American Institute of Mathematical Sciences

Advanced
March  2022, 18(2): 843-872. doi: 10.3934/jimo.2020181

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 Published  March 2022 Early access  December 2020

Fund Project: 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

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.

Keywords: Supply chain coordination, demand information updating, hybrid option-buyback contract, Pareto improvement.
Mathematics Subject Classification: Primary: 90B50; Secondary: 91A35, 91A80.
Citation: Honglin Yang, Jiawu Peng. Coordinating a supply chain with demand information updating. Journal of Industrial and Management Optimization, 2022, 18 (2) : 843-872. doi: 10.3934/jimo.2020181
References:
[1]

H. V. Arani, M. Rabbani and H. Rafiei, A revenue-sharing option contract toward coordination of supply chains, International Journal of Production Economics, 178 (2016), 42-56. 

[2]

I. Baltas and A. N. Yannacopoulos, Uncertainty and inside information, Journal of Dynamics and Games, 3 (2016), 1-24.  doi: 10.3934/jdg.2016001.

[3]

I. Baltas, A. Xepapadeas and A. N. Yannacopoulos, Robust portfolio decisions for financial institutions, Journal of Dynamics and Games, 5 (2018), 61-94.  doi: 10.3934/jdg.2018006.

[4]

D. Barnes-Schuster, Y. Bassok and R. Anupind, Coordination and flexibility in supply contracts with option, Manufacturing & Service Operations Management, 4 (2002), 171-207.  doi: 10.1287/msom.4.3.171.7754.

[5]

S. B$\ddot{o}$ckem and U. Schiller, Option contracts in supply chains, Journal of Economics & Management Strategy, 17 (2008), 219-245. 

[6]

J. Buzacott, H. M. Yan and H. Q. Zhang, Risk analysis of commitment–option contracts with forecast updates, IIE transactions, 43 (2011), 415-431.  doi: 10.1080/0740817X.2010.532851.

[7]

M. A. Begen, H. Pun and X. H. Yan, Supply and demand uncertain reduction efforts and cost comparison, International Journal of Production Economics, 180 (2010), 125-134. 

[8]

G. P. Cachon, Supply chain coordination with contracts, Handbooks in Operations Research and Management Science: Supply Chain Management, North-Holland, 11 (2003), 227-339.  doi: 10.1016/S0927-0507(03)11006-7.

[9]

G. Cachon and R. Swinney, Purchasing, pricing and quick response in the presence of strategic consumers, Management Science, 55 (2009), 497-511.  doi: 10.1287/mnsc.1080.0948.

[10]

G. Cachon and R. Swinney, The value of fast fashion: Quick response, enhanced design, and strategic consumer behavior, Management Science, 57 (2011), 778-795. 

[11]

T. M. Choi, D. Li and H. Yan, Optimal two-stage ordering policy with Bayesian information updating, Journal of the Operational Research Society, 54 (2003), 846-859.  doi: 10.1057/palgrave.jors.2601584.

[12]

T. M. Choi, D. Li and H. Yan, Quick response policy with Bayesian information updates, European Journal of Operational Research, 170 (2006), 788-808.  doi: 10.1016/j.ejor.2004.07.049.

[13]

H. Chen, J. Chen and Y. F. Chen, A coordination mechanism for a supply chain with demand information updating, International Journal of Production Economics, 103 (2006), 347-361.  doi: 10.1016/j.ijpe.2005.09.002.

[14]

H. Chen, Y. F. Chen, C. H. Chiu, T. M. Choi and S. Sethi, Coordination mechanism for the supply chain with lead-time consideration and price-dependent demand, European Journal of Operational Research, 203 (2010), 70-80. 

[15]

A. Cheaitou and R. Cheaytou, A two-stage capacity reservation supply contract with risky supplier and forecast updating, International Journal of Production Economics, 209 (2019), 42-60.  doi: 10.1016/j.ijpe.2018.01.019.

[16]

K. L. Donohue, Efficient supply contracts for fashion goods with forecast updating and two production modes, Management Science, 46 (2000), 1397-1411.  doi: 10.1287/mnsc.46.11.1397.12088.

[17]

G. D. Eppen and A. V. Iyer, Backup-agreements in fashion buying-the value of upstream flexibility, Management Science, 43 (1997), 1469-1484.  doi: 10.1287/mnsc.43.11.1469.

[18]

M. L. Fisher and A. Raman, Reducing the cost of demand uncertainty through accurate response to early sales, Operations Research, 44 (1996), 87-99.  doi: 10.1287/opre.44.1.87.

[19]

H Gurnani and C. S. Tang, Note: optimal ordering decisions with uncertain cost and demand forecast updating, Management Science, 45 (1999), 1456-1462.  doi: 10.1287/mnsc.45.10.1456.

[20]

H. Huang, S. P. Sethi and H. Yan, Purchase contract management with demand forecast updating, IIE Transactions, 37 (2005), 775-785. 

[21]

K. J$\ddot{o}$rnsten, S. L. Nonas, L. Sandal and J. Ubøe, Mixed contracts for the newsvendor problem with real options and discrete demand, Omega, 41 (2013), 809-881. 

[22]

P. Kouvelis and W. Zhao, Supply chain contract design under financial constraints and bankruptcy costs, Management Science, 62 (2015), 2341-2357.  doi: 10.1287/mnsc.2015.2248.

[23]

H. Li, P. Ritchken and Y. Wang., Option and forward contracting with asymmetric information: Valuation issues in supply chains, European Journal of Operational Research, 197 (2009), 134-148.  doi: 10.1016/j.ejor.2008.06.021.

[24]

T. Y. Li, G. W. Fang and M. Baykal-Gürsoy, Two-stage inventory management with financing under demand updates, International Journal of Production Economics, (2020), 107915. doi: 10.1016/j.ijpe.2020.107915.

[25]

L. J. Ma, Y. X. Zhao, W. L. Xue, T. C. E. Cheng and H. M. Yan, Loss-averse newsvendor model with two ordering opportunities and market information updating, International Journal of Production Economics, 140 (2012), 912-921.  doi: 10.1016/j.ijpe.2012.07.012.

[26]

N. C. Petruzzi and M. Dada, Pricing and the newsvendor problem: A review with extensions, Operations Research, 47 (1999), 183-194.  doi: 10.1287/opre.47.2.183.

[27]

T. Pfeiffer, A Comparison of simple two-part supply chain contracts, International Journal of Production Economics, 180 (2016), 114-124.  doi: 10.1016/j.ijpe.2016.06.023.

[28]

M. Pervin, S. K. Roy and G. W. Weber, Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Annals of Operations Research, 260 (2018), 437-460.  doi: 10.1007/s10479-016-2355-5.

[29]

M. Pervin, S. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price- and stock-dependent demand: A trade-credit policy, Journal of Industrial and Management Optimization, 15 (2019), 1345-1373.  doi: 10.3934/jimo.2018098.

[30]

M. Pervin, S. K. Roy and G. W. Weber, Deteriorating inventory with preservation technology under price- and stock-sensitive demand, Journal of Industrial and Management Optimization, 16 (2020), 1585-1612.  doi: 10.3934/jimo.2019019.

[31]

E. Savku and G. W. Weber, A stochastic maximum principle for a Markov regime-switching jump-diffusion model with delay and an application to Finance, Journal of Optimization Theory and Applications, 179 (2018), 696-721.  doi: 10.1007/s10957-017-1159-3.

[32]

E. Savku and G. W. Weber, Stochastic differential games for optimal investment problems in a Markov regime-switching jump-diffusion market, Annals of Operations Research, (2020). doi: 10.1007/s10479-020-03768-5.

[33]

J. Spengler, Vertical integration and antitrust policy, Journal of Political Economy, 58 (1950), 347-352.  doi: 10.1086/256964.

[34]

B. Shen, T. M. Choi and S. Minner, A review on supply chain contracting with information considerations: Information updating and information asymmetry, International Journal of Production Research, 57 (2019), 4898-4936. 

[35]

A. A. Tsay, The quantity flexibility contract and supplier-customer incentives, Management Science, 45 (1999), 1339-1358.  doi: 10.1287/mnsc.45.10.1339.

[36]

J. Wu, Quantity flexibility contracts under Bayesian updating, Computers & Operations Research, 32 (2005), 1267-1288.  doi: 10.1016/j.cor.2003.11.004.

[37]

Q. Z. Wang and D. B. Tsao, Supply contract with bidirectional options: The buyer's perspective, International Journal of Production Economics, 101 (2006), 30-52.  doi: 10.1016/j.ijpe.2005.05.005.

[38]

X. T. Wang, P. Ma and Y. L. Zhang, Pricing and inventory strategies under quick response with strategic and myopic customer, International Transactions in Operational Research, 27 (2020), 1729-1750.  doi: 10.1111/itor.12453.

[39]

X. M. Yan and Y. Wang, A newsvendor model with capital constraint and demand forecast updating, International Journal of Production Research, 52 (2014), 5012-5040. 

[40]

D. Q. Yang, T. J. Xiao, T. M. Choi and T. C. E. Cheng, Optimal reservation pricing strategy for a fashion supply chain with forecast update and asymmetric cost information, International Journal of Production Research, 56 (2018), 1960-1981.  doi: 10.1080/00207543.2014.998789.

[41]

Y. W. Zhou and S. D. Wang, Supply chain coordination for newsvendor-type products with two ordering opportunities and demand information update, Journal of the Operational Research Society, 63 (2012), 1655-1678.  doi: 10.1057/jors.2011.152.

[42]

M. M. Zheng, K. Wu and Y. Shu, Newsvendor problems with demand forecast updating and supply constraints, Computers & Operations Research, 67 (2016), 193-206.  doi: 10.1016/j.cor.2015.10.007.

[43]

Y. X. Zhao, T. M. Choi, T. C. E. Cheng and S. Y. Wang, Supply option contracts with spot market and demand information updating, European Journal of Operational Research, 266 (2018), 1062-1071.  doi: 10.1016/j.ejor.2017.11.001.

[44]

Y. Z. Zhai, W. L. Xue and L. J. Ma., Commitment decisions with demand information updating and a capital-constrained supplier, International Transactions in Operational Research, 27 (2020), 2294-2316.  doi: 10.1111/itor.12722.

show all references

References:
[1]

H. V. Arani, M. Rabbani and H. Rafiei, A revenue-sharing option contract toward coordination of supply chains, International Journal of Production Economics, 178 (2016), 42-56. 

[2]

I. Baltas and A. N. Yannacopoulos, Uncertainty and inside information, Journal of Dynamics and Games, 3 (2016), 1-24.  doi: 10.3934/jdg.2016001.

[3]

I. Baltas, A. Xepapadeas and A. N. Yannacopoulos, Robust portfolio decisions for financial institutions, Journal of Dynamics and Games, 5 (2018), 61-94.  doi: 10.3934/jdg.2018006.

[4]

D. Barnes-Schuster, Y. Bassok and R. Anupind, Coordination and flexibility in supply contracts with option, Manufacturing & Service Operations Management, 4 (2002), 171-207.  doi: 10.1287/msom.4.3.171.7754.

[5]

S. B$\ddot{o}$ckem and U. Schiller, Option contracts in supply chains, Journal of Economics & Management Strategy, 17 (2008), 219-245. 

[6]

J. Buzacott, H. M. Yan and H. Q. Zhang, Risk analysis of commitment–option contracts with forecast updates, IIE transactions, 43 (2011), 415-431.  doi: 10.1080/0740817X.2010.532851.

[7]

M. A. Begen, H. Pun and X. H. Yan, Supply and demand uncertain reduction efforts and cost comparison, International Journal of Production Economics, 180 (2010), 125-134. 

[8]

G. P. Cachon, Supply chain coordination with contracts, Handbooks in Operations Research and Management Science: Supply Chain Management, North-Holland, 11 (2003), 227-339.  doi: 10.1016/S0927-0507(03)11006-7.

[9]

G. Cachon and R. Swinney, Purchasing, pricing and quick response in the presence of strategic consumers, Management Science, 55 (2009), 497-511.  doi: 10.1287/mnsc.1080.0948.

[10]

G. Cachon and R. Swinney, The value of fast fashion: Quick response, enhanced design, and strategic consumer behavior, Management Science, 57 (2011), 778-795. 

[11]

T. M. Choi, D. Li and H. Yan, Optimal two-stage ordering policy with Bayesian information updating, Journal of the Operational Research Society, 54 (2003), 846-859.  doi: 10.1057/palgrave.jors.2601584.

[12]

T. M. Choi, D. Li and H. Yan, Quick response policy with Bayesian information updates, European Journal of Operational Research, 170 (2006), 788-808.  doi: 10.1016/j.ejor.2004.07.049.

[13]

H. Chen, J. Chen and Y. F. Chen, A coordination mechanism for a supply chain with demand information updating, International Journal of Production Economics, 103 (2006), 347-361.  doi: 10.1016/j.ijpe.2005.09.002.

[14]

H. Chen, Y. F. Chen, C. H. Chiu, T. M. Choi and S. Sethi, Coordination mechanism for the supply chain with lead-time consideration and price-dependent demand, European Journal of Operational Research, 203 (2010), 70-80. 

[15]

A. Cheaitou and R. Cheaytou, A two-stage capacity reservation supply contract with risky supplier and forecast updating, International Journal of Production Economics, 209 (2019), 42-60.  doi: 10.1016/j.ijpe.2018.01.019.

[16]

K. L. Donohue, Efficient supply contracts for fashion goods with forecast updating and two production modes, Management Science, 46 (2000), 1397-1411.  doi: 10.1287/mnsc.46.11.1397.12088.

[17]

G. D. Eppen and A. V. Iyer, Backup-agreements in fashion buying-the value of upstream flexibility, Management Science, 43 (1997), 1469-1484.  doi: 10.1287/mnsc.43.11.1469.

[18]

M. L. Fisher and A. Raman, Reducing the cost of demand uncertainty through accurate response to early sales, Operations Research, 44 (1996), 87-99.  doi: 10.1287/opre.44.1.87.

[19]

H Gurnani and C. S. Tang, Note: optimal ordering decisions with uncertain cost and demand forecast updating, Management Science, 45 (1999), 1456-1462.  doi: 10.1287/mnsc.45.10.1456.

[20]

H. Huang, S. P. Sethi and H. Yan, Purchase contract management with demand forecast updating, IIE Transactions, 37 (2005), 775-785. 

[21]

K. J$\ddot{o}$rnsten, S. L. Nonas, L. Sandal and J. Ubøe, Mixed contracts for the newsvendor problem with real options and discrete demand, Omega, 41 (2013), 809-881. 

[22]

P. Kouvelis and W. Zhao, Supply chain contract design under financial constraints and bankruptcy costs, Management Science, 62 (2015), 2341-2357.  doi: 10.1287/mnsc.2015.2248.

[23]

H. Li, P. Ritchken and Y. Wang., Option and forward contracting with asymmetric information: Valuation issues in supply chains, European Journal of Operational Research, 197 (2009), 134-148.  doi: 10.1016/j.ejor.2008.06.021.

[24]

T. Y. Li, G. W. Fang and M. Baykal-Gürsoy, Two-stage inventory management with financing under demand updates, International Journal of Production Economics, (2020), 107915. doi: 10.1016/j.ijpe.2020.107915.

[25]

L. J. Ma, Y. X. Zhao, W. L. Xue, T. C. E. Cheng and H. M. Yan, Loss-averse newsvendor model with two ordering opportunities and market information updating, International Journal of Production Economics, 140 (2012), 912-921.  doi: 10.1016/j.ijpe.2012.07.012.

[26]

N. C. Petruzzi and M. Dada, Pricing and the newsvendor problem: A review with extensions, Operations Research, 47 (1999), 183-194.  doi: 10.1287/opre.47.2.183.

[27]

T. Pfeiffer, A Comparison of simple two-part supply chain contracts, International Journal of Production Economics, 180 (2016), 114-124.  doi: 10.1016/j.ijpe.2016.06.023.

[28]

M. Pervin, S. K. Roy and G. W. Weber, Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Annals of Operations Research, 260 (2018), 437-460.  doi: 10.1007/s10479-016-2355-5.

[29]

M. Pervin, S. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price- and stock-dependent demand: A trade-credit policy, Journal of Industrial and Management Optimization, 15 (2019), 1345-1373.  doi: 10.3934/jimo.2018098.

[30]

M. Pervin, S. K. Roy and G. W. Weber, Deteriorating inventory with preservation technology under price- and stock-sensitive demand, Journal of Industrial and Management Optimization, 16 (2020), 1585-1612.  doi: 10.3934/jimo.2019019.

[31]

E. Savku and G. W. Weber, A stochastic maximum principle for a Markov regime-switching jump-diffusion model with delay and an application to Finance, Journal of Optimization Theory and Applications, 179 (2018), 696-721.  doi: 10.1007/s10957-017-1159-3.

[32]

E. Savku and G. W. Weber, Stochastic differential games for optimal investment problems in a Markov regime-switching jump-diffusion market, Annals of Operations Research, (2020). doi: 10.1007/s10479-020-03768-5.

[33]

J. Spengler, Vertical integration and antitrust policy, Journal of Political Economy, 58 (1950), 347-352.  doi: 10.1086/256964.

[34]

B. Shen, T. M. Choi and S. Minner, A review on supply chain contracting with information considerations: Information updating and information asymmetry, International Journal of Production Research, 57 (2019), 4898-4936. 

[35]

A. A. Tsay, The quantity flexibility contract and supplier-customer incentives, Management Science, 45 (1999), 1339-1358.  doi: 10.1287/mnsc.45.10.1339.

[36]

J. Wu, Quantity flexibility contracts under Bayesian updating, Computers & Operations Research, 32 (2005), 1267-1288.  doi: 10.1016/j.cor.2003.11.004.

[37]

Q. Z. Wang and D. B. Tsao, Supply contract with bidirectional options: The buyer's perspective, International Journal of Production Economics, 101 (2006), 30-52.  doi: 10.1016/j.ijpe.2005.05.005.

[38]

X. T. Wang, P. Ma and Y. L. Zhang, Pricing and inventory strategies under quick response with strategic and myopic customer, International Transactions in Operational Research, 27 (2020), 1729-1750.  doi: 10.1111/itor.12453.

[39]

X. M. Yan and Y. Wang, A newsvendor model with capital constraint and demand forecast updating, International Journal of Production Research, 52 (2014), 5012-5040. 

[40]

D. Q. Yang, T. J. Xiao, T. M. Choi and T. C. E. Cheng, Optimal reservation pricing strategy for a fashion supply chain with forecast update and asymmetric cost information, International Journal of Production Research, 56 (2018), 1960-1981.  doi: 10.1080/00207543.2014.998789.

[41]

Y. W. Zhou and S. D. Wang, Supply chain coordination for newsvendor-type products with two ordering opportunities and demand information update, Journal of the Operational Research Society, 63 (2012), 1655-1678.  doi: 10.1057/jors.2011.152.

[42]

M. M. Zheng, K. Wu and Y. Shu, Newsvendor problems with demand forecast updating and supply constraints, Computers & Operations Research, 67 (2016), 193-206.  doi: 10.1016/j.cor.2015.10.007.

[43]

Y. X. Zhao, T. M. Choi, T. C. E. Cheng and S. Y. Wang, Supply option contracts with spot market and demand information updating, European Journal of Operational Research, 266 (2018), 1062-1071.  doi: 10.1016/j.ejor.2017.11.001.

[44]

Y. Z. Zhai, W. L. Xue and L. J. Ma., Commitment decisions with demand information updating and a capital-constrained supplier, International Transactions in Operational Research, 27 (2020), 2294-2316.  doi: 10.1111/itor.12722.

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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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
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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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} $.
Table 3.  Optimal quantity decisions with uniformly distribution
Optimal decisions Centralized supply chain Decentralized supply chain
$ \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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Optimal decisions Centralized supply chain Decentralized supply chain
$ \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}} $
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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$ \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
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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$ \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} $
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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$ \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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