An integrated multi-product, multi-buyer supply chain under penalty, green, and quality control polices and a vendor managed inventory with consignment stock agreement: the outer approximation with equality relaxation and augmented penalty algorithm.

*(English)*Zbl 1460.90012Summary: It is of great importance to develop an optimal supply chain (SC) batch-sizing policy that collectively embodies green policies and a vendor-managed inventory (VMI) with consignment stock (CS) agreement. This article provides a mathematical model that includes the buyers’ total cost (TC) and the vendor’s TC in an SC under penalty, green, and quality control (QC) policies and a VMI-CS agreement. The proposed model is a multiproduct, multi-buyer model and has real stochastic constraints. Moreover, the model differentiates between the holding costs for financial and nonfinancial components, in which the first includes the investment in the space and the second includes the cost due to physical storage, movement, and insurance of the products. Financial components are carried by the vendor on implementation of the VMI-CS agreement, while holding costs for stocking items in the buyers’ warehouses are carried by the buyers as nonfinancial components. The objective is to determine the optimal batch-sizing policy with the minimum TC in the integrated SC that finds both the number of the vendor’s batches for each of the transported products and the volume of the batches transported to the buyers so as to minimize the TC of the integrated SC while the stochastic constraints are satisfied. Because of the complexity of the optimization model and mathematical formulations, an outer approximation with equality relaxation and augmented penalty algorithm is presented to determine the optimal batch-sizing policy. With application of this technique, the large-scale and hard-to-solve mixed-integer nonlinear programming problem is minimized. The optimality criteria results obtained in numerical examples and sensitivity analysis demonstrate the excellent performance of the method used. Finally, managerial insights, analytical results, and future research directions are provided.

##### MSC:

90B05 | Inventory, storage, reservoirs |

90B06 | Transportation, logistics and supply chain management |

90B30 | Production models |

##### Keywords:

batch-sizing; supply chain; integration; consignment stock; vendor-managed inventory; outer approximation with equality relaxation and augmented penalty##### Software:

DICOPT
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\textit{A. Gharaei} et al., Appl. Math. Modelling 69, 223--254 (2019; Zbl 1460.90012)

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