Nonconvex piecewise linear knapsack problems
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.
If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it.
References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Paul H. Zipkin, 1980. "Simple Ranking Methods for Allocation of One Resource," Management Science, INFORMS, vol. 26(1), pages 34-43, January.
- Gabriel R. Bitran & Arnoldo C. Hax, 1981. "Disaggregation and Resource Allocation Using Convex Knapsack Problems with Bounded Variables," Management Science, INFORMS, vol. 27(4), pages 431-441, April.
- Kameshwaran, S. & Narahari, Y. & Rosa, Charles H. & Kulkarni, Devadatta M. & Tew, Jeffrey D., 2007. "Multiattribute electronic procurement using goal programming," European Journal of Operational Research, Elsevier, vol. 179(2), pages 518-536, June.
- Richard Bellman, 1957. "On a Dynamic Programming Approach to the Caterer Problem--I," Management Science, INFORMS, vol. 3(3), pages 270-278, April.
- Dudzinski, Krzysztof & Walukiewicz, Stanislaw, 1987. "Exact methods for the knapsack problem and its generalizations," European Journal of Operational Research, Elsevier, vol. 28(1), pages 3-21, January.
- Keely L. Croxton & Bernard Gendron & Thomas L. Magnanti, 2003. "A Comparison of Mixed-Integer Programming Models for Nonconvex Piecewise Linear Cost Minimization Problems," Management Science, INFORMS, vol. 49(9), pages 1268-1273, September.
- Bretthauer, Kurt M. & Shetty, Bala, 2002. "The nonlinear knapsack problem - algorithms and applications," European Journal of Operational Research, Elsevier, vol. 138(3), pages 459-472, May.
When requesting a correction, please mention this item's handle: RePEc:eee:ejores:v:192:y:2009:i:1:p:56-68. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Shamier, Wendy)
If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.
If references are entirely missing, you can add them using this form.
If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.
If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.
Please note that corrections may take a couple of weeks to filter through the various RePEc services.