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October  2018, 14(4): 1521-1544. doi: 10.3934/jimo.2018019

Optimal pricing and inventory management for a loss averse firm when facing strategic customers

Ruopeng Wang 1,2, , Jinting Wang 1,, and  Chang Sun 1,

1. 

Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China
2. 

Department of Mathematics and Physics, Beijing Institute of Petrochemical Technology, Beijing 102617, China

* Corresponding author: Jinting Wang

Received  February 2017 Revised  July 2017 Published  January 2018

Fund Project: This work is supported in part by the National Natural Science Foundation of China (Grant nos. 71571014,71390334.)

This paper considers the joint inventory and pricing decision problem that a loss averse firm with reference point selling seasonal products to strategic consumers with risk preference and decreasing value. Consumers can decide whether to buy at the full price in stage 1, or to wait till stage 2 for the salvage price. They may not get the product if the product is sold out in stage 2. The firm aims to choose a base stock policy and find an optimal price to maximize its expected utility, while consumers aim to decide whether to buy or wait strategically for optimizing their payoffs. We formulate the problem as a Stackelberg game between the firm and the strategic consumers in which the firm is the leader. By deriving the rational expectation equilibrium, we find both the optimal stocking level and the full price in our model are lower than those in the classical model without strategic consumers, by which leads to a lower profit. Furthermore, it is shown that the reimbursement contract cannot alleviate the impact of strategic behavior of customers while the firm's profit can be improved by the price commitment strategy in most cases. Numerical studies are carried out to investigate the impact of strategic customer behavior and system parameters on the firm's optimal decisions.

Keywords: Inventory, loss-aversion, reference point, rational expectation (RE) equilibrium, prospect theory.
Mathematics Subject Classification: Primary: 90B05, 90B60; Secondary: 91B42.
Citation: Ruopeng Wang, Jinting Wang, Chang Sun. Optimal pricing and inventory management for a loss averse firm when facing strategic customers. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1521-1544. doi: 10.3934/jimo.2018019
References:
[1]

S. Anily and R. Hassin, Pricing, replenishment, and timing of selling in a market with heterogeneous customers, International Journal of Production Economics, 145 (2013), 672-682.

[2]

Y. Aviv and A. Pazgal, Optimal pricing of seasonal products in the presence of forward-looking consumers, Manuf. Serv. Oper. Manag., 10 (2008), 339-359.

[3]

O. Baron, M. Hu, S. Najafi-Asadolahi and Q. Qian, Newsvendor selling to loss-averse consumers with stochastic reference points, Manuf. Serv. Oper. Manag., 17 (2015), 456-469.

[4]

O. Besbes and I. Lobel, Intertemporal price discrimination: Structure and computation of optimal policies, Management Science, 61 (2015), 92-110.

[5]

J. I. Bulow, Durable-goods monopolist, J. Political Econom, 90 (1982), 314-332.

[6]

G. P. Cachon and R. Swinney, Purchasing, pricing, and quick response in the presence of strategic consumer behavior, Management Science, 55 (2009), 497-511.

[7]

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

[8]

R. H. Coase, Durability and monopoly, J. Law Econom., 15 (1972), 143-149.

[9]

J. R. Correa, R. Montoya and C. Thraves, Contingent preannounced pricing policies with strategic consumers, Operations Research, 64 (2016), 251-272. doi: 10.1287/opre.2015.1452.

[10]

J. Du, J. Zhang and G. Hua, Pricing and inventory management in the presence of strategic customer with risk preference and decreasing value, International Journal of Production Economics, 164 (2015), 160-166.

[11]

L. Eeckhoudt, C. Gollier and H. Schlesinger, The risk-averse (and prudent) newsboy, Management Science, 41 (1995), 786-794.

[12]

X. D. He and X. Y. Zhou, Portfolio choice under cumulative prospect theory: An analytical treatment, Management Science, 57 (2011), 315-331.

[13]

X. D. He and X. Y. Zhou, Myopic loss aversion, reference point, and money illusion, Quant. Finance, 14 (2014), 1541-1554. doi: 10.1080/14697688.2014.917805.

[14]

P. Heidhues and B. Koszegi, The Impact of Consumer Loss Averse on Pricing, Woking paper, University of California, Berkeley, 2005.
[15]

P. Heidhues and B. Koszegi, Competition and price variation when consumers are loss averse, Amer. Econom. Rev., 98 (2008), 1245-1268.

[16]

P. Heidhues and B. Koszegi, Regular prices and sales, Theor. Econom., 9 (2014), 217-251. doi: 10.3982/TE1274.

[17]

D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-291.

[18]

B. Keren and J. S. Pliskin, A benchmark solution for the risk-averse newsvendor problem, Eur. J. Oper. Res., 37 (2006), 1463-1650.

[19]

V. Kobberling and P. P. Wakker, An index of loss aversion, J. of Economic Theory, 122 (2005), 119-131. doi: 10.1016/j.jet.2004.03.009.

[20]

B. Koszegi and M. Rabin, A model of reference-dependent preferences, Quart. J. Econ., 121 (2006), 1133-1165.

[21]

J. Li, N. Granedos and S. Netessine, Are consumers strategic? structural estimation from the air travel industry, Management Science, 60 (2014), 2114-2137.

[22]

Q. Liu and G. J. van Ryzin, Strategic capacity rationing to induce early purchases, Management Science, 54 (2008), 1115-1131.

[23]

X. Y. Long and J. Nasiry, Prospect theory explain newsvendor behavior: The role of reference points, Management Science, 60 (2014), 1057-1062.

[24]

J. F. Muth, Rational expectations and the theory of price movements, Econometrica, 29 (1961), 315-335.

[25]

M. Nagarajan and S. Shechter, Prospect theory and the newsvendor problem, Management Science, 60 (2014), 1057-1062.

[26]

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, 253 (2016), 1-24.

[27]

M. Pervin, S. K. Roy and G. C. Mahata, An inventory model with demand declining market for deteriorating items under trade credit policy, International Journal of Management Science and Engineering Management, 11 (2016), 243-251.

[28]

M. Pervin, S. K. Roy and G. W. Weber, A Two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items, Numerical Algebra, Control and Optimization, 7 (2017), 21-50. doi: 10.3934/naco.2017002.

[29] Porteus and L. Evan, Foundations of Stochastic Inventory Theory, Stanford University Press, 2002.
[30]

P. Ray and M. Jenamani, Mean-variance analysis of souring decision under disruption risk, Eur. J. Oper. Res., 250 (2015), 679-689. doi: 10.1016/j.ejor.2015.09.028.

[31]

J. Rubio-Herrero, M. Baykal-Gursoy and A. Jaskiewicz, A price setting newsvendor problem under mean-variance criteria, Eur. J. Oper. Res., 247 (2015), 575-587. doi: 10.1016/j.ejor.2015.06.006.

[32]

Z. J. Shen and X. Su, Customer behavior modeling in revenue management and auctions: A review and new research opportunities, Production and Operations Management, 16 (2007), 713-728.

[33]

Y. Shi, X. Y. Cui, J. Yao and D. Li, Dynamic trading with reference point adaptation and loss aversion, Operations Research, 63 (2015), 789-806. doi: 10.1287/opre.2015.1399.

[34]

X. Su, Intertemporal pricing with strategic customer behavior, Management Science, 53 (2007), 726-741.

[35]

X. Su and F. Zhang, Strategic customer behavior, commitment, and supply chain performance, Management Science, 54 (2008), 1759-1773.

[36]

N. L. Stokey, Rational expectation and durable goods pricing, The Bell Journal of Economics, 12 (1981), 112-128. doi: 10.2307/3003511.

[37]

C. X. Wang, S. Webster and N. Surech, Would a risk-averse newsvendor order less at a highter selling price?, Eur. J. Oper. Res., 196 (2009), 544-553. doi: 10.1016/j.ejor.2008.04.002.

[38]

J. Wu, J. Li, S. Y. Wang and T. C. E. Cheng, Mean-variance analysis of the newsvendor model with stockout cost, Omega, 37 (2009), 724-730.

[39]

R. Yin, Y. Aviv, A. Pazgal and C. S. Tang, Optimal markdown pricing: Implications of inventory display formats in the presence of strategic customers, Management Science, 55 (2009), 1391-1408.

[40]

P. H. Zipkin, Foundations of Inventory Management, McGraw-Hill/Irwin, 2000.

show all references

References:
[1]

S. Anily and R. Hassin, Pricing, replenishment, and timing of selling in a market with heterogeneous customers, International Journal of Production Economics, 145 (2013), 672-682.

[2]

Y. Aviv and A. Pazgal, Optimal pricing of seasonal products in the presence of forward-looking consumers, Manuf. Serv. Oper. Manag., 10 (2008), 339-359.

[3]

O. Baron, M. Hu, S. Najafi-Asadolahi and Q. Qian, Newsvendor selling to loss-averse consumers with stochastic reference points, Manuf. Serv. Oper. Manag., 17 (2015), 456-469.

[4]

O. Besbes and I. Lobel, Intertemporal price discrimination: Structure and computation of optimal policies, Management Science, 61 (2015), 92-110.

[5]

J. I. Bulow, Durable-goods monopolist, J. Political Econom, 90 (1982), 314-332.

[6]

G. P. Cachon and R. Swinney, Purchasing, pricing, and quick response in the presence of strategic consumer behavior, Management Science, 55 (2009), 497-511.

[7]

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

[8]

R. H. Coase, Durability and monopoly, J. Law Econom., 15 (1972), 143-149.

[9]

J. R. Correa, R. Montoya and C. Thraves, Contingent preannounced pricing policies with strategic consumers, Operations Research, 64 (2016), 251-272. doi: 10.1287/opre.2015.1452.

[10]

J. Du, J. Zhang and G. Hua, Pricing and inventory management in the presence of strategic customer with risk preference and decreasing value, International Journal of Production Economics, 164 (2015), 160-166.

[11]

L. Eeckhoudt, C. Gollier and H. Schlesinger, The risk-averse (and prudent) newsboy, Management Science, 41 (1995), 786-794.

[12]

X. D. He and X. Y. Zhou, Portfolio choice under cumulative prospect theory: An analytical treatment, Management Science, 57 (2011), 315-331.

[13]

X. D. He and X. Y. Zhou, Myopic loss aversion, reference point, and money illusion, Quant. Finance, 14 (2014), 1541-1554. doi: 10.1080/14697688.2014.917805.

[14]

P. Heidhues and B. Koszegi, The Impact of Consumer Loss Averse on Pricing, Woking paper, University of California, Berkeley, 2005.
[15]

P. Heidhues and B. Koszegi, Competition and price variation when consumers are loss averse, Amer. Econom. Rev., 98 (2008), 1245-1268.

[16]

P. Heidhues and B. Koszegi, Regular prices and sales, Theor. Econom., 9 (2014), 217-251. doi: 10.3982/TE1274.

[17]

D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-291.

[18]

B. Keren and J. S. Pliskin, A benchmark solution for the risk-averse newsvendor problem, Eur. J. Oper. Res., 37 (2006), 1463-1650.

[19]

V. Kobberling and P. P. Wakker, An index of loss aversion, J. of Economic Theory, 122 (2005), 119-131. doi: 10.1016/j.jet.2004.03.009.

[20]

B. Koszegi and M. Rabin, A model of reference-dependent preferences, Quart. J. Econ., 121 (2006), 1133-1165.

[21]

J. Li, N. Granedos and S. Netessine, Are consumers strategic? structural estimation from the air travel industry, Management Science, 60 (2014), 2114-2137.

[22]

Q. Liu and G. J. van Ryzin, Strategic capacity rationing to induce early purchases, Management Science, 54 (2008), 1115-1131.

[23]

X. Y. Long and J. Nasiry, Prospect theory explain newsvendor behavior: The role of reference points, Management Science, 60 (2014), 1057-1062.

[24]

J. F. Muth, Rational expectations and the theory of price movements, Econometrica, 29 (1961), 315-335.

[25]

M. Nagarajan and S. Shechter, Prospect theory and the newsvendor problem, Management Science, 60 (2014), 1057-1062.

[26]

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, 253 (2016), 1-24.

[27]

M. Pervin, S. K. Roy and G. C. Mahata, An inventory model with demand declining market for deteriorating items under trade credit policy, International Journal of Management Science and Engineering Management, 11 (2016), 243-251.

[28]

M. Pervin, S. K. Roy and G. W. Weber, A Two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items, Numerical Algebra, Control and Optimization, 7 (2017), 21-50. doi: 10.3934/naco.2017002.

[29] Porteus and L. Evan, Foundations of Stochastic Inventory Theory, Stanford University Press, 2002.
[30]

P. Ray and M. Jenamani, Mean-variance analysis of souring decision under disruption risk, Eur. J. Oper. Res., 250 (2015), 679-689. doi: 10.1016/j.ejor.2015.09.028.

[31]

J. Rubio-Herrero, M. Baykal-Gursoy and A. Jaskiewicz, A price setting newsvendor problem under mean-variance criteria, Eur. J. Oper. Res., 247 (2015), 575-587. doi: 10.1016/j.ejor.2015.06.006.

[32]

Z. J. Shen and X. Su, Customer behavior modeling in revenue management and auctions: A review and new research opportunities, Production and Operations Management, 16 (2007), 713-728.

[33]

Y. Shi, X. Y. Cui, J. Yao and D. Li, Dynamic trading with reference point adaptation and loss aversion, Operations Research, 63 (2015), 789-806. doi: 10.1287/opre.2015.1399.

[34]

X. Su, Intertemporal pricing with strategic customer behavior, Management Science, 53 (2007), 726-741.

[35]

X. Su and F. Zhang, Strategic customer behavior, commitment, and supply chain performance, Management Science, 54 (2008), 1759-1773.

[36]

N. L. Stokey, Rational expectation and durable goods pricing, The Bell Journal of Economics, 12 (1981), 112-128. doi: 10.2307/3003511.

[37]

C. X. Wang, S. Webster and N. Surech, Would a risk-averse newsvendor order less at a highter selling price?, Eur. J. Oper. Res., 196 (2009), 544-553. doi: 10.1016/j.ejor.2008.04.002.

[38]

J. Wu, J. Li, S. Y. Wang and T. C. E. Cheng, Mean-variance analysis of the newsvendor model with stockout cost, Omega, 37 (2009), 724-730.

[39]

R. Yin, Y. Aviv, A. Pazgal and C. S. Tang, Optimal markdown pricing: Implications of inventory display formats in the presence of strategic customers, Management Science, 55 (2009), 1391-1408.

[40]

P. H. Zipkin, Foundations of Inventory Management, McGraw-Hill/Irwin, 2000.
Figure 1.  Two-stage decision model
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Figure 2.  Impact of customer's risk preference on the firm's decisions in different $\alpha$ and $\delta$
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Figure 3.  Effect of decreasing rate on the firm's decision variables in different values $\alpha$ and $\lambda$
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Figure 4.  The firm's expected profit changes with $\alpha$ and $\lambda$
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Figure 5.  The pattern of the firm's expected profit.
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Figure 6.  The pattern of the firm's expected profit
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Table 1.  Classification of literature on pricing and inventory control with strategic customers
Contributions Risk preference of Customers Decreasing value Loss aversion
Su & Zhang (2008) - - -
Liu & Van (2008) $\surd$ - -
Aviv & Pazgal (2008) - $\surd$ -
Du, Zhang & Hua (2015) $\surd$ $\surd$ -
This paper $\color{red}\surd$ $\color{red}\surd$ $\color{red}\surd$
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Contributions Risk preference of Customers Decreasing value Loss aversion
Su & Zhang (2008) - - -
Liu & Van (2008) $\surd$ - -
Aviv & Pazgal (2008) - $\surd$ -
Du, Zhang & Hua (2015) $\surd$ $\surd$ -
This paper $\color{red}\surd$ $\color{red}\surd$ $\color{red}\surd$
Table 2.  Parameters and notations
Notation Description
$p_{0}^{*}$, $p_{\alpha}^{*}$ and $p_{r}^{*}$ The full price of unit product in classical model, the model with strategic customers and the model with reimbursement contract, respectively in period 1
$Q_{0}^{*}$, $Q_{\alpha}^{*}$ and $Q_{r}^{*}$ The stocking quantity in classical model, the model with strategic customers and the model with reimbursement contract, respectively
$Q$, $p$ Decision variables denoting stocking quantity and full price, respectively
$D$ Nonnegative and independent random variable, which indicates customers' demand
$F(x)$ Cumulative distribution function, characterizing the demand, and tail distribution is $\overline{F}(x)]=1-F(x)$
$G(x)$ Partial expectation of random $D$, which is defined as $G(x)=\int_{0}^{x}Df(D)dD$
$s$ Salvage price in period 2
$c$ Unit procurement cost of the product to the firm
$V$ Customers' valuation for the unit production
$r$ Customers' reservation price or maximum price which the customers are willing to pay
$\xi_{r}$ The firm's belief over customers' reservation price
$\xi_{prob}$ Customers' belief from obtaining the product on the salvage market
$\delta$ The decreasing rate ($0<\delta\leq1$)
$\lambda$ Customers' risk preference ($\lambda>0$)
$\alpha$ The firm's loss aversion ($\alpha\geq1$)
$E(\cdot)$ Expectation operator
$U(\cdot)$ Utility function of the firm
$x^{+}$ and $x^{-}$ The maximum and minimal function between $0$ and $x$, respectively. $x^{+}=\max\{0, x\}$ and $x^{-}=\min\{0, x\}$
Table Options

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Notation Description
$p_{0}^{*}$, $p_{\alpha}^{*}$ and $p_{r}^{*}$ The full price of unit product in classical model, the model with strategic customers and the model with reimbursement contract, respectively in period 1
$Q_{0}^{*}$, $Q_{\alpha}^{*}$ and $Q_{r}^{*}$ The stocking quantity in classical model, the model with strategic customers and the model with reimbursement contract, respectively
$Q$, $p$ Decision variables denoting stocking quantity and full price, respectively
$D$ Nonnegative and independent random variable, which indicates customers' demand
$F(x)$ Cumulative distribution function, characterizing the demand, and tail distribution is $\overline{F}(x)]=1-F(x)$
$G(x)$ Partial expectation of random $D$, which is defined as $G(x)=\int_{0}^{x}Df(D)dD$
$s$ Salvage price in period 2
$c$ Unit procurement cost of the product to the firm
$V$ Customers' valuation for the unit production
$r$ Customers' reservation price or maximum price which the customers are willing to pay
$\xi_{r}$ The firm's belief over customers' reservation price
$\xi_{prob}$ Customers' belief from obtaining the product on the salvage market
$\delta$ The decreasing rate ($0<\delta\leq1$)
$\lambda$ Customers' risk preference ($\lambda>0$)
$\alpha$ The firm's loss aversion ($\alpha\geq1$)
$E(\cdot)$ Expectation operator
$U(\cdot)$ Utility function of the firm
$x^{+}$ and $x^{-}$ The maximum and minimal function between $0$ and $x$, respectively. $x^{+}=\max\{0, x\}$ and $x^{-}=\min\{0, x\}$
Table 3.  Numerical results for various systems of expected profit when $s = 2$ and $V = 15$
$\alpha=1$ $\alpha=2$ $\alpha=3$
$\delta=0.25$ $\delta=0.5$ $\delta=1$ $ \delta=0.25 $ $\delta=0.5$ $\delta=1$ $\delta=0.25$ $\delta=0.5$ $\delta=1$
465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 $\Pi_{0}^{*}$
$\lambda=0.5$ $\color{red} {\pmb {406.6204}} $ 298.9421 162.4174 $\color{red}{\pmb {408.1164}} $ 304.5730 175.2384 $\color{red}{\pmb {409.3841}} $ 309.4728 185.7115 $\Pi_{\alpha}^{*}$
$\color{green}{\pmb {403.3333}} $ 403.3333 403.3333 $\color{green}{\pmb {402.8578}} $ 402.8578 402.8578 $\color{green}{\pmb {401.5556}} $ 401.5556 401.5556 $\Pi_{p}^{*}$
465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 $\Pi_{0}^{*}$
$c=4$ $\lambda=1$ 394.9592 260.6531 94.1742 395.8801 264.8910 105.3170 396.6050 268.5129 113.9793 $\Pi_{\alpha}^{*}$
403.3333 403.3333 403.3333 402.8578 402.8578 402.8578 401.5556 401.5556 401.5556 $\Pi_{p}^{*}$
465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 $\Pi_{0}^{*}$
$\lambda=2$ 388.0647 235.1442 46.7179 388.5513 237.8978 54.9701 388.8571 240.1336 60.9781 $\Pi_{\alpha}^{*}$
403.3333 403.3333 403.3333 402.8578 402.8578 402.8578 401.5556 401.5556 401.5556 $\Pi_{p}^{*}$
384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 $\Pi_{0}^{*}$
$\lambda=0.5$ $\color{red}{\pmb {338.1945}} $ 256.3412 154.3494 $\color{red}{\pmb {340.4831}} $ 264.0884 169.3020 $\color{red}{\pmb {341.9950}} $ 270.0823 180.6548 $\Pi_{\alpha}^{*}$
$\color{green}{\pmb {333.3333}} $ 333.3333 333.3333 $\color{green}{\pmb {332.4099}} $ 332.4099 332.4099 $\color{green}{\pmb {330}} $ 330 330 $\Pi_{p}^{*}$
384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 $\Pi_{0}^{*}$
$c=5$ $\lambda=1$ 323.4039 212.4169 84.3076 324.9323 218.7253 96.7836 325.7676 223.4203 105.7881 $\Pi_{\alpha}^{*}$
333.3333 333.3333 333.3333 332.4099 332.4099 332.4099 330 330 330 $\Pi_{p}^{*}$
384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 $\Pi_{0}^{*}$
$\lambda=2$ 313.9245 180.8531 36.9033 314.7191 185.1787 45.1564 314.8643 188.0670 50.7402 $\Pi_{\alpha}^{*}$
333.3333 333.3333 333.3333 332.4099 332.4099 332.4099 330 330 330 $\Pi_{p}^{*}$
311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 $\Pi_{0}^{*}$
$\lambda=0.5$ $\color{red}{\pmb {275.9692}} $ 214.7582 138.5999 $\color{red}{\pmb {278.5443}} $ 223.2811 153.8583 $\color{red}{\pmb {279.5104}} $ 228.8967 164.4633 $\Pi_{\alpha}^{*}$
$\color{green}{\pmb {270}} $ 270 270 $\color{green}{\pmb {268.5185}} $ 268.5185 268.5185 $\color{green}{\pmb {264.8633}} $ 264.8633 264.8633 $\Pi_{p}^{*}$
311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 $\Pi_{0}^{*}$
$c=6$ $\lambda=1$ 259.6189 169.7678 71.4952 261.4923 177.1413 83.8399 261.8907 181.7852 92.0999 $\Pi_{\alpha}^{*}$
270 270 270 268.5185 268.5185 268.5185 264.8633 264.8633 264.8633 $\Pi_{p}^{*}$
311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 $\Pi_{0}^{*}$
$\lambda=2$ 248.2615 135.5270 27.4710 249.1956 140.7354 34.7699 248.7501 143.5081 39.4236 $\Pi_{\alpha}^{*}$
270 270 270 268.5185 268.5185 268.5185 264.8633 264.8633 264.8633 $\Pi_{p}^{*}$
96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 $\Pi_{0}^{*}$
$\lambda=0.5$ $\color{red}{\pmb {32.8174}} $ 29.5585 24.7304 $\color{red}{\pmb {31.7544}} $ $\color{red}{\pmb {29.7348}} $ 26.4622 $\color{red}{\pmb {29.1689}} $ $\color{red}{\pmb {27.8619}} $ $\color{red}{\pmb {25.6263}} $ $\Pi_{\alpha}^{*}$
$\color{green}{\pmb {30}} $ 30 30 $\color{green}{\pmb {28.2369}} $ $\color{green}{\pmb {28.2369}} $ 28.2369 $\color{green}{\pmb {25.4313}} $ $\color{green}{\pmb {25.4313}} $ $\color{green}{\pmb {25.4313}} $ $\Pi_{p}^{*}$
96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 $\Pi_{0}^{*}$
$c=12$ $\lambda=1$ 27.4769 17.5997 8.61672 27.3621 19.2783 10.8693 $\color{red}{\pmb {25.6164}} $ 19.0687 11.674 $\Pi_{\alpha}^{*}$
30 30 30 28.2369 28.2369 28.2369 $\color{green}{\pmb {25.4313}} $ 25.4313 25.4313 $\Pi_{p}^{*}$
96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 $\Pi_{0}^{*}$
$\lambda=2$ 20.6633 6.42676 0.8171 20.8133 7.6716 1.2734 19.6287 7.9764 1.5628 $\Pi_{\alpha}^{*}$
30 30 30 28.2369 28.2369 28.2369 25.4313 25.4313 25.4313 $\Pi_{p}^{*}$
Note: The expected profits are the classical inventory model, the proposed model and the model under price commitment strategy in turn.
We mark by red and green color when the expected profit of our model is larger than that of under price commitment strategy model.
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$\alpha=1$ $\alpha=2$ $\alpha=3$
$\delta=0.25$ $\delta=0.5$ $\delta=1$ $ \delta=0.25 $ $\delta=0.5$ $\delta=1$ $\delta=0.25$ $\delta=0.5$ $\delta=1$
465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 $\Pi_{0}^{*}$
$\lambda=0.5$ $\color{red} {\pmb {406.6204}} $ 298.9421 162.4174 $\color{red}{\pmb {408.1164}} $ 304.5730 175.2384 $\color{red}{\pmb {409.3841}} $ 309.4728 185.7115 $\Pi_{\alpha}^{*}$
$\color{green}{\pmb {403.3333}} $ 403.3333 403.3333 $\color{green}{\pmb {402.8578}} $ 402.8578 402.8578 $\color{green}{\pmb {401.5556}} $ 401.5556 401.5556 $\Pi_{p}^{*}$
465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 $\Pi_{0}^{*}$
$c=4$ $\lambda=1$ 394.9592 260.6531 94.1742 395.8801 264.8910 105.3170 396.6050 268.5129 113.9793 $\Pi_{\alpha}^{*}$
403.3333 403.3333 403.3333 402.8578 402.8578 402.8578 401.5556 401.5556 401.5556 $\Pi_{p}^{*}$
465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 465.3846 $\Pi_{0}^{*}$
$\lambda=2$ 388.0647 235.1442 46.7179 388.5513 237.8978 54.9701 388.8571 240.1336 60.9781 $\Pi_{\alpha}^{*}$
403.3333 403.3333 403.3333 402.8578 402.8578 402.8578 401.5556 401.5556 401.5556 $\Pi_{p}^{*}$
384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 $\Pi_{0}^{*}$
$\lambda=0.5$ $\color{red}{\pmb {338.1945}} $ 256.3412 154.3494 $\color{red}{\pmb {340.4831}} $ 264.0884 169.3020 $\color{red}{\pmb {341.9950}} $ 270.0823 180.6548 $\Pi_{\alpha}^{*}$
$\color{green}{\pmb {333.3333}} $ 333.3333 333.3333 $\color{green}{\pmb {332.4099}} $ 332.4099 332.4099 $\color{green}{\pmb {330}} $ 330 330 $\Pi_{p}^{*}$
384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 $\Pi_{0}^{*}$
$c=5$ $\lambda=1$ 323.4039 212.4169 84.3076 324.9323 218.7253 96.7836 325.7676 223.4203 105.7881 $\Pi_{\alpha}^{*}$
333.3333 333.3333 333.3333 332.4099 332.4099 332.4099 330 330 330 $\Pi_{p}^{*}$
384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 384.6154 $\Pi_{0}^{*}$
$\lambda=2$ 313.9245 180.8531 36.9033 314.7191 185.1787 45.1564 314.8643 188.0670 50.7402 $\Pi_{\alpha}^{*}$
333.3333 333.3333 333.3333 332.4099 332.4099 332.4099 330 330 330 $\Pi_{p}^{*}$
311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 $\Pi_{0}^{*}$
$\lambda=0.5$ $\color{red}{\pmb {275.9692}} $ 214.7582 138.5999 $\color{red}{\pmb {278.5443}} $ 223.2811 153.8583 $\color{red}{\pmb {279.5104}} $ 228.8967 164.4633 $\Pi_{\alpha}^{*}$
$\color{green}{\pmb {270}} $ 270 270 $\color{green}{\pmb {268.5185}} $ 268.5185 268.5185 $\color{green}{\pmb {264.8633}} $ 264.8633 264.8633 $\Pi_{p}^{*}$
311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 $\Pi_{0}^{*}$
$c=6$ $\lambda=1$ 259.6189 169.7678 71.4952 261.4923 177.1413 83.8399 261.8907 181.7852 92.0999 $\Pi_{\alpha}^{*}$
270 270 270 268.5185 268.5185 268.5185 264.8633 264.8633 264.8633 $\Pi_{p}^{*}$
311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 311.5384 $\Pi_{0}^{*}$
$\lambda=2$ 248.2615 135.5270 27.4710 249.1956 140.7354 34.7699 248.7501 143.5081 39.4236 $\Pi_{\alpha}^{*}$
270 270 270 268.5185 268.5185 268.5185 264.8633 264.8633 264.8633 $\Pi_{p}^{*}$
96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 $\Pi_{0}^{*}$
$\lambda=0.5$ $\color{red}{\pmb {32.8174}} $ 29.5585 24.7304 $\color{red}{\pmb {31.7544}} $ $\color{red}{\pmb {29.7348}} $ 26.4622 $\color{red}{\pmb {29.1689}} $ $\color{red}{\pmb {27.8619}} $ $\color{red}{\pmb {25.6263}} $ $\Pi_{\alpha}^{*}$
$\color{green}{\pmb {30}} $ 30 30 $\color{green}{\pmb {28.2369}} $ $\color{green}{\pmb {28.2369}} $ 28.2369 $\color{green}{\pmb {25.4313}} $ $\color{green}{\pmb {25.4313}} $ $\color{green}{\pmb {25.4313}} $ $\Pi_{p}^{*}$
96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 $\Pi_{0}^{*}$
$c=12$ $\lambda=1$ 27.4769 17.5997 8.61672 27.3621 19.2783 10.8693 $\color{red}{\pmb {25.6164}} $ 19.0687 11.674 $\Pi_{\alpha}^{*}$
30 30 30 28.2369 28.2369 28.2369 $\color{green}{\pmb {25.4313}} $ 25.4313 25.4313 $\Pi_{p}^{*}$
96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 96.1538 $\Pi_{0}^{*}$
$\lambda=2$ 20.6633 6.42676 0.8171 20.8133 7.6716 1.2734 19.6287 7.9764 1.5628 $\Pi_{\alpha}^{*}$
30 30 30 28.2369 28.2369 28.2369 25.4313 25.4313 25.4313 $\Pi_{p}^{*}$
Note: The expected profits are the classical inventory model, the proposed model and the model under price commitment strategy in turn.
We mark by red and green color when the expected profit of our model is larger than that of under price commitment strategy model.
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