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: August 2018

Revised: March 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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