Integer and combinatorial optimization
Integer and combinatorial optimization
Knapsack problems: algorithms and computer implementations
Knapsack problems: algorithms and computer implementations
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Encyclopedia of Operations Research and Management Science
Encyclopedia of Operations Research and Management Science
The Lagrangian Relaxation Method for Solving Integer Programming Problems
Management Science
A virtual pegging approach to the max-min optimization of the bi-criteria knapsack problem
International Journal of Computer Mathematics
An incomplete m-exchange algorithm for solving the large-scale multi-scenario knapsack problem
Computers and Operations Research
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We are concerned with a variation of the standard 0-1 knapsack problem, where the values of items differ under possible S scenarios. By applying the 'pegging test' the ordinary knapsack problem can be reduced, often significantly, in size; but this is not directly applicable to our problem. We introduce a kind of surrogate relaxation to derive upper and lower bounds quickly, and show that, with this preprocessing, the similar pegging test can be applied to our problem. The reduced problem can be solved to optimality by the branch-and-bound algorithm. Here, we make use of the surrogate variables to evaluate the upper bound at each branch-and-bound node very quickly by solving a continuous knapsack problem. Through numerical experiments we show that the developed method finds upper and lower bounds of very high accuracy in a few seconds, and solves larger instances to optimality faster than the previously published algorithms.