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**Efficient algorithms for solving multiconstraint zero-one knapsack problems to optimality.**
*(English)*
Zbl 0571.90065

The multiconstraint 0-1 knapsack problem is encountered when one has to decide how to use a knapsack with multiple resource constraints. Even though the single constraint version of this problem has received a lot of attention, the multiconstraint knapsack problem has been seldom addressed.

This paper deals with developing an effective solution procedure for the multiconstraint knapsack problem. Various relaxations of the problem are suggested and theoretical relations between these relaxations are pointed out. Detailed computational experiments are carried out to compare bounds produced by these relaxations. New algorithms for obtaining surrogate bounds are developed and tested. Rules for reducing problem size are suggested and shown to be effective through computational tests. Different separation, branching and bounding rules are compared using an experimental branch and bound code. An efficient branch and bound procedure is developed, tested and compared with two previously developed optimal algorithms. Solution times with the new procedure are found to be considerably lower. This procedure can also be used as a heuristic for large problems by early termination of the search tree. This scheme was tested and found to be very effective.

This paper deals with developing an effective solution procedure for the multiconstraint knapsack problem. Various relaxations of the problem are suggested and theoretical relations between these relaxations are pointed out. Detailed computational experiments are carried out to compare bounds produced by these relaxations. New algorithms for obtaining surrogate bounds are developed and tested. Rules for reducing problem size are suggested and shown to be effective through computational tests. Different separation, branching and bounding rules are compared using an experimental branch and bound code. An efficient branch and bound procedure is developed, tested and compared with two previously developed optimal algorithms. Solution times with the new procedure are found to be considerably lower. This procedure can also be used as a heuristic for large problems by early termination of the search tree. This scheme was tested and found to be very effective.

### MSC:

90C10 | Integer programming |

90C09 | Boolean programming |

65K05 | Numerical mathematical programming methods |

### Keywords:

multiconstraint 0-1 knapsack problem; multiple resource constraints; solution procedure; relaxations; computational experiments; surrogate bounds; branch and bound### Software:

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\textit{B. Gavish} and \textit{H. Pirkul}, Math. Program. 31, 78--105 (1985; Zbl 0571.90065)

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### References:

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