American Institute of Mathematical Sciences

Advanced
September  2021, 17(5): 2903-2924. doi: 10.3934/jimo.2020100

Quality competition and coordination in a VMI supply chain with two risk-averse manufacturers

Bin Chen 1, , Wenying Xie 2, , Fuyou Huang 1,, and  Juan He 2,

1. 

Institute of Transportation Development Strategy & Planning of Sichuan Province, Chengdu 610041, China
2. 

School of Transportation and Logistics, Southwest Jiaotong University, Chengdu 610036, China

* Corresponding author: Fuyou Huang

Received  November 2019 Revised  March 2020 Published  September 2021 Early access  May 2020

Fund Project: The paper is supported by the Science and Technology Project of Sichuan Province (Grant No.20CXTD0081) and the Science and Technology Project of Transportation Department of Sichuan Province (Grant No.2019-D-05)

Quality competition and risk aversion have become more and more common in today's many industries, making it a challenge to supply chain management and coordination. This paper considers a vendor-managed inventory (VMI) supply chain comprising two risk-averse manufacturers who sell their competing products through a common retailer. Market demand shared by each manufacturer is dependent on the quality level of its own product as well as on the competitor's product quality. The Conditional Value-at-Risk (CVaR) criterion is employed to formulate the risk aversion of manufacturers. This study first develops basic models without coordination mechanism and analyzes the effect of the quality sensitivity, competition intensity, risk aversion degree and cost coefficient of quality improvement on equilibrium decisions and supply chain efficiency. Further, a combined contract composed of option and cost-sharing is proposed to investigate the supply chain coordination issue. The results reveal that the combined contract can coordinate the supply chain and achieve a win-win outcome only when the manufacturers are low in risk aversion, and the system-wide profit of the supply chain can be allocated arbitrarily only by the option price. Also, this research examines the effect of the quality sensitivity, competition intensity, risk aversion degree and cost coefficient of quality improvement on the feasible region of option price.

Keywords: Supply chain coordination, quality competition, VMI, option contract, CVaR.
Mathematics Subject Classification: Primary: 90B50; Secondary: 91B30.
Citation: Bin Chen, Wenying Xie, Fuyou Huang, Juan He. Quality competition and coordination in a VMI supply chain with two risk-averse manufacturers. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2903-2924. doi: 10.3934/jimo.2020100
References:
[1]

R. D. Banker, I. Khosla and K. K. Sinha, Quality and competition, Manag. Sci., 44 (1998), 1179-1192.  doi: 10.1287/mnsc.44.9.1179.

[2]

D. Barnes-Schuster, Y. Bassok and R. Anupindi, Coordination and flexibility in supply contracts with options, Manufacturing Service Oper. Manag., 4 (2002), 171-207.  doi: 10.1287/msom.4.3.171.7754.

[3]

J. Cai, X. Hu, Y. Han, H. Cheng and W. Huang, Supply chain coordination with an option contract under vendor-managed inventory, Int. Trans. Oper. Res., 23 (2016), 1163-1183.  doi: 10.1111/itor.12172.

[4]

J. Cai, M. Zhong, J. Shang and W. Huang, Coordinating VMI supply chain under yield uncertainty: Option contract, subsidy contract, and replenishment tactic, Internat. J. Prod. Econ., 185 (2017), 196-210.  doi: 10.1016/j.ijpe.2016.12.032.

[5]

Z. Chang, S. Song, Y. Zhang, J.-Y. Ding, R. Zhang and R. Chiong, Distributionally robust single machine scheduling with risk aversion, European J. Oper. Res., 256 (2017), 261-274.  doi: 10.1016/j.ejor.2016.06.025.

[6]

G. H. Chao, S. M. R. Iravani and R. C. Savaskan, Quality improvement incentives and product recall cost sharing contracts, Manag. Sci., 55 (2009), 1122-1138.  doi: 10.1287/mnsc.1090.1008.

[7]

J.-M. Chen, I.-C. Lin and H.-L. Cheng, Channel coordination under consignment and vendor-managed inventory in a distribution system, Transpor. Res. Part E., 46 (2010), 831-843.  doi: 10.1016/j.tre.2010.05.007.

[8]

X. Chen, G. Hao and L. Li, Channel coordination with a loss-averse retailer and option contracts, Internat. J. Prod. Econ., 150 (2014), 52-57.  doi: 10.1016/j.ijpe.2013.12.004.

[9]

X. Chen, S. Shum and D. Simchi-Levi, Stable and coordinating contracts for a supply chain with multiple risk-averse suppliers, Prod. Oper. Manag., 23 (2014), 379-392.  doi: 10.1111/poms.12073.

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Y. Chen, M. Xu and Z. G. Zhang, Technical note–A risk-averse newsvendor model under the CVaR criterion, Oper. Res., 57 (2009), 1040-1044.  doi: 10.1287/opre.1080.0603.

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J. Dai and W. Meng, A risk-averse newsvendor model under marketing-dependency and price-dependency, Internat. J. Prod. Econ., 160 (2015), 220-229.  doi: 10.1016/j.ijpe.2014.11.006.

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Y. Dai, S. X. Zhou and Y. Xu, Competitive and collaborative quality and warranty management in supply chains, Prod. Oper. Manag., 21 (2012), 129-144.  doi: 10.1111/j.1937-5956.2011.01217.x.

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M. A. Darwish and O. M. Odah, Vendor managed inventory model for single-vendor multi-retailer supply chains, European J. Oper. Res., 204 (2010), 473-484.  doi: 10.1016/j.ejor.2009.11.023.

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Y. Dong, M. Dresner and Y. Yao, Beyond information sharing: An empirical analysis of vendor-managed inventory, Prod. Oper. Manag., 23 (2014), 817-828.  doi: 10.1111/poms.12085.

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X. Gan, S. P. Sethi and H. Yan, Channel coordination with a risk-neutral supplier and a downside-risk-averse retailer, Prod. Oper. Manag., 14 (2005), 80-89.  doi: 10.1111/j.1937-5956.2005.tb00011.x.

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N. Gans, Customer loyalty and supplier quality competition, Manag. Sci., 48 (2002), 207-221.  doi: 10.1287/mnsc.48.2.207.256.

[21]

R. Guan and X. Zhao, On contracts for VMI program with continuous review $(r, Q)$ policy, European J. Oper. Res., 207 (2010), 656-667.  doi: 10.1016/j.ejor.2010.04.037.

[22]

H. Gurnani, M. Erkoc and Y. Luo, Impact of product pricing and timing of investment decisions on supply chain co-opetition, European J. Oper. Res., 180 (2007), 228-248.  doi: 10.1016/j.ejor.2006.02.047.

[23]

M. Hariga, M. Gumus and A. Daghfous, Storage constrained vendor managed inventory models with unequal shipment frequencies, Omega, 48 (2014), 94-106.  doi: 10.1016/j.omega.2013.11.003.

[24]

J. He, C. Ma and K. Pan, Capacity investment in supply chain with risk averse supplier under risk diversification contract, Transport. Res. Part E., 106 (2017), 255-275.  doi: 10.1016/j.tre.2017.08.005.

[25]

B. Hu, X. Chen, F. T. S. Chan and C. Meng, Portfolio procurement policies for budget-constrained supply chains with option contracts and external financing, J. Ind. Manag. Optim., 14 (2018), 1105-1122.  doi: 10.3934/jimo.2018001.

[26]

F. Huang, J. He and J. Wang, Coordination of VMI supply chain with a loss-averse manufacturer under quality-dependency and marketing-dependency, J. Ind. Manag. Optim., 15 (2019), 1753-1772.  doi: 10.3934/jimo.2018121.

[27]

C. H. Huynh and W. Pan, Operational strategies for supplier and retailer with risk preference under VMI contract, Internat. J. Prod. Econ., 169 (2015), 413-421.  doi: 10.1016/j.ijpe.2015.07.026.

[28]

S. Khalilpourazari, A. Mirzazadeh, G.-W. Weber and S. H. R. Pasandideh, A robust fuzzy approach for constrained multi-product economic production quantity with imperfect items and rework process, Optimization, 69 (2020), 63-90.  doi: 10.1080/02331934.2019.1630625.

[29]

S. Khalilpourazari and S. H. R. Pasandideh, Bi-objective optimization of multi-product EPQ model with backorders, rework process and random defective rate, 12th International Conference on Industrial Engineering (ICIE), Tehran, Iran, 2016, 36–40. doi: 10.1109/INDUSENG.2016.7519346.

[30]

S. Khalilpourazari and S. H. R. Pasandideh, Modeling and optimization of multi-item multi-constrained EOQ model for growing items, Knowledge-Based Systems, 164 (2019), 150-162.  doi: 10.1016/j.knosys.2018.10.032.

[31]

S. Khalilpourazari, S. H. R. Pasandideh and A. Ghodratnama, Robust possibilistic programming for multi-item EOQ model with defective supply batches: Whale optimization and water cycle algorithms, Neural Comput. Appl., 31 (2019), 6587-6614.  doi: 10.1007/s00521-018-3492-3.

[32]

S. Khalilpourazari, S. H. R. Pasandideh and S. T. A. Niaki, Optimizing a multi-item economic order quantity problem with imperfect items, inspection errors, and backorders, Soft Comput., 23 (2019), 11671-11698.  doi: 10.1007/s00500-018-03718-1.

[33]

B. Li, P.-W. Hou, P. Chen and Q.-H. Li, Pricing strategy and coordination in a dual channel supply chain with a risk-averse retailer, Internat. J. Prod. Econ., 178 (2016), 154-168.  doi: 10.1016/j.ijpe.2016.05.010.

[34]

Y. T. Lin, A. K. Parlaktürk and J. M. Swaminathan, Vertical integration under competition: Forward, backward, or no integration?, Prod. Oper. Manag., 23 (2014), 19-35.  doi: 10.1111/poms.12030.

[35]

L. Ma, F. Liu, S. Li and H. Yan, Channel bargaining with risk-averse retailer, Internat. J. Prod. Econ., 139 (2012), 155-167.  doi: 10.1016/j.ijpe.2010.08.016.

[36]

B. K. Mishra and S. Raghunathan, Retailer- vs. vendor-managed inventory and brand competition, Manag. Sci., 50 (2004), 445-457.  doi: 10.1287/mnsc.1030.0174.

[37]

N. M. Modak, S. Panda and S. S. Sana, Three-echelon supply chain coordination considering duopolistic retailers with perfect quality products, Internat. J. Prod. Econ., 182 (2016), 564-578.  doi: 10.1016/j.ijpe.2015.05.021.

[38]

Z. J. Ren and Y.-P. Zhou, Call center outsourcing: Coordinating staffing level and service quality, Manag. Sci., 54 (2008), 369-383.  doi: 10.1287/mnsc.1070.0820.

[39]

R. T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distributions, J. Banking Finance, 26 (2002), 1443-1471.  doi: 10.1016/S0378-4266(02)00271-6.

[40]

W. Shang and L. Yang, Contract negotiation and risk preferences in dual-channel supply chain coordination, Internat. J. Prod. Res., 53 (2015), 4837-4856.  doi: 10.1080/00207543.2014.998785.

[41]

A. A. Taleizadeh, M. S. Moshtagh and I. Moon, Pricing, product quality, and collection optimization in a decentralized closed-loop supply chain with different channel structures: Game theoretical approach, J. Cleaner Prod., 189 (2018), 406-431.  doi: 10.1016/j.jclepro.2018.02.209.

[42]

M. Wu, S. X. Zhu and R. H. Teunter, A risk-averse competitive newsvendor problem under the CVaR criterion, Internat. J. Prod. Econ., 156 (2014), 13-23.  doi: 10.1016/j.ijpe.2014.05.009.

[43]

G. Xie, S. Wang and K. K. Lai, Quality improvement in competing supply chains, Internat. J. Prod. Econ., 134 (2011), 262-270.  doi: 10.1016/j.ijpe.2011.07.007.

[44]

G. Xie, W. Yue, S. Wang and K. K. Lai, Quality investment and price decision in a risk-averse supply chain, European J. Oper. Res., 214 (2011), 403-410.  doi: 10.1016/j.ejor.2011.04.036.

[45]

K. Xu and M. T. Leung, Stocking policy in a two-party vendor managed channel with space restrictions, Internat. J. Prod. Econ., 117 (2009), 271-285.  doi: 10.1016/j.ijpe.2008.11.003.

[46]

L. Yang, M. Xu, G. Yu and H. Zhang, Supply chain coordination with CVaR criterion, Asia-Pac. J. Oper. Res., 26 (2009), 135-160.  doi: 10.1142/S0217595909002109.

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show all references

References:
[1]

R. D. Banker, I. Khosla and K. K. Sinha, Quality and competition, Manag. Sci., 44 (1998), 1179-1192.  doi: 10.1287/mnsc.44.9.1179.

[2]

D. Barnes-Schuster, Y. Bassok and R. Anupindi, Coordination and flexibility in supply contracts with options, Manufacturing Service Oper. Manag., 4 (2002), 171-207.  doi: 10.1287/msom.4.3.171.7754.

[3]

J. Cai, X. Hu, Y. Han, H. Cheng and W. Huang, Supply chain coordination with an option contract under vendor-managed inventory, Int. Trans. Oper. Res., 23 (2016), 1163-1183.  doi: 10.1111/itor.12172.

[4]

J. Cai, M. Zhong, J. Shang and W. Huang, Coordinating VMI supply chain under yield uncertainty: Option contract, subsidy contract, and replenishment tactic, Internat. J. Prod. Econ., 185 (2017), 196-210.  doi: 10.1016/j.ijpe.2016.12.032.

[5]

Z. Chang, S. Song, Y. Zhang, J.-Y. Ding, R. Zhang and R. Chiong, Distributionally robust single machine scheduling with risk aversion, European J. Oper. Res., 256 (2017), 261-274.  doi: 10.1016/j.ejor.2016.06.025.

[6]

G. H. Chao, S. M. R. Iravani and R. C. Savaskan, Quality improvement incentives and product recall cost sharing contracts, Manag. Sci., 55 (2009), 1122-1138.  doi: 10.1287/mnsc.1090.1008.

[7]

J.-M. Chen, I.-C. Lin and H.-L. Cheng, Channel coordination under consignment and vendor-managed inventory in a distribution system, Transpor. Res. Part E., 46 (2010), 831-843.  doi: 10.1016/j.tre.2010.05.007.

[8]

X. Chen, G. Hao and L. Li, Channel coordination with a loss-averse retailer and option contracts, Internat. J. Prod. Econ., 150 (2014), 52-57.  doi: 10.1016/j.ijpe.2013.12.004.

[9]

X. Chen, S. Shum and D. Simchi-Levi, Stable and coordinating contracts for a supply chain with multiple risk-averse suppliers, Prod. Oper. Manag., 23 (2014), 379-392.  doi: 10.1111/poms.12073.

[10]

Y. Chen, M. Xu and Z. G. Zhang, Technical note–A risk-averse newsvendor model under the CVaR criterion, Oper. Res., 57 (2009), 1040-1044.  doi: 10.1287/opre.1080.0603.

[11]

J. Dai and W. Meng, A risk-averse newsvendor model under marketing-dependency and price-dependency, Internat. J. Prod. Econ., 160 (2015), 220-229.  doi: 10.1016/j.ijpe.2014.11.006.

[12]

Y. Dai, S. X. Zhou and Y. Xu, Competitive and collaborative quality and warranty management in supply chains, Prod. Oper. Manag., 21 (2012), 129-144.  doi: 10.1111/j.1937-5956.2011.01217.x.

[13]

M. A. Darwish and O. M. Odah, Vendor managed inventory model for single-vendor multi-retailer supply chains, European J. Oper. Res., 204 (2010), 473-484.  doi: 10.1016/j.ejor.2009.11.023.

[14]

B. De Vos, B. Raa and S. De Vuyst, A savings analysis of horizontal collaboration among VMI suppliers, J. Ind. Manag. Optim., 15 (2019), 1733-1751.  doi: 10.3934/jimo.2018120.

[15]

Y. Dong, M. Dresner and Y. Yao, Beyond information sharing: An empirical analysis of vendor-managed inventory, Prod. Oper. Manag., 23 (2014), 817-828.  doi: 10.1111/poms.12085.

[16]

F. El Ouardighi, Supply quality management with optimal wholesale price and revenue sharing contracts: A two-stage game approach, Internat. J. Prod. Econ., 156 (2014), 260-268.  doi: 10.1016/j.ijpe.2014.06.006.

[17]

F. El Ouardighi and B. Kim, Supply quality management with wholesale price and revenue-sharing contracts under horizontal competition, European J. Oper. Res., 206 (2010), 329-340.  doi: 10.1016/j.ejor.2010.02.035.

[18]

S. Eskandarzadeh, K. Eshghi and M. Bahramgiri, Risk shaping in production planning problem with pricing under random yield, European J. Oper. Res., 253 (2016), 108-120.  doi: 10.1016/j.ejor.2016.02.032.

[19]

X. Gan, S. P. Sethi and H. Yan, Channel coordination with a risk-neutral supplier and a downside-risk-averse retailer, Prod. Oper. Manag., 14 (2005), 80-89.  doi: 10.1111/j.1937-5956.2005.tb00011.x.

[20]

N. Gans, Customer loyalty and supplier quality competition, Manag. Sci., 48 (2002), 207-221.  doi: 10.1287/mnsc.48.2.207.256.

[21]

R. Guan and X. Zhao, On contracts for VMI program with continuous review $(r, Q)$ policy, European J. Oper. Res., 207 (2010), 656-667.  doi: 10.1016/j.ejor.2010.04.037.

[22]

H. Gurnani, M. Erkoc and Y. Luo, Impact of product pricing and timing of investment decisions on supply chain co-opetition, European J. Oper. Res., 180 (2007), 228-248.  doi: 10.1016/j.ejor.2006.02.047.

[23]

M. Hariga, M. Gumus and A. Daghfous, Storage constrained vendor managed inventory models with unequal shipment frequencies, Omega, 48 (2014), 94-106.  doi: 10.1016/j.omega.2013.11.003.

[24]

J. He, C. Ma and K. Pan, Capacity investment in supply chain with risk averse supplier under risk diversification contract, Transport. Res. Part E., 106 (2017), 255-275.  doi: 10.1016/j.tre.2017.08.005.

[25]

B. Hu, X. Chen, F. T. S. Chan and C. Meng, Portfolio procurement policies for budget-constrained supply chains with option contracts and external financing, J. Ind. Manag. Optim., 14 (2018), 1105-1122.  doi: 10.3934/jimo.2018001.

[26]

F. Huang, J. He and J. Wang, Coordination of VMI supply chain with a loss-averse manufacturer under quality-dependency and marketing-dependency, J. Ind. Manag. Optim., 15 (2019), 1753-1772.  doi: 10.3934/jimo.2018121.

[27]

C. H. Huynh and W. Pan, Operational strategies for supplier and retailer with risk preference under VMI contract, Internat. J. Prod. Econ., 169 (2015), 413-421.  doi: 10.1016/j.ijpe.2015.07.026.

[28]

S. Khalilpourazari, A. Mirzazadeh, G.-W. Weber and S. H. R. Pasandideh, A robust fuzzy approach for constrained multi-product economic production quantity with imperfect items and rework process, Optimization, 69 (2020), 63-90.  doi: 10.1080/02331934.2019.1630625.

[29]

S. Khalilpourazari and S. H. R. Pasandideh, Bi-objective optimization of multi-product EPQ model with backorders, rework process and random defective rate, 12th International Conference on Industrial Engineering (ICIE), Tehran, Iran, 2016, 36–40. doi: 10.1109/INDUSENG.2016.7519346.

[30]

S. Khalilpourazari and S. H. R. Pasandideh, Modeling and optimization of multi-item multi-constrained EOQ model for growing items, Knowledge-Based Systems, 164 (2019), 150-162.  doi: 10.1016/j.knosys.2018.10.032.

[31]

S. Khalilpourazari, S. H. R. Pasandideh and A. Ghodratnama, Robust possibilistic programming for multi-item EOQ model with defective supply batches: Whale optimization and water cycle algorithms, Neural Comput. Appl., 31 (2019), 6587-6614.  doi: 10.1007/s00521-018-3492-3.

[32]

S. Khalilpourazari, S. H. R. Pasandideh and S. T. A. Niaki, Optimizing a multi-item economic order quantity problem with imperfect items, inspection errors, and backorders, Soft Comput., 23 (2019), 11671-11698.  doi: 10.1007/s00500-018-03718-1.

[33]

B. Li, P.-W. Hou, P. Chen and Q.-H. Li, Pricing strategy and coordination in a dual channel supply chain with a risk-averse retailer, Internat. J. Prod. Econ., 178 (2016), 154-168.  doi: 10.1016/j.ijpe.2016.05.010.

[34]

Y. T. Lin, A. K. Parlaktürk and J. M. Swaminathan, Vertical integration under competition: Forward, backward, or no integration?, Prod. Oper. Manag., 23 (2014), 19-35.  doi: 10.1111/poms.12030.

[35]

L. Ma, F. Liu, S. Li and H. Yan, Channel bargaining with risk-averse retailer, Internat. J. Prod. Econ., 139 (2012), 155-167.  doi: 10.1016/j.ijpe.2010.08.016.

[36]

B. K. Mishra and S. Raghunathan, Retailer- vs. vendor-managed inventory and brand competition, Manag. Sci., 50 (2004), 445-457.  doi: 10.1287/mnsc.1030.0174.

[37]

N. M. Modak, S. Panda and S. S. Sana, Three-echelon supply chain coordination considering duopolistic retailers with perfect quality products, Internat. J. Prod. Econ., 182 (2016), 564-578.  doi: 10.1016/j.ijpe.2015.05.021.

[38]

Z. J. Ren and Y.-P. Zhou, Call center outsourcing: Coordinating staffing level and service quality, Manag. Sci., 54 (2008), 369-383.  doi: 10.1287/mnsc.1070.0820.

[39]

R. T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distributions, J. Banking Finance, 26 (2002), 1443-1471.  doi: 10.1016/S0378-4266(02)00271-6.

[40]

W. Shang and L. Yang, Contract negotiation and risk preferences in dual-channel supply chain coordination, Internat. J. Prod. Res., 53 (2015), 4837-4856.  doi: 10.1080/00207543.2014.998785.

[41]

A. A. Taleizadeh, M. S. Moshtagh and I. Moon, Pricing, product quality, and collection optimization in a decentralized closed-loop supply chain with different channel structures: Game theoretical approach, J. Cleaner Prod., 189 (2018), 406-431.  doi: 10.1016/j.jclepro.2018.02.209.

[42]

M. Wu, S. X. Zhu and R. H. Teunter, A risk-averse competitive newsvendor problem under the CVaR criterion, Internat. J. Prod. Econ., 156 (2014), 13-23.  doi: 10.1016/j.ijpe.2014.05.009.

[43]

G. Xie, S. Wang and K. K. Lai, Quality improvement in competing supply chains, Internat. J. Prod. Econ., 134 (2011), 262-270.  doi: 10.1016/j.ijpe.2011.07.007.

[44]

G. Xie, W. Yue, S. Wang and K. K. Lai, Quality investment and price decision in a risk-averse supply chain, European J. Oper. Res., 214 (2011), 403-410.  doi: 10.1016/j.ejor.2011.04.036.

[45]

K. Xu and M. T. Leung, Stocking policy in a two-party vendor managed channel with space restrictions, Internat. J. Prod. Econ., 117 (2009), 271-285.  doi: 10.1016/j.ijpe.2008.11.003.

[46]

L. Yang, M. Xu, G. Yu and H. Zhang, Supply chain coordination with CVaR criterion, Asia-Pac. J. Oper. Res., 26 (2009), 135-160.  doi: 10.1142/S0217595909002109.

[47]

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Figure 1.  Structure of the supply chain
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Figure 2.  Effect of the option price on profit allocation under channel coordination
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Table 1.  Differences between this paper and other relevant papers
Literature [1] [16] [24] [26] [37] [43] [46] This paper
Product quality Yes Yes No Yes Yes Yes No Yes
Competition Yes No No No No Yes No Yes
Stochastic demand No No Yes Yes No No Yes Yes
Risk aversion No No Yes No No No Yes Yes
Coordination No Yes Yes Yes Yes No Yes Yes
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Literature [1] [16] [24] [26] [37] [43] [46] This paper
Product quality Yes Yes No Yes Yes Yes No Yes
Competition Yes No No No No Yes No Yes
Stochastic demand No No Yes Yes No No Yes Yes
Risk aversion No No Yes No No No Yes Yes
Coordination No Yes Yes Yes Yes No Yes Yes
Table 2.  Notations
Symbol Definition
$ i $ Index for product, $ i = 1, 2 $
$ {d_i} $ Quality dependent deterministic demand for the $ i $th product
$ {D_i} $ Demand faced by the retailer for the $ i $th product
$ {c_i} $ Production cost per unit of the $ i $th product
$ {w_i} $ Wholesale price per unit of the $ i $th product
$ {p_i} $ Retail price per unit of the $ i $th product
$ {\upsilon _i} $ Salvage value per unit of the $ i $th product
$ {x_i} $ Random demand faced by the retailer for the $ i $th product
$ {L_i}, {U_i} $ Lower bound and upper bound on $ {x_i} $
$ {f_i}({x_i}) $ Probability density function of the random variable $ {x_i} $
$ {F_i}({x_i}) $ Cumulative distribution function of the random variable $ {x_i} $
$ {a_i} $ Initial market size of the $ i $th product
$ \alpha $ Demand sensitivity of product $ i $'s own quality improvement level
$ \beta $ Competition intensity
$ {s_i} $ Quality improvement level of the $ i $th product
$ {Q_i} $ Production quantity for the $ i $th product
$ {\eta _i} $ Risk aversion coefficient of the $ i $th manufacturer
$ {k_i} $ Cost coefficient of investment in quality improvement of product $ i $
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Symbol Definition
$ i $ Index for product, $ i = 1, 2 $
$ {d_i} $ Quality dependent deterministic demand for the $ i $th product
$ {D_i} $ Demand faced by the retailer for the $ i $th product
$ {c_i} $ Production cost per unit of the $ i $th product
$ {w_i} $ Wholesale price per unit of the $ i $th product
$ {p_i} $ Retail price per unit of the $ i $th product
$ {\upsilon _i} $ Salvage value per unit of the $ i $th product
$ {x_i} $ Random demand faced by the retailer for the $ i $th product
$ {L_i}, {U_i} $ Lower bound and upper bound on $ {x_i} $
$ {f_i}({x_i}) $ Probability density function of the random variable $ {x_i} $
$ {F_i}({x_i}) $ Cumulative distribution function of the random variable $ {x_i} $
$ {a_i} $ Initial market size of the $ i $th product
$ \alpha $ Demand sensitivity of product $ i $'s own quality improvement level
$ \beta $ Competition intensity
$ {s_i} $ Quality improvement level of the $ i $th product
$ {Q_i} $ Production quantity for the $ i $th product
$ {\eta _i} $ Risk aversion coefficient of the $ i $th manufacturer
$ {k_i} $ Cost coefficient of investment in quality improvement of product $ i $
Table 3.  Effect of the quality sensitivity on the expected profits and supply chain efficiency
$ \alpha $ $ E\pi _r^{wp*} $ $ E\pi _m^{wp*} $ $ E\pi _{sc}^{wp*} $ $ E\pi _{sc}^{I*} $ $ {E_f} $
1 24133 7722 39577 46960 84.28%
2 26053 7962 41977 49960 84.02%
3 28933 8362 45657 54960 83.07%
4 32773 8922 51576 61960 81.69%
5 37573 9642 56857 70960 80.13%
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$ \alpha $ $ E\pi _r^{wp*} $ $ E\pi _m^{wp*} $ $ E\pi _{sc}^{wp*} $ $ E\pi _{sc}^{I*} $ $ {E_f} $
1 24133 7722 39577 46960 84.28%
2 26053 7962 41977 49960 84.02%
3 28933 8362 45657 54960 83.07%
4 32773 8922 51576 61960 81.69%
5 37573 9642 56857 70960 80.13%
Table 4.  Effect of the competition intensity on the expected profits and supply chain efficiency
$ \beta $ $ E\pi _r^{wp*} $ $ E\pi _m^{wp*} $ $ E\pi _{sc}^{wp*} $ $ {E_f} $
1 37573 9642 56857 80.13%
2 39973 9402 58777 82.83%
3 42373 9002 60377 85.09%
4 44773 8442 61657 86.89%
5 47173 7722 62617 88.24%
6 49573 6842 63257 89.14%
7 51973 5802 63577 89.60%
8 54373 4602 63577 89.60%
9 56773 3242 63257 89.14%
10 59173 1722 62617 88.24%
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$ \beta $ $ E\pi _r^{wp*} $ $ E\pi _m^{wp*} $ $ E\pi _{sc}^{wp*} $ $ {E_f} $
1 37573 9642 56857 80.13%
2 39973 9402 58777 82.83%
3 42373 9002 60377 85.09%
4 44773 8442 61657 86.89%
5 47173 7722 62617 88.24%
6 49573 6842 63257 89.14%
7 51973 5802 63577 89.60%
8 54373 4602 63577 89.60%
9 56773 3242 63257 89.14%
10 59173 1722 62617 88.24%
Table 5.  Effect of the risk aversion degree on the expected profits and supply chain efficiency
$ \eta $ $ E\pi _r^{wp*} $ $ E\pi _m^{wp*} $ $ E\pi _{sc}^{wp*} $ $ {E_f} $
1 38148 9833 57814 81.47%
0.95 37861 9738 57337 80.80%
0.9 37573 9642 56857 80.13%
0.85 37286 9546 56378 79.45%
0.8 36998 9451 55900 78.78%
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$ \eta $ $ E\pi _r^{wp*} $ $ E\pi _m^{wp*} $ $ E\pi _{sc}^{wp*} $ $ {E_f} $
1 38148 9833 57814 81.47%
0.95 37861 9738 57337 80.80%
0.9 37573 9642 56857 80.13%
0.85 37286 9546 56378 79.45%
0.8 36998 9451 55900 78.78%
Table 6.  Effect of the cost coefficient of quality improvement on the expected profits and supply chain efficiency
$ k $ $ E\pi _r^{wp*} $ $ E\pi _m^{wp*} $ $ E\pi _{sc}^{wp*} $ $ E\pi _{sc}^{I*} $ $ {E_f} $
0.1 51973 11562 75097 95960 78.23%
0.2 37573 9642 56857 70960 80.13%
0.3 32773 9002 50777 62627 81.08%
0.4 30373 8682 47737 58460 81.66%
0.5 28933 8490 45913 55960 82.05%
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$ k $ $ E\pi _r^{wp*} $ $ E\pi _m^{wp*} $ $ E\pi _{sc}^{wp*} $ $ E\pi _{sc}^{I*} $ $ {E_f} $
0.1 51973 11562 75097 95960 78.23%
0.2 37573 9642 56857 70960 80.13%
0.3 32773 9002 50777 62627 81.08%
0.4 30373 8682 47737 58460 81.66%
0.5 28933 8490 45913 55960 82.05%
Table 7.  Effect of the quality sensitivity on the feasible region of option price
$ \alpha $ 1 2 3 4 5
feasible region of $ o $ $ [3.76, 4.16] $ [3.75, 4.17] [3.75, 4.20] [3.74, 4.23] [3.73, 4.27]
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$ \alpha $ 1 2 3 4 5
feasible region of $ o $ $ [3.76, 4.16] $ [3.75, 4.17] [3.75, 4.20] [3.74, 4.23] [3.73, 4.27]
Table 8.  Effect of the competition intensity on the feasible region of option price
$ \beta $ 1 2 3 4 5
feasible region of $ o $ [3.73, 4.27] [3.73, 4.23] [3.72, 4.20] [3.72, 4.17] [3.71, 4.16]
$ \beta $ 6 7 8 9 10
feasible region of $ o $ [3.69, 4.16] [3.68, 4.19] [3.66, 4.26] [3.63, 4.38] [3.59, 4.59]
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$ \beta $ 1 2 3 4 5
feasible region of $ o $ [3.73, 4.27] [3.73, 4.23] [3.72, 4.20] [3.72, 4.17] [3.71, 4.16]
$ \beta $ 6 7 8 9 10
feasible region of $ o $ [3.69, 4.16] [3.68, 4.19] [3.66, 4.26] [3.63, 4.38] [3.59, 4.59]
Table 9.  Effect of the risk aversion degree on the feasible region of option price
$ \eta $ 0.94 0.92 0.90 0.88 0.86
feasible region of $ o $ [3.24, 3.97] [3.49, 4.12] [3.73, 4.27] [3.98, 4.42] [4.24, 4.57]
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$ \eta $ 0.94 0.92 0.90 0.88 0.86
feasible region of $ o $ [3.24, 3.97] [3.49, 4.12] [3.73, 4.27] [3.98, 4.42] [4.24, 4.57]
Table 10.  Effect of the cost coefficient of quality improvement on the feasible region of option price
$ k $ 0.1 0.2 0.3 0.4 0.5
feasible region of $ o $ [3.72, 4.33] [3.73, 4.27] [3.74, 4.24] [3.75, 4.22] [3.75, 4.21]
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$ k $ 0.1 0.2 0.3 0.4 0.5
feasible region of $ o $ [3.72, 4.33] [3.73, 4.27] [3.74, 4.24] [3.75, 4.22] [3.75, 4.21]
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