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doi: 10.3934/jimo.2019077

Bargaining equilibrium in a two-echelon supply chain with a capital-constrained retailer

1. 

School of Business Administration, Hunan University, Changsha, Hunan Province 410082, China

2. 

School of Business, State University of New York at Oswego, Oswego, NY 13126, USA

3. 

School of Marketing and Logistics Management, Nanjing University of Finance and Economics, Nanjing, Jiangsu 210023, China

* Corresponding author: ottoyang@126.com (Honglin Yang)

Received  August 2018 Revised  March 2019 Published  July 2019

Fund Project: This work was supported by the National Natural Science Foundation of China under Grant Nos. 71571065, 71790593 and 71521061

We investigate the bargaining equilibrium in a two-echelon supply chain consisting of a supplier and a capital-constrained retailer. The newsvendor-like retailer can borrow from a bank or use the supplier's trade credit to fund his business. In the presence of bankruptcy risk for both the supplier and retailer, with a wholesale price contract, we model the player's strategic interactions under the Nash and Rubinstein bargaining games. In both financing schemes, the Nash bargaining game overcomes the double marginalization effect under the Stackelberg game and achieves supply chain coordination. The Rubinstein bargaining game realizes the Pareto improvement of the supply chain. The player with stronger bargaining power always prefers to initially offer a contract under the Rubinstein bargaining game to obtain greater expected profit. Furthermore, we characterize the conditions under which bargaining power and discount factor affect the bargaining equilibrium. We numerically verify our theoretical results.

Citation: Honglin Yang, Qiang Yan, Hong Wan, Wenyan Zhuo. Bargaining equilibrium in a two-echelon supply chain with a capital-constrained retailer. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019077
References:
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S. Z. Alparslan-GökS. Miquel and S. H. Tijs, Cooperation under interval uncertainty, Mathematical Methods of Operations Research, 69 (2009), 99-109. doi: 10.1007/s00186-008-0211-3. Google Scholar

[2]

O. BaronO. Berman and D. Wu, Bargaining within the supply chain and its implications in an industry, Decision Sciences, 47 (2016), 193-218. Google Scholar

[3]

M. Berlin, Trade credit: Why do production firms act as financial intermediaries?, Business Review, Q3 (2003), 21-28. Google Scholar

[4]

K. BinmoreA. Rubinstein and A. Wolinsky, The Nash bargaining solution in economic modelling, Rand Journal of Economics, 17 (1986), 176-188. doi: 10.2307/2555382. Google Scholar

[5]

M. J. BrennanV. Maksimovic and J. Zechner, Vendor financing, Journal of Finance, 43 (1988), 1127-1141. doi: 10.1111/j.1540-6261.1988.tb03960.x. Google Scholar

[6]

D. B. Bradley and M. J. Rubach, Trade credit and small businesses: A cause of business failures?, University of Central Arkansas, (2002).Google Scholar

[7]

J. A. Buzacott and R. Q. Zhang, Inventory management with asset-based financing, Management Science, 50 (2004), 1274-1292. doi: 10.1287/mnsc.1040.0278. Google Scholar

[8]

X. Chen and A. Wang, Trade credit contract with limited liability in the supply chain with budget constraints, Annals of Operations Research, 196 (2012), 153-165. doi: 10.1007/s10479-012-1119-0. Google Scholar

[9]

X. Chen, A model of trade credit in a capital-constrained distribution channel, International Journal of Production Economics, 159 (2015), 347-357. doi: 10.1007/s10479-014-1602-x. Google Scholar

[10]

M. S. ChernQ. PanJ. T. TengY. L. Chan and S. C. Chen, Stackelberg solution in a vendor-buyer supply chain model with permissible delay in payments, International Journal of Production Economics, 144 (2013), 397-404. doi: 10.1016/j.ijpe.2013.03.008. Google Scholar

[11]

K. J. Chung, The simplified solution procedures for the optimal replenishment decisions under two levels of trade credit policy depending on the order quantity in a supply chain system, Expert Systems with Applications, 38 (2011), 13482-13486. doi: 10.1016/j.eswa.2011.04.094. Google Scholar

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M. Dada and Q. Hu, Financing newsvendor inventory, Operations Research Letters, 36 (2008), 569-573. doi: 10.1016/j.orl.2008.06.004. Google Scholar

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M. DraganskaD. Klapper and S. B. Villas-Boas, A larger slice or a larger pie? An empirical investigation of bargaining power in the distribution channel, Marketing Science, 29 (2010), 57-74. Google Scholar

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G. Ertek and P. M. Griffin, Supplier-and buyer-driven channels in a two-stage supply chain, IIE Transactions, 34 (2002), 691-700. doi: 10.1080/07408170208928905. Google Scholar

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X. FengI. Moon and K. Ryu, Supply chain coordination under budget constraints, Computers & Industrial Engineering, 88 (2015), 487-500. doi: 10.1016/j.cie.2015.08.005. Google Scholar

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Q. FengG. Lai and L. X. Lu, Dynamic bargaining in a supply chain with asymmetric demand information, Management Science, 61 (2014), 301-315. Google Scholar

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L. Guo, The benefits of downstream information acquisition, Marketing Science, 28 (2009), 457-471. Google Scholar

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Z. HuaS. Li and L. Liang, Impact of demand uncertainty on supply chain cooperation of single-period products, International Journal of Production Economics, 100 (2006), 268-284. doi: 10.1016/j.ijpe.2004.11.007. Google Scholar

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G. Iyer and J. M. Villas-Boas, A bargaining theory of distribution channels, Journal of Marketing Research, 40 (2003), 80-100. doi: 10.1509/jmkr.40.1.80.19134. Google Scholar

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B. JingX. Chen and G. Cai, Equilibrium financing in a distribution channel with capital constraint, Production and Operations Management, 21 (2012), 1090-1101. doi: 10.1111/j.1937-5956.2012.01328.x. Google Scholar

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B. Jing and A. Seidmann, Finance sourcing in a supply chain, Decision Support Systems, 58 (2014), 15-20. Google Scholar

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P. Kouvelis and W. Zhao, Financing the newsvendor: supplier vs. bank, and the structure of optimal trade credit contracts, Operations Research, 60 (2012), 566-580. doi: 10.1287/opre.1120.1040. Google Scholar

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P. Kouvelis and W. Zhao, Supply chain contract design under financial constraints and bankruptcy costs, Management Science, 62 (2015), 2341-2357. doi: 10.1287/mnsc.2015.2248. Google Scholar

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P. Kouvelis and W. Zhao, Who should finance the supply chain? Impact of credit ratings on supply chain decisions, Manufacturing & Service Operations Management, 20 (2017), 19-35. doi: 10.1287/msom.2017.0669. Google Scholar

[25]

V. B. Kreng and S. J. Tan, The optimal replenishment decisions under two levels of trade credit policy depending on the order quantity, Expert Systems with Applications, 37 (2010), 5514-5522. doi: 10.1016/j.eswa.2009.12.014. Google Scholar

[26]

M. A. Lariviere and E. L. Porteus, Selling to the newsvendor: An analysis of price-only contracts, Manufacturing & Service Operations Management, 73 (2001), 293-305. doi: 10.1287/msom.3.4.293.9971. Google Scholar

[27]

C. H. Lee and B. D. Rhee, Trade credit for supply chain coordination, European Journal of Operational Research, 214 (2011), 136-146. doi: 10.1016/j.ejor.2011.04.004. Google Scholar

[28]

R. LiK. SkouriJ. T. Teng and W. G. Yang, Seller's optimal replenishment policy and payment term among advance, cash, and credit payments, International Journal of Production Economics, 197 (2018), 35-42. doi: 10.1016/j.ijpe.2017.12.015. Google Scholar

[29]

R. Lotfi, G. W. Weber, S. M. Sajadifar and N. Mardani, Interdependent demand in the two-period newsvendor problem, Journal of Industrial & Management Optimization, , (2018), 777–792. doi: 10.3934/jimo.2018143. Google Scholar

[30]

L. MaF. LiuS. Li and H. Yan, Channel bargaining with risk-averse retailer, International Journal of Production Economics, 139 (2012), 155-167. doi: 10.1016/j.ijpe.2010.08.016. Google Scholar

[31]

G. C. Mahata, An EPQ-based inventory model for exponentially deteriorating items under retailer partial trade credit policy in supply chain, Expert Systems with Applications, 39 (2012), 3537-3550. doi: 10.1016/j.eswa.2011.09.044. Google Scholar

[32]

A. MirzazadehM. M. Seyyed Esfahani and S. M. T. Fatemi Ghomi, An inventory model under uncertain inflationary conditions, finite production rate and inflation-dependent demand rate for deteriorating items with shortages, International Journal of Systems Science, 40 (2009), 21-31. doi: 10.1080/00207720802088264. Google Scholar

[33]

A. Mirzazadeh, A comparison of the mathematical modeling methods in the inventory systems under uncertain conditions, International Journal of Engineering Science and Technology, 3 (2011), 6131-6142. Google Scholar

[34] A. Muthoo, Bargaining Theory with Appliactions, Cambridge University Press, 1999. Google Scholar
[35]

J. F. Nash and Jr ., The bargaining problem, Econometrica, 18 (1950), 155-162. doi: 10.2307/1907266. Google Scholar

[36]

E. Özceylan and T. Paksoy, A mixed integer programming model for a closed-loop supply-chain network, International Journal of Production Research, 51 (2013), 718-734. Google Scholar

[37]

T. PaksoyÍ. KaraoǧlanH. GökçenP. M. Pardalos and B. Torǧul, An experimental research on closed loop supply chain management with Internet of things, Journal of Economics Bibliography, 3 (2016), 1-20. Google Scholar

[38]

M. PervinS. K. Roy and G. W. Weber, An integrated inventory model with variable holding cost under two levels of trade-credit policy, Numerical Algebra, Control & Optimization, 8 (2018), 169-191. doi: 10.3934/naco.2018010. Google Scholar

[39]

M. Pervin, S. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price-and stock-dependent demand: A trade-credit policy, Journal of Industrial & Management Optimization, (2018), 658–662. doi: 10.3934/naco.2017002. Google Scholar

[40]

Q. QingT. Deng and H. Wang, Capacity allocation under downstream competition and bargaining, European Journal of Operational Research, 261 (2017), 97-107. doi: 10.1016/j.ejor.2017.01.031. Google Scholar

[41]

S. K. Roy, Game Theory Under MCDM and Fuzzy Set Theory: Some Problems in Multi-Criteria Decision Making Using Game Theoretic Approach, VDM Publishing, 2010.Google Scholar

[42]

A. Rubinstein, Perfect equilibrium in a bargaining model, Econometrica, 50 (1982), 97-109. doi: 10.2307/1912531. Google Scholar

[43]

N. S. Raghavan and V. K. Mishra, Short-term financing in a cash-constrained supply chain, International Journal of Production Economics, 134 (2011), 407-412. Google Scholar

[44]

D. SeifertR. W. Seifert and M. Protopappa-Sieke, A review of trade credit literature: Opportunities for research in operations?, European Journal of Operational Research, 231 (2013), 245-256. doi: 10.1016/j.ejor.2013.03.016. Google Scholar

[45]

J. Sutton, Non-cooperative bargaining theory: An introduction, The Review of Economic Studies, 53 (1986), 709-724. doi: 10.2307/2297715. Google Scholar

[46]

A. A. TaleizadehD. W. PenticoM. S. Jabalameli and M. Aryanezhad, An EOQ model with partial delayed payment and partial backordering, Omega, 41 (2013), 354-368. doi: 10.1016/j.omega.2012.03.008. Google Scholar

[47]

T. I. Tunca and W. Zhu, Buyer intermediation in supplier finance, Management Science, 64 (2017), 5461-5959. doi: 10.1287/mnsc.2017.2863. Google Scholar

[48]

A. A. Vasin and P. A. Kartunova, Auctions of homogeneous goods: Game-theoretic analysis, Contributions to Game Theory and Management, 8 (2015), 315-335. Google Scholar

[49]

A. VasinM. Dolmatova and G. W. Weber, Supply function equilibria for uniform price auction in oligopolistic markets, Central European Journal of Operations Research, 24 (2016), 819-831. doi: 10.1007/s10100-015-0390-y. Google Scholar

[50]

A. VasinH. GaoM. Dolmatova and G. W. Weber, Optimization of transmission network for homogeneous good market, Optimization, 66 (2017), 2125-2134. doi: 10.1080/02331934.2016.1247269. Google Scholar

[51]

D. WuO. Baron and O. Berman, Bargaining in competing supply chains with uncertainty, European Journal of Operational Research, 197 (2009), 548-556. doi: 10.1016/j.ejor.2008.06.032. Google Scholar

[52]

D. D. Wu, Bargaining in supply chain with price and promotional effort dependent demand, Mathematical and Computer Modelling, 58 (2013), 1659-1669. doi: 10.1016/j.mcm.2010.12.035. Google Scholar

[53]

X. Xu and J. R. Birge, Production and financing decisions: Modeling and analysis, (2004). doi: 10.2139/ssrn.652562. Google Scholar

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N. YanB. SunH. Zhang and C. Liu, A partial credit guarantee contract in a capital-constrained supply chain: Financing equilibrium and coordinating strategy, International Journal of Production Economics, 173 (2016), 122-133. doi: 10.1016/j.ijpe.2015.12.005. Google Scholar

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[57]

H. YangW. Zhuo and L. Shao, Equilibrium evolution in a two-echelon supply chain with financially constrained retailers: The impact of equity financing, International Journal of Production Economics, 185 (2017), 139-149. doi: 10.1016/j.ijpe.2016.12.027. Google Scholar

[58]

H. Yang, H. Dai and W. Zhuo, Permissible delay period and pricing decisions in a two-echelon supply chain, Applied Economics Letters, 24 (2017), 820-825. doi: 10.1080/13504851.2016.1231889. Google Scholar

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show all references

References:
[1]

S. Z. Alparslan-GökS. Miquel and S. H. Tijs, Cooperation under interval uncertainty, Mathematical Methods of Operations Research, 69 (2009), 99-109. doi: 10.1007/s00186-008-0211-3. Google Scholar

[2]

O. BaronO. Berman and D. Wu, Bargaining within the supply chain and its implications in an industry, Decision Sciences, 47 (2016), 193-218. Google Scholar

[3]

M. Berlin, Trade credit: Why do production firms act as financial intermediaries?, Business Review, Q3 (2003), 21-28. Google Scholar

[4]

K. BinmoreA. Rubinstein and A. Wolinsky, The Nash bargaining solution in economic modelling, Rand Journal of Economics, 17 (1986), 176-188. doi: 10.2307/2555382. Google Scholar

[5]

M. J. BrennanV. Maksimovic and J. Zechner, Vendor financing, Journal of Finance, 43 (1988), 1127-1141. doi: 10.1111/j.1540-6261.1988.tb03960.x. Google Scholar

[6]

D. B. Bradley and M. J. Rubach, Trade credit and small businesses: A cause of business failures?, University of Central Arkansas, (2002).Google Scholar

[7]

J. A. Buzacott and R. Q. Zhang, Inventory management with asset-based financing, Management Science, 50 (2004), 1274-1292. doi: 10.1287/mnsc.1040.0278. Google Scholar

[8]

X. Chen and A. Wang, Trade credit contract with limited liability in the supply chain with budget constraints, Annals of Operations Research, 196 (2012), 153-165. doi: 10.1007/s10479-012-1119-0. Google Scholar

[9]

X. Chen, A model of trade credit in a capital-constrained distribution channel, International Journal of Production Economics, 159 (2015), 347-357. doi: 10.1007/s10479-014-1602-x. Google Scholar

[10]

M. S. ChernQ. PanJ. T. TengY. L. Chan and S. C. Chen, Stackelberg solution in a vendor-buyer supply chain model with permissible delay in payments, International Journal of Production Economics, 144 (2013), 397-404. doi: 10.1016/j.ijpe.2013.03.008. Google Scholar

[11]

K. J. Chung, The simplified solution procedures for the optimal replenishment decisions under two levels of trade credit policy depending on the order quantity in a supply chain system, Expert Systems with Applications, 38 (2011), 13482-13486. doi: 10.1016/j.eswa.2011.04.094. Google Scholar

[12]

M. Dada and Q. Hu, Financing newsvendor inventory, Operations Research Letters, 36 (2008), 569-573. doi: 10.1016/j.orl.2008.06.004. Google Scholar

[13]

M. DraganskaD. Klapper and S. B. Villas-Boas, A larger slice or a larger pie? An empirical investigation of bargaining power in the distribution channel, Marketing Science, 29 (2010), 57-74. Google Scholar

[14]

G. Ertek and P. M. Griffin, Supplier-and buyer-driven channels in a two-stage supply chain, IIE Transactions, 34 (2002), 691-700. doi: 10.1080/07408170208928905. Google Scholar

[15]

X. FengI. Moon and K. Ryu, Supply chain coordination under budget constraints, Computers & Industrial Engineering, 88 (2015), 487-500. doi: 10.1016/j.cie.2015.08.005. Google Scholar

[16]

Q. FengG. Lai and L. X. Lu, Dynamic bargaining in a supply chain with asymmetric demand information, Management Science, 61 (2014), 301-315. Google Scholar

[17]

L. Guo, The benefits of downstream information acquisition, Marketing Science, 28 (2009), 457-471. Google Scholar

[18]

Z. HuaS. Li and L. Liang, Impact of demand uncertainty on supply chain cooperation of single-period products, International Journal of Production Economics, 100 (2006), 268-284. doi: 10.1016/j.ijpe.2004.11.007. Google Scholar

[19]

G. Iyer and J. M. Villas-Boas, A bargaining theory of distribution channels, Journal of Marketing Research, 40 (2003), 80-100. doi: 10.1509/jmkr.40.1.80.19134. Google Scholar

[20]

B. JingX. Chen and G. Cai, Equilibrium financing in a distribution channel with capital constraint, Production and Operations Management, 21 (2012), 1090-1101. doi: 10.1111/j.1937-5956.2012.01328.x. Google Scholar

[21]

B. Jing and A. Seidmann, Finance sourcing in a supply chain, Decision Support Systems, 58 (2014), 15-20. Google Scholar

[22]

P. Kouvelis and W. Zhao, Financing the newsvendor: supplier vs. bank, and the structure of optimal trade credit contracts, Operations Research, 60 (2012), 566-580. doi: 10.1287/opre.1120.1040. Google Scholar

[23]

P. Kouvelis and W. Zhao, Supply chain contract design under financial constraints and bankruptcy costs, Management Science, 62 (2015), 2341-2357. doi: 10.1287/mnsc.2015.2248. Google Scholar

[24]

P. Kouvelis and W. Zhao, Who should finance the supply chain? Impact of credit ratings on supply chain decisions, Manufacturing & Service Operations Management, 20 (2017), 19-35. doi: 10.1287/msom.2017.0669. Google Scholar

[25]

V. B. Kreng and S. J. Tan, The optimal replenishment decisions under two levels of trade credit policy depending on the order quantity, Expert Systems with Applications, 37 (2010), 5514-5522. doi: 10.1016/j.eswa.2009.12.014. Google Scholar

[26]

M. A. Lariviere and E. L. Porteus, Selling to the newsvendor: An analysis of price-only contracts, Manufacturing & Service Operations Management, 73 (2001), 293-305. doi: 10.1287/msom.3.4.293.9971. Google Scholar

[27]

C. H. Lee and B. D. Rhee, Trade credit for supply chain coordination, European Journal of Operational Research, 214 (2011), 136-146. doi: 10.1016/j.ejor.2011.04.004. Google Scholar

[28]

R. LiK. SkouriJ. T. Teng and W. G. Yang, Seller's optimal replenishment policy and payment term among advance, cash, and credit payments, International Journal of Production Economics, 197 (2018), 35-42. doi: 10.1016/j.ijpe.2017.12.015. Google Scholar

[29]

R. Lotfi, G. W. Weber, S. M. Sajadifar and N. Mardani, Interdependent demand in the two-period newsvendor problem, Journal of Industrial & Management Optimization, , (2018), 777–792. doi: 10.3934/jimo.2018143. Google Scholar

[30]

L. MaF. LiuS. Li and H. Yan, Channel bargaining with risk-averse retailer, International Journal of Production Economics, 139 (2012), 155-167. doi: 10.1016/j.ijpe.2010.08.016. Google Scholar

[31]

G. C. Mahata, An EPQ-based inventory model for exponentially deteriorating items under retailer partial trade credit policy in supply chain, Expert Systems with Applications, 39 (2012), 3537-3550. doi: 10.1016/j.eswa.2011.09.044. Google Scholar

[32]

A. MirzazadehM. M. Seyyed Esfahani and S. M. T. Fatemi Ghomi, An inventory model under uncertain inflationary conditions, finite production rate and inflation-dependent demand rate for deteriorating items with shortages, International Journal of Systems Science, 40 (2009), 21-31. doi: 10.1080/00207720802088264. Google Scholar

[33]

A. Mirzazadeh, A comparison of the mathematical modeling methods in the inventory systems under uncertain conditions, International Journal of Engineering Science and Technology, 3 (2011), 6131-6142. Google Scholar

[34] A. Muthoo, Bargaining Theory with Appliactions, Cambridge University Press, 1999. Google Scholar
[35]

J. F. Nash and Jr ., The bargaining problem, Econometrica, 18 (1950), 155-162. doi: 10.2307/1907266. Google Scholar

[36]

E. Özceylan and T. Paksoy, A mixed integer programming model for a closed-loop supply-chain network, International Journal of Production Research, 51 (2013), 718-734. Google Scholar

[37]

T. PaksoyÍ. KaraoǧlanH. GökçenP. M. Pardalos and B. Torǧul, An experimental research on closed loop supply chain management with Internet of things, Journal of Economics Bibliography, 3 (2016), 1-20. Google Scholar

[38]

M. PervinS. K. Roy and G. W. Weber, An integrated inventory model with variable holding cost under two levels of trade-credit policy, Numerical Algebra, Control & Optimization, 8 (2018), 169-191. doi: 10.3934/naco.2018010. Google Scholar

[39]

M. Pervin, S. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price-and stock-dependent demand: A trade-credit policy, Journal of Industrial & Management Optimization, (2018), 658–662. doi: 10.3934/naco.2017002. Google Scholar

[40]

Q. QingT. Deng and H. Wang, Capacity allocation under downstream competition and bargaining, European Journal of Operational Research, 261 (2017), 97-107. doi: 10.1016/j.ejor.2017.01.031. Google Scholar

[41]

S. K. Roy, Game Theory Under MCDM and Fuzzy Set Theory: Some Problems in Multi-Criteria Decision Making Using Game Theoretic Approach, VDM Publishing, 2010.Google Scholar

[42]

A. Rubinstein, Perfect equilibrium in a bargaining model, Econometrica, 50 (1982), 97-109. doi: 10.2307/1912531. Google Scholar

[43]

N. S. Raghavan and V. K. Mishra, Short-term financing in a cash-constrained supply chain, International Journal of Production Economics, 134 (2011), 407-412. Google Scholar

[44]

D. SeifertR. W. Seifert and M. Protopappa-Sieke, A review of trade credit literature: Opportunities for research in operations?, European Journal of Operational Research, 231 (2013), 245-256. doi: 10.1016/j.ejor.2013.03.016. Google Scholar

[45]

J. Sutton, Non-cooperative bargaining theory: An introduction, The Review of Economic Studies, 53 (1986), 709-724. doi: 10.2307/2297715. Google Scholar

[46]

A. A. TaleizadehD. W. PenticoM. S. Jabalameli and M. Aryanezhad, An EOQ model with partial delayed payment and partial backordering, Omega, 41 (2013), 354-368. doi: 10.1016/j.omega.2012.03.008. Google Scholar

[47]

T. I. Tunca and W. Zhu, Buyer intermediation in supplier finance, Management Science, 64 (2017), 5461-5959. doi: 10.1287/mnsc.2017.2863. Google Scholar

[48]

A. A. Vasin and P. A. Kartunova, Auctions of homogeneous goods: Game-theoretic analysis, Contributions to Game Theory and Management, 8 (2015), 315-335. Google Scholar

[49]

A. VasinM. Dolmatova and G. W. Weber, Supply function equilibria for uniform price auction in oligopolistic markets, Central European Journal of Operations Research, 24 (2016), 819-831. doi: 10.1007/s10100-015-0390-y. Google Scholar

[50]

A. VasinH. GaoM. Dolmatova and G. W. Weber, Optimization of transmission network for homogeneous good market, Optimization, 66 (2017), 2125-2134. doi: 10.1080/02331934.2016.1247269. Google Scholar

[51]

D. WuO. Baron and O. Berman, Bargaining in competing supply chains with uncertainty, European Journal of Operational Research, 197 (2009), 548-556. doi: 10.1016/j.ejor.2008.06.032. Google Scholar

[52]

D. D. Wu, Bargaining in supply chain with price and promotional effort dependent demand, Mathematical and Computer Modelling, 58 (2013), 1659-1669. doi: 10.1016/j.mcm.2010.12.035. Google Scholar

[53]

X. Xu and J. R. Birge, Production and financing decisions: Modeling and analysis, (2004). doi: 10.2139/ssrn.652562. Google Scholar

[54]

N. YanB. SunH. Zhang and C. Liu, A partial credit guarantee contract in a capital-constrained supply chain: Financing equilibrium and coordinating strategy, International Journal of Production Economics, 173 (2016), 122-133. doi: 10.1016/j.ijpe.2015.12.005. Google Scholar

[55]

S. YangK. S. Hong and C. Lee, Supply chain coordination with stock- dependent demand rate and credit incentives, International Journal of Production Economics, 261 (2017), 97-107. doi: 10.1016/j.ijpe.2013.06.014. Google Scholar

[56]

H. YangW. ZhuoY. Zha and H. Wan, Two-period supply chain with flexible trade credit contract, Expert Systems with Applications, 66 (2016), 95-105. doi: 10.1016/j.eswa.2016.08.056. Google Scholar

[57]

H. YangW. Zhuo and L. Shao, Equilibrium evolution in a two-echelon supply chain with financially constrained retailers: The impact of equity financing, International Journal of Production Economics, 185 (2017), 139-149. doi: 10.1016/j.ijpe.2016.12.027. Google Scholar

[58]

H. Yang, H. Dai and W. Zhuo, Permissible delay period and pricing decisions in a two-echelon supply chain, Applied Economics Letters, 24 (2017), 820-825. doi: 10.1080/13504851.2016.1231889. Google Scholar

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Figure 1.  Mobike’s financing and operation
Figure 2.  $ \pi_{Bj}^{R*} $ and $ \delta_j $
Figure 3.  $ w $ and $ \beta $
Figure 4.  Relationship of $ \pi_{is}^{N*} $ and $ \pi_{is}^{S*} $ with $ \beta $ with BCF and TCF
Figure 5.  Relationship of $ \pi_{ir}^{N*} $ and $ \pi_{is}^{S*} $ with $ \beta $ with BCF and TCF
Table 1.  Notations and explanations
Notation Explanation
$ D $ Uncertain demand.
$ p $ Retailer's retail price per unit, where $ p=1 $.
$ c $ Supplier's production cost per unit, where $ 0 <c<1 $.
$ q^0 $ Supply chain's optimal order quantity in the centralized case.
$ r_B $ Bank's interest rate.
$ f(\cdot) $ Probability density function of $ D $.
$ F(\cdot) $ Cumulative distribution function of $ D $.
$ w_i^K $ Supplier's wholesale price per unit purchased under game $ K $,
where $ K= S,N,R $ denotes Stackelberg, Nash bargaining and
Rubinstein bargaining game, respectively. The subscript $ i=B,T $
denotes BCF and TCF, respectively. (Decision variable).
$ q_i^K $ Retailer's order quantity under game $ K $, where $ K=S,N,R $ and
$ i=B,T $. (Decision variable).
$ \pi_{ij}^{K} $ Player $ j' $s expected profit under game $ K $, where $ K=S,N,R $ and
$ i=B,T $. The subscript $ j=s,r $ denotes the supplier and retailer, respectively.
$ \beta $ Retailer's bargaining power under Nash bargaining game with TCF.
$ \Pi $ Supply chain's optimal expected profit.
Notation Explanation
$ D $ Uncertain demand.
$ p $ Retailer's retail price per unit, where $ p=1 $.
$ c $ Supplier's production cost per unit, where $ 0 <c<1 $.
$ q^0 $ Supply chain's optimal order quantity in the centralized case.
$ r_B $ Bank's interest rate.
$ f(\cdot) $ Probability density function of $ D $.
$ F(\cdot) $ Cumulative distribution function of $ D $.
$ w_i^K $ Supplier's wholesale price per unit purchased under game $ K $,
where $ K= S,N,R $ denotes Stackelberg, Nash bargaining and
Rubinstein bargaining game, respectively. The subscript $ i=B,T $
denotes BCF and TCF, respectively. (Decision variable).
$ q_i^K $ Retailer's order quantity under game $ K $, where $ K=S,N,R $ and
$ i=B,T $. (Decision variable).
$ \pi_{ij}^{K} $ Player $ j' $s expected profit under game $ K $, where $ K=S,N,R $ and
$ i=B,T $. The subscript $ j=s,r $ denotes the supplier and retailer, respectively.
$ \beta $ Retailer's bargaining power under Nash bargaining game with TCF.
$ \Pi $ Supply chain's optimal expected profit.
Table 2.  $ \pi_{is}^{R*} $ and $ \delta_j $ given $ c = 0.2 $
$ \delta_s \backslash \delta_r $ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 B 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76
T 47.35 47.35 47.35 47.35 47.35 47.35 47.35 47.35 47.35
0.2 B 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76
T 46.89 46.89 46.89 46.89 46.89 46.89 46.89 46.89 46.89
0.3 B 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76
T 46.43 46.43 46.43 46.43 46.43 46.43 46.43 46.43 46.43
0.4 B 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76
T 45.97 45.97 45.97 45.97 45.97 45.97 45.97 45.97 45.97
0.5 B 34.38 34.38 34.38 34.38 34.38 34.38 34.76 34.76 34.76
T 45.51 45.51 45.51 45.51 45.51 45.51 45.51 45.51 45.51
0.6 B 31.70 31.70 31.70 31.70 31.70 31.70 32.97 34.76 34.76
T 45.05 45.05 45.05 45.05 45.05 45.05 45.05 45.05 45.05
0.7 B 29.01 29.01 29.01 29.01 29.01 29.01 29.01 32.60 34.76
T 44.59 44.59 44.59 44.59 44.59 44.59 44.59 44.59 44.59
0.8 B 26.33 26.33 26.33 26.33 26.33 26.33 26.33 26.56 34.15
T 44.13 44.13 44.13 44.13 44.13 44.13 44.13 44.13 44.13
0.9 B 23.64 23.64 23.64 23.64 23.64 23.64 23.64 23.64 25.16
T 43.67 43.67 43.67 43.67 43.67 43.67 43.67 43.67 43.67
Note:The symbols "B" and "T" denote "BCF" and "TCF", respectively.
$ \delta_s \backslash \delta_r $ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 B 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76
T 47.35 47.35 47.35 47.35 47.35 47.35 47.35 47.35 47.35
0.2 B 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76
T 46.89 46.89 46.89 46.89 46.89 46.89 46.89 46.89 46.89
0.3 B 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76
T 46.43 46.43 46.43 46.43 46.43 46.43 46.43 46.43 46.43
0.4 B 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76
T 45.97 45.97 45.97 45.97 45.97 45.97 45.97 45.97 45.97
0.5 B 34.38 34.38 34.38 34.38 34.38 34.38 34.76 34.76 34.76
T 45.51 45.51 45.51 45.51 45.51 45.51 45.51 45.51 45.51
0.6 B 31.70 31.70 31.70 31.70 31.70 31.70 32.97 34.76 34.76
T 45.05 45.05 45.05 45.05 45.05 45.05 45.05 45.05 45.05
0.7 B 29.01 29.01 29.01 29.01 29.01 29.01 29.01 32.60 34.76
T 44.59 44.59 44.59 44.59 44.59 44.59 44.59 44.59 44.59
0.8 B 26.33 26.33 26.33 26.33 26.33 26.33 26.33 26.56 34.15
T 44.13 44.13 44.13 44.13 44.13 44.13 44.13 44.13 44.13
0.9 B 23.64 23.64 23.64 23.64 23.64 23.64 23.64 23.64 25.16
T 43.67 43.67 43.67 43.67 43.67 43.67 43.67 43.67 43.67
Note:The symbols "B" and "T" denote "BCF" and "TCF", respectively.
Table 3.  $ \pi_{is}^{R*} $ and $ \delta_j $ given $ c = 0.65 $
$ \delta_s \backslash \delta_r $ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 B 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19
T 6.36 6.43 6.49 6.56 6.63 6.70 6.77 6.85 6.92
0.2 B 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19
T 5.71 5.83 5.96 6.09 6.22 6.36 6.51 6.67 6.83
0.3 B 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19
T 5.05 5.21 5.38 5.57 5.76 5.97 6.20 6.45 6.71
0.4 B 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19
T 4.37 4.56 4.77 5.00 5.25 5.53 5.83 6.18 6.56
0.5 B 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19
T 3.68 3.89 4.12 4.37 4.67 5.00 5.38 5.83 6.36
0.6 B 4.82 4.82 4.82 4.82 4.82 4.82 4.83 5.19 5.19
T 2.98 3.18 3.41 3.68 4.00 4.37 4.83 5.38 6.09
0.7 B 4.46 4.46 4.46 4.46 4.46 4.46 4.46 4.77 5.19
T 2.26 2.44 2.66 2.92 3.23 3.62 4.12 4.77 5.67
0.8 B 4.10 4.10 4.10 4.10 4.10 4.10 4.10 4.10 5.00
T -0.03 -0.03 -0.03 -0.03 -0.03 -0.03 -0.03 -0.03 -0.03
0.9 B 3.74 3.74 3.74 3.74 3.74 3.74 3.74 3.74 3.74
T -0.91 -0.91 -0.91 -0.91 -0.91 -0.91 -0.91 -0.91 -0.91
$ \delta_s \backslash \delta_r $ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 B 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19
T 6.36 6.43 6.49 6.56 6.63 6.70 6.77 6.85 6.92
0.2 B 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19
T 5.71 5.83 5.96 6.09 6.22 6.36 6.51 6.67 6.83
0.3 B 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19
T 5.05 5.21 5.38 5.57 5.76 5.97 6.20 6.45 6.71
0.4 B 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19
T 4.37 4.56 4.77 5.00 5.25 5.53 5.83 6.18 6.56
0.5 B 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19
T 3.68 3.89 4.12 4.37 4.67 5.00 5.38 5.83 6.36
0.6 B 4.82 4.82 4.82 4.82 4.82 4.82 4.83 5.19 5.19
T 2.98 3.18 3.41 3.68 4.00 4.37 4.83 5.38 6.09
0.7 B 4.46 4.46 4.46 4.46 4.46 4.46 4.46 4.77 5.19
T 2.26 2.44 2.66 2.92 3.23 3.62 4.12 4.77 5.67
0.8 B 4.10 4.10 4.10 4.10 4.10 4.10 4.10 4.10 5.00
T -0.03 -0.03 -0.03 -0.03 -0.03 -0.03 -0.03 -0.03 -0.03
0.9 B 3.74 3.74 3.74 3.74 3.74 3.74 3.74 3.74 3.74
T -0.91 -0.91 -0.91 -0.91 -0.91 -0.91 -0.91 -0.91 -0.91
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