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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

Shouyu Ma 1,, , Zied Jemai 2, , Evren Sahin 3, and  Yves Dallery 3,

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.

Keywords: Inventory control, Newsboy Problem, price dependent demand, multiple discounts.
Mathematics Subject Classification: Primary: 90B05; Secondary: 90B50.
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 and Management Optimization, 2018, 14 (3) : 931-951. doi: 10.3934/jimo.2017083
References:
[1]

F. J. Arcelus, S. Kumar and G. Srinivasan, Channel coordination with manufacturer's return policies within a newsvendor framework, 4OR, 9 (2011), 279-297.  doi: 10.1007/s10288-011-0160-1.

[2]

F. Y. Chen, H. Yan and L. Yao, A newsvendor pricing game, IEEE Transactions on Systems, Man, and Cybernetics, 34 (2004), 450-456.  doi: 10.1109/TSMCA.2004.826290.

[3]

W. Chung, S. Talluri and R. Narasimhan, Optimal pricing and inventory strategies with multiple price markdowns over time, European Journal of Operational Research, 243 (2014), 130-141.  doi: 10.1016/j.ejor.2014.11.020.

[4]

G. Gallego and I. Moon, The distribution free newsboy problem: review and extensions, The Journal of the Operational Research Society, 44 (1993), 825-834. 

[5]

S. Karlin and C. R. Carr, Prices and Optimal Inventory Policy Studies in Applied Probability and Management Science. Stanford University Press, 1962.

[6]

M. Khouja, The newsboy problem under progressive multiple discounts, European Journal of Operational Research, 84 (1995), 458-466.  doi: 10.1016/0377-2217(94)00053-F.

[7]

M. Khouja, The newsboy problem with progressive retailer discounts and supplier quantity discounts, Decision Sciences, 27 (1996), 589-599. 

[8]

M. Khouja, Optimal ordering, discounting, and pricing in the single-period problem, International Jounal of Production Economics, 65 (2000), 201-216.  doi: 10.1016/S0925-5273(99)00027-4.

[9]

M. Khouja and A. Mehrez, A multi-product constrained newsboy problem with progressive multiple discounts, Computers and Industrial Engineering, 30 (1996), 95-101.  doi: 10.1016/0360-8352(95)00025-9.

[10]

A. Lau and H. Lau, The newsboy problem with price-dependent demand distribution, IIE Transactions, 20 (1998), 168-175.  doi: 10.1080/07408178808966166.

[11]

E. S. Mills, Uncertainty and price theory, the Quarterly Journal of Economics, 73 (1959), 116-130.  doi: 10.2307/1883828.

[12]

L. H. Polatoglu, Optimal order quantity and pricing decisions in single-period inventory systems, International Journal of Production Economics, 23 (1991), 175-185.  doi: 10.1016/0925-5273(91)90060-7.

[13]

Y. Qin, R. Wang, A.J. Vakharia, Y. Chen and M.M. H. Seref, The newsvendor problem: Review and directions for future research, European Journal of Operational Research, 213 (2011), 361-374.  doi: 10.1016/j.ejor.2010.11.024.

[14]

S.A. Raza, A distribution free approach to newsvendor problem with pricing, 4OR, 12 (2014), 335-358.  doi: 10.1007/s10288-013-0249-9.

[15]

S.S. Sana, Price sensitive demand with random sales price-a newsboy problem, International Journal of Systems Science, 43 (2012), 491-498.  doi: 10.1080/00207721.2010.517856.

[16]

K.-H. Wang and C.-T. Tung, Construction of a model towards {EOQ} and pricing strategy for gradually obsolescent products, Applied Mathematics and Computation, 217 (2011), 6926-6933.  doi: 10.1016/j.amc.2011.01.100.

[17]

L. R. Weatherford and P. E. Pfeifer, The economic value of using advance booking of orders, Omega, 22 (1994), 105-111.  doi: 10.1016/0305-0483(94)90011-6.

[18]

H. Yu and J. Zhai, The distribution-free newsvendor problem with balking and penalties for balking and stockout, Journal of Systems Science and Systems Engineering, 23 (2014), 153-175.  doi: 10.1007/s11518-014-5246-9.

[19]

Y. Zhang, X. Yang and B. Li, Distribution-free solutions to the extended multi-period newsboy problem, Journal of Industrial and Management Optimization, 13 (2017), 633-647.  doi: 10.3934/jimo.2016037.

show all references

References:
[1]

F. J. Arcelus, S. Kumar and G. Srinivasan, Channel coordination with manufacturer's return policies within a newsvendor framework, 4OR, 9 (2011), 279-297.  doi: 10.1007/s10288-011-0160-1.

[2]

F. Y. Chen, H. Yan and L. Yao, A newsvendor pricing game, IEEE Transactions on Systems, Man, and Cybernetics, 34 (2004), 450-456.  doi: 10.1109/TSMCA.2004.826290.

[3]

W. Chung, S. Talluri and R. Narasimhan, Optimal pricing and inventory strategies with multiple price markdowns over time, European Journal of Operational Research, 243 (2014), 130-141.  doi: 10.1016/j.ejor.2014.11.020.

[4]

G. Gallego and I. Moon, The distribution free newsboy problem: review and extensions, The Journal of the Operational Research Society, 44 (1993), 825-834. 

[5]

S. Karlin and C. R. Carr, Prices and Optimal Inventory Policy Studies in Applied Probability and Management Science. Stanford University Press, 1962.

[6]

M. Khouja, The newsboy problem under progressive multiple discounts, European Journal of Operational Research, 84 (1995), 458-466.  doi: 10.1016/0377-2217(94)00053-F.

[7]

M. Khouja, The newsboy problem with progressive retailer discounts and supplier quantity discounts, Decision Sciences, 27 (1996), 589-599. 

[8]

M. Khouja, Optimal ordering, discounting, and pricing in the single-period problem, International Jounal of Production Economics, 65 (2000), 201-216.  doi: 10.1016/S0925-5273(99)00027-4.

[9]

M. Khouja and A. Mehrez, A multi-product constrained newsboy problem with progressive multiple discounts, Computers and Industrial Engineering, 30 (1996), 95-101.  doi: 10.1016/0360-8352(95)00025-9.

[10]

A. Lau and H. Lau, The newsboy problem with price-dependent demand distribution, IIE Transactions, 20 (1998), 168-175.  doi: 10.1080/07408178808966166.

[11]

E. S. Mills, Uncertainty and price theory, the Quarterly Journal of Economics, 73 (1959), 116-130.  doi: 10.2307/1883828.

[12]

L. H. Polatoglu, Optimal order quantity and pricing decisions in single-period inventory systems, International Journal of Production Economics, 23 (1991), 175-185.  doi: 10.1016/0925-5273(91)90060-7.

[13]

Y. Qin, R. Wang, A.J. Vakharia, Y. Chen and M.M. H. Seref, The newsvendor problem: Review and directions for future research, European Journal of Operational Research, 213 (2011), 361-374.  doi: 10.1016/j.ejor.2010.11.024.

[14]

S.A. Raza, A distribution free approach to newsvendor problem with pricing, 4OR, 12 (2014), 335-358.  doi: 10.1007/s10288-013-0249-9.

[15]

S.S. Sana, Price sensitive demand with random sales price-a newsboy problem, International Journal of Systems Science, 43 (2012), 491-498.  doi: 10.1080/00207721.2010.517856.

[16]

K.-H. Wang and C.-T. Tung, Construction of a model towards {EOQ} and pricing strategy for gradually obsolescent products, Applied Mathematics and Computation, 217 (2011), 6926-6933.  doi: 10.1016/j.amc.2011.01.100.

[17]

L. R. Weatherford and P. E. Pfeifer, The economic value of using advance booking of orders, Omega, 22 (1994), 105-111.  doi: 10.1016/0305-0483(94)90011-6.

[18]

H. Yu and J. Zhai, The distribution-free newsvendor problem with balking and penalties for balking and stockout, Journal of Systems Science and Systems Engineering, 23 (2014), 153-175.  doi: 10.1007/s11518-014-5246-9.

[19]

Y. Zhang, X. Yang and B. Li, Distribution-free solutions to the extended multi-period newsboy problem, Journal of Industrial and Management Optimization, 13 (2017), 633-647.  doi: 10.3934/jimo.2016037.

Figure 1.  sequence of events for a selling season
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Figure 2.  Expected profit $E(\pi(Q^{*}))$, as a function of the discount number, for normally distributed demand
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Figure 3.  Expected profit $E(\pi(Q^{*}))$, as a function of the intial price
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Figure 4.  discount schemes
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Figure 5.  The value of ($E(\pi(Q^{*}))-E_\sigma$), as a function of discount number, with normal distribution
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Figure 6.  The value of ($E(\pi(Q^{*}))-E_\sigma$), as a function of discount number, with uniform distribution
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Figure 7.  Expected profit as function of discount number n
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Figure 8.  Discount percentages at $v_0=6$ for different schemes
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Figure 9.  Expected profit as function of initial price
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Table 1.  Comparison with the work of Khouja(1995, 2000)
parameterprice-demand relation demand distribution discount prices
[6] fixed general known
[8] additive uniform and normal linear
our paper additive and multiplicative general all types
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parameterprice-demand relation demand distribution discount prices
[6] fixed general known
[8] additive uniform and normal linear
our paper additive and multiplicative general all types
Table 2.  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
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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
Table 3.  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
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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
Table 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
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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
Table 5.  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
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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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