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

Honglin Yang; Qiang Yan; Hong Wan; Wenyan Zhuo

doi:10.3934/jimo.2019077

Article Contents

2020, Volume 16, Issue 6: 2723-2741. Doi: 10.3934/jimo.2019077

This issue Previous Article How's the performance of the optimized portfolios by safety-first rules: Theory with empirical comparisons Next Article Corporate and personal credit scoring via fuzzy non-kernel SVM with fuzzy within-class scatter

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

Received: July 31, 2018

Revised: February 28, 2019

Early access: July 2019

Published: October 2020

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

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 90B50; Secondary: 91A35, 91A80.

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Figure 1. Mobike’s financing and operation

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Figure 2. $ \pi_{Bj}^{R*} $ and $ \delta_j $

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Figure 3. $ w $ and $ \beta $

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Figure 4. Relationship of $ \pi_{is}^{N*} $ and $ \pi_{is}^{S*} $ with $ \beta $ with BCF and TCF

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Figure 5. Relationship of $ \pi_{ir}^{N*} $ and $ \pi_{is}^{S*} $ with $ \beta $ with BCF and TCF

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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.

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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.

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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

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References

[1]	S. Z. Alparslan-Gök, S. 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.
[2]	O. Baron, O. Berman and D. Wu, Bargaining within the supply chain and its implications in an industry, Decision Sciences, 47 (2016), 193-218.
[3]	M. Berlin, Trade credit: Why do production firms act as financial intermediaries?, Business Review, Q3 (2003), 21-28.
[4]	K. Binmore, A. Rubinstein and A. Wolinsky, The Nash bargaining solution in economic modelling, Rand Journal of Economics, 17 (1986), 176-188. doi: 10.2307/2555382.
[5]	M. J. Brennan, V. Maksimovic and J. Zechner, Vendor financing, Journal of Finance, 43 (1988), 1127-1141. doi: 10.1111/j.1540-6261.1988.tb03960.x.
[6]	D. B. Bradley and M. J. Rubach, Trade credit and small businesses: A cause of business failures?, University of Central Arkansas, (2002).
[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.
[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.
[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.
[10]	M. S. Chern, Q. Pan, J. T. Teng, Y. 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.
[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.
[12]	M. Dada and Q. Hu, Financing newsvendor inventory, Operations Research Letters, 36 (2008), 569-573. doi: 10.1016/j.orl.2008.06.004.
[13]	M. Draganska, D. 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.
[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.
[15]	X. Feng, I. 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.
[16]	Q. Feng, G. Lai and L. X. Lu, Dynamic bargaining in a supply chain with asymmetric demand information, Management Science, 61 (2014), 301-315.
[17]	L. Guo, The benefits of downstream information acquisition, Marketing Science, 28 (2009), 457-471.
[18]	Z. Hua, S. 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.
[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.
[20]	B. Jing, X. 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.
[21]	B. Jing and A. Seidmann, Finance sourcing in a supply chain, Decision Support Systems, 58 (2014), 15-20.
[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.
[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.
[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.
[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.
[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.
[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.
[28]	R. Li, K. Skouri, J. 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.
[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.
[30]	L. Ma, F. Liu, S. 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.
[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.
[32]	A. Mirzazadeh, M. 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.
[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.
[34]	A. Muthoo, Bargaining Theory with Appliactions, Cambridge University Press, 1999.
[35]	J. F. Nash and Jr ., The bargaining problem, Econometrica, 18 (1950), 155-162. doi: 10.2307/1907266.
[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.
[37]	T. Paksoy, Í. Karaoǧlan, H. Gökçen, P. 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.
[38]	M. Pervin, S. 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.
[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.
[40]	Q. Qing, T. 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.
[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.
[42]	A. Rubinstein, Perfect equilibrium in a bargaining model, Econometrica, 50 (1982), 97-109. doi: 10.2307/1912531.
[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.
[44]	D. Seifert, R. 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.
[45]	J. Sutton, Non-cooperative bargaining theory: An introduction, The Review of Economic Studies, 53 (1986), 709-724. doi: 10.2307/2297715.
[46]	A. A. Taleizadeh, D. W. Pentico, M. 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.
[47]	T. I. Tunca and W. Zhu, Buyer intermediation in supplier finance, Management Science, 64 (2017), 5461-5959. doi: 10.1287/mnsc.2017.2863.
[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.
[49]	A. Vasin, M. 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.
[50]	A. Vasin, H. Gao, M. Dolmatova and G. W. Weber, Optimization of transmission network for homogeneous good market, Optimization, 66 (2017), 2125-2134. doi: 10.1080/02331934.2016.1247269.
[51]	D. Wu, O. 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.
[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.
[53]	X. Xu and J. R. Birge, Production and financing decisions: Modeling and analysis, (2004). doi: 10.2139/ssrn.652562.
[54]	N. Yan, B. Sun, H. 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.
[55]	S. Yang, K. 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.
[56]	H. Yang, W. Zhuo, Y. 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.
[57]	H. Yang, W. 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.
[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.
[59]	S. Zhang and X. Li, Managerial ties, firm resources, and performance of cluster firms, Asia Pacific Journal of Management, 25 (2008), 615-633. doi: 10.1007/s10490-008-9090-7.
[60]	F. Zhong, J. Xie, X. Zhao and Z. J. M. Shen, On efficiency of multistage channel with bargaining over wholesale prices, Naval Research Logistics, 63 (2016), 449-459. doi: 10.1002/nav.21713.
[61]	W. Zhuo, L. Shao and H. Yang, Mean-Variance analysis of option contracts in a two-echelon supply chain, European Journal of Operational Research, 271 (2018), 535-547. doi: 10.1016/j.ejor.2018.05.033.