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**A piecewise nonlinear optimization for a production-inventory model under maintenance, variable setup costs, and trade credits.**
*(English)*
Zbl 1325.90034

Summary: This study investigates a production-inventory model considering system maintenance, variable setup costs, and trade credits. In production systems, manufacturers usually carry out system maintenance when systems are in an out-of-control state. We also consider setup costs because these costs may decrease over time, for example, when manufacturers effectively improve production efficiency because of the effect of the learning curve. The model considers trade credits because suppliers commonly provide credit periods to manufacturers. This study determines the optimal replenishment frequencies that minimize total costs. We provide lemmas for optimality, develop a piecewise nonlinear optimization algorithm to solve the problems described, and verify the model using a practical case in the automotive parts industry. Based on numerical experiments, we discuss how system parameters affect the decision behaviors of manufacturers. The results of this study can serve as references for business managers and administrators.

### MSC:

90B30 | Production models |

90B05 | Inventory, storage, reservoirs |

90C30 | Nonlinear programming |

90B25 | Reliability, availability, maintenance, inspection in operations research |

### Keywords:

production and inventory; piecewise nonlinear optimization; maintenance; variable setup cost; trade credit
Full Text:
DOI

### References:

[1] | Chen, J. M., & Chen, T. H. (2007). The profit maximization model for a multi-item distribution channel. Transportation Research. Part E, Logistics and Transportation Review, 43, 338-354. |

[2] | Chen, T. H. (2010). The optimal ordering and advertising policy for a single-period commodity in a supply chain. Journal of the Chinese Institute of Industrial Engineers, 27(5), 363-371. |

[3] | Chen, T. H. (2011). Coordinating the ordering and advertising policies for a single-period commodity in a two-level supply chain. Computers & Industrial Engineering, 61(4), 1268-1274. |

[4] | Chuang, C. J., Ho, C. H., Ouyang, L. Y., & Wu, C. W. (2013). An integrated inventory model with order-size-dependent trade credit and quality improvement. Procedia Computer Science, 17, 365-372. |

[5] | Chung, K. J., & Huang, Y. F. (2003). The optimal cycle time for EPQ inventory model under permissible delay in payments. International Journal of Production Economics, 84, 307-318. |

[6] | Chung, K. J. (2010). The optimal inventory policy for EPQ model under trade credit. International Journal of Systems Science, 41, 1115-1120. · Zbl 1202.90007 |

[7] | Goyal, S. K. (1985). Economics order quantity under conditions of permissible delay in payments. Journal of the Operational Research Society, 36, 335-338. · Zbl 0568.90025 |

[8] | Huang, Y. F. (2003). Optimal retailer’s ordering policies in the EOQ model under trade credit financing. Journal of the Operational Research Society, 54, 1011-1015. · Zbl 1097.90501 |

[9] | Huang, Y. F. (2006). An inventory model under two levels of trade credit and limited storage space derived without derivatives. Apply Mathematics Modelling, 30, 418-436. · Zbl 1182.90007 |

[10] | Huang, Y. F. (2007a). Economic order quantity under conditionally permissible delay in payments. European Journal of Operational Research, 176, 911-924. · Zbl 1103.90017 |

[11] | Huang, Y. F. (2007b). Optimal retailer’s replenishment decisions in the EPQ model under two levels of trade credit policy. European Journal of Operational Research, 176, 1577-1591. · Zbl 1110.90007 |

[12] | Huang, Y. F., & Huang, H. F. (2008). Optimal inventory replenishment policy for the EPQ model under trade credit derived without derivatives. International Journal of Systems Science, 39(5), 539-546. · Zbl 1166.90300 |

[13] | Kreng, V. B., & Tan, S. J. (2010). The optimal replenishment decisions under two levels of trade credit policy depending on the order quantity. Expert Systems with Applications, 37, 5514-5522. |

[14] | Liao, J. J. (2008). An EOQ model with noninstantaneous receipt and exponentially deteriorating items under two-level trade credit. International Journal of Production Economics, 113, 852-861. |

[15] | Lin, Y. j., Ouyang, L. Y., & Dang, Y. F. (2013). A joint optimal ordering and delivery policy for an integrated supplier-retailer inventory model with trade credit and defective items. Applied Mathematics and Computation, 218, 7498-7514. · Zbl 1242.90015 |

[16] | Ouyang, L. Y., & Chang, C. T. (2013). Optimal production lot with imperfect production process under permissible delay in payments and complete backlogging. International Journal of Production Economics, 144(2), 610-617. |

[17] | Ouyang, L. Y., Chang, C. T., & Teng, J. T. (2005). An EOQ model for deteriorating items under trade credits. Journal of the Operational Research Society, 56, 719-726. · Zbl 1095.90007 |

[18] | Ouyang, L. Y., Chang, C. T., & Teng, J. T. (2006). A study on an inventory model for non-instantaneous deteriorating item with permissible delay in payments. Computers & Industrial Engineering, 51, 637-651. |

[19] | Ouyang, L. Y., Ho, C. H., & Su, C. H. (2008). Optimal strategy for an integrated system with variable production rate when the freight rate and trade credit are both linked to the order quantity. International Journal of Production Economics, 115, 151-162. |

[20] | Ouyang, L. Y., Teng, J. T., Goyal, S. K., & Yang, C. T. (2009). An economic order quantity model for deteriorating items with partially permissible delay in payments linked to order quantity. European Journal of Operational Research, 194, 418-431. · Zbl 1154.90309 |

[21] | Porteus, E. L. (1986). Optimal lot sizing, process quality improvement and setup cost reduction. Operations Research, 34, 137-144. · Zbl 0591.90043 |

[22] | Rosenblatt, M. J., & Lee, H. L. (1986). Economic production cycle with imperfect production process. IIE Transactions, 18, 48-55. |

[23] | Russell, R. S., & Taylor, B. W. III. (2000). Operations management (3rd ed.). New York: Prentice Hall. |

[24] | Sheen, G. J., & Tsao, Y. C. (2007). Channel coordination, trade credit and quantity discounts for freight cost. Transportation Research. Part E, Logistics and Transportation Review, 43, 112-128. |

[25] | Tsao, Y. C. (2009). Production and payment policies for an imperfect manufacturing system with machine maintenance and credit policies. International Journal of Technology Management, 48(2), 240-257. |

[26] | Tsao, Y. C. (2010). Two-phase pricing and inventory management for deteriorating and fashion goods under trade credit. Mathematical Methods of Operations Research, 72(1), 107-127. · Zbl 1194.90010 |

[27] | Tsao, Y. C. (2011). Managing a retail-competition distribution channel with incentive policies. Applied Mathematical Modelling, 35(9), 4140-4148. · Zbl 1225.91033 |

[28] | Tsao, Y. C., & Sheen, G. J. (2007). Joint pricing and replenishment decisions for deteriorating items with lot-size and time dependent purchasing cost under credit period. International Journal of Systems Science, 38, 549-561. · Zbl 1148.90316 |

[29] | Tsao, Y. C., & Sheen, G. J. (2008). Dynamic pricing, promotion and replenishment policies for a deteriorating item under permissible delay in payments. Computers & Operations Research, 35, 3562-3580. · Zbl 1140.91358 |

[30] | Tsao, Y. C., Chen, T. H., & Zhang, Q. H. (2013). Effects of maintenance policy on an imperfect production system under trade credit. International Journal of Production Research, 51, 1549-1562. |

[31] | Teng, J. T. (2002). On the economic order quantity under conditions of permissible delay in payments. Journal of the Operational Research Society, 53, 915-918. · Zbl 1098.90006 |

[32] | Teng, J. T. (2009). Optimal ordering policies for a retailer who offers distinct trade credits to its good and bad credit customers. International Journal of Production Economics, 119, 415-423. |

[33] | Teng, J. T., & Chang, C. T. (2009). Optimal manufacturer’s replenishment policies in the EPQ model under two levels of trade credit policy. European Journal of Operational Research, 195, 358-363. · Zbl 1159.90009 |

[34] | Teng, J. T., & Goyal, S. K. (2009). Comment on ‘Optimal inventory replenishment policy for the EPQ model under trade credit derived without derivatives’. International Journal of Systems Science, 40, 1095-1098. · Zbl 1173.90318 |

[35] | Yang, P. C., & Wee, H. M. (2002). A single-vendor and multiple-buyers production-inventory policy for a deteriorating item. European Journal of Operational Research, 143, 570-581. · Zbl 1082.90515 |

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