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October  2019, 15(4): 1631-1651. doi: 10.3934/jimo.2018115

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

 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 Published  August 2018

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.

Citation: Bibhas C. Giri, Bhaba R. Sarker. Coordinating a multi-echelon supply chain under production disruption and price-sensitive stochastic demand. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1631-1651. doi: 10.3934/jimo.2018115
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##### References:
Impact of $\alpha$ on the manufacturer's decisions
Impact of $\beta$ on the supplier's profit
Impact of $\beta$ on the supply chain's total profit
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
 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
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
 $\xi$ $\eta$ $\tilde{p}^d\tilde{\Pi}_r$ $\tilde{Q}_d\tilde{\Pi}_m$ $\tilde{R}^d\tilde{\Pi}_s$ 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
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
 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
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
 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
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
 $e$ 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
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
 $a$ 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
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
 $\sigma$ 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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