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

Bibhas C. Giri; Bhaba R. Sarker

doi:10.3934/jimo.2018115

Article Contents

2019, Volume 15, Issue 4: 1631-1651. Doi: 10.3934/jimo.2018115

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

Bibhas C. Giri^1, and
Bhaba R. Sarker^2,

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

Early access: August 2018

Published: July 2019

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Abstract

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.

Keywords:

Mathematics Subject Classification: 90B05.

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Figure 1. Impact of $\alpha$ on the manufacturer's decisions

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Figure 2. Impact of $\beta$ on the supplier's profit

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Figure 3. Impact of $\beta$ on the supply chain's total profit

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Table 1. 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

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Table 2. 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

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Table 3. 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

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Table 4. 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

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Table 5. 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

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Table 6. 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

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Table 7. 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

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