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doi: 10.3934/jimo.2018115

## Coordinating a multi-echelon supply chain under production disruption and price-sensitive stochastic demand

 1 Department of Mathematics, Jadavpur University, Kolkata, India 2 Department of Mechanical and Industrial Engineering, Louisiana State University, Baton Rouge LA 70803, USA

Received  February 2017 Revised  March 2018 Published  August 2018

This paper considers a three-echelon supply chain system with one raw-material supplier, one manufacturer and one retailer in which both the manufacturer and the raw-material supplier are exposed to the risk of production disruptions. The market demand is assumed to be uncertain but sensitive to the retail price. The objective is to determine the optimal lot sizes of the supplier and the manufacturer, and the selling price of the retailer when the wholesale prices of the upstream entities are prescribed and the retailer's order quantity is chosen before the actual demand is realized. As the benchmark case, the expected total profit of the centralized channel is maximized. The decentralized supply chain is coordinated under pairwise and spanning revenue sharing mechanisms. Numerical study shows that disruptions have remarkable impact on supply chain decisions.

Citation: Bibhas C. Giri, Bhaba R. Sarker. Coordinating a multi-echelon supply chain under production disruption and price-sensitive stochastic demand. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018115
##### References:
 [1] N. Azad, H. Davoudpour, G. K. D. Saharidis and M. Shiripour, A new model to mitigating random disruption risks of facility and transportation in supply chain network design, International Journal of Advanced Manufacturing Technology, 70 (2014), 1757-1774. doi: 10.1007/s00170-013-5404-0. [2] P. Azimi, M. R. Ghanbari and H. Mohammadi, Simulation modeling for analysis of a (Q, r) inventory system under supply disruption and customer differentiation with partial backordering, Modelling and Simulation in Engineering, 2012 (2012), Article ID 103258, 10 pages. doi: 10.1155/2012/103258. [3] G. P. Cachon and M. A. Lariviere, Supply chain coordination with revenue-sharing contracts: Strengths and limitations, Management Science, 51 (2005), 30-44. doi: 10.1287/mnsc.1040.0215. [4] S. S. Chauhan and J. M. Proth, Analysis of a supply chain partnership with revenue sharing, International Journal of Production Economics, 97 (2005), 44-51. doi: 10.1016/j.ijpe.2004.05.006. [5] C. Chen and Y. Fan, Bioethanol supply chain system planning under supply and demand uncertainties, Transportation Research Part E: Logistics and Transportation Review, 48 (2012), 150-164. doi: 10.1016/j.tre.2011.08.004. [6] J. Chen and L. Xu, Coordination of the supply chain of seasonal products, IEEE Transactions on Systems, Man and Cybernetics-Part A: Systems and Humans, 31 (2001), 524-532. [7] M. Dada, N. C. Petruzzi and L. B. Schwarz, A newsvendor's procurement problem when suppliers are unreliable, Manufacturing, & Service Operations Management, 9 (2007), 9-32. doi: 10.1287/msom.1060.0128. [8] S. Dani and A. Deep, Fragile food supply chains - Reacting to risks, International Journal of Logistics Research and Applications, 13 (2010), 395-410. [9] I. Giannoccaro and P. Pontrandolfo, Supply chain coordination by revenue-sharing contracts, International Journal of Production Economics, 89 (2004), 131-139. [10] Q. Gu and T. Gao, Production disruption management for R/M integrated supply chain using system dynamics methodology, International Journal of Sustainable Engineering, 10 (2017), 44-57. doi: 10.1080/19397038.2016.1250838. [11] M. G. Güler and M. E. Keskin, On the coordination under random yield and random demand, Expert Systems with Applications, 40 (2013), 3688-3695. [12] R. Güllü, E. Önol and N. Erkip, Analysis of an inventory system under supply uncertainty, International Journal of Production Economics, 59 (1999), 377-385. [13] Y. He and X. Zhao, Coordination in multi-echelon supply chain under supply and demand uncertainty, International Journal of Production Economics, 139 (2012), 106-115. doi: 10.1016/j.ijpe.2011.04.021. [14] W. J. Hopp, S. M. R. Iravani and Z. Liu, Mitigating the impact of disruptions in supply chains, In Supply Chain Disruptions: Theory and Practice of Managing Risk, edited by H. Gurnani, A. Mehrotra and S. Ray, Springer, (2011), 21-49. doi: 10.1007/978-0-85729-778-5_2. [15] C. Hsieh and C. Wu, Capacity allocation, ordering, and pricing decisions in a supply chain with demand and supply uncertainties, European Journal of Operational Research, 184 (2008), 667-684. doi: 10.1016/j.ejor.2006.11.004. [16] F. Hu, C. C. Lim and Z. Lu, Coordination of supply chains with a flexible ordering policy under yield and demand uncertainty, International Journal of Production Economics, 146 (2013), 686-693. doi: 10.1016/j.ijpe.2013.08.024. [17] J. Huang, M. Leng and M. Parlar, Demand functions in decision modeling: A comprehensive survey and research directions, Decision Sciences, 44 (2013), 557-609. doi: 10.1111/deci.12021. [18] S. H. Huang and P. C. Lin, A modified ant colony optimization algorithm for multi-item inventory routing problems with demand uncertainty, Transportation Research Part E: Logistics and Transportation Review, 46 (2010), 598-611. doi: 10.1016/j.tre.2010.01.006. [19] A. V. Iyer and M. E. Bergen, Quick response in manufacturer-retailer channels, Management Science, 43 (1997), 559-570. doi: 10.1287/mnsc.43.4.559. [20] P. C. Jones, T. J. Lowe, R. D. Traub and G. Kegler, Matching supply and demand: the value of a second chance in producing hybrid seed corn, Manufacturing & Service Operations Management, 3 (2001), 122-137. doi: 10.1287/msom.3.2.122.9992. [21] B. Kazaz, Production planning under yield and demand uncertainty with yield-dependent cost and price, Manufacturing & Service Operations Management, 6 (2004), 209-224. doi: 10.1287/msom.1030.0024. [22] A. H. L. Lau and H. S. Lau, The effects of reducing demand uncertainty in a manufacturer-retailer channel for single-period products, Computers & Operations Research, 29 (2002), 1583-1602. [23] S. X. Li, Z. M. Huang and A. Ashley, Inventory, channel coordination and bargaining in a manufacturer-retailer system, Annals of Operations Research, 68 (1996), 47-60. doi: 10.1007/BF02205448. [24] Q. Li and S. Zheng, Joint inventory replenishment and pricing control for systems with uncertain yield and demand, Operations Research, 54 (2006), 696-705. doi: 10.1287/opre.1060.0273. [25] S. Liu, K. C. So and F. Zhang, The effect of supply reliability in a retail setting with joint marketing and inventory decision, Manufacturing & Services Operations Management, 12 (2010), 19-32. [26] M. K. Mantrala and K. Raman, Demand uncertainty and supplier's returns policies for a multi-store style-good retailer, European Journal of Operational Research, 115 (1999), 270-284. doi: 10.1016/S0377-2217(98)00302-6. [27] P. L. Meena, S. P. Sarmah and A. Sarkar, Sourcing decisions under risks of catastrophic event disruptions, Transportation Research Part E: Logistics and Transportation Review, 47 (2014), 1058-1074. doi: 10.1016/j.tre.2011.03.003. [28] Y. Merzifonluoglu, Risk averse supply portfolio selection with supply, demand and spot market volatility, Omega, 57 (2015), 40-53. doi: 10.1016/j.omega.2015.03.006. [29] C. L. Munson and M. J. Rosenblatt, Coordinating a three-level supply chain with quantity discounts, IIE Transations, 33 (2001), 371-384. doi: 10.1080/07408170108936836. [30] S. K. Paul, R. Sarker and D. Essam, Managing disruption in an imperfect production-inventory system, Computers & Industrial Engineering, 84 (2015), 101-112. [31] S. B. Petkov and C. D. Maranas, Multi period planning and scheduling of multi product batch plants under demand uncertainty, Industrial & Engineering Chemical Research, 36 (1997), 48-64. [32] N. Petruzzi and M. Dada, Pricing and the newsvendor problem: A review with extensions, Operations Research, 47 (1999), 184-194. doi: 10.1287/opre.47.2.183. [33] B. Rhee, J. A. A. Veen, V. Venugopal and V. R. Nalla, A new revenue sharing mechanism for coordinating multi-echelon supply chains, Operations Research Letters, 38 (2010), 296-301. doi: 10.1016/j.orl.2010.03.004. [34] T. Sawik, A portfolio approach to supply chain disruption management, International Journal of Production Research, 55 (2017), 1970-1991. doi: 10.1080/00207543.2016.1249432. [35] A. J. Schmitt and L. V. Snyder, Infinite-horizon models for inventory control under yield uncertainty and disruptions, Computers & Operations Research, 39 (2012), 850-862. doi: 10.1016/j.cor.2010.08.004. [36] D. A. Serel, Inventory and pricing decisions in a single-period problem involving risky supply, International Journal of Production Economics, 116 (2008), 115-128. doi: 10.1016/j.ijpe.2008.07.012. [37] A. R. Singh, P. K. Mishra, R. Jain R and M. K. Khurana, Design of global supply chain network with operational risks, International Journal of Advanced Manufacturing Technology, 60 (2012), 273-290. doi: 10.1007/s00170-011-3615-9. [38] B. Tomlin, On the value of mitigation and contingency strategies for managing supply chain disruption risks, Management Science, 52 (2006), 639-657. doi: 10.1287/mnsc.1060.0515. [39] Z. Wan, H. Wu and L. Dai, A polymorphic uncertain equilibrium model and its deterministic equivalent formulation for decentralized supply chain management, Applied Mathematical Modelling, 58 (2018), 281-299. doi: 10.1016/j.apm.2017.06.028. [40] Y. Wang, Joint pricing-product decisions in supply chains of complementary products with uncertain demand, Operations Research, 54 (2006), 1110-1127. doi: 10.1287/opre.1060.0326. [41] Y. Wang and Y. Gerchak, Periodic review production models with variable capacity, random yield, and uncertain demand, Management Science, 42 (1996), 130-137. doi: 10.1287/mnsc.42.1.130. [42] Y. Wang, L. Jiang and Z. Shen, Channel performance under consignment contract with revenue sharing, Management Science, 50 (2004), 34-47. doi: 10.1287/mnsc.1030.0168. [43] M. C. Wilson, The impact of transportation disruptions on supply chain performance, Transportation Research Part E: Logistics Transportation Review, 43 (2007), 295-320. doi: 10.1016/j.tre.2005.09.008. [44] W. M. Yeo and X. Yuan, Optimal inventory policy with supply uncertainty and demand cancellation, European Journal of Operational Research, 211 (2011), 26-34. doi: 10.1016/j.ejor.2010.10.031. [45] X. Zhang, S. Huang and Z. Wan, Optimal pricing and ordering in global supply chain management with constraints under random demand, Applied Mathematical Modelling, 40 (2016), 10105-10130. doi: 10.1016/j.apm.2016.06.054. [46] A. Z. Zeng and Y. Xia, Building a mutually beneficial partnership to ensure backup supply, Omega, 52 (2015), 77-91. doi: 10.1016/j.omega.2014.10.008.

show all references

##### References:
 [1] N. Azad, H. Davoudpour, G. K. D. Saharidis and M. Shiripour, A new model to mitigating random disruption risks of facility and transportation in supply chain network design, International Journal of Advanced Manufacturing Technology, 70 (2014), 1757-1774. doi: 10.1007/s00170-013-5404-0. [2] P. Azimi, M. R. Ghanbari and H. Mohammadi, Simulation modeling for analysis of a (Q, r) inventory system under supply disruption and customer differentiation with partial backordering, Modelling and Simulation in Engineering, 2012 (2012), Article ID 103258, 10 pages. doi: 10.1155/2012/103258. [3] G. P. Cachon and M. A. Lariviere, Supply chain coordination with revenue-sharing contracts: Strengths and limitations, Management Science, 51 (2005), 30-44. doi: 10.1287/mnsc.1040.0215. [4] S. S. Chauhan and J. M. Proth, Analysis of a supply chain partnership with revenue sharing, International Journal of Production Economics, 97 (2005), 44-51. doi: 10.1016/j.ijpe.2004.05.006. [5] C. Chen and Y. Fan, Bioethanol supply chain system planning under supply and demand uncertainties, Transportation Research Part E: Logistics and Transportation Review, 48 (2012), 150-164. doi: 10.1016/j.tre.2011.08.004. [6] J. Chen and L. Xu, Coordination of the supply chain of seasonal products, IEEE Transactions on Systems, Man and Cybernetics-Part A: Systems and Humans, 31 (2001), 524-532. [7] M. Dada, N. C. Petruzzi and L. B. Schwarz, A newsvendor's procurement problem when suppliers are unreliable, Manufacturing, & Service Operations Management, 9 (2007), 9-32. doi: 10.1287/msom.1060.0128. [8] S. Dani and A. Deep, Fragile food supply chains - Reacting to risks, International Journal of Logistics Research and Applications, 13 (2010), 395-410. [9] I. Giannoccaro and P. Pontrandolfo, Supply chain coordination by revenue-sharing contracts, International Journal of Production Economics, 89 (2004), 131-139. [10] Q. Gu and T. Gao, Production disruption management for R/M integrated supply chain using system dynamics methodology, International Journal of Sustainable Engineering, 10 (2017), 44-57. doi: 10.1080/19397038.2016.1250838. [11] M. G. Güler and M. E. Keskin, On the coordination under random yield and random demand, Expert Systems with Applications, 40 (2013), 3688-3695. [12] R. Güllü, E. Önol and N. Erkip, Analysis of an inventory system under supply uncertainty, International Journal of Production Economics, 59 (1999), 377-385. [13] Y. He and X. Zhao, Coordination in multi-echelon supply chain under supply and demand uncertainty, International Journal of Production Economics, 139 (2012), 106-115. doi: 10.1016/j.ijpe.2011.04.021. [14] W. J. Hopp, S. M. R. Iravani and Z. Liu, Mitigating the impact of disruptions in supply chains, In Supply Chain Disruptions: Theory and Practice of Managing Risk, edited by H. Gurnani, A. Mehrotra and S. Ray, Springer, (2011), 21-49. doi: 10.1007/978-0-85729-778-5_2. [15] C. Hsieh and C. Wu, Capacity allocation, ordering, and pricing decisions in a supply chain with demand and supply uncertainties, European Journal of Operational Research, 184 (2008), 667-684. doi: 10.1016/j.ejor.2006.11.004. [16] F. Hu, C. C. Lim and Z. Lu, Coordination of supply chains with a flexible ordering policy under yield and demand uncertainty, International Journal of Production Economics, 146 (2013), 686-693. doi: 10.1016/j.ijpe.2013.08.024. [17] J. Huang, M. Leng and M. Parlar, Demand functions in decision modeling: A comprehensive survey and research directions, Decision Sciences, 44 (2013), 557-609. doi: 10.1111/deci.12021. [18] S. H. Huang and P. C. Lin, A modified ant colony optimization algorithm for multi-item inventory routing problems with demand uncertainty, Transportation Research Part E: Logistics and Transportation Review, 46 (2010), 598-611. doi: 10.1016/j.tre.2010.01.006. [19] A. V. Iyer and M. E. Bergen, Quick response in manufacturer-retailer channels, Management Science, 43 (1997), 559-570. doi: 10.1287/mnsc.43.4.559. [20] P. C. Jones, T. J. Lowe, R. D. Traub and G. Kegler, Matching supply and demand: the value of a second chance in producing hybrid seed corn, Manufacturing & Service Operations Management, 3 (2001), 122-137. doi: 10.1287/msom.3.2.122.9992. [21] B. Kazaz, Production planning under yield and demand uncertainty with yield-dependent cost and price, Manufacturing & Service Operations Management, 6 (2004), 209-224. doi: 10.1287/msom.1030.0024. [22] A. H. L. Lau and H. S. Lau, The effects of reducing demand uncertainty in a manufacturer-retailer channel for single-period products, Computers & Operations Research, 29 (2002), 1583-1602. [23] S. X. Li, Z. M. Huang and A. Ashley, Inventory, channel coordination and bargaining in a manufacturer-retailer system, Annals of Operations Research, 68 (1996), 47-60. doi: 10.1007/BF02205448. [24] Q. Li and S. Zheng, Joint inventory replenishment and pricing control for systems with uncertain yield and demand, Operations Research, 54 (2006), 696-705. doi: 10.1287/opre.1060.0273. [25] S. Liu, K. C. So and F. Zhang, The effect of supply reliability in a retail setting with joint marketing and inventory decision, Manufacturing & Services Operations Management, 12 (2010), 19-32. [26] M. K. Mantrala and K. Raman, Demand uncertainty and supplier's returns policies for a multi-store style-good retailer, European Journal of Operational Research, 115 (1999), 270-284. doi: 10.1016/S0377-2217(98)00302-6. [27] P. L. Meena, S. P. Sarmah and A. Sarkar, Sourcing decisions under risks of catastrophic event disruptions, Transportation Research Part E: Logistics and Transportation Review, 47 (2014), 1058-1074. doi: 10.1016/j.tre.2011.03.003. [28] Y. Merzifonluoglu, Risk averse supply portfolio selection with supply, demand and spot market volatility, Omega, 57 (2015), 40-53. doi: 10.1016/j.omega.2015.03.006. [29] C. L. Munson and M. J. Rosenblatt, Coordinating a three-level supply chain with quantity discounts, IIE Transations, 33 (2001), 371-384. doi: 10.1080/07408170108936836. [30] S. K. Paul, R. Sarker and D. Essam, Managing disruption in an imperfect production-inventory system, Computers & Industrial Engineering, 84 (2015), 101-112. [31] S. B. Petkov and C. D. Maranas, Multi period planning and scheduling of multi product batch plants under demand uncertainty, Industrial & Engineering Chemical Research, 36 (1997), 48-64. [32] N. Petruzzi and M. Dada, Pricing and the newsvendor problem: A review with extensions, Operations Research, 47 (1999), 184-194. doi: 10.1287/opre.47.2.183. [33] B. Rhee, J. A. A. Veen, V. Venugopal and V. R. Nalla, A new revenue sharing mechanism for coordinating multi-echelon supply chains, Operations Research Letters, 38 (2010), 296-301. doi: 10.1016/j.orl.2010.03.004. [34] T. Sawik, A portfolio approach to supply chain disruption management, International Journal of Production Research, 55 (2017), 1970-1991. doi: 10.1080/00207543.2016.1249432. [35] A. J. Schmitt and L. V. Snyder, Infinite-horizon models for inventory control under yield uncertainty and disruptions, Computers & Operations Research, 39 (2012), 850-862. doi: 10.1016/j.cor.2010.08.004. [36] D. A. Serel, Inventory and pricing decisions in a single-period problem involving risky supply, International Journal of Production Economics, 116 (2008), 115-128. doi: 10.1016/j.ijpe.2008.07.012. [37] A. R. Singh, P. K. Mishra, R. Jain R and M. K. Khurana, Design of global supply chain network with operational risks, International Journal of Advanced Manufacturing Technology, 60 (2012), 273-290. doi: 10.1007/s00170-011-3615-9. [38] B. Tomlin, On the value of mitigation and contingency strategies for managing supply chain disruption risks, Management Science, 52 (2006), 639-657. doi: 10.1287/mnsc.1060.0515. [39] Z. Wan, H. Wu and L. Dai, A polymorphic uncertain equilibrium model and its deterministic equivalent formulation for decentralized supply chain management, Applied Mathematical Modelling, 58 (2018), 281-299. doi: 10.1016/j.apm.2017.06.028. [40] Y. Wang, Joint pricing-product decisions in supply chains of complementary products with uncertain demand, Operations Research, 54 (2006), 1110-1127. doi: 10.1287/opre.1060.0326. [41] Y. Wang and Y. Gerchak, Periodic review production models with variable capacity, random yield, and uncertain demand, Management Science, 42 (1996), 130-137. doi: 10.1287/mnsc.42.1.130. [42] Y. Wang, L. Jiang and Z. Shen, Channel performance under consignment contract with revenue sharing, Management Science, 50 (2004), 34-47. doi: 10.1287/mnsc.1030.0168. [43] M. C. Wilson, The impact of transportation disruptions on supply chain performance, Transportation Research Part E: Logistics Transportation Review, 43 (2007), 295-320. doi: 10.1016/j.tre.2005.09.008. [44] W. M. Yeo and X. Yuan, Optimal inventory policy with supply uncertainty and demand cancellation, European Journal of Operational Research, 211 (2011), 26-34. doi: 10.1016/j.ejor.2010.10.031. [45] X. Zhang, S. Huang and Z. Wan, Optimal pricing and ordering in global supply chain management with constraints under random demand, Applied Mathematical Modelling, 40 (2016), 10105-10130. doi: 10.1016/j.apm.2016.06.054. [46] A. Z. Zeng and Y. Xia, Building a mutually beneficial partnership to ensure backup supply, Omega, 52 (2015), 77-91. doi: 10.1016/j.omega.2014.10.008.
Impact of $\alpha$ on the manufacturer's decisions
Impact of $\beta$ on the supplier's profit
Impact of $\beta$ on the supply chain's total profit
Effects of $\alpha$ and $\beta$ on the manufacturer's and the supplier's decentralized decisions
 when $p^d$ (= 24.60) is known when $Q^d$ (= 19.42) is known $\alpha$ $Q^d$ $\Pi_m$ $\beta$ $R^d$ $\Pi_s$ 0.0 16.79 50.36 0.0 19.42 38.84 0.2 19.42 38.33 0.2 22.62 29.94 0.4 21.28 29.82 0.4 25.20 25.45 0.6 22.03 21.98 0.6 26.27 21.91 0.8 22.43 14.33 0.8 26.87 18.65 1.0 22.69 6.76 1.0 27.24 15.51
 when $p^d$ (= 24.60) is known when $Q^d$ (= 19.42) is known $\alpha$ $Q^d$ $\Pi_m$ $\beta$ $R^d$ $\Pi_s$ 0.0 16.79 50.36 0.0 19.42 38.84 0.2 19.42 38.33 0.2 22.62 29.94 0.4 21.28 29.82 0.4 25.20 25.45 0.6 22.03 21.98 0.6 26.27 21.91 0.8 22.43 14.33 0.8 26.87 18.65 1.0 22.69 6.76 1.0 27.24 15.51
Optimal results for different values of $\xi$ and $\eta$ in the decentralized system
 $\xi$ $\eta$ $\tilde{p}^d\tilde{\Pi}_r$ $\tilde{Q}_d\tilde{\Pi}_m$ $\tilde{R}^d\tilde{\Pi}_s$ Total profit 0.95 0.90 23.58 89.98 24.12 42.50 30.11 31.18 163.66 0.92 23.58 89.98 24.60 46.76 30.71 27.24 163.98 0.94 23.58 89.98 25.12 51.07 30.71 23.06 164.11 0.97 0.90 23.03 96.38 25.89 39.40 32.32 32.78 168.56 0.92 23.03 96.38 26.41 43.83 32.97 28.80 169.01 0.94 23.03 96.38 26.97 48.32 33.67 24.74 169.44 0.99 0.90 22.49 103.11 27.80 35.89 34.71 34.48 173.48 0.92 22.49 103.11 28.35 40.51 35.39 30.35 173.97 0.94 22.49 103.11 28.96 45.19 36.16 26.14 174.44
 $\xi$ $\eta$ $\tilde{p}^d\tilde{\Pi}_r$ $\tilde{Q}_d\tilde{\Pi}_m$ $\tilde{R}^d\tilde{\Pi}_s$ Total profit 0.95 0.90 23.58 89.98 24.12 42.50 30.11 31.18 163.66 0.92 23.58 89.98 24.60 46.76 30.71 27.24 163.98 0.94 23.58 89.98 25.12 51.07 30.71 23.06 164.11 0.97 0.90 23.03 96.38 25.89 39.40 32.32 32.78 168.56 0.92 23.03 96.38 26.41 43.83 32.97 28.80 169.01 0.94 23.03 96.38 26.97 48.32 33.67 24.74 169.44 0.99 0.90 22.49 103.11 27.80 35.89 34.71 34.48 173.48 0.92 22.49 103.11 28.35 40.51 35.39 30.35 173.97 0.94 22.49 103.11 28.96 45.19 36.16 26.14 174.44
A comparison of results in different scenarios of pairwise RS contract
 Decentralized model Retailer's profit Manufacturer's profit Supplier's profit Total profit without RS contract 87.03 38.33 27.48 152.84 Pairwise RS contract $\xi =0.95, \eta = 0.90$ 89.98 42.50 31.18 163.66 $\xi =0.95, \eta = 1.0$ 89.98 64.38 11.21 165.57 $\xi =1.0, \eta = 0.90$ 106.60 33.96 35.35 175.91
 Decentralized model Retailer's profit Manufacturer's profit Supplier's profit Total profit without RS contract 87.03 38.33 27.48 152.84 Pairwise RS contract $\xi =0.95, \eta = 0.90$ 89.98 42.50 31.18 163.66 $\xi =0.95, \eta = 1.0$ 89.98 64.38 11.21 165.57 $\xi =1.0, \eta = 0.90$ 106.60 33.96 35.35 175.91
Optimal results in Example 2
 Model scenario Retailer's profit Manufacturer's profit Supplier's profit Total profit Centralized - - - 463.60 Decentralized without RS contract 167.97 73.05 52.35 293.37 Decentralized with pairwise RS $\xi = 0.95, \eta = 0.90$ 172.28 81.26 58.84 312.38 Decentralized with spanning RS $\xi_1 = 0.05, \xi_2 = 0.02$ 178.30 81.92 62.27 322.49
 Model scenario Retailer's profit Manufacturer's profit Supplier's profit Total profit Centralized - - - 463.60 Decentralized without RS contract 167.97 73.05 52.35 293.37 Decentralized with pairwise RS $\xi = 0.95, \eta = 0.90$ 172.28 81.26 58.84 312.38 Decentralized with spanning RS $\xi_1 = 0.05, \xi_2 = 0.02$ 178.30 81.92 62.27 322.49
Optimal results for different values of $e$ in the decentralized system
 $e$ Retailer($p^d, \Pi_r^d$) Manufacturer($Q^d, \Pi_m^d$) Supplier($R^d, \Pi_s^d$) Total profit 3.0 (31.5,167.97) (37.01, 73.05) (46.21, 52.35) 293.37 3.1 (31.0,119.06) (27.54, 54.37) (34.38, 38.96) 212.39 3.2 (30.55, 84.52) (20.47, 40.41) (25.31, 28.68) 153.61 3.3 (30.13, 60.08) (15.22, 30.05) (19.0, 21.53) 111.66 3.4 (29.75, 42.77) (11.31, 22.32) (14.12, 16.0) 81.09 3.5 (29.40, 30.48) (8.39, 16.58) (10.47, 11.87) 58.92
 $e$ Retailer($p^d, \Pi_r^d$) Manufacturer($Q^d, \Pi_m^d$) Supplier($R^d, \Pi_s^d$) Total profit 3.0 (31.5,167.97) (37.01, 73.05) (46.21, 52.35) 293.37 3.1 (31.0,119.06) (27.54, 54.37) (34.38, 38.96) 212.39 3.2 (30.55, 84.52) (20.47, 40.41) (25.31, 28.68) 153.61 3.3 (30.13, 60.08) (15.22, 30.05) (19.0, 21.53) 111.66 3.4 (29.75, 42.77) (11.31, 22.32) (14.12, 16.0) 81.09 3.5 (29.40, 30.48) (8.39, 16.58) (10.47, 11.87) 58.92
Optimal results for different values of $a$ in the decentralized system
 $a$ Retailer($p^d, \Pi_r^d$) Manufacturer($Q^d, \Pi_m^d$) Supplier($R^d, \Pi_s^d$) Total profit 5000 (31.5,167.97) (37.01, 73.05) (46.21, 52.35) 293.37 6000 (31.5,201.56) (44.41, 87.66) (55.45, 62.83) 352.05 7000 (31.5,235.16) (51.81,102.28) (64.68, 73.30) 410.74 8000 (31.5,268.75) (59.21,116.89) (73.92, 83.77) 469.41 9000 (31.5,302.34) (66.61,131.50) (83.16, 94.24) 528.08 10000 (31.5,335.94) (74.01,146.11) (92.40,104.71) 586.76
 $a$ Retailer($p^d, \Pi_r^d$) Manufacturer($Q^d, \Pi_m^d$) Supplier($R^d, \Pi_s^d$) Total profit 5000 (31.5,167.97) (37.01, 73.05) (46.21, 52.35) 293.37 6000 (31.5,201.56) (44.41, 87.66) (55.45, 62.83) 352.05 7000 (31.5,235.16) (51.81,102.28) (64.68, 73.30) 410.74 8000 (31.5,268.75) (59.21,116.89) (73.92, 83.77) 469.41 9000 (31.5,302.34) (66.61,131.50) (83.16, 94.24) 528.08 10000 (31.5,335.94) (74.01,146.11) (92.40,104.71) 586.76
Optimal results for different values of $\sigma$ in the decentralized system
 $\sigma$ Retailer's profit Manufacturer's profit Supplier's profit Total profit 51 167.97 73.05 52.35 293.37 53 160.95 69.96 50.14 281.05 55 153.97 66.81 47.88 268.66 57 147.04 63.75 45.69 256.48 59 140.19 60.74 43.53 244.46
 $\sigma$ Retailer's profit Manufacturer's profit Supplier's profit Total profit 51 167.97 73.05 52.35 293.37 53 160.95 69.96 50.14 281.05 55 153.97 66.81 47.88 268.66 57 147.04 63.75 45.69 256.48 59 140.19 60.74 43.53 244.46
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