doi: 10.3934/jimo.2020111

Two-period pricing and ordering decisions of perishable products with a learning period for demand disruption

1. 

College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, China

2. 

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

*Corresponding author: Dong Lei

Received  November 2019 Revised  March 2020 Published  June 2020

In this paper, we develop a two-period inventory model of perishable products with considering the random demand disruption. Faced with the random demand disruption, the firm has two order opportunities: the initial order at the beginning of selling season (i.e., Period 1) is intended to learn the real information of the disrupted demand. When the information of disruption is realized, the firm places the second order, and also decides how many unsold units should be carried into the rest of selling season (i.e., Period 2). The firm may offer two products of different perceived quality in Period 2, and therefore it must trade-off between the quantity of carry-over units and the quantity of young units when the carry-over units cannibalize the sales of young units. Meanwhile, there is both price competition and substitutability between young and old units. We find that the quantity of young units ordered in Period 2 decreases with the quality of units ordered in Period 1, while the pricing of young units is independent of the quality level of old units. However, both the surplus inventory level and the pricing of old units monotonically increase with their quality. We also investigate the influence of two demand disruption scenarios on the optimal order quantity and the optimal pricing when considering different quality situations. We find that in the continuous random disruption scenario, the information value of disruption to the firm is only related to the disruption mean, while in the discrete random disruption scenario, it is related to both unit purchase cost of young units and the disruption levels.

Citation: Kebing Chen, Haijie Zhou, Dong Lei. Two-period pricing and ordering decisions of perishable products with a learning period for demand disruption. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020111
References:
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E. P. ChewC. Lee and R. Liu, Joint inventory allocation and pricing decisions for perishable products, International Journal of Production Economics, 120 (2009), 139-150.  doi: 10.1016/j.ijpe.2008.07.018.  Google Scholar

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K. B. Chen and T. J. Xiao, Production planning and backup sourcing strategy of a buyer-dominant supply chain with random yield and demand, International Journal of Systems Science, 46 (2015), 2799-2817.  doi: 10.1080/00207721.2013.879234.  Google Scholar

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P. Chintapalli, Simultaneous pricing and inventory management of deteriorating perishable products, Annals of Operations Research, 229 (2015), 287–301. doi: 10.1007/s10479-014-1753-9.  Google Scholar

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T. FanC. Xu and F. Tao, Dynamic pricing and replenishment policy for fresh produce, Computers & Industrial Engineering, 139 (2020), 106-127.   Google Scholar

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L. FengJ. Zhang and W. Tang, Dynamic joint pricing and production policy for perishable products, International Transactions in Operational Research, 25 (2018), 2031-2051.  doi: 10.1111/itor.12239.  Google Scholar

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L. FengY. L. Chan and L. E. Cárdenas-Barrón, Pricing and lot-sizing polices for perishable goods when the demand depends on selling price, displayed stocks, and expiration date, International Journal of Production Economics, 185 (2017), 11-20.   Google Scholar

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M. E. Ferguson and O. Koenigsberg, How should a firm manage deteriorating inventory?, Production and Operations Management, 16 (2007), 306-321.  doi: 10.1111/j.1937-5956.2007.tb00261.x.  Google Scholar

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Y. He and S. Wang, Analysis of production-inventory system for deteriorating items with demand disruption, International Journal of Production Research, 50 (2012), 4580-4592.  doi: 10.1080/00207543.2011.615351.  Google Scholar

[25]

Z. HeG. HanT. C. E. ChengB. Fan and J. Dong, Evolutionary food quality and location strategies for restaurants in competitive online-to-offline food ordering and delivery markets: An agent-based approach, International Journal of Production Economics, 215 (2019), 61-72.  doi: 10.1016/j.ijpe.2018.05.008.  Google Scholar

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S. HuangC. Yang and X. Zhang, Pricing and production decisions in dual-channel supply chains with demand disruptions, Computers & Industrial Engineering, 62 (2012), 70-83.  doi: 10.1016/j.cie.2011.08.017.  Google Scholar

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[29]

A. Kara and I. Dogan, Reinforcement learning approaches for specifying ordering policies of perishable inventory systems, Expert Systems with Applications, 91 (2018), 150-158.  doi: 10.1016/j.eswa.2017.08.046.  Google Scholar

[30]

M. LashgariA. A. Taleizadeh and S. S. Sana, An inventory control problem for deteriorating items with back-ordering and financial considerations under two levels of trade credit linked to order quantity, Journal of Industrial & Management Optimization, 12 (2016), 1091-1119.  doi: 10.3934/jimo.2016.12.1091.  Google Scholar

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[32]

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R. Li and J. T. Teng, Pricing and lot-sizing decisions for perishable goods when demand depends on selling price, reference price, product freshness, and displayed stocks, European Journal of Operational Research, 270 (2018), 1099-1108.  doi: 10.1016/j.ejor.2018.04.029.  Google Scholar

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S. K. LiJ. X. Zhang and W. S. Tang, Joint dynamic pricing and inventory control policy for a stochastic inventory system with perishable products, International Journal of Production Research, 53 (2015), 2937-3950.  doi: 10.1080/00207543.2014.961206.  Google Scholar

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T. Li and H. Zhang, Information sharing in a supply chain with a make-to-stock manufacturer, Omega, 50 (2015), 115-125.  doi: 10.1016/j.omega.2014.08.001.  Google Scholar

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Y. LiA. Lim and B. Rodrigues, Note-Pricing and inventory control for a perishable product, Manufacturing & Service Operations Management, 11 (2009), 538-542.   Google Scholar

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W. LiuY. LiuD. ZhuY. Wang and Z. Liang, The influences of demand disruption on logistics service supply chain coordination: A comparison of three coordination modes, International Journal of Production Economics, 179 (2016), 59-76.  doi: 10.1016/j.ijpe.2016.05.022.  Google Scholar

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I. Mallidis, D. Vlachos, V. Yakavenka and Z. Eleni, Development of a single period inventory planning model for perishable product redistribution, Annals of Operations Research, (2018), 1–17. doi: 10.1007/s10479-018-2948-2.  Google Scholar

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show all references

References:
[1]

Z. AzadiS. D. EksiogluB. Eksioglu and G. Palak, Stochastic optimization models for joint pricing and inventory replenishment of perishable products, Computers & Industrial Engineering, 127 (2019), 625-642.  doi: 10.1016/j.cie.2018.11.004.  Google Scholar

[2]

Y. Aviv, The effect of collaborative forecasting on supply chain performance, Management science, 47 (2001), 1331-1440.  doi: 10.1287/mnsc.47.10.1326.10260.  Google Scholar

[3]

İ. S. BakalZ. P. Bayındır and D. E. Emer, Value of disruption information in an EOQ environment, European J. Oper. Res., 263 (2017), 446-460.  doi: 10.1016/j.ejor.2017.04.045.  Google Scholar

[4]

M. A. BegenH. Pun and X. Yan, Supply and demand uncertainty reduction efforts and cost comparison, International Journal of Production Economics, 180 (2016), 125-134.  doi: 10.1016/j.ijpe.2016.07.013.  Google Scholar

[5]

A. Bensoussan, Q. Feng, S. Luo and S.P. Sethi, Evaluating long-term service performance under short-term forecast updates, International Journal of Production Research, (2003), 1–14. Google Scholar

[6]

E. Cao, C. Wan and M. Lai, Coordination of a supply chain with one manufacturer and multiple competing retailers under simultaneous demand and cost disruptions, International Journal of Production Economics, 141 (2013), 425–433. Google Scholar

[7]

E. P. ChewC. Lee and R. Liu, Joint inventory allocation and pricing decisions for perishable products, International Journal of Production Economics, 120 (2009), 139-150.  doi: 10.1016/j.ijpe.2008.07.018.  Google Scholar

[8]

J. ChenM. DongY. Rong and L. Yang, Dynamic pricing for deteriorating products with menu cost, Omega, 75 (2018), 13-26.  doi: 10.1016/j.omega.2017.02.001.  Google Scholar

[9]

K. B. Chen and P. Zhuang, Disruption management for a dominant retailer with constant demand-stimulating service cost, Computers & Industrial Engineering, 61 (2011), 936-946.  doi: 10.1016/j.cie.2011.06.006.  Google Scholar

[10]

K. B. Chen and T. J. Xiao, Production planning and backup sourcing strategy of a buyer-dominant supply chain with random yield and demand, International Journal of Systems Science, 46 (2015), 2799-2817.  doi: 10.1080/00207721.2013.879234.  Google Scholar

[11]

K. B. Chen, R. Xu and H. Fang, Information disclosure model under supply chain competition with asymmetric demand disruption, Asia-Pacific Journal of Operational Research, 33 (2016), 1650043, 35pp. doi: 10.1142/S0217595916500433.  Google Scholar

[12]

Z. X. Chen, Optimization of production inventory with pricing and promotion effort for a single-vendor multi-buyer system of perishable products, International Journal of Production Economics, 203 (2018), 333-349.   Google Scholar

[13]

P. Chintapalli, Simultaneous pricing and inventory management of deteriorating perishable products, Annals of Operations Research, 229 (2015), 287–301. doi: 10.1007/s10479-014-1753-9.  Google Scholar

[14]

J. Danusantoso and S. A. Moses, Disruption management in a two-period three-tier electronics supply chain, Cogent Business & Management, 3 (2016), 1137138. doi: 10.1080/23311975.2015.1137138.  Google Scholar

[15]

P. S. DesaiO. Koenigsberg and D. Purohit, Research note-the role of production lead time and demand uncertainty in marketing durable goods, Management Science, 53 (2007), 150-158.  doi: 10.1287/mnsc.1060.0599.  Google Scholar

[16]

L. DuongL. Wood and W. Wang, A review and reflection on inventory management of perishable products in a single-echelon model, International Journal of Operational Research, 31 (2018), 313-329.  doi: 10.1504/IJOR.2018.089734.  Google Scholar

[17]

C. Y. Dye, Optimal joint dynamic pricing, advertising and inventory control model for perishable items with psychic stock effect, European Journal of Operational Research, 283 (2020), 576–587. doi: 10.1016/j.ejor.2019.11.008.  Google Scholar

[18]

A. Ehrenberg and G. Goodhardt, New brands: Near-instant loyalty, Journal of Targeting, Measurement & Analysis for Marketing, 16 (2001), 607–617. Google Scholar

[19]

W. Elmaghraby and P. Keskinocak, Dynamic pricing in the presence of inventory considerations: Research overview, current practices, and future directions, Management Science, 49 (2003), 1287-1309.   Google Scholar

[20]

T. FanC. Xu and F. Tao, Dynamic pricing and replenishment policy for fresh produce, Computers & Industrial Engineering, 139 (2020), 106-127.   Google Scholar

[21]

L. FengJ. Zhang and W. Tang, Dynamic joint pricing and production policy for perishable products, International Transactions in Operational Research, 25 (2018), 2031-2051.  doi: 10.1111/itor.12239.  Google Scholar

[22]

L. FengY. L. Chan and L. E. Cárdenas-Barrón, Pricing and lot-sizing polices for perishable goods when the demand depends on selling price, displayed stocks, and expiration date, International Journal of Production Economics, 185 (2017), 11-20.   Google Scholar

[23]

M. E. Ferguson and O. Koenigsberg, How should a firm manage deteriorating inventory?, Production and Operations Management, 16 (2007), 306-321.  doi: 10.1111/j.1937-5956.2007.tb00261.x.  Google Scholar

[24]

Y. He and S. Wang, Analysis of production-inventory system for deteriorating items with demand disruption, International Journal of Production Research, 50 (2012), 4580-4592.  doi: 10.1080/00207543.2011.615351.  Google Scholar

[25]

Z. HeG. HanT. C. E. ChengB. Fan and J. Dong, Evolutionary food quality and location strategies for restaurants in competitive online-to-offline food ordering and delivery markets: An agent-based approach, International Journal of Production Economics, 215 (2019), 61-72.  doi: 10.1016/j.ijpe.2018.05.008.  Google Scholar

[26]

A. Herbon, Potential additional profits of selling a perishable product due to implementing price discrimination versus implementation costs, International Transactions in Operational Research, 26 (2019), 1402-1421.  doi: 10.1111/itor.12426.  Google Scholar

[27]

S. HuangC. Yang and X. Zhang, Pricing and production decisions in dual-channel supply chains with demand disruptions, Computers & Industrial Engineering, 62 (2012), 70-83.  doi: 10.1016/j.cie.2011.08.017.  Google Scholar

[28]

X. JiJ. Sun and Z. Wang, Turn bad into good: Using transshipment-before-buyback for disruptions of stochastic demand, International Journal of Production Economics, 185 (2017), 150-161.  doi: 10.1016/j.ijpe.2016.12.019.  Google Scholar

[29]

A. Kara and I. Dogan, Reinforcement learning approaches for specifying ordering policies of perishable inventory systems, Expert Systems with Applications, 91 (2018), 150-158.  doi: 10.1016/j.eswa.2017.08.046.  Google Scholar

[30]

M. LashgariA. A. Taleizadeh and S. S. Sana, An inventory control problem for deteriorating items with back-ordering and financial considerations under two levels of trade credit linked to order quantity, Journal of Industrial & Management Optimization, 12 (2016), 1091-1119.  doi: 10.3934/jimo.2016.12.1091.  Google Scholar

[31]

C. Y. Lee and R. Yang, Supply chain contracting with competing suppliers under asymmetric information, IIE Transactions, 45 (2013), 25-52.  doi: 10.1080/0740817X.2012.662308.  Google Scholar

[32]

B. Li, C. Yang and S. Huang, Study on supply chain disruption management under service level dependent demand, Journal of Networks, 9 (2014), 1432. Google Scholar

[33]

R. Li and J. T. Teng, Pricing and lot-sizing decisions for perishable goods when demand depends on selling price, reference price, product freshness, and displayed stocks, European Journal of Operational Research, 270 (2018), 1099-1108.  doi: 10.1016/j.ejor.2018.04.029.  Google Scholar

[34]

S. K. LiJ. X. Zhang and W. S. Tang, Joint dynamic pricing and inventory control policy for a stochastic inventory system with perishable products, International Journal of Production Research, 53 (2015), 2937-3950.  doi: 10.1080/00207543.2014.961206.  Google Scholar

[35]

T. Li and H. Zhang, Information sharing in a supply chain with a make-to-stock manufacturer, Omega, 50 (2015), 115-125.  doi: 10.1016/j.omega.2014.08.001.  Google Scholar

[36]

Y. LiA. Lim and B. Rodrigues, Note-Pricing and inventory control for a perishable product, Manufacturing & Service Operations Management, 11 (2009), 538-542.   Google Scholar

[37]

W. LiuY. LiuD. ZhuY. Wang and Z. Liang, The influences of demand disruption on logistics service supply chain coordination: A comparison of three coordination modes, International Journal of Production Economics, 179 (2016), 59-76.  doi: 10.1016/j.ijpe.2016.05.022.  Google Scholar

[38]

I. Mallidis, D. Vlachos, V. Yakavenka and Z. Eleni, Development of a single period inventory planning model for perishable product redistribution, Annals of Operations Research, (2018), 1–17. doi: 10.1007/s10479-018-2948-2.  Google Scholar

[39]

S. Minner and S. Transchel, Order variability in perishable product supply chains, European Journal of Operational Research, 260 (2017), 93-107.  doi: 10.1016/j.ejor.2016.12.016.  Google Scholar

[40]

S. Minner and S. Transchel, Periodic review inventory-control for perishable products under service-level constraints, OR spectrum, 32 (2010), 979-996.  doi: 10.1007/s00291-010-0196-1.  Google Scholar

[41]

C. Muriana, An EOQ model for perishable products with fixed shelf life under stochastic demand conditions, European Journal of Operational Research, 255 (2016), 388–396. doi: 10.1016/j.ejor.2016.04.036.  Google Scholar

[42]

X. Qi, J. F. Bard and G. Yu, Supply chain coordination with demand disruptions, Omega, 32 (2004), 301–312. doi: 10.1016/j.omega.2003.12.002.  Google Scholar

[43]

P. E. Rossi and G. M. Allenby, Bayesian statistics and marketing, Marketing Science, 49 (2003), 230-230.   Google Scholar

[44]

M. R. G. Samani and S. M. Hosseini-Motlagh, An enhanced procedure for managing blood supply chain under disruptions and uncertainties, Annals of Operations Research, 283 (2019), 1413-1462.  doi: 10.1007/s10479-018-2873-4.  Google Scholar

[45]

H. Scarf, Bayes solutions of the statistical inventory problem, Annals of Mathematical Statistics, 30 (1959), 490-508.  doi: 10.1214/aoms/1177706264.  Google Scholar

[46]

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, (2018), 1–39. Google Scholar

[47]

N. TashakkorS. H. Mirmohammadi and M. Iranpoor, Joint optimization of dynamic pricing and replenishment cycle considering variable non-instantaneous deterioration and stock-dependent demand, Computers & Industrial Engineering, 123 (2018), 232-241.  doi: 10.1016/j.cie.2018.06.029.  Google Scholar

[48]

T. S. Vaughan, A model of the perishable inventory system with reference to consumer-realized product expiration, Journal of the Operational Research Society, 45 (1994), 519-528.   Google Scholar

[49]

T. J. Xiao and X. T. Qi, Price competition, cost and demand disruptions and coordination of a supply chain with one manufacturer and two competing retailers, Omega, 36 (2008), 741-753.  doi: 10.1016/j.omega.2006.02.008.  Google Scholar

[50]

X. Xu and X. Cai, Price and delivery-time competition of perishable products: Existence and uniqueness of Nash equilibrium, Journal of Industrial & Management Optimization, 4 (2008), 843-859.  doi: 10.3934/jimo.2008.4.843.  Google Scholar

[51]

M. Xue and G. Zhu, Partial myopia vs. forward-looking behaviors in a dynamic pricing and replenishment model for perishable items, Journal of Industrial & Management Optimization, (2019). doi: 10.3934/jimo.2019126.  Google Scholar

[52]

G. Yi, X. Chen and C. Tan, Optimal pricing of perishable products with replenishment policy in the presence of strategic consumers, Journal of Industrial & Management Optimization, 15 (2019), 1579–1597. doi: 10.3934/jimo.2018112.  Google Scholar

[53]

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Figure 1.  Graphical representation of the two-period inventory system
Figure 2.  The optimal solution regions on the disrupted demand, where M is sufficiently large number
Figure 3.  Optimal order quantities versus low disruption amount
Figure 4.  Optimal prices versus low disruption amount
Figure 5.  Order quantity, $ \tilde{x}_{1H} $ versus disruption amounts
Figure 6.  Order quantity, $ \tilde{x}_{1L} $ versus disruption amounts
Figure 7.  Optimal pricing, $ \tilde{P}_{1H} $ versus disruption amounts
Figure 8.  Optimal pricing, $ \tilde{P}_{1L} $ versus disruption amounts
Figure 9.  Optimal expected profit, $ \textbf{E}[\tilde{\Pi}_{12}^{B*}] $ versus disruption amounts
Figure 10.  Continuous disruption versus discrete disruption $ \Delta R_H = 8 $, $ \Delta R_L = -8 $ and $ \beta = 0.5 $
Figure 11.  Continuous disruption versus discrete disruption $ \Delta R_H = 8 $, $ \Delta R_L = -8 $ and $ \overline{\Delta R} = 0 $
Table 1.  Notations used in the stochastic model
Parameters
$ R $ Market potential in Period 1, $ R>c $
$ \Delta R $ Random disrupted amount in Period 1, i.e., continuous or
discrete random variable
$ \overline{\Delta R} $ Mean of random disrupted amount $ \Delta R $
$ \overline R $ Determined market potential in Period 2, $ \overline R>c $
$ h $ Unit cost to carry old products
$ q $ Perceived quality of old units when the young units go
on sale, $ 0\leq q \leq1 $
$ c $ Unit cost to purchase young products
$ c_u $ Unit penalty cost for a unit increased quantity, $ c_u \geq 0 $
$ c_s $ Unit penalty cost for a unit decreased quantity, $ c_s \geq 0 $
$ P_1 $ Retail price of young products in Period 1
$ P_2^n $ Retail price of young products in Period 2
$ P_2^o $ Retail price of old products in Period 2
Decision Variables
$ x_i $ Order quantity of young products in Period $ i $, $ i=1, 2 $
$ E $ Reorder point, or ending-stock level at the end of Period 1,
where $ E \geq 0 $
Functions
$ \Pi_2(x_2, E) $ Profit of Period 2 given the surplus inventory $ E $ and the
reorder quantity $ x_2 $
$ \Pi_{12}(x_1) $ Expected profit of both periods given the initial order
quantity $ x_1 $
Parameters
$ R $ Market potential in Period 1, $ R>c $
$ \Delta R $ Random disrupted amount in Period 1, i.e., continuous or
discrete random variable
$ \overline{\Delta R} $ Mean of random disrupted amount $ \Delta R $
$ \overline R $ Determined market potential in Period 2, $ \overline R>c $
$ h $ Unit cost to carry old products
$ q $ Perceived quality of old units when the young units go
on sale, $ 0\leq q \leq1 $
$ c $ Unit cost to purchase young products
$ c_u $ Unit penalty cost for a unit increased quantity, $ c_u \geq 0 $
$ c_s $ Unit penalty cost for a unit decreased quantity, $ c_s \geq 0 $
$ P_1 $ Retail price of young products in Period 1
$ P_2^n $ Retail price of young products in Period 2
$ P_2^o $ Retail price of old products in Period 2
Decision Variables
$ x_i $ Order quantity of young products in Period $ i $, $ i=1, 2 $
$ E $ Reorder point, or ending-stock level at the end of Period 1,
where $ E \geq 0 $
Functions
$ \Pi_2(x_2, E) $ Profit of Period 2 given the surplus inventory $ E $ and the
reorder quantity $ x_2 $
$ \Pi_{12}(x_1) $ Expected profit of both periods given the initial order
quantity $ x_1 $
Table 2.  Optimal decisions and profits of Period 2 based on the perceived quality of old inventory
Cases $ x_2^* $ $ E^* $ $ P_2^{n*} $ $ P_2^{o*} $ $ \Pi_2(x_2^*, E^*) $
$ q \leq \frac{h}{c} $ $ \frac{\overline R-c}{2} $ $ 0 $ $ \frac{\overline R+c}{2} $ NA} $ \frac{(\overline R-c)^2}{4} $
$ \frac{h}{c}<q<\frac{\overline R-c+h}{\overline R} $ $ \frac{(1-q)\overline R-c+h}{2(1-q)} $ $ \frac{cq-h}{2q(1-q)} $ $ \frac{\overline R+c}{2} $ $ \frac{q\overline R+h}{2} $ $ \frac{{\overline R}^2-2\overline Rc}{4}+\frac{2qch-qc^2-h^2}{4q(q-1)} $
$ q \geq \frac{\overline R-c+h}{\overline R} $ $ 0 $ $ \frac{q\overline R-h}{2q} $ NA} $ \frac{q\overline R+h}{2} $ $ \frac{(q\overline R-h)^2}{4q} $
Cases $ x_2^* $ $ E^* $ $ P_2^{n*} $ $ P_2^{o*} $ $ \Pi_2(x_2^*, E^*) $
$ q \leq \frac{h}{c} $ $ \frac{\overline R-c}{2} $ $ 0 $ $ \frac{\overline R+c}{2} $ NA} $ \frac{(\overline R-c)^2}{4} $
$ \frac{h}{c}<q<\frac{\overline R-c+h}{\overline R} $ $ \frac{(1-q)\overline R-c+h}{2(1-q)} $ $ \frac{cq-h}{2q(1-q)} $ $ \frac{\overline R+c}{2} $ $ \frac{q\overline R+h}{2} $ $ \frac{{\overline R}^2-2\overline Rc}{4}+\frac{2qch-qc^2-h^2}{4q(q-1)} $
$ q \geq \frac{\overline R-c+h}{\overline R} $ $ 0 $ $ \frac{q\overline R-h}{2q} $ NA} $ \frac{q\overline R+h}{2} $ $ \frac{(q\overline R-h)^2}{4q} $
Table 3.  Optimal decisions and profits of Period 1 based on the perceived quality of old inventory
$ q $ $ E^* $ $ x_1^* $ $ P_1^* $ $ \Pi_{12}(x_1^*) $
$ q \leq \frac{h}{c} $ 0 $ \frac{R-c}{2} $ $ \frac{R+c}{2} $ $ \frac{(1+\rho)(R-c)^2}{4} $
$ \frac{h}{c}<q<\frac{R-c+h}{R} $ $ \frac{cq-h}{2q(1-q)} $ $ \frac{R-c+E^*}{2} $ $ \frac{R+c-E^*}{2} $ $ \frac{(R-c-E^*)^2}{4}-cE^*+\rho\Pi_2^*(E^*) $
$ q \geq \frac{R-c+h}{R} $ $ \frac{qR-h}{2q} $ $ \frac{R-c+E^*}{2} $ $ \frac{R+c-E^*}{2} $ $ \frac{(R-c-E^*)^2}{4}-cE^*+\rho\Pi_2^*(E^*) $
$ q $ $ E^* $ $ x_1^* $ $ P_1^* $ $ \Pi_{12}(x_1^*) $
$ q \leq \frac{h}{c} $ 0 $ \frac{R-c}{2} $ $ \frac{R+c}{2} $ $ \frac{(1+\rho)(R-c)^2}{4} $
$ \frac{h}{c}<q<\frac{R-c+h}{R} $ $ \frac{cq-h}{2q(1-q)} $ $ \frac{R-c+E^*}{2} $ $ \frac{R+c-E^*}{2} $ $ \frac{(R-c-E^*)^2}{4}-cE^*+\rho\Pi_2^*(E^*) $
$ q \geq \frac{R-c+h}{R} $ $ \frac{qR-h}{2q} $ $ \frac{R-c+E^*}{2} $ $ \frac{R+c-E^*}{2} $ $ \frac{(R-c-E^*)^2}{4}-cE^*+\rho\Pi_2^*(E^*) $
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