..."
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Cited by 9 (0 self)
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Despite the slowdown in the economy, advertisement revenue remains a significant source of income for many Internetbased organizations. Banner advertisements form a critical component of this income, accounting for 40 to 50 percent of the total revenue. There are considerable gains to be realized through the efficient scheduling of banner advertisements. This problem has been observed to be intractable via traditional optimization techniques, and has received only limited attention in the literature. This paper presents a procedure to generate advertisement schedules under the most commonly used advertisement pricing scheme—the CPM model. The solution approach is based on Lagrangean decomposition and is seen to provide extremely good advertisement schedules in a relatively short period of time, taking only a few hundred seconds of elapsed time on a 450 MHz PC compared to a few thousand seconds of CPU time on a workstation that other approaches need. Additionally, this approach can be incorporated into an actual implementation with minimal alterations and hence is of particular interest.
The Multidimensional Knapsack Problem: Structure and Algorithms
, 2007
"... We study the multidimensional knapsack problem, present some theoretical and empirical results about its structure, and evaluate different Integer Linear Programming (ILP) based, metaheuristic, and collaborative approaches for it. We start by considering the distances between optimal solutions to th ..."
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Cited by 8 (1 self)
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We study the multidimensional knapsack problem, present some theoretical and empirical results about its structure, and evaluate different Integer Linear Programming (ILP) based, metaheuristic, and collaborative approaches for it. We start by considering the distances between optimal solutions to the LPrelaxation and the original problem and then introduce a new core concept for the MKP, which we study extensively. The empirical analysis is then used to develop new concepts for solving the MKP using ILPbased and memetic algorithms. Different collaborative combinations of the presented methods are discussed and evaluated. Further computational experiments with longer runtimes are also performed in order to compare the solutions of our approaches to the best known solutions of another so far leading approach for common MKP benchmark instances. The extensive computational experiments show the effectiveness of the proposed methods, which yield highly competitive results in significantly shorter runtimes than previously described approaches.