# Nonconvex piecewise linear knapsack problems

Kameshwaran, S and Narahari, Y
(2009)
*Nonconvex piecewise linear knapsack problems.*
European Journal of Operational Research, 192 (1).
pp. 56-68.

## Abstract

This paper considers the minimization version of a class of nonconvex knapsack problems with piecewise linear cost structure. The items to be included in the knapsack have a divisible quantity and a cost function. An item can be included partially in the given quantity range and the cost is a nonconvex piecewise linear function of quantity. Given a demand, the optimization problem is to choose an optimal quantity for each item such that the demand is satisfied and the total cost is minimized. This problem and its close variants are encountered in manufacturing planning, supply chain design, volume discount procurement auctions, and many other contemporary applications. Two separate mixed integer linear programming formulations of this problem are proposed and are compared with existing formulations. Motivated by different scenarios in which the problem is useful, the following algorithms are developed: (1) a fast polynomial time, near-optimal heuristic using convex envelopes; (2) exact pseudo-polynomial time dynamic programming algorithms; (3) a 2-approximation algorithm; and (4) a fully polynomial time approximation scheme. A comprehensive test suite is developed to generate representative problem instances with different characteristics. Extensive computational experiments show that the proposed formulations and algorithms are faster than the existing techniques.

ISB Creiators: |
ISB Creators ORCiD Kameshwaran, S UNSPECIFIED |
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Item Type: | Article |

Uncontrolled Keywords: | Approximation algorithm; Convex envelope; Dynamic programming; Fully polynomial time approximation scheme; Linear relaxation; Multiple choice knapsack problem; Piecewise linear knapsack problem; Precedence constrained knapsack problem |

Subjects: | Business Analytics Business and Management |

Depositing User: | Ilammaran A |

Date Deposited: | 04 Nov 2014 15:40 |

Last Modified: | 04 Nov 2014 18:40 |

URI: | http://eprints.exchange.isb.edu/id/eprint/170 |

Publisher URL: | http://dx.doi.org./10.1016/j.ejor.2007.08.044 |

Publisher OA policy: | http://www.sherpa.ac.uk/romeo/issn/0377-2217/ |

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