Abstract
In this paper we consider the problem in which crude oil is shipped from platforms to terminals using oil tankers at minimum transportation cost. This subproblem, which arises in petroleum supply chain models, can be surprisingly expensive to solve with a straightforward formulation involving inventory balances. We present a reformulation that has a special structure defined in this paper as Cascading Knapsack Inequalities. This is used as the basis for deriving tight reformulations for special cases with a limited number of classes of tankers. Numerical results are presented to demonstrate that significant computational savings can be accomplished with the proposed reformulations.
Similar content being viewed by others
References
Agra, A., & Constantino, M. F. (2007). Lifting two-integer knapsack inequalities. Mathematical Programming, 109(1), 115–154.
Ahuja, R., Magnanti, T., & Orlin, J. (1993). Network flows: theory, algorithms and applications. Englewood Cliffs: Prentice Hall.
Alex, H., Steffen, H., & Martin, S. (2007). Multicommodity flows over time: efficient algorithms and complexity. Theoretical Computer Science, 379(3), 387–404.
Ashford, R. (2007). Mixed integer programming: a historical perspective with xpress-mp. Annals of Operation Research, 149(1), 5–17.
Atamturk, A., & Savelsbergh, M. (2005). Integer-programming software systems. Annals of Operation Research, 140(1), 67–124.
Balas, E. (1975). Facets of knapsack polytope. Mathematical Programming, 8(2), 146–164.
Bixby, R., & Rothberg, E. (2007). Progress in computational mixed integer programming—a look back from the other side of the tipping point. Annals of Operation Research, 149(1), 37–41.
Brahimi, N., Dauzere-Peres, S., Najib, N. M., & Nordli, A. (2006). Single item lot sizing problems. European Journal of Operational Research, 168(1), 1–16.
Guignard, M. M., & Spielberg, K. (1972). Mixed-integer algorithms for (0,1) knapsack problem. IBM Journal of Research and Development, 16(4), 424–430.
Hirschberg, D. S., & Wong, C. K. (1976). A polynomial-time algorithm for the knapsack problem with two variables. Journal of the ACM, 23(1), 147–154.
ILOG, Inc. (2007). ILOG CPLEX C++ API 11.0 Reference Manual.
Meserve, B. E. (1993). Fundamental concepts of algebra. New York: Dover.
Miller, A. J., & Wolsey, L. A. (2003). Tight mip formulations for multi-item discrete lot-sizing problems. Operations Research, 51(4), 557–565.
Parsons, J. A. (1969). Branch and bound algorithms—knapsack problem. Journal of Systems Management, 20(9), 35–37.
Rocha, R., Grossmann, I. E., & Poggi de Aragão, M. V. S. (2009). Petroleum allocation at petrobras: mathematical model and a solution algorithm. Computers and Chemical Engineering, 33(12), 2123–2133.
van Hoesel, S., Kuik, R., Salomon, M., & van Wassenhove, L. N. (1994). The single-item discrete lotsizing and scheduling problem: optimization by linear and dynamic programming. Discrete Applied Mathematics, 48(3), 289–303.
Ziegler, G. M. (1998). Lectures on polytopes (2nd ed.). New York: Springer.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rocha, R., Grossmann, I.E. & Poggi de Aragão, M.V.S. Cascading Knapsack Inequalities: reformulation of a crude oil distribution problem. Ann Oper Res 203, 217–234 (2013). https://doi.org/10.1007/s10479-011-0857-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-011-0857-8