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Annals of Operations Research

, Volume 238, Issue 1–2, pp 329–354 | Cite as

Partial up-stream advanced payment and partial down-stream delayed payment in a three-level supply chain

  • Mohsen Lashgari
  • Ata Allah TaleizadehEmail author
  • Abbas Ahmadi
Article

Abstract

In a competitive market, the retailers, in order to encourage the customers to increase their orders, give them the opportunity to pay a fraction of the purchasing cost after delivery of the ordered items (i.e., down-stream partial delayed payment). On the other hand, the suppliers, in order to reduce the risk of cancellations of orders from buyers, may ask the retailers to pay a portion of the purchasing cost before delivery of products (i.e., up-stream partial prepayment). In this paper, an EOQ model with down-stream partial delayed payment and up-stream partial prepayment under three different scenarios (without shortage, with full backordering and with partial backordering) is presented. In order to find the optimal solutions of the models developed for different scenarios, the convexity of the objective functions (i.e., total cost functions) are proved and then closed-form optimal solutions are derived. Also, a solution algorithm is proposed for the model of the third scenario. To demonstrate the applicability of the framework, some numerical examples are presented. Finally, sensitivity analyses are made on several key parameters, in order to gain some managerial insight.

Keywords

Inventory Backordering Partial delayed payment  Partial prepayment Partial backordering 

Notes

Acknowledgments

This research was supported by the Iran National Science Foundation (INSF). [grant number INSF-94001671].

References

  1. Aggarwal, S., & Jaggi, C. (1995). Ordering policies of deteriorating items under permissible delay in payments. Journal of the operational Research Society, 5, 658–662.Google Scholar
  2. Bhunia, A., Jaggi, C. K., Sharma, A., & Sharma, R. (2014). A two-warehouse inventory model for deteriorating items under permissible delay in payment with partial backlogging. Applied Mathematics and Computation, 232, 1125–1137.CrossRefGoogle Scholar
  3. Bhunia, A. K., & Shaikh, A. A. (2015). An application of PSO in a two-warehouse inventory model for deteriorating item under permissible delay in payment with different inventory policies. Applied Mathematics and Computation, 256, 831–850.CrossRefGoogle Scholar
  4. Bhunia, A., Shaikh, A., & Gupta, R. (2015). A study on two-warehouse partially backlogged deteriorating inventory models under inflation via particle swarm optimisation. International Journal of Systems Science, 46(6), 1036–1050.CrossRefGoogle Scholar
  5. Bregman, R. L. (1992). The effect of the timing of disbursements on order quantities. Journal of the Operational Research Society, 10, 971–977.Google Scholar
  6. Cárdenas-Barrón, L. E., & Sana, S. S. (2015). Multi-item EOQ inventory model in a two-layer supply chain while demand varies with promotional effort. Applied Mathematical Modelling, 39, 6725–6737.CrossRefGoogle Scholar
  7. Chang, C. T., Ouyang, L.-Y., & Teng, J.-T. (2003). An EOQ model for deteriorating items under supplier credits linked to ordering quantity. Applied Mathematical Modelling, 27(12), 983–996.CrossRefGoogle Scholar
  8. Ghoreishi, M., Weber, G. W., & Mirzazadeh, A. (2015). An inventory model for non-instantaneous deteriorating items with partial backlogging, permissible delay in payments, inflation-and selling price-dependent demand and customer returns. Annals of Operations Research, 226(1), 221–238.CrossRefGoogle Scholar
  9. Goyal, S. K. (1985). Economic order quantity under conditions of permissible delay in payments. Journal of the operational research society, 4, 335–338.Google Scholar
  10. Gupta, R., Bhunia, A., & Goyal, S. (2009). An application of genetic algorithm in solving an inventory model with advance payment and interval valued inventory costs. Mathematical and Computer Modelling, 49(5), 893–905.CrossRefGoogle Scholar
  11. Harris, F. W. (1990). How many parts to make at once. Operations Research, 38(6), 947–950.CrossRefGoogle Scholar
  12. Huang, Y. F. (2003). Optimal retailer’s ordering policies in the EOQ model under trade credit financing. Journal of the Operational Research Society, 54(9), 1011–1015.CrossRefGoogle Scholar
  13. Jamal, A., Sarker, B., & Wang, S. (1997). An ordering policy for deteriorating items with allowable shortage and permissible delay in payment. Journal of the Operational Research Society, 8, 826–833.Google Scholar
  14. Kaanodiya, K., & Pachauri, R. (2012). An EPQ model under two levels of trade credit and limited storage space. International Journal of Industrial Engineering Computations, 3(3), 445–462.CrossRefGoogle Scholar
  15. Lashgari, M., Taleizadeh, A. A., & Sana, S. S. (2016). An inventory control problem for deteriorating items with back-ordering and financial consideration under two levels of trade credit linked to order quantity. Journal of Industrial and management optimization, 12(3), 1091–1119.CrossRefGoogle Scholar
  16. Lau, H. S., & Lau, A. H.-L. (1993). The effect of cost disbursement timings in inventory control. Journal of the Operational Research Society, 44(7), 739–740.CrossRefGoogle Scholar
  17. 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(2), 852–861.CrossRefGoogle Scholar
  18. Maiti, A., Maiti, M., & Maiti, M. (2009). Inventory model with stochastic lead-time and price dependent demand incorporating advance payment. Applied Mathematical Modelling, 33(5), 2433–2443.CrossRefGoogle Scholar
  19. Maiti, A. K., Bhunia, A. K., & Maiti, M. (2007). Some inventory problems via genetic algorithms. Ph.D. Thesis, Vidyasagar University, India.Google Scholar
  20. Min, J., Zhou, Y.-W., & Zhao, J. (2010). An inventory model for deteriorating items under stock-dependent demand and two-level trade credit. Applied Mathematical Modelling, 34(11), 3273–3285.CrossRefGoogle Scholar
  21. 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.CrossRefGoogle Scholar
  22. Ouyang, L. Y., Ho, C.-H., Su, C.-H., & Yang, C.-T. (2015). An integrated inventory model with capacity constraint and order-size dependent trade credit. Computers & Industrial Engineering.Google Scholar
  23. Ouyang, L. Y., Teng, J.-T., & Chen, L.-H. (2006). Optimal ordering policy for deteriorating items with partial backlogging under permissible delay in payments. Journal of Global Optimization, 34(2), 245–271.CrossRefGoogle Scholar
  24. Pentico, D. W., & Drake, M. J. (2009). The deterministic EOQ with partial backordering: A new approach. European Journal of Operational Research, 194, 102–113.CrossRefGoogle Scholar
  25. Salameh, M., Abboud, N., El-Kassar, A., & Ghattas, R. (2003). Continuous review inventory model with delay in payments. International Journal of Production Economics, 85(1), 91–95.CrossRefGoogle Scholar
  26. Sana, S. S., & Chaudhuri, K. (2008). A deterministic EOQ model with delays in payments and price-discount offers. European Journal of Operational Research, 184(2), 509–533.CrossRefGoogle Scholar
  27. Seifert, D., Seifert, R. W., & Protopappa-Sieke, M. (2013). A review of trade credit literature: Opportunities for research in operations. European Journal of Operational Research, 231(2), 245–256.CrossRefGoogle Scholar
  28. Shah, N. H. (2015). Manufacturer-retailer inventory model for deteriorating items with price-sensitive credit-linked demand under two-level trade credit financing and profit sharing contract. Cogent Engineering, 2(1).Google Scholar
  29. Singh, C., & Singh, S. (2015). Progressive trade credit policy in a supply chain with and without stock-out for supplier’s lead time under inflationary and fuzzy environment. Systems Science & Control Engineering: An Open Access Journal(just-accepted), 1–33.Google Scholar
  30. Singh, T., & Pattanayak, H. (2015). An ordering policy with time-proportional deterioration, linear demand and permissible delay in payment. In L. C. Jain, H. S. Behera, J. K. Mandal & D. P. Mohapatra (Eds.), Computational intelligence in data mining-volume 3. (pp. 649–658), Springer.Google Scholar
  31. Taleizadeh, A. A. (2014a). An economic order quantity model for deteriorating item in a purchasing system with multiple prepayments. Applied Mathematical Modelling, 38(23), 5357–5366.CrossRefGoogle Scholar
  32. Taleizadeh, A. A. (2014b). An EOQ model with partial backordering and advance payments for an evaporating item. International Journal of Production Economics, 155, 185–193.CrossRefGoogle Scholar
  33. Taleizadeh, A. A., & Nematollahi, M. (2014). An inventory control problem for deteriorating items with back-ordering and financial considerations. Applied Mathematical Modelling, 38(1), 93–109.CrossRefGoogle Scholar
  34. Taleizadeh, A. A., Niaki, S. T. A., & Nikousokhan, R. (2011). Constraint multiproduct joint-replenishment inventory control problem using uncertain programming. Applied Soft Computing, 11(8), 5143–5154.CrossRefGoogle Scholar
  35. Taleizadeh, A. A., Pentico, D. W., Jabalameli, M. S., & Aryanezhad, M. (2013a). An economic order quantity model with multiple partial prepayments and partial backordering. Mathematical and Computer Modelling, 57(3), 311–323.CrossRefGoogle Scholar
  36. Taleizadeh, A. A., Pentico, D. W., Jabalameli, M. S., & Aryanezhad, M. (2013b). An EOQ model with partial delayed payment and partial backordering. Omega, 41(2), 354–368.CrossRefGoogle Scholar
  37. Taleizadeh, A. A., Wee, H.-M., & Jolai, F. (2013). Revisiting a fuzzy rough economic order quantity model for deteriorating items considering quantity discount and prepayment. Mathematical and Computer Modelling, 57(5), 1466–1479.CrossRefGoogle Scholar
  38. 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(2), 415–423.CrossRefGoogle Scholar
  39. Teng, J. T., Lou, K.-R., & Wang, L. (2014). Optimal trade credit and lot size policies in economic production quantity models with learning curve production costs. International Journal of Production Economics, 155, 318–323.CrossRefGoogle Scholar
  40. Teng, J. T., Min, J., & Pan, Q. (2012). Economic order quantity model with trade credit financing for non-decreasing demand. Omega, 40(3), 328–335.CrossRefGoogle Scholar
  41. Wu, J., & Chan, Y.-L. (2014). Lot-sizing policies for deteriorating items with expiration dates and partial trade credit to credit-risk customers. International Journal of Production Economics, 155, 292–301.CrossRefGoogle Scholar
  42. Wu, X. l., & Zhou, J.-C. (2015). Retailer’s optimal ordering secision with trade credit financing. In Paper presented at the proceedings of the 5th international Asia conference on industrial engineering and management innovation (IEMI2014).Google Scholar
  43. Yadav, D., Singh, S., & Kumari, R. (2015). Retailer’s optimal policy under inflation in fuzzy environment with trade credit. International Journal of Systems Science, 46(4), 754–762.CrossRefGoogle Scholar
  44. Pourmohammad-Zia, N., & Taleizadeh, A. A. (2015). A lot-sizing model with backordering under hybrid linked-to-order multiple advance payments and delayed payment. Transportation Research Part E: Logistics and Transportation Review, 82, 19–37.CrossRefGoogle Scholar
  45. Zhang, A. X. (1996). Optimal advance payment scheme involving fixed per-payment costs. Omega, 24(5), 577–582.CrossRefGoogle Scholar
  46. Zhang, Q., Dong, M., Luo, J., & Segerstedt, A. (2014a). Supply chain coordination with trade credit and quantity discount incorporating default risk. International Journal of Production Economics, 153, 352–360.Google Scholar
  47. Zhang, Q., Tsao, Y. C., & Chen, T. H. (2014b). Economic order quantity under advance payment. Applied Mathematical Modelling, 38(24), 5910–5921.Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Mohsen Lashgari
    • 1
  • Ata Allah Taleizadeh
    • 2
    Email author
  • Abbas Ahmadi
    • 1
  1. 1.School of Industrial EngineeringIran University of Science and TechnologyTehranIran
  2. 2.School of Industrial Engineering, College of EngineeringUniversity of TehranTehranIran

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