• Title/Summary/Keyword: Knapsack Problem

### The Generalized Multiple-Choice Multi-Divisional Linear Programming Knapsack Problem (일반 다중선택 다분할 선형계획 배낭문제)

• Won, Joong-Yeon
• Journal of Korean Institute of Industrial Engineers
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• v.40 no.4
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• pp.396-403
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• 2014
• The multi-divisional knapsack problem is defined as a binary knapsack problem where each mutually exclusive division has its own capacity. In this paper, we present an extension of the multi-divisional knapsack problem that has generalized multiple-choice constraints. We explore the linear programming relaxation (P) of this extended problem and identify some properties of problem (P). Then, we develop a transformation which converts the problem (P) into an LP knapsack problem and derive the optimal solutions of problem (P) from those of the converted LP knapsack problem. The solution procedures have a worst case computational complexity of order $O(n^2{\log}\;n)$, where n is the total number of variables. We illustrate a numerical example and discuss some variations of problem (P).

### The multi-divisional linear knapsack problem (다분할(多分割) 선형배낭문제(線型背囊問題))

• Won, Joong-Yeon;Chung, Sung-Jin
• Journal of Korean Institute of Industrial Engineers
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• v.17 no.1
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• pp.127-130
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• 1991
• The multi-divisional knapsack problem is defined as a binary knapsack problem where each of mutually exclusive divisions has its own capacity. We consider the relaxed LP problem and develop a transformation which converts the multi-divisional linear knapsack problem into smaller size linear knapsack problems. Solution procedures and a numerical example are presented.

### On a Two Dimensional Linear Programming Knapsack Problem with the Extended GUB Constrain (확장된 일반상한제약을 갖는 이차원 선형계획 배낭문제 연구)

• Won, Joong-Yeon
• Journal of Korean Institute of Industrial Engineers
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• v.27 no.1
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• pp.25-29
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• 2001
• We present a two dimensional linear programming knapsack problem with the extended GUB constraint. The presented problem is an extension of the cardinality constrained linear programming knapsack problem. We identify some new properties of the problem and derive a solution algorithm based on the parametric analysis for the knapsack right-hand-side. The solution algorithm has a worst case time complexity of order O($n^2logn$). A numerical example is given.

### Cover Inequalities for the Robust Knapsack Problem

• Park, Kyung-Chul
• Management Science and Financial Engineering
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• v.14 no.1
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• pp.91-96
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• 2008
• Robust knapsack problem appears when dealing with data uncertainty on the knapsack constraint. This note presents a generalization of the cover inequality for the problem with its lifting procedure. Specifically, we show that the lifting can be done in a polynomial time as in the usual knapsack problem. The results can serve as a building block in devising an efficient branch-and-cut algorithm for the general robust (0, 1) IP problem.

### An Algorithm for a Cardinality Constrained Linear Programming Knapsack Problem (선수제약 선형배낭문제의 해법연구)

• 원중연
• Journal of the Society of Korea Industrial and Systems Engineering
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• v.19 no.40
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• pp.137-142
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• 1996
• An algorithm for solving the cardinality constrained linear programming knapsack problem is presented. The algorithm has a convenient structure for a branch-and-bound approach to the integer version, especially to the 0-1 collapsing knapsack problem. A numerical example is given.

### On the Separation of the Rank-1 Chvatal-Gomory Inequalities for the Fixed-Charge 0-1 Knapsack Problem (고정비용 0-1 배낭문제에 대한 크바탈-고모리 부등식의 분리문제에 관한 연구)

• Park, Kyung-Chul;Lee, Kyung-Sik
• Journal of the Korean Operations Research and Management Science Society
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• v.36 no.2
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• pp.43-50
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• 2011
• We consider the separation problem of the rank-1 Chvatal-Gomory (C-G) inequalities for the 0-1 knapsack problem with the knapsack capacity defined by an additional binary variable, which we call the fixed-charge 0-1 knapsack problem. We analyze the structural properties of the optimal solutions to the separation problem and show that the separation problem can be solved in pseudo-polynomial time. By using the result, we also show that the existence of a pseudo-polynomial time algorithm for the separation problem of the rank-1 C-G inequalities of the ordinary 0-1 knapsack problem.

### The Maximin Linear Programming Knapsack Problem With Extended GUB Constraints (확장된 일반상한제약을 갖는 최대최소 선형계획 배낭문제)

• 원중연
• Journal of the Korean Operations Research and Management Science Society
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• v.26 no.3
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• pp.95-104
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• 2001
• In this paper, we consider a maximin version of the linear programming knapsack problem with extended generalized upper bound (GUB) constraints. We solve the problem efficiently by exploiting its special structure without transforming it into a standard linear programming problem. We present an O(n$^3$) algorithm for deriving the optimal solution where n is the total number of problem variables. We illustrate a numerical example.

### Separation Heuristic for the Rank-1 Chvatal-Gomory Inequalities for the Binary Knapsack Problem (이진배낭문제의 크바탈-고모리 부등식 분리문제에 대한 발견적 기법)

• Lee, Kyung-Sik
• Journal of Korean Institute of Industrial Engineers
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• v.38 no.2
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• pp.74-79
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• 2012
• An efficient separation heuristic for the rank-1 Chvatal-Gomory cuts for the binary knapsack problem is proposed. The proposed heuristic is based on the decomposition property of the separation problem for the fixedcharge 0-1 knapsack problem characterized by Park and Lee [14]. Computational tests on the benchmark instances of the generalized assignment problem show that the proposed heuristic procedure can generate strong rank-1 C-G cuts more efficiently than the exact rank-1 C-G cut separation and the exact knapsack facet generation.

### An Efficient Algorithm for the Generalized Continuous Multiple Choice linear Knapsack Problem (일반연속 다중선택 선형배낭문제의 효율적인 해법연구)

• Won, Joong-Yeon
• Journal of Korean Institute of Industrial Engineers
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• v.23 no.4
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• pp.661-667
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• 1997
• We consider a generalized problem of the continuous multiple choice knapsack problem and study on the LP relaxation of the candidate problems which are generated in the branch and bound algorithm for solving the generalized problem. The LP relaxed candidate problem is called the generalized continuous multiple choice linear knapsack problem and characterized by some variables which are partitioned into continuous multiple choice constraints and the others which only belong to simple upper bound constraints. An efficient algorithm of order O($n^2logn$) is developed by exploiting some structural properties and applying binary search to ordered solution sets, where n is the total number of variables. A numerical example is presented.

### A Novel Genetic Algorithm for Multiconstrained Knapsack Problem (다중제약 배낭문제를 위한 새로운 유전 알고리즘)

• Lee, Sang-Uk;Seok, Sang-Mun;Lee, Ju-Sang;Jang, Seok-Cheol;An, Byeong-Ha
• Proceedings of the Korean Operations and Management Science Society Conference
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• pp.773-774
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• 2005
• The knapsack problem (KP) is one of the traditional optimization problems. Specially, multiconstrained knapsack problem (MKP) is well-known NP-hard problem. Many heuristic algorithms and evolutionary algorithms have tackled this problem and shown good performance. This paper presents a novel genetic algorithm for the multiconstrained knapsack problem. The proposed algorithm is called 'Adaptive Link Adjustment'. It is based on integer random key representation and uses additional ${\alpha}$ and ${\beta}$-process as well as selection, crossover and mutation. The experiment results show that it can be archive good performance.