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July  2022, 18(4): 2827-2845. doi: 10.3934/jimo.2021093

## Risk minimization inventory model with a profit target and option contracts under spot price uncertainty

 1 School of Economics and Management, Southwest University of Science and Technology, Mianyang, 621010, China 2 Zhongshan Institute, University of Electronic Science and Technology of China, Zhongshan, 528402, China 3 School of Marketing and Logistics Management, Nanjing University of Finance & Economics, Nanjing, 210023, China

*Corresponding author: Nana Wan

Received  June 2020 Revised  March 2021 Published  July 2022 Early access  May 2021

This paper aims to analyze the inventory purchasing model for a manufacturer with an objective of minimizing risk and a constraint on profit target, where the manufacturer buys the components from the supplier or in the spot market and tailors them into the final products to meet a deterministic demand. This paper develops the mean-variance optimization models without and with option contracts, and conducts numerical examples to explore how the target profit level, the spot price uncertainty and option contracts affect the manufacturer's optimal solutions and the level of risk. It is shown that without and with option contracts the manufacturer's level of risk is non-decreasing in the target profit level. With (without) option contracts, the manufacturer suffers a zero risk from a higher spot price uncertainty if the profit target is low, whereas suffers a lower (higher) risk from a higher spot price uncertainty if the profit target is high. Finally, the level of risk faced by the manufacturer is not higher with option contracts than without them. This paper facilitates the application of option contracts in inventory purchasing management with a spot market for the risk minimization manufacturer with a profit target consideration. New insights are also provided for the manufacturer to set an appropriate profit target for an affordable level of risk, and establish the risk observation mechanism for hedging against the spot price volatility effectively.

Citation: Nana Wan, Li Li, Xiaozhi Wu, Jianchang Fan. Risk minimization inventory model with a profit target and option contracts under spot price uncertainty. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2827-2845. doi: 10.3934/jimo.2021093
##### References:
 [1] J. Arnold and S. Minner, Financial and operational instruments for commodity procurement in quantity competition, International Journal of Production Economics, 131 (2011), 96-106.  doi: 10.1016/j.ijpe.2010.02.007. [2] J. Buzacott, H. Yan and H. Zhang, Risk analysis of commitment-option contracts with forecast updates, IIE Transactions, 43 (2011), 415–431. doi: 10.1080/0740817X.2010.532851. [3] T.-M. Choi, Multi-period risk minimization purchasing models for fashion products with interest rate, budget, and profit target considerations, Annals of Operations Research, 237 (2016), 77-98.  doi: 10.1007/s10479-013-1453-x. [4] C.-H. Chiu and T.-M. Choi, Supply chain risk analysis with mean-variance models: A technical review, Annals of Operations Research, 240 (2016), 489-507.  doi: 10.1007/s10479-013-1386-4. [5] T.-M. Choi and C.-H. Chiu, Mean-downside-risk and mean-variance newsvendor models: Implications for sustainable fashion retailing, International Journal of Production Economics, 135, (2012), 552–560. doi: 10.1016/j.ijpe.2010.10.004. [6] T.-M. Choi, C.-H. Chiu and P.-L. Fu, Periodic review multiperiod inventory control under a mean–variance optimization objective, IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 41 (2011), 678-682.  doi: 10.1109/TSMCA.2010.2089515. [7] C.-H. Chiu, T.-M. Choi, G. Hao and X. Li, Innovative menu of contracts for coordinating a supply chain with multiple mean-variance retailers, European Journal of Operational Research, 246 (2015), 815-826.  doi: 10.1016/j.ejor.2015.05.035. [8] T.-M. Choi, D. Li and H. Yan, Mean–variance analysis for the newsvendor problem, IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 38 (2008), 1169-1180.  doi: 10.1109/TSMCA.2008.2001057. [9] T.-M. Choi, D. Li and H. Yan, Mean–variance analysis of a single supplier and retailer supply chain under a returns policy, European Journal of Operational Research, 184 (2008), 356-376.  doi: 10.1016/j.ejor.2006.10.051. [10] G. S. Day, A. J. Fein and G. Ruppersberger, Shakeouts in digital markets: Lessons from B2B exchanges, California Management Review, 45 (2003), 131-150.  doi: 10.2307/41166169. [11] Q. Fu, C.-Y. Lee and C.-P. Teo, Procurement management using option contracts: Random spot price and the portfolio effect, IIE transactions, 42 (2010), 793-811.  doi: 10.1080/07408171003670983. [12] Q. Fu, S. X. Zhou, X. Chao and C.-Y. Lee, Combined pricing and portfolio option procurement, Production and Operations Management, 21 (2012), 361-377.  doi: 10.1111/j.1937-5956.2011.01255.x. [13] Y. Feng and Q. Wu, Option contract design and risk analysis: Supplier's Perspective, Asia-Pacific Journal of Operational Research, 35 (2018), 1850017. doi: 10.1142/S0217595918500173. [14] K. Inderfurth and P. Kelle, Capacity reservation under spot market price uncertainty, International Journal of Production Economics, 133 (2011), 272-279.  doi: 10.1016/j.ijpe.2010.04.022. [15] J. Luo and X. Chen, Risk hedging via option contracts in a random yield supply chain, Annals of Operations Research, 257 (2017), 697-719.  doi: 10.1007/s10479-015-1964-8. [16] M. Liu, E. Cao and C. K. Salifou, Pricing strategies of a dual-channel supply chain with risk aversion, Transportation Research Part E: Logistics and Transportation Review, 90 (2016), 108-120.  doi: 10.1016/j.tre.2015.11.007. [17] Q. Li, B. Li, P. Chen and P. Hou, Dual-channel supply chain decisions under asymmetric information with a risk-averse retailer, Annals of Operations Research, 257 (2017), 423-447.  doi: 10.1007/s10479-015-1852-2. [18] C.-Y. Lee, X. Li and Y. Xie, Procurement risk management using capacitated option contracts with fixed ordering costs, IIE Transactions, 45 (2013), 845-864.  doi: 10.1080/0740817X.2012.745203. [19] Z. Liu and A. Nagurney, Supply chain outsourcing under exchange rate risk and competition, Omega, 39 (2011), 539-549.  doi: 10.1016/j.omega.2010.11.003. [20] Y. Merzifonluoglu, Integrated demand and procurement portfolio management with spot market volatility and option contracts, European Journal of Operational Research, 258 (2017), 181-192.  doi: 10.1016/j.ejor.2016.08.052. [21] V. Martínez-de-Albéniz and D. Simchi-Levi, Competition in the supply option market, Operations Research, 57 (2009), 1082-1097.  doi: 10.1287/opre.1090.0735. [22] S. Ma, Z. Yin and X. Guan, The role of spot market in a decentralised supply chain under random yield, International Journal of Production Research, 51 (2013), 6410-6434.  doi: 10.1080/00207543.2013.813987. [23] V. Nagali, J. Hwang, D. Sanghera, M. Gaskins, M. Pridgen, T. Thurston, P. Mackenroth, D. Branvold, P. Scholler and G. Shoemaker, Procurement risk management (PRM) at Hewlett-Packard company, INFORMS Journal on Applied Analytics, 38 (2008), 51-60.  doi: 10.1287/inte.1070.0333. [24] B. Shen, T.-M. Choi, Y. Wang and C. K. Y. Lo, The coordination of fashion supply chains with a risk-averse supplier under the markdown money policy, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 43 (2013), 266-276.  doi: 10.1109/TSMCA.2012.2204739. [25] S. Spinler, A. Huchzermeier and P. Kleindorfer, Risk hedging via options contracts for physical delivery, OR Spectrum, 25 (2003), 379-395.  doi: 10.1007/s00291-003-0128-4. [26] M. J. Sobel and R. Q. Zhang, Inventory policies for systems with stochastic and deterministic demand, Operations Research, 49 (2001), 157-162.  doi: 10.1287/opre.49.1.157.11197. [27] N. Wan and X. Chen, Multi-period dual-sourcing replenishment problem with option contracts and a spot market, Industrial Management & Data Systems, 118 (2018), 782–805. doi: 10.1108/IMDS-07-2017-0291. [28] Y. Wei and T.-M. Choi, Mean-variance analysis of supply chains under wholesale pricing and profit sharing schemes, European Journal of Operational Research, 204 (2010), 255-262.  doi: 10.1016/j.ejor.2009.10.016. [29] D. J. Wu and P. R. Kleindorfer, Competitive options, supply contracting, and electronic markets, Management Science, 51 (2005), 452-466.  doi: 10.1287/mnsc.1040.0341. [30] D. J. Wu, P. R. Kleindorfer and J. E. Zhang, Optimal bidding and contracting strategies for capital-intensive goods, European Journal of Operational Research, 137 (2002), 657-676.  doi: 10.1016/S0377-2217(01)00093-5. [31] J. Wu, J. Li, S. Wang and T. C. E. Cheng, Mean–variance analysis of the newsvendor model with stockout cost, Omega, 37 (2009), 724-730.  doi: 10.1016/j.omega.2008.02.005. [32] J. Wu, L. Li and L. D. Xu, A randomized pricing decision support system in electronic commerce, Decision Support Systems, 58 (2014), 43-52.  doi: 10.1016/j.dss.2013.01.015. [33] J. Xu, G. Feng, W. Jiang and S. Wang, Optimal procurement of long-term contracts in the presence of imperfect spot market, \emphOmega, 52 (2015), 42-52. doi: 10.1016/j.omega.2014.10.003. [34] Y. Zhao, T.-M. Choi, T. C. E. Cheng and S. Wang, Mean-risk analysis of wholesale price contracts with stochastic price-dependent demand, Annals of Operations Research, 257 (2017), 491-518.  doi: 10.1007/s10479-014-1689-0. [35] Y. Zhao, T.-M. Choi, T. C. E. Cheng and S. 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. [36] W. Zhuo, L. Shao and H. Yang, Mean–variance analysis of option contracts in a two-echelon supply chain, European Journal of Operational Research, 271 (2018), 535-547.  doi: 10.1016/j.ejor.2018.05.033. [37] S. Zhao and Q. Zhu, A risk-averse marketing strategy and its effect on coordination activities in a remanufacturing supply chain under market fluctuation, Journal of Cleaner Production, 171 (2018), 1290-1299.  doi: 10.1016/j.jclepro.2017.10.107.

show all references

##### References:
 [1] J. Arnold and S. Minner, Financial and operational instruments for commodity procurement in quantity competition, International Journal of Production Economics, 131 (2011), 96-106.  doi: 10.1016/j.ijpe.2010.02.007. [2] J. Buzacott, H. Yan and H. Zhang, Risk analysis of commitment-option contracts with forecast updates, IIE Transactions, 43 (2011), 415–431. doi: 10.1080/0740817X.2010.532851. [3] T.-M. Choi, Multi-period risk minimization purchasing models for fashion products with interest rate, budget, and profit target considerations, Annals of Operations Research, 237 (2016), 77-98.  doi: 10.1007/s10479-013-1453-x. [4] C.-H. Chiu and T.-M. Choi, Supply chain risk analysis with mean-variance models: A technical review, Annals of Operations Research, 240 (2016), 489-507.  doi: 10.1007/s10479-013-1386-4. [5] T.-M. Choi and C.-H. Chiu, Mean-downside-risk and mean-variance newsvendor models: Implications for sustainable fashion retailing, International Journal of Production Economics, 135, (2012), 552–560. doi: 10.1016/j.ijpe.2010.10.004. [6] T.-M. Choi, C.-H. Chiu and P.-L. Fu, Periodic review multiperiod inventory control under a mean–variance optimization objective, IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 41 (2011), 678-682.  doi: 10.1109/TSMCA.2010.2089515. [7] C.-H. Chiu, T.-M. Choi, G. Hao and X. Li, Innovative menu of contracts for coordinating a supply chain with multiple mean-variance retailers, European Journal of Operational Research, 246 (2015), 815-826.  doi: 10.1016/j.ejor.2015.05.035. [8] T.-M. Choi, D. Li and H. Yan, Mean–variance analysis for the newsvendor problem, IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 38 (2008), 1169-1180.  doi: 10.1109/TSMCA.2008.2001057. [9] T.-M. Choi, D. Li and H. Yan, Mean–variance analysis of a single supplier and retailer supply chain under a returns policy, European Journal of Operational Research, 184 (2008), 356-376.  doi: 10.1016/j.ejor.2006.10.051. [10] G. S. Day, A. J. Fein and G. Ruppersberger, Shakeouts in digital markets: Lessons from B2B exchanges, California Management Review, 45 (2003), 131-150.  doi: 10.2307/41166169. [11] Q. Fu, C.-Y. Lee and C.-P. Teo, Procurement management using option contracts: Random spot price and the portfolio effect, IIE transactions, 42 (2010), 793-811.  doi: 10.1080/07408171003670983. [12] Q. Fu, S. X. Zhou, X. Chao and C.-Y. Lee, Combined pricing and portfolio option procurement, Production and Operations Management, 21 (2012), 361-377.  doi: 10.1111/j.1937-5956.2011.01255.x. [13] Y. Feng and Q. Wu, Option contract design and risk analysis: Supplier's Perspective, Asia-Pacific Journal of Operational Research, 35 (2018), 1850017. doi: 10.1142/S0217595918500173. [14] K. Inderfurth and P. Kelle, Capacity reservation under spot market price uncertainty, International Journal of Production Economics, 133 (2011), 272-279.  doi: 10.1016/j.ijpe.2010.04.022. [15] J. Luo and X. Chen, Risk hedging via option contracts in a random yield supply chain, Annals of Operations Research, 257 (2017), 697-719.  doi: 10.1007/s10479-015-1964-8. [16] M. Liu, E. Cao and C. K. Salifou, Pricing strategies of a dual-channel supply chain with risk aversion, Transportation Research Part E: Logistics and Transportation Review, 90 (2016), 108-120.  doi: 10.1016/j.tre.2015.11.007. [17] Q. Li, B. Li, P. Chen and P. Hou, Dual-channel supply chain decisions under asymmetric information with a risk-averse retailer, Annals of Operations Research, 257 (2017), 423-447.  doi: 10.1007/s10479-015-1852-2. [18] C.-Y. Lee, X. Li and Y. Xie, Procurement risk management using capacitated option contracts with fixed ordering costs, IIE Transactions, 45 (2013), 845-864.  doi: 10.1080/0740817X.2012.745203. [19] Z. Liu and A. Nagurney, Supply chain outsourcing under exchange rate risk and competition, Omega, 39 (2011), 539-549.  doi: 10.1016/j.omega.2010.11.003. [20] Y. Merzifonluoglu, Integrated demand and procurement portfolio management with spot market volatility and option contracts, European Journal of Operational Research, 258 (2017), 181-192.  doi: 10.1016/j.ejor.2016.08.052. [21] V. Martínez-de-Albéniz and D. Simchi-Levi, Competition in the supply option market, Operations Research, 57 (2009), 1082-1097.  doi: 10.1287/opre.1090.0735. [22] S. Ma, Z. Yin and X. Guan, The role of spot market in a decentralised supply chain under random yield, International Journal of Production Research, 51 (2013), 6410-6434.  doi: 10.1080/00207543.2013.813987. [23] V. Nagali, J. Hwang, D. Sanghera, M. Gaskins, M. Pridgen, T. Thurston, P. Mackenroth, D. Branvold, P. Scholler and G. Shoemaker, Procurement risk management (PRM) at Hewlett-Packard company, INFORMS Journal on Applied Analytics, 38 (2008), 51-60.  doi: 10.1287/inte.1070.0333. [24] B. Shen, T.-M. Choi, Y. Wang and C. K. Y. Lo, The coordination of fashion supply chains with a risk-averse supplier under the markdown money policy, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 43 (2013), 266-276.  doi: 10.1109/TSMCA.2012.2204739. [25] S. Spinler, A. Huchzermeier and P. Kleindorfer, Risk hedging via options contracts for physical delivery, OR Spectrum, 25 (2003), 379-395.  doi: 10.1007/s00291-003-0128-4. [26] M. J. Sobel and R. Q. Zhang, Inventory policies for systems with stochastic and deterministic demand, Operations Research, 49 (2001), 157-162.  doi: 10.1287/opre.49.1.157.11197. [27] N. Wan and X. Chen, Multi-period dual-sourcing replenishment problem with option contracts and a spot market, Industrial Management & Data Systems, 118 (2018), 782–805. doi: 10.1108/IMDS-07-2017-0291. [28] Y. Wei and T.-M. Choi, Mean-variance analysis of supply chains under wholesale pricing and profit sharing schemes, European Journal of Operational Research, 204 (2010), 255-262.  doi: 10.1016/j.ejor.2009.10.016. [29] D. J. Wu and P. R. Kleindorfer, Competitive options, supply contracting, and electronic markets, Management Science, 51 (2005), 452-466.  doi: 10.1287/mnsc.1040.0341. [30] D. J. Wu, P. R. Kleindorfer and J. E. Zhang, Optimal bidding and contracting strategies for capital-intensive goods, European Journal of Operational Research, 137 (2002), 657-676.  doi: 10.1016/S0377-2217(01)00093-5. [31] J. Wu, J. Li, S. Wang and T. C. E. Cheng, Mean–variance analysis of the newsvendor model with stockout cost, Omega, 37 (2009), 724-730.  doi: 10.1016/j.omega.2008.02.005. [32] J. Wu, L. Li and L. D. Xu, A randomized pricing decision support system in electronic commerce, Decision Support Systems, 58 (2014), 43-52.  doi: 10.1016/j.dss.2013.01.015. [33] J. Xu, G. Feng, W. Jiang and S. Wang, Optimal procurement of long-term contracts in the presence of imperfect spot market, \emphOmega, 52 (2015), 42-52. doi: 10.1016/j.omega.2014.10.003. [34] Y. Zhao, T.-M. Choi, T. C. E. Cheng and S. Wang, Mean-risk analysis of wholesale price contracts with stochastic price-dependent demand, Annals of Operations Research, 257 (2017), 491-518.  doi: 10.1007/s10479-014-1689-0. [35] Y. Zhao, T.-M. Choi, T. C. E. Cheng and S. 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. [36] W. Zhuo, L. Shao and H. Yang, Mean–variance analysis of option contracts in a two-echelon supply chain, European Journal of Operational Research, 271 (2018), 535-547.  doi: 10.1016/j.ejor.2018.05.033. [37] S. Zhao and Q. Zhu, A risk-averse marketing strategy and its effect on coordination activities in a remanufacturing supply chain under market fluctuation, Journal of Cleaner Production, 171 (2018), 1290-1299.  doi: 10.1016/j.jclepro.2017.10.107.
The timeline of the event
The impact of the target profit level
The impact of the spot price uncertainty when $K>\left(r-w\right)D$
Notations
 Notations Descriptions $p_{s}$ Random spot price. $f(p_{s})$, $F(p_{s})$ PDF and CDF function of $p_{s}$. $\mu_{s}$, $\sigma_{s}$ Mean and standard deviation of $p_{s}$. $D$ Deterministic demand. $r$ Unit retail price of the final product. $\omega$ Unit wholesale price of the component. $o$ Unit option price. $e$ Unit exercise price. $K$ The target profit level. $q_{0}$ The firm order quantity without option contracts. $q_{1}$ The firm order quantity with option contracts. $q_{2}$ The options order quantity $H$ $\int_{e}^{+\infty}\left(p_{s}-e\right)f\left(p_{s}\right){\rm d}p_{s}$ $J$ $\int_{e}^{+\infty}\left(p_{s}-\mu_{s}\right)\left(p_{s}-e\right)f\left(p_{s}\right){\rm d}p_{s}$ $L$ $\int_{e}^{+\infty}\left(p_{s}-e\right)^{2}f\left(p_{s}\right){\rm d}p_{s}$ $E(\cdot)$ Expected value. $V(\cdot)$ Variance value.
 Notations Descriptions $p_{s}$ Random spot price. $f(p_{s})$, $F(p_{s})$ PDF and CDF function of $p_{s}$. $\mu_{s}$, $\sigma_{s}$ Mean and standard deviation of $p_{s}$. $D$ Deterministic demand. $r$ Unit retail price of the final product. $\omega$ Unit wholesale price of the component. $o$ Unit option price. $e$ Unit exercise price. $K$ The target profit level. $q_{0}$ The firm order quantity without option contracts. $q_{1}$ The firm order quantity with option contracts. $q_{2}$ The options order quantity $H$ $\int_{e}^{+\infty}\left(p_{s}-e\right)f\left(p_{s}\right){\rm d}p_{s}$ $J$ $\int_{e}^{+\infty}\left(p_{s}-\mu_{s}\right)\left(p_{s}-e\right)f\left(p_{s}\right){\rm d}p_{s}$ $L$ $\int_{e}^{+\infty}\left(p_{s}-e\right)^{2}f\left(p_{s}\right){\rm d}p_{s}$ $E(\cdot)$ Expected value. $V(\cdot)$ Variance value.
The impact of the wholesale price when $K>\left(r-w\right)D$
 $w$ Wholesale price contracts Option contracts $q^*_0$ $V\left[\Pi \left(q^*_0\right)\right]$ $q^*_1$ $q^*_2$ $V\left[\Pi \left(q^*_1,q^*_2\right)\right]$ 9.9 114.75 14222.00 86.66 30.37 12959.00 10.0 116.67 18148.10 75.91 43.33 15385.30 10.1 118.64 22709.90 64.74 56.36 17595.60 10.2 120.69 27966.70 53.77 68.78 19508.20 10.3 122.81 33983.80 43.51 80.10 21087.80 10.4 125.00 40833.30 34.24 96.07 22337.60
 $w$ Wholesale price contracts Option contracts $q^*_0$ $V\left[\Pi \left(q^*_0\right)\right]$ $q^*_1$ $q^*_2$ $V\left[\Pi \left(q^*_1,q^*_2\right)\right]$ 9.9 114.75 14222.00 86.66 30.37 12959.00 10.0 116.67 18148.10 75.91 43.33 15385.30 10.1 118.64 22709.90 64.74 56.36 17595.60 10.2 120.69 27966.70 53.77 68.78 19508.20 10.3 122.81 33983.80 43.51 80.10 21087.80 10.4 125.00 40833.30 34.24 96.07 22337.60
The impact of the option price when $K>\left(r-w\right)D$
 $o$ Wholesale price contracts Option contracts $q^*_0$ $V\left[\Pi \left(q^*_0\right)\right]$ $q^*_1$ $q^*_2$ $V\left[\Pi \left(q^*_1,q^*_2\right)\right]$ 2.9 116.67 18148.10 68.18 50.65 13999.33 3.0 116.67 18148.10 75.91 43.33 15385.30 3.1 116.67 18148.10 85.73 33.49 16620.79 3.2 116.67 18148.10 97.34 21.31 17563.07 3.3 116.67 18148.10 110.04 7.44 18078.89 3.4 116.67 18148.10 N.A. N.A. N.A.
 $o$ Wholesale price contracts Option contracts $q^*_0$ $V\left[\Pi \left(q^*_0\right)\right]$ $q^*_1$ $q^*_2$ $V\left[\Pi \left(q^*_1,q^*_2\right)\right]$ 2.9 116.67 18148.10 68.18 50.65 13999.33 3.0 116.67 18148.10 75.91 43.33 15385.30 3.1 116.67 18148.10 85.73 33.49 16620.79 3.2 116.67 18148.10 97.34 21.31 17563.07 3.3 116.67 18148.10 110.04 7.44 18078.89 3.4 116.67 18148.10 N.A. N.A. N.A.
The impact of the exercise price when $K>\left(r-w\right)D$
 $e$ Wholesale price contracts Option contracts $q^*_0$ $V\left[\Pi \left(q^*_0\right)\right]$ $q^*_1$ $q^*_2$ $V\left[\Pi \left(q^*_1,q^*_2\right)\right]$ 7.8 116.67 18148.10 63.96 54.52 13533.20 8.0 116.67 18148.10 75.91 43.33 15385.30 8.2 116.67 18148.10 89.21 30.03 16847.18 8.4 116.67 18148.10 102.50 15.95 17776.12 8.6 116.67 18148.10 114.54 2.47 18138.83 8.8 116.67 18148.10 N.A. N.A. N.A
 $e$ Wholesale price contracts Option contracts $q^*_0$ $V\left[\Pi \left(q^*_0\right)\right]$ $q^*_1$ $q^*_2$ $V\left[\Pi \left(q^*_1,q^*_2\right)\right]$ 7.8 116.67 18148.10 63.96 54.52 13533.20 8.0 116.67 18148.10 75.91 43.33 15385.30 8.2 116.67 18148.10 89.21 30.03 16847.18 8.4 116.67 18148.10 102.50 15.95 17776.12 8.6 116.67 18148.10 114.54 2.47 18138.83 8.8 116.67 18148.10 N.A. N.A. N.A
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