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July  2018, 14(3): 931-951. doi: 10.3934/jimo.2017083

## Analysis of the Newsboy Problem subject to price dependent demand and multiple discounts

 1 School of business administration, Zhongnan university of economics and law, 182 Nanhu Avenue, East Lake High-tech Development Zone, Wuhan 430073, China 2 OASIS -ENIT, University of Tunis El Manar, BP 37, LE BELVEDERE 1002 TUNIS, Tunisia 3 LGI, Centrale Supelec, Paris Saclay University, Grande Voie des Vignes, 92295 CHATNAY-MALABRY CEDEX, France

* Corresponding author: Shouyu Ma

Received  March 2016 Revised  August 2017 Published  July 2018 Early access  September 2017

Fund Project: The first author is supported by the China Scholarship Council.

Existing papers on the Newsboy Problem that deal with price dependent demand and multiple discounts often analyze those two problems separately. This paper considers a setting where price dependence and multiple discounts are observed simultaneously, as is the case of the apparel industry. Henceforth, we analyze the optimal order quantity, initial selling price and discount scheme in the News-Vendor Problem context. The term of discount scheme is often used to specify the number of discounts as well as the discount percentages. We present a solution procedure of the problem with general demand distributions and two types of price-dependent demand: additive case and multiplicative case. We provide interesting insights based on a numerical study. An approximation method is proposed which confirms our numerical results.

Citation: Shouyu Ma, Zied Jemai, Evren Sahin, Yves Dallery. Analysis of the Newsboy Problem subject to price dependent demand and multiple discounts. Journal of Industrial & Management Optimization, 2018, 14 (3) : 931-951. doi: 10.3934/jimo.2017083
##### References:

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##### References:
sequence of events for a selling season
Expected profit $E(\pi(Q^{*}))$, as a function of the discount number, for normally distributed demand
Expected profit $E(\pi(Q^{*}))$, as a function of the intial price
discount schemes
The value of ($E(\pi(Q^{*}))-E_\sigma$), as a function of discount number, with normal distribution
The value of ($E(\pi(Q^{*}))-E_\sigma$), as a function of discount number, with uniform distribution
Expected profit as function of discount number n
Discount percentages at $v_0=6$ for different schemes
Expected profit as function of initial price
Comparison with the work of Khouja(1995, 2000)
 parameter price-demand relation demand distribution discount prices [6] fixed general known [8] additive uniform and normal linear our paper additive and multiplicative general all types
 parameter price-demand relation demand distribution discount prices [6] fixed general known [8] additive uniform and normal linear our paper additive and multiplicative general all types
The optimal order initial price, order quantity and expected profit for different combinations of n, b, $\sigma_0$ for normally distributed demand
 test n b $\sigma_0$ $v^*_{0}$ $Q^*$ $E(\pi(Q^*, v_0^*))$ 1 4 6 2 10.20 55.8 249.0 2 4 6 4 10.18 55.9 246.9 3 4 6 6 10.24 56.1 245.0 4 4 6 8 10.23 56.9 243.4 5 4 8 2 8.54 50.4 153.3 6 4 8 4 8.58 49.8 151.6 7 4 8 6 8.59 49.6 150.2 8 4 8 8 8.57 50.0 148.6 9 4 10 2 6.60 46.3 95.0 10 4 10 4 6.64 44.5 94.3 11 4 10 6 6.64 44.3 93.6 12 4 10 8 6.61 44.6 92.2 13 5 6 2 11.41 56.6 263.9 14 5 6 4 11.51 56.4 262.0 15 5 6 6 11.47 56.7 260.2 16 5 6 8 11.54 57.4 258.2 17 5 8 2 8.81 51.9 159.8 18 5 8 4 8.71 50.9 158.6 19 5 8 6 8.75 50.8 157.4 20 5 8 8 8.81 51.2 155.8 21 5 10 2 7.09 45.7 100.1 22 5 10 4 7.06 45.0 99.8 23 5 10 6 7.01 45.1 98.8 24 5 10 8 7.09 45.3 97.6 25 6 6 2 11.90 57.6 271.5 26 6 6 4 11.90 57.2 270.0 27 6 6 6 11.88 57.5 268.3 28 6 6 8 12.0 58.2 266.3 29 6 8 2 8.91 52.6 164.5 30 6 8 4 8.91 51.5 163.7 31 6 8 6 8.94 51.6 162.6 32 6 8 8 8.91 52.1 161.0 33 6 10 2 7.16 44.8 103.8 34 6 10 4 7.18 45.7 103.3 35 6 10 6 7.19 45.8 102.3 36 6 10 8 7.18 46.1 100.0
 test n b $\sigma_0$ $v^*_{0}$ $Q^*$ $E(\pi(Q^*, v_0^*))$ 1 4 6 2 10.20 55.8 249.0 2 4 6 4 10.18 55.9 246.9 3 4 6 6 10.24 56.1 245.0 4 4 6 8 10.23 56.9 243.4 5 4 8 2 8.54 50.4 153.3 6 4 8 4 8.58 49.8 151.6 7 4 8 6 8.59 49.6 150.2 8 4 8 8 8.57 50.0 148.6 9 4 10 2 6.60 46.3 95.0 10 4 10 4 6.64 44.5 94.3 11 4 10 6 6.64 44.3 93.6 12 4 10 8 6.61 44.6 92.2 13 5 6 2 11.41 56.6 263.9 14 5 6 4 11.51 56.4 262.0 15 5 6 6 11.47 56.7 260.2 16 5 6 8 11.54 57.4 258.2 17 5 8 2 8.81 51.9 159.8 18 5 8 4 8.71 50.9 158.6 19 5 8 6 8.75 50.8 157.4 20 5 8 8 8.81 51.2 155.8 21 5 10 2 7.09 45.7 100.1 22 5 10 4 7.06 45.0 99.8 23 5 10 6 7.01 45.1 98.8 24 5 10 8 7.09 45.3 97.6 25 6 6 2 11.90 57.6 271.5 26 6 6 4 11.90 57.2 270.0 27 6 6 6 11.88 57.5 268.3 28 6 6 8 12.0 58.2 266.3 29 6 8 2 8.91 52.6 164.5 30 6 8 4 8.91 51.5 163.7 31 6 8 6 8.94 51.6 162.6 32 6 8 8 8.91 52.1 161.0 33 6 10 2 7.16 44.8 103.8 34 6 10 4 7.18 45.7 103.3 35 6 10 6 7.19 45.8 102.3 36 6 10 8 7.18 46.1 100.0
Optimal epected profit for different discount schemes
 scheme coe optimal expected profit linear 0 158.5 1 -0.03 144.9 2 -0.02 151.1 3 -0.01 155.8 4 0.01 159.1 5 0.02 157.8 6 0.03 153.4
 scheme coe optimal expected profit linear 0 158.5 1 -0.03 144.9 2 -0.02 151.1 3 -0.01 155.8 4 0.01 159.1 5 0.02 157.8 6 0.03 153.4
Expected profit function for uniform and normal distributions
 Distribution $U[\mu_0-\sigma_0, \mu_0+\sigma_0]$ $N(\mu_0, \sigma_0)$ Condition for $\epsilon=0$ $\forall j, \sigma_0\leq \frac{\mu_{j}-\mu_{j-1}}{2}$ $\forall j, \sigma_0\leq \frac{\mu_{j}-\mu_{j-1}}{4}$ $E(\pi(Q^*))$ $E_\sigma+E_v$ $E_\sigma+E_v$ $E(\pi(Q^*))$ for linear case equation 4.11 equation 4.11 $E_v$ equation 4.8 equation 4.8 $E_\sigma$ equation 4.9 equation 4.10
 Distribution $U[\mu_0-\sigma_0, \mu_0+\sigma_0]$ $N(\mu_0, \sigma_0)$ Condition for $\epsilon=0$ $\forall j, \sigma_0\leq \frac{\mu_{j}-\mu_{j-1}}{2}$ $\forall j, \sigma_0\leq \frac{\mu_{j}-\mu_{j-1}}{4}$ $E(\pi(Q^*))$ $E_\sigma+E_v$ $E_\sigma+E_v$ $E(\pi(Q^*))$ for linear case equation 4.11 equation 4.11 $E_v$ equation 4.8 equation 4.8 $E_\sigma$ equation 4.9 equation 4.10
Expected profit function for uniform and normal distributions
 Distribution $U[\mu_0-\sigma_0, \mu_0+\sigma_0]$ $N(\mu_0, \sigma_0)$ Condition that $\epsilon=0$ $\forall j, \sigma_0\leq\frac{\mu_{j}-\mu_{j-1}}{2}$ $\forall j, \sigma_0\leq\frac{\mu_{j}-\mu_{j-1}}{4}$ $E(\pi(Q^*))$ $E_\sigma+E_v$ $E_\sigma+E_v$ Exponential case equation 5.8 equation 5.8 $E_v$ equation 5.5 equation 5.5 $E_\sigma$ equation 5.6 equation 5.7
 Distribution $U[\mu_0-\sigma_0, \mu_0+\sigma_0]$ $N(\mu_0, \sigma_0)$ Condition that $\epsilon=0$ $\forall j, \sigma_0\leq\frac{\mu_{j}-\mu_{j-1}}{2}$ $\forall j, \sigma_0\leq\frac{\mu_{j}-\mu_{j-1}}{4}$ $E(\pi(Q^*))$ $E_\sigma+E_v$ $E_\sigma+E_v$ Exponential case equation 5.8 equation 5.8 $E_v$ equation 5.5 equation 5.5 $E_\sigma$ equation 5.6 equation 5.7
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