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

Shouyu Ma; Zied Jemai; Evren Sahin; Yves Dallery

doi:10.3934/jimo.2017083

Article Contents

2018, Volume 14, Issue 3: 931-951. Doi: 10.3934/jimo.2017083

This issue Previous Article Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems Next Article Solutions for bargaining games with incomplete information: General type space and action space

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
* Corresponding author: Shouyu Ma

Received: February 29, 2016

Revised: July 31, 2017

Early access: August 2017

Published: June 2018

The first author is supported by the China Scholarship Council.

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Abstract

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:

Mathematics Subject Classification: Primary: 90B05; Secondary: 90B50.

Citation:

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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)

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

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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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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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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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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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