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doi: 10.3934/jimo.2021052

Fresh produce price-setting newsvendor with bidirectional option contracts

1. 

School of Business, Sichuan Agricultural University, Chengdu, 611830, China

2. 

School of Management and Economics, University of Electronic Science and Technology of China, Chengdu, 611731, China

* Corresponding author: Xu Chen

Received  August 2020 Revised  December 2020 Early access  March 2021

This paper examines a newsvendor problem for fresh produce with bidirectional option contracts, in which the stochastic demand is price-dependent. The bidirectional option, which may be exercised as either a call or put option, provides the newsvendor the flexibility to increase or decrease the initial order after real demand is realized, respectively. The condition of the fresh produce may deteriorate during circulation. The optimal order and pricing decisions for the newsvendor are analytically derived with the bidirectional option and circulation loss. Comparative statics analysis show that the optimal total order quantity and optimal retail price of the newsvendor decrease with the option price but increase with the exercise price. In addition, numerical examples show that the optimal total order quantity and optimal retail price of the newsvendor increase with the circulation loss. The optimal option order quantity first decreases then increases with the exercise price. The optimal firm order quantity first increases then decreases with the circulation loss. The maximum profit of the newsvendor decreases with the option price and circulation loss but increases with the exercise price. Furthermore, the values of bidirectional option contracts are more significant when the demand uncertainty and the circulation loss become more volatile.

Citation: Chong Wang, Xu Chen. Fresh produce price-setting newsvendor with bidirectional option contracts. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021052
References:
[1]

R. AkkermanP. Farahani and M. Grunow, Quality, safety and sustainability in food distribution: a review of quantitative operations management approaches and challenges, OR Spectrum, 32 (2010), 863-904.  doi: 10.1007/s00291-010-0223-2.  Google Scholar

[2]

M. Bagnoli and T. Bergstrom, Log-concave probability and its applications, Economic Theory, 26 (2005), 445-469.  doi: 10.1007/s00199-004-0514-4.  Google Scholar

[3]

D. Barnes-SchusterY. Bassok and R. Anupindi, Coordination and flexibility in supply contracts with options, Manufacturing & Service Operations Management, 4 (2002), 171-240.  doi: 10.1287/msom.4.3.171.7754.  Google Scholar

[4]

J. Blackburn and G. Scudder, Supply chain strategies for perishable products: The case of fresh produce, Production and Operations Management, 18 (2009), 129-137.  doi: 10.1111/j.1937-5956.2009.01016.x.  Google Scholar

[5]

X. CaiJ. ChenY. Xiao and al. et, Optimization and coordination of fresh product supply chains with freshness-keeping effort, Production and Operations Management, 19 (2010), 261-278.  doi: 10.1111/j.1937-5956.2009.01096.x.  Google Scholar

[6]

F. Chen and M. Parlar, Value of a put option to the risk-averse newsvendor, IIE Transactions, 39 (2007), 481-500.  doi: 10.1080/07408170600941607.  Google Scholar

[7]

X. Chen and N. Wan, Multiperiod portfolio procurement problem with option contracts, IEEE Transactions on Engineering Management, (2019), 1–17. doi: 10.1109/TEM. 2019.2923526.  Google Scholar

[8]

X. ChenN. Wan and X. Wang, Flexibility and coordination in a supply chain with bidirectional option contracts and service requirement, International Journal of Production Economics, 193 (2017), 183-192.  doi: 10.1016/j.ijpe.2017.07.013.  Google Scholar

[9]

X. ChenG. Hao and L. Li, Channel coordination with a loss-averse newsvendor and option contracts, International Journal of Production Economics, 150 (2014), 52-57.   Google Scholar

[10]

X. Chen and Z. J. Shen, An analysis of a supply chain with options contracts and service requirements, IIE Transactions, 44 (2012), 805-819.  doi: 10.1080/0740817X.2011.649383.  Google Scholar

[11]

F. Fang and A. Whinston, Option contracts and capacity management-enabling price discrimination under demand uncertainty, Production and Operations Management, 16 (2007), 125-137.  doi: 10.1111/j.1937-5956.2007.tb00170.x.  Google Scholar

[12]

Y. FengY. MuB. Hu and al. et, Commodity options purchasing and credit financing under capital constraint, International Journal of Production Economics, 153 (2014), 230-237.  doi: 10.1016/j.ijpe.2014.03.003.  Google Scholar

[13]

Q. FuS. X. ZhouX. Chao and al. et, Combined pricing and portfolio option procurement, Production and Operations Management, 21 (2012), 361-377.  doi: 10.1111/j.1937-5956.2011.01255.x.  Google Scholar

[14]

A. Gomez_Padilla and T. Mishina, Supply contract with options, International Journal of Production Economics, 122 (2009), 312-318.   Google Scholar

[15]

X. Hou and D. Liu, Building fresh product supply chain cooperation in a typical wholesale market, Journal of the Operational Research Society, 68 (2017), 566-576.  doi: 10.1057/s41274-016-0025-2.  Google Scholar

[16]

Z. HuaS. Li and L. Liang, Impact of demand uncertainty on supply chain cooperation of single-period products, International Journal of Production Economics, 100 (2006), 268-284.  doi: 10.1016/j.ijpe.2004.11.007.  Google Scholar

[17]

K. InderfurthP. Kelle and R. Kleber, Dual sourcing using capacity reservation and spot market: Optimal procurement policy and heuristic parameter determination, European Journal of Operational Research, 225 (2013), 298-309.  doi: 10.1016/j.ejor.2012.08.025.  Google Scholar

[18]

C. Y. LeeX. Li and M. Yu, The loss-averse newsvendor problem with supply options, Naval Research Logistics, 62 (2015), 46-59.  doi: 10.1002/nav.21613.  Google Scholar

[19]

J. LiX. Cai and Y. Zeng, Cost allocation for less-than-truckload collaboration among perishable product retailers, OR Spectrum, 38 (2016), 81-117.  doi: 10.1007/s00291-015-0424-9.  Google Scholar

[20]

J. C. LiY. W. Zhou and W. Huang, Production and procurement strategies for seasonal product supply chain under yield uncertainty with commitment-option contracts, International Journal of Production Economics, 183 (2017), 208-222.  doi: 10.1016/j.ijpe.2016.10.019.  Google Scholar

[21]

C. LiuZ. JiangL. Liu and al. et, Solutions for flexible container leasing contracts with options under capacity and order constraints, International Journal of Production Economics, 141 (2013), 403-413.  doi: 10.1016/j.ijpe.2012.09.005.  Google Scholar

[22]

G. LiuJ. Zhang and W. Tang, Joint dynamic pricing and investment strategy for perishable foods with price-quality dependent demand, Annals of Operations Research, 226 (2015), 397-416.  doi: 10.1007/s10479-014-1671-x.  Google Scholar

[23]

T. J. Lowe and P. V. Preckel, Decision technologies for agribusiness problems: A brief review of selected literature and a call for research, Manufacturing & Service Operations Management, 6 (2014), 201-208.  doi: 10.1287/msom.1040.0051.  Google Scholar

[24]

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.  Google Scholar

[25]

S. MarzbanM. Mahootchi and A. A. Khamseh, Developing a multi-period robust optimization model considering American style options, Annals of Operations Research, 233 (2015), 305-320.  doi: 10.1007/s10479-013-1461-x.  Google Scholar

[26]

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.  Google Scholar

[27]

J. M. Milner and M. J. Rosenblatt, Flexible supply contracts for short life-cycle goods: The buyer's perspective, Naval Research Logistics, 49 (2002), 25-45.  doi: 10.1002/nav.10002.  Google Scholar

[28]

V. NagaliJ. HwangD. Sanghera and al. et, Procurement risk management (PRM) at Hewlett-Packard Company, Interfaces, 38 (2008), 51-60.  doi: 10.1287/inte.1070.0333.  Google Scholar

[29]

I. Nosoohi and A. S. Nookabadi, Outsource planning through option contracts with demand and cost uncertainty, European Journal of Operational Research, 250 (2016), 131-142.  doi: 10.1016/j.ejor.2015.10.030.  Google Scholar

[30]

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.  Google Scholar

[31]

W. E. Soto-SilvaE. Nadal-RoigM. C. González-Araya and al. et, Operational research models applied to the fresh fruit supply chain, European Journal of Operational Research, 251 (2016), 345-355.  doi: 10.1016/j.ejor.2015.08.046.  Google Scholar

[32]

S. Treville, N. Schürhoff and L. Trigeorgis, et al., Optimal sourcing and lead-time reduction under evolutionary demand risk, Production and Operations Management, 23 (2014), 2103- 2117. Google Scholar

[33]

C. Wang and X. Chen, Optimal ordering policy for a price-setting newsvendor with option contracts under demand uncertainty, International Journal of Production Research, 53 (2015), 6279-6293.  doi: 10.1080/00207543.2015.1053577.  Google Scholar

[34]

C. Wang and X. Chen, Option pricing and coordination in the fresh produce supply chain with portfolio contracts, Annals of Operations Research, 248 (2017), 471-491.  doi: 10.1007/s10479-016-2167-7.  Google Scholar

[35]

C. Wang and X. Chen, Joint order and pricing decisions for fresh produce with put option contracts, Journal of the Operational Research Society, 69 (2018), 474-484.   Google Scholar

[36]

Q. Wang and D. B. Tsao, Supply contract with bidirectional options: The buyer's perspective, International Journal of Production Economics, 101 (2006), 30-52.   Google Scholar

[37]

X. WangZ. P. Fan and Z. Liu, Optimal markdown policy of perishable food under the consumer price fairness perception, International Journal of Production Research, 54 (2016), 5811-5828.  doi: 10.1080/00207543.2016.1179810.  Google Scholar

[38]

X. WangF. LiL. Liang and al. et, Pre-purchasing with option contract and coordination in a relief supply chain, International Journal of Production Economics, 167 (2015), 170-176.  doi: 10.1016/j.ijpe.2015.05.031.  Google Scholar

[39]

Wilson, New rules on security of gas supply, 2016. Available from: http://www.europarl.europa.eu/RegData/etudes/ BRIE/2016/593487/EPRS_BRI%282016%29593487_EN.pdf. Google Scholar

[40]

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.  Google Scholar

[41]

Y. Xiao and J. Chen, Supply chain management of fresh products with producer transportation, Decision Sciences, 43 (2012), 785-815.  doi: 10.1111/j.1540-5915.2012.00375.x.  Google Scholar

[42]

H. Xu, Managing production and procurement through option contracts in supply chains with random yield, International Journal of Production Economics, 126 (2010), 306-313.   Google Scholar

[43]

W. XueL. Ma and H. Shen, Optimal inventory and hedging decisions with CVaR consideration, International Journal of Production Economics, 162 (2015), 70-82.  doi: 10.1016/j.ijpe.2015.01.011.  Google Scholar

[44]

J. ZhangG. LiuQ. Zhang and al. et, Coordinating a supply chain for deteriorating items with a revenue sharing and cooperative investment contract, Omega, 56 (2015), 37-49.  doi: 10.1016/j.omega.2015.03.004.  Google Scholar

[45]

Y. ZhaoL. MaG. Xie and al. et, Coordination of supply chains with bidirectional option contracts, European Journal of Operational Research, 229 (2013), 375-381.   Google Scholar

show all references

References:
[1]

R. AkkermanP. Farahani and M. Grunow, Quality, safety and sustainability in food distribution: a review of quantitative operations management approaches and challenges, OR Spectrum, 32 (2010), 863-904.  doi: 10.1007/s00291-010-0223-2.  Google Scholar

[2]

M. Bagnoli and T. Bergstrom, Log-concave probability and its applications, Economic Theory, 26 (2005), 445-469.  doi: 10.1007/s00199-004-0514-4.  Google Scholar

[3]

D. Barnes-SchusterY. Bassok and R. Anupindi, Coordination and flexibility in supply contracts with options, Manufacturing & Service Operations Management, 4 (2002), 171-240.  doi: 10.1287/msom.4.3.171.7754.  Google Scholar

[4]

J. Blackburn and G. Scudder, Supply chain strategies for perishable products: The case of fresh produce, Production and Operations Management, 18 (2009), 129-137.  doi: 10.1111/j.1937-5956.2009.01016.x.  Google Scholar

[5]

X. CaiJ. ChenY. Xiao and al. et, Optimization and coordination of fresh product supply chains with freshness-keeping effort, Production and Operations Management, 19 (2010), 261-278.  doi: 10.1111/j.1937-5956.2009.01096.x.  Google Scholar

[6]

F. Chen and M. Parlar, Value of a put option to the risk-averse newsvendor, IIE Transactions, 39 (2007), 481-500.  doi: 10.1080/07408170600941607.  Google Scholar

[7]

X. Chen and N. Wan, Multiperiod portfolio procurement problem with option contracts, IEEE Transactions on Engineering Management, (2019), 1–17. doi: 10.1109/TEM. 2019.2923526.  Google Scholar

[8]

X. ChenN. Wan and X. Wang, Flexibility and coordination in a supply chain with bidirectional option contracts and service requirement, International Journal of Production Economics, 193 (2017), 183-192.  doi: 10.1016/j.ijpe.2017.07.013.  Google Scholar

[9]

X. ChenG. Hao and L. Li, Channel coordination with a loss-averse newsvendor and option contracts, International Journal of Production Economics, 150 (2014), 52-57.   Google Scholar

[10]

X. Chen and Z. J. Shen, An analysis of a supply chain with options contracts and service requirements, IIE Transactions, 44 (2012), 805-819.  doi: 10.1080/0740817X.2011.649383.  Google Scholar

[11]

F. Fang and A. Whinston, Option contracts and capacity management-enabling price discrimination under demand uncertainty, Production and Operations Management, 16 (2007), 125-137.  doi: 10.1111/j.1937-5956.2007.tb00170.x.  Google Scholar

[12]

Y. FengY. MuB. Hu and al. et, Commodity options purchasing and credit financing under capital constraint, International Journal of Production Economics, 153 (2014), 230-237.  doi: 10.1016/j.ijpe.2014.03.003.  Google Scholar

[13]

Q. FuS. X. ZhouX. Chao and al. et, Combined pricing and portfolio option procurement, Production and Operations Management, 21 (2012), 361-377.  doi: 10.1111/j.1937-5956.2011.01255.x.  Google Scholar

[14]

A. Gomez_Padilla and T. Mishina, Supply contract with options, International Journal of Production Economics, 122 (2009), 312-318.   Google Scholar

[15]

X. Hou and D. Liu, Building fresh product supply chain cooperation in a typical wholesale market, Journal of the Operational Research Society, 68 (2017), 566-576.  doi: 10.1057/s41274-016-0025-2.  Google Scholar

[16]

Z. HuaS. Li and L. Liang, Impact of demand uncertainty on supply chain cooperation of single-period products, International Journal of Production Economics, 100 (2006), 268-284.  doi: 10.1016/j.ijpe.2004.11.007.  Google Scholar

[17]

K. InderfurthP. Kelle and R. Kleber, Dual sourcing using capacity reservation and spot market: Optimal procurement policy and heuristic parameter determination, European Journal of Operational Research, 225 (2013), 298-309.  doi: 10.1016/j.ejor.2012.08.025.  Google Scholar

[18]

C. Y. LeeX. Li and M. Yu, The loss-averse newsvendor problem with supply options, Naval Research Logistics, 62 (2015), 46-59.  doi: 10.1002/nav.21613.  Google Scholar

[19]

J. LiX. Cai and Y. Zeng, Cost allocation for less-than-truckload collaboration among perishable product retailers, OR Spectrum, 38 (2016), 81-117.  doi: 10.1007/s00291-015-0424-9.  Google Scholar

[20]

J. C. LiY. W. Zhou and W. Huang, Production and procurement strategies for seasonal product supply chain under yield uncertainty with commitment-option contracts, International Journal of Production Economics, 183 (2017), 208-222.  doi: 10.1016/j.ijpe.2016.10.019.  Google Scholar

[21]

C. LiuZ. JiangL. Liu and al. et, Solutions for flexible container leasing contracts with options under capacity and order constraints, International Journal of Production Economics, 141 (2013), 403-413.  doi: 10.1016/j.ijpe.2012.09.005.  Google Scholar

[22]

G. LiuJ. Zhang and W. Tang, Joint dynamic pricing and investment strategy for perishable foods with price-quality dependent demand, Annals of Operations Research, 226 (2015), 397-416.  doi: 10.1007/s10479-014-1671-x.  Google Scholar

[23]

T. J. Lowe and P. V. Preckel, Decision technologies for agribusiness problems: A brief review of selected literature and a call for research, Manufacturing & Service Operations Management, 6 (2014), 201-208.  doi: 10.1287/msom.1040.0051.  Google Scholar

[24]

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.  Google Scholar

[25]

S. MarzbanM. Mahootchi and A. A. Khamseh, Developing a multi-period robust optimization model considering American style options, Annals of Operations Research, 233 (2015), 305-320.  doi: 10.1007/s10479-013-1461-x.  Google Scholar

[26]

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.  Google Scholar

[27]

J. M. Milner and M. J. Rosenblatt, Flexible supply contracts for short life-cycle goods: The buyer's perspective, Naval Research Logistics, 49 (2002), 25-45.  doi: 10.1002/nav.10002.  Google Scholar

[28]

V. NagaliJ. HwangD. Sanghera and al. et, Procurement risk management (PRM) at Hewlett-Packard Company, Interfaces, 38 (2008), 51-60.  doi: 10.1287/inte.1070.0333.  Google Scholar

[29]

I. Nosoohi and A. S. Nookabadi, Outsource planning through option contracts with demand and cost uncertainty, European Journal of Operational Research, 250 (2016), 131-142.  doi: 10.1016/j.ejor.2015.10.030.  Google Scholar

[30]

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.  Google Scholar

[31]

W. E. Soto-SilvaE. Nadal-RoigM. C. González-Araya and al. et, Operational research models applied to the fresh fruit supply chain, European Journal of Operational Research, 251 (2016), 345-355.  doi: 10.1016/j.ejor.2015.08.046.  Google Scholar

[32]

S. Treville, N. Schürhoff and L. Trigeorgis, et al., Optimal sourcing and lead-time reduction under evolutionary demand risk, Production and Operations Management, 23 (2014), 2103- 2117. Google Scholar

[33]

C. Wang and X. Chen, Optimal ordering policy for a price-setting newsvendor with option contracts under demand uncertainty, International Journal of Production Research, 53 (2015), 6279-6293.  doi: 10.1080/00207543.2015.1053577.  Google Scholar

[34]

C. Wang and X. Chen, Option pricing and coordination in the fresh produce supply chain with portfolio contracts, Annals of Operations Research, 248 (2017), 471-491.  doi: 10.1007/s10479-016-2167-7.  Google Scholar

[35]

C. Wang and X. Chen, Joint order and pricing decisions for fresh produce with put option contracts, Journal of the Operational Research Society, 69 (2018), 474-484.   Google Scholar

[36]

Q. Wang and D. B. Tsao, Supply contract with bidirectional options: The buyer's perspective, International Journal of Production Economics, 101 (2006), 30-52.   Google Scholar

[37]

X. WangZ. P. Fan and Z. Liu, Optimal markdown policy of perishable food under the consumer price fairness perception, International Journal of Production Research, 54 (2016), 5811-5828.  doi: 10.1080/00207543.2016.1179810.  Google Scholar

[38]

X. WangF. LiL. Liang and al. et, Pre-purchasing with option contract and coordination in a relief supply chain, International Journal of Production Economics, 167 (2015), 170-176.  doi: 10.1016/j.ijpe.2015.05.031.  Google Scholar

[39]

Wilson, New rules on security of gas supply, 2016. Available from: http://www.europarl.europa.eu/RegData/etudes/ BRIE/2016/593487/EPRS_BRI%282016%29593487_EN.pdf. Google Scholar

[40]

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.  Google Scholar

[41]

Y. Xiao and J. Chen, Supply chain management of fresh products with producer transportation, Decision Sciences, 43 (2012), 785-815.  doi: 10.1111/j.1540-5915.2012.00375.x.  Google Scholar

[42]

H. Xu, Managing production and procurement through option contracts in supply chains with random yield, International Journal of Production Economics, 126 (2010), 306-313.   Google Scholar

[43]

W. XueL. Ma and H. Shen, Optimal inventory and hedging decisions with CVaR consideration, International Journal of Production Economics, 162 (2015), 70-82.  doi: 10.1016/j.ijpe.2015.01.011.  Google Scholar

[44]

J. ZhangG. LiuQ. Zhang and al. et, Coordinating a supply chain for deteriorating items with a revenue sharing and cooperative investment contract, Omega, 56 (2015), 37-49.  doi: 10.1016/j.omega.2015.03.004.  Google Scholar

[45]

Y. ZhaoL. MaG. Xie and al. et, Coordination of supply chains with bidirectional option contracts, European Journal of Operational Research, 229 (2013), 375-381.   Google Scholar

Figure 1.  Sequence of events and decisions
Figure 2.  Effect of CV on the optimal decisions and maximum expected profits of the NV
Figure 3.  Effect of CV on the optimal order quantities of the M-NV
Figure 4.  Effect of $ o $ on the M-NV's optimal decisions and maximum expected profits
Figure 5.  Effect of $ o $ on the optimal order quantities of the M-NV
Figure 6.  Effect of $ e $ on the optimal decisions and maximum expected profits of the M-NV
Figure 7.  Effect of $ e $ on the optimal order quantities of the M-NV
Figure 8.  Effect of $ \beta $ on the optimal decisions and maximum expected profits of the NVs
Figure 9.  Effect of $ \beta $ on the optimal order quantities of the M-NV
Table 1.  Notation
Symbol Description
$ \varepsilon $ The random demand, $ \varepsilon \in[A,B] $, and $ E(\varepsilon)=\mu $;
$ f(x) $ The PDF function of $ \varepsilon $;
$ F(x) $ The CDF function of $ \varepsilon $;
$ D(p,\varepsilon) $ The demand function;
$ \beta $ The circulation loss of fresh produce, and $ 0<\beta<1 $;
$ o $ The per unit charge for purchasing bidirectional options (option price);
$ c $ The per unit charge for firm orders (wholesale price);
$ e $ The per unit charge/compensation for exercising bidirectional options (exercise price);
$ s $ The per unit penalty cost for unfilled demand;
$ p $ The per unit selling price of product;
$ q_c $ The firm order quantity of the M-NV;
$ q_o $ The option order quantity of the M-NV;
$ q $ The total order quantity of the M-NV, note $ q=q_c+q_o $;
$ \pi(\cdot) $ The profit of the M-NV;
N Subscript, denoting the case where no bidirectional options are offered for the F-NV.
Symbol Description
$ \varepsilon $ The random demand, $ \varepsilon \in[A,B] $, and $ E(\varepsilon)=\mu $;
$ f(x) $ The PDF function of $ \varepsilon $;
$ F(x) $ The CDF function of $ \varepsilon $;
$ D(p,\varepsilon) $ The demand function;
$ \beta $ The circulation loss of fresh produce, and $ 0<\beta<1 $;
$ o $ The per unit charge for purchasing bidirectional options (option price);
$ c $ The per unit charge for firm orders (wholesale price);
$ e $ The per unit charge/compensation for exercising bidirectional options (exercise price);
$ s $ The per unit penalty cost for unfilled demand;
$ p $ The per unit selling price of product;
$ q_c $ The firm order quantity of the M-NV;
$ q_o $ The option order quantity of the M-NV;
$ q $ The total order quantity of the M-NV, note $ q=q_c+q_o $;
$ \pi(\cdot) $ The profit of the M-NV;
N Subscript, denoting the case where no bidirectional options are offered for the F-NV.
Table 2.  Effect of bidirectional option contracts
Optimal total order quantity Optimal option order quantity Optimal price Maximum expected profit
Fresh produce F-NV 342 32.80 3829
Fresh produce M-NV 403 102 33.07 4370
Optimal total order quantity Optimal option order quantity Optimal price Maximum expected profit
Fresh produce F-NV 342 32.80 3829
Fresh produce M-NV 403 102 33.07 4370
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