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2022, Volume 18, Issue 2: 843-872. Doi: 10.3934/jimo.2020181

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Coordinating a supply chain with demand information updating

Honglin Yang and
Jiawu Peng

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

Received: July 2020

Revised: October 2020

Early access: December 2020

Published: March 2022

The authors would like to thank the editor and the anonymous referees for their helpful comments and suggestions that greatly improved the quality of this paper. The research is supported by the National Natural Science Foundation of China under Grant Nos. 72071072 and the Postgraduate Scientific Research Innovation Project of Hunan Province under Grant Nos.CX20200456

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Abstract

We investigate how to coordinate a two-echelon supply chain in which a supplier builds production capacity in advance and a manufacturer makes the ordering decision based on updated demand information. By combining European call option and buyback mechanisms, we propose a new hybrid option-buyback contract to coordinate such a supply chain with demand information updating. We construct a two-stage optimization model in that the supplier offers option price and the manufacturer decides initial ordering quantity in the first stage, then the supplier offers exercise price and buyback price and the manufacturer decides final ordering quantity in the second stage after demand information is updated. In both the centralized and decentralized settings, we analytically derive the optimal equilibrium solutions of two-stage ordering quantity. Particularly, we obtain closed-form formulae to describe the members' optimal behavior with a bivariate uniformly distribution. We prove that the proposed contracts can realize the perfect coordination of the supply chain and analyze how the proposed contracts affect the members' decisions. The theoretical results show that, by tuning the option price or buyback price, the supply chain profit can be arbitrarily split between the members, which is a desired property for supply chain coordination. Compared with the standard option and buyback contract, the proposed contract results in a greater supply chain profit and achieves Pareto improvement for the supply chain members. Furthermore, extending the baseline model focusing on price-independent demand to the case of price-dependent demand, we show that the proposed contract still can achieve supply chain coordination. Numerical examples are also conducted to complement the theoretical results.

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

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Figure 1. The sequence of events

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Figure 2. Pareto improvement area

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Figure 3. The impacts of $ c_e $ and $ w_e $

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Figure 4. The impacts of $ b $ and $ p $

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Figure 5. The impacts of $ c_1 $ and $ c_2 $

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Figure 6. The impacts of $ v_1 $ and $ v_2 $

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Table 1. Notations and explanations

Notation	Explanation
$ x $	Random demand
$ p $	Retail price
$ c_e $	Option price (decision variable)
$ w_e $	Exercise price (decision variable)
$ \lambda $	Revenue sharing ratio.
$ b $	Buyback price (decision variable)
$ Q_k^j $	Capacity reservation quantity of the key component for the manufacturer,
	where $ j=c,d $ and $ k=sc,m,s $(decision variable)
$ Q_e^j $	Final order quantity of the manufacturer, where $ j=c,d $ (decision variable)
$ I $	Demand information
$ c_1 $	Capacity investment cost of the supplier
$ c_2 $	Production cost of the supplier
$ v_1 $	Salvage value of excess capacity
$ v_2 $	Salvage value of leftover inventory
$ f(\cdot),F(\cdot) $	Prior probability density and distribution functions of $ x $
$ g(\cdot),G(\cdot) $	Probability density and distribution functions of $ I $
$ h(x\|i) $	Probability density function of $ x $ given $ I=i $
$ H(x\|i) $	Cumulative distribution functions for $ x $ given $ I=i $
$ I $	Total profit

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Table 2. The optimal decisions of the supply chain with demand information updating

Supply chain	Final order quantity	Capacity reservation quantity
Centralized	$ Q_e^{c*}=Q_e^c\|i, \ if \ i\leq i_{Q_{sc}^c} $,	$ \int_{i_{Q_{sc}^{c}}}^{+\infty}[H(Q_e^c\|i)-H(Q_{sc}^{c}\|i)]dG(i)=\frac{c_1-v_1}{p-v_2} $.
	$ Q_e^{c*}=Q_{sc}^c, \ if \ i \ i_{Q_{sc}^c} $,
	where $ H(Q_e^c\|i)=\frac{p-c_2-v_1}{p-v_2} $}.
Decentralized	$ Q_e^{d*}=Q_e^d\|i, \ if \ i\leq i_{Q_{m}^d} $,	$ \int_{i_{Q_m^{d}}}^{+\infty}[H(Q_e^d\|i)-H(Q_m^{d}\|i)]dG(i)=\frac{c_e}{p-b}. $
	$ Q_e^{d*}=Q_{m}^d, \ if \ i \ i_{Q_{m}^d} $,
	where $ H(Q_e^d\|i)=\frac{p-w_e}{p-b} $.

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Table 3. Optimal quantity decisions with uniformly distribution

Optimal decisions	Centralized supply chain		Decentralized supply chain
Optimal decisions	$ \gamma-\frac{\alpha}{2}\leq i \leq i_{Q_{sc}^c} $	$ {i_{Q_{sc}^c}} \le \gamma + \frac{\alpha }{2}$	$ \gamma-\frac{\alpha}{2}\leq i \leq i_{Q_{m}^d} $	$ {i_{Q_m^d}} \le \gamma + \frac{\alpha }{2}$
Exercise quantity	$ i-\frac{\beta}{2}+\frac{p-c_2-v_1}{p-v_2}\beta $	$ Q_{sc}^c $	$ i-\frac{\beta}{2}+\frac{p-w_e}{p-b}\beta $	$ Q_{m}^d $
Option quantity	$ Q_{sc}^c=\gamma+\frac{\alpha-\beta}{2}+\frac{p-c_2-v_1}{p-v_2}\beta-\sqrt{\frac{2(c_1-v_1)\alpha\beta}{p-v_2}} $		$ Q_{m}^d=\gamma+\frac{\alpha-\beta}{2}+\frac{p-w_e}{p-b}\beta-\sqrt{\frac{2c_e\alpha\beta}{p-b}} $

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Table 4. Optimal quantity and expected profits with $ \{c_e,w_e,b\}\in M $

$ \lambda $	$ c_e $	$ w_e $	$ b $	$ Q_m^{d*} $	$ Q_e^{d*}\|i $	$ \Pi^d $	$ \Pi_s^d $	$ \Pi_m^d $	$ \Pi_s^d/\Pi^d $	$ \Pi_m^d/\Pi^d $
0.1	1.5	96.5	93	1414	$ i $	2.1355	1.92919	0.2135	0.9	0.1
0.2	3.0	93.0	86	1414	$ i $	2.1355	1.7084	0.4271	0.8	0.2
0.3	4.5	89.5	79	1414	$ i $	2.1355	1.4948	0.6406	0.7	0.3
0.4	6.0	86.0	72	1414	$ i $	2.1355	1.2813	0.8542	0.6	0.4
0.5	7.5	82.5	65	1414	$ i $	2.1355	1.0677	1.0677	0.5	0.5
0.6	9.0	79.0	58	1414	$ i $	2.1355	0.8542	1.2813	0.4	0.6
0.7	10.5	75.5	51	1414	$ i $	2.1355	0.6406	1.4948	0.3	0.7
0.8	12.0	72.0	44	1414	$ i $	2.1355	0.4271	1.7084	0.2	0.8
0.9	13.5	68.5	37	1414	$ i $	2.1355	0.2135	1.9219	0.1	0.9

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Table 5. Optimal quantity and expected profits with $ \{c_e,w_e\} $

$ \lambda $	$ c_e $	$ w_e $	$ Q_m^{A*} $	$ Q_e^{A*}\|i $	$ \Pi^A $	$ \Pi_s^A $	$ Gap_s^A $	$ \Pi_m^A $	$ Gap_m^A $
0.1	1.5	96.5	1454	i-360	1.7374	1.5729	0.181	0.1645	0.025
0.2	3.0	93.0	1418	i-320	1.8526	1.5062	0.105	0.3463	0.042
0.3	4.5	89.5	1399	i-280	1.9339	1.3937	0.052	0.5402	0.052
0.4	6.0	86.0	1389	i-240	1.9954	1.2512	0.015	0.7441	0.057
0.5	7.5	82.5	1386	i-200	2.0425	1.0855	-0.009	0.9570	0.058
0.6	9.0	79.0	1386	i-160	2.0782	0.9001	-0.023	1.1781	0.054
0.7	10.5	75.5	1390	i-120	2.1043	0.6974	-0.029	1.4069	0.046
0.8	12.0	72.0	1396	i-80	2.1220	0.4791	-0.027	1.6429	0.034
0.9	13.5	68.5	1404	i-40	2.1322	0.2463	-0.017	1.8859	0.019
$ \mathbf{1} $	$ \mathbf{5} $	$ \mathbf{55} $	$ \mathbf{1414} $	$ \mathbf{i} $	$ \mathbf{2.1355} $	$ \mathbf{0} $	$ \mathbf{0} $	$ \mathbf{2.1355} $	$ \mathbf{0} $

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Table 6. Optimal quantity and expected profits with $ \{c_e,w_e\} $

$ \lambda $	$ w $	$ b $	$ Q_s^{B*} $	$ Q_m^{B*}\|i $	$ \Pi^B $	$ \Pi_s^B $	$ Gap_s^B $	$ \Pi_m^B $	$ Gap_m^B $
0.1	95	93	1335	i+171	1.8209	1.4827	0.228	0.3382	-0.065
0.2	95	86	1358	i-114	2.0587	1.5237	0.096	0.5350	-0.056
0.3	95	79	1328	i-210	1.9339	1.4712	0.012	0.5270	0.059
0.4	95	72	1302	i-257	1.9386	1.4283	-0.076	0.5102	0.179
0.5	95	65	1279	i-285	1.8924	1.3988	-0.172	0.4936	0.298
0.6	95	58	1258	i-305	1.8560	1.3791	-0.273	0.4769	0.418
0.7	95	51	1236	i-318	1.8259	1.3668	-0.378	0.4591	0.538
0.8	95	44	1213	i-329	1.7990	1.3601	-0.485	0.4388	0.661
0.9	95	37	1187	i-337	1.7724	1.3581	-0.595	0.4142	0.784

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