• 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).

• 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.

• Journal of the Korean Operations Research and Management Science Society
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• v.28 no.4
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• pp.191-198
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• 2003

The generalized knapsack problem or gknap is the combinatorial optimization problem of optimizing a nonnegative linear function over the integral hull of the intersection of a polynomially separable 0-1 polytope and a knapsack constraint. The knapsack, the restricted shortest path, and the constrained spanning tree problem are a partial list of gknap. More interesting1y, all the problem that are known to have a fully polynomial approximation scheme, or FPTAS are gknap. We establish some necessary and sufficient conditions for a gknap to admit an FPTAS. To do so, we recapture the standard scaling and approximate binary search techniques in the framework of gknap. This also enables us to find a weaker sufficient condition than the strong NP-hardness that a gknap does not have an FPTAS. Finally, we apply the conditions to explore the fully polynomial approximability of the constrained spanning problem whose fully polynomial approximability is still open.

• Proceedings of the Korean Operations and Management Science Society Conference
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• 2004.10a
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• pp.225-228
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• 2004

The generalized knapsack problem, or gknap is the combinatorial optimization problem of optimizing a nonnegative linear functional over the integral hull of the intersection of a polynomially separable 0 - 1 polytope and a knapsack constraint. Among many potential applications, the knapsack, the restricted shortest path, and the restricted spanning tree problem are such examples. We prove via the ellipsoid method the equivalence between the fully polynomial approximability and a certain pseudo-polynomial separability of the gknap polytope.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.21 no.45
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• pp.291-299
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• 1998

We present a variant for the generalized continuous multiple-choice knapsack problem[1], which additionally has the well-known generalized lower bound constraints. The presented problem is characterized by some variables which only belong to the simple upper bound constraints and the others which are partitioned into both the continuous multiple-choice constraints and the generalized lower bound constraints. By exploiting some extended structural properties, an efficient algorithm of order Ο($n^2$1og n) is developed, where n is the total number of variables. A numerical example is presented.

• Proceedings of the Korean Operations and Management Science Society Conference
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• 2003.11a
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• pp.93-96
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• 2003

The generalized knapsack problem, or gknap is the combinatorial optimization problem of optimizing a nonnegative linear functional over the integral hull of the intersection of a polynomially separable 0 - 1 polytope and a knapsack constraint. Among many potential applications, the knapsack, the restricted shortest path, and the restricted spanning tree problem are such examples. We establish some necessary and sufficient conditions for a gknap to admit a fully polynomial approximation scheme, or FPTAS, To do so, we recapture the scaling and approximate binary search techniques in the framework of gknap. This also enables us to find a condition that a gknap does not have an FP-TAS. This condition is more general than the strong NP-hardness.

• Journal of Korean Institute of Industrial Engineers
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• v.37 no.3
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• pp.258-263
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• 2011

We study on a generalization of the two dimensional linear programming knapsack problem with the extended GUB constraint, which was presented in paper Won(2001). We identify some new parametric properties of the generalized problem and derive a solution algorithm based on the identified parametric properties. The solution algorithm has a worst case time complexity of order O($n^2logn$), where n is the total number of variables. We illustrate a numerical example.

• 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.

• 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.

• Journal of the Korean Operations Research and Management Science Society
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• v.15 no.2
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• pp.33-44
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• 1990

An efficient algorithm is developed for the linear programming relaxation of generalized multiple choice knaspack problem. The generalized multiple choice knaspack problem is an extension of the multiple choice knaspack problem whose relaxed LP problem has been studied extensively. In the worst case, the computational coimplexity of the proposed algorithm is of order 0(n. $n_{max}$)$^{2}$), where n is the total number of variables and $n_{max}$ denotes the cardinality of the largest multiple choice set. The algorithm can be easily embedded in a branch-and-bound procedure for the generalized multiple choice knapsack problem. A numerical example is presented and computational aspects are discussed.sed.