##
**Benders decomposition with alternative multiple cuts for a multi-product closed-loop supply chain network design model.**
*(English)*
Zbl 1135.90362

Summary: In this article, we consider a multi-product closed-loop supply chain network design problem where we locate collection centers and remanufacturing facilities while coordinating the forward and reverse flows in the network so as to minimize the processing, transportation, and fixed location costs. The problem of interest is motivated by the practice of an original equipment manufacturer in the automotive industry that provides service parts for vehicle maintenance and repair. We provide an effective problem formulation that is amenable to efficient Benders reformulation and an exact solution approach.

More specifically, we develop an efficient dual solution approach to generate strong Benders cuts, and, in addition to the classical single Benders cut approach, we propose three different approaches for adding multiple Benders cuts. These cuts are obtained via dual problem disaggregation based either on the forward and reverse flows, or the products, or both. We present computational results which illustrate the superior performance of the proposed solution methodology with multiple Benders cuts in comparison to the branch-and-cut approach as well as the traditional Benders decomposition approach with a single cut. In particular, we observe that the use of multiple Benders cuts generates stronger lower bounds and promotes faster convergence to optimality. We also observe that if the model parameters are such that the different costs are not balanced, but, rather, are biased towards one of the major cost categories (processing, transportation or fixed location costs), the time required to obtain the optimal solution decreases considerably when using the proposed solution methodology as well as the branch-and-cut approach.

More specifically, we develop an efficient dual solution approach to generate strong Benders cuts, and, in addition to the classical single Benders cut approach, we propose three different approaches for adding multiple Benders cuts. These cuts are obtained via dual problem disaggregation based either on the forward and reverse flows, or the products, or both. We present computational results which illustrate the superior performance of the proposed solution methodology with multiple Benders cuts in comparison to the branch-and-cut approach as well as the traditional Benders decomposition approach with a single cut. In particular, we observe that the use of multiple Benders cuts generates stronger lower bounds and promotes faster convergence to optimality. We also observe that if the model parameters are such that the different costs are not balanced, but, rather, are biased towards one of the major cost categories (processing, transportation or fixed location costs), the time required to obtain the optimal solution decreases considerably when using the proposed solution methodology as well as the branch-and-cut approach.

### MSC:

90B80 | Discrete location and assignment |

90C57 | Polyhedral combinatorics, branch-and-bound, branch-and-cut |

90B10 | Deterministic network models in operations research |

90B30 | Production models |

90B50 | Management decision making, including multiple objectives |

PDFBibTeX
XMLCite

\textit{H. Üster} et al., Nav. Res. Logist. 54, No. 8, 890--907 (2007; Zbl 1135.90362)

Full Text:
DOI

### References:

[1] | Beamon, Prod Plann Contr 15 pp 270– (2004) |

[2] | Benders, Numer Math 4 pp 238– (1962) |

[3] | Bloemhof-Ruwaard, Lect Notes Econ Math Syst 480 pp 23– (1999) |

[4] | Brown, Manag Sci 33 pp 1469– (1987) |

[5] | , , (Editors), Reverse logistics: Quantitative models for closed-loop supply chains, Springer, Berlin, 2004. |

[6] | Reverse logistics network structures and design. ERIM Report Series ERS-2001-52-LIS, Erasmus Research Institute of Management, September 2001. Available at http://ssrn.com/abstract=370907. |

[7] | Fleischmann, Prod Oper Manag 10 pp 156– (2001) |

[8] | Fleischmann, Eur J Oper Res 103 pp 1– (1997) |

[9] | Fleischmann, OMEGA 28 pp 653– (2000) |

[10] | Geoffrion, Manag Sci 20 pp 822– (1974) |

[11] | Jayaraman, Eur J Oper Res 133 pp 394– (2001) |

[12] | Krikke, Int J Prod Res 41 pp 3689– (2003) |

[13] | Lu, Comput Oper Res 34 pp 299– (2007) |

[14] | Magnanti, Oper Res 29 pp 464– (1981) |

[15] | Pirkul, Transport Sci 30 pp 291– (1996) |

[16] | Pyke, Eur J Oper Res 74 pp 18– (1994) |

[17] | , , , ” A generic network design for a closed-loop supply chain using genetic algorithm,” in: , ,, , , , , , , , , , , , , (Editors), GECCO (2), Vol. 3103 of Lecture notes in computer science, New York, Springer, 2004, pp. 1214–1225. |

[18] | van Roy, Oper Res 34 pp 145– (1986) |

[19] | Wentges, Math Meth Oper Res 44 pp 267– (1996) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.