Interdependent demand in the two-period newsvendor problem

Reza Lotfi; Gerhard-Wilhelm Weber; S. Mehdi Sajadifar; Nooshin Mardani

doi:10.3934/jimo.2018143

Article Contents

2020, Volume 16, Issue 1: 117-140. Doi: 10.3934/jimo.2018143

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Interdependent demand in the two-period newsvendor problem

1.
Department of Industrial Engineering, Yazd University, Yazd, Iran
2.
Poznan University of Technology, Faculty of Engineering, Management, Poznan, Poland, IAM, METU, Ankara, Turkey
3.
Department of Industrial Engineering, University of Science and Culture, Tehran, Iran
4.
Department of Environment, College of Agriculture, Takestan Branch, Islamic Azad University, Takestan, Iran

* Corresponding author:Rezalotfi@stu.yazd.ac.ir
* Corresponding author:Rezalotfi@stu.yazd.ac.ir

Received: March 2017

Revised: May 2018

Early access: September 2018

Published: October 2019

Abstract / Introduction Full Text(HTML) Figure(5) / Table(7) Related Papers Cited by

Abstract

The newsvendor problem is a classical task in inventory management. The present paper considers a two-period newsvendor problem where demand of different periods is interdependent (not independent), and seeks to follow this approach to develop a two-period newsvendor problem with unsatisfied demand or unsold quantity. Concerning the complexity of solution of multiple integrals, the problem is assessed for only two periods. In the course of a numerical solution, the probability distribution function of demand pertaining to each period is assumed to be given (in the form of a bivariate normal distribution). The optimal solution is presented in the form of the initial inventory level that maximizes the expected profit. Finally, all model parameters are subjected to a sensitivity analysis. This model can be used in a number of applications, such as procurement of raw materials in projects (e.g., construction, bridge-building and molding) where demand of different periods is interdependent. Proposed model takes into account interdependent demand oughts to provide a better solution than a model based on independent demand.

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Mathematics Subject Classification: 90B05.

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Figure 1. Process of implementation of the conceptual model

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Figure 2. Chart of Differences between the proposed model and Matsuyama [17]

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Figure 3. Chart of the ratio ($\alpha $)

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Figure 4. Chart of the ratio ($\beta $)

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Figure 5. Chart of the ratio ($\delta $)

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Table 1. Classification of the literature

Reference	Fuzzy	Single-period	Multi-period	Multi-product	Risk	Demand	Product	Market	Discount
Bouakiz and Sobel [5]			1		1	Independent
Perakis and Sood [24]			1			Independent	Perishable	Competitive
Matsuyama [17]			1			Independent
Mileff and Nehéz [18]			1	1		Independent
Burnetas et al. [6]		1				Independent			Incremental
Altintas et al. [2]			1			Independent			All-Unit
Wang and Webster [29]		1			1	Independent
Behret and Kahraman [4]	1		1			Independent
Chen and Ho [8]	1	1				Independent
Zhang [32]				1		Independent			All-Units
Zhang and Du [30]				1		Independent
Zhang and Hua [31]				1		Independent
Huang et al. [13]				1		Independent
Sana [27]		1				Independent
Chen and Ho [9]	1	1				Independent
Ray and Jenamani [26]		1			1	Independent
Ding and Gao [10]				1		Independent
Kamburowski [14]		1				Independent
Kamburowski [14]		1				Independent
Pal and Sana [22]		1				Independent
Hanasusanto et al. [12]				1	1	Interdependent product
Alwan et al. [3]		1				Interdependent
Summary	3	10	6	7	4	20 Independent 2 Interdependent	1 Perishable	1 Competitive	1 Incremental 2 All-Unit
The present study			1			Interdependent Demand

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Table 2. Conceptual Model

Description	Period $j$
Status of demand	$L\leq x_j\leq l_j\leq N$	$L\leq l_j\leq x_j\leq N$
Sale income	$q_jx_j$	$q_jl_j$
Buying cost	$p_jl_j$	$p_jl_j$
Unsold	$\left(l_j-x_j\right)$	0
Stocked amount	$\alpha (l_j-x_j)$	0
Holding cost of amount unsold	${{s}}_j\alpha (l_j -x_j )$	0
Unsatisfied demand	0	$\beta \left({x_j -{l}}_j\right)$
Penalty for unsatisfied demand	0	$\pi (x_j -l_j )$
Order of period j+1	$l_{j+1}-\alpha (l_j -x_j )$	$l_{j+1}+\beta ({x_j-l}_j)$

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Table 3. Differences between the proposed model and [17]

Problem	Expected Profit ($H^$) of Proposed Model Corrolation = -0.5*	Expected Profit of Matsuyama [17] Correlation = 0	Gap
P1	866.59	790.38	8.79%
P2	2173.1	1983	8.75%
P3	3486.3	3181.7	8.74%
P4	4801.6	4382.3	8.73%
P5	5459.7	4983	8.73%
P6	6118	5583.9	8.73%
		Mean(Gap)	8.75%
		Variance(Gap)	0.0000061%

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Table 4. Sensitivity analysis of the proposed model

Parameter	Result of differentiation	Proof
$\alpha $	$\frac{\partial }{\partial \alpha }H\left(l_1, l_2, x_1, x_2\right)=\frac{p_2-s_1}{\left\|p_2-s_1\right\|}$	Appendix 3 Proof 2
$\beta $	$\frac{\partial }{\partial \beta }H\left(l_1, l_2, x_1, x_2\right)=\frac{\left(\delta q_1+\left(1-\delta \right)q_2\right)-p_2}{\left\|\left(\delta q_1+\left(1-\delta \right)q_2\right)-p_2\right\|}$	Appendix 3 Proof 2
$\delta $	$\frac{\partial }{\partial \delta }H\left(l_1, l_2, x_1, x_2\right)=\frac{q_1-q_2}{\left\|q_1-q_2\right\|}$	Appendix 3 Proof 2
$\pi $	$\frac{\partial }{\partial \pi }H\left(l_1, l_2, x_1, x_2\right)< 0, \;\;\;\;\forall \pi $	Appendix 3 Proof 2
$q_1$	$\frac{\partial }{\partial q_1}H\left(l_1, l_2, x_1, x_2\right)>0 \;\;\;\; \forall \ q_1$	Appendix 3 Proof 3
$q_2$	$\frac{\partial }{\partial q_2}H\left(l_1, l_2, x_1, x_2\right)>0 \;\;\;\; \forall \ q_2$	Appendix 3 Proof 3
$p_1$	$\frac{\partial }{\partial p_1}H\left(l_1, l_2, x_1, x_2\right)<0, \;\;\;\; \forall \ p_1\ \ \ \ \ $	Appendix 3 Proof 4
$p_2$	$\frac{\partial }{\partial p_2}H\left(l_1, l_2, x_1, x_2\right) =-l_2+\alpha \left(l_1-\mu _1\right)$ $+\left(\alpha -\beta \right)\int^{\infty }_{ -\infty }{\int^{\infty }_{l_1}{\left({x_1-l}_1\right)f\left(x_1, x_2\right)dx_1dx_2}}$	Appendix 3 Proof 4

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Table 5. Sensitivity analysis on expected profit ($H$) of the ratio ($0 \leq \alpha \leq 1$)

$\alpha $	$H^*$	$l^*_1$	$l^*_2$
20%	861.47	190.73	240.88
40%	862.27	195.58	240.88
60%	863.3	201.73	240.88
80%	864.65	209.82	240.88
100%	866.59	220.99	240.88

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Table 6. Sensitivity analysis on expected profit ($H$) of the ratio ($0 \leq \beta \leq 1$)

$\beta $	$H^*$	$l^*_1$	$l^*_2$
20%	858.4	270.2	240.88
40%	859.07	266.14	240.88
60%	860.13	259.76	240.88
80%	862.04	248.29	240.88
100%	866.59	220.99	240.88

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Table 7. Sensitivity analysis on expected profit ($H$) of the ratio ($0 \leq \delta \leq 1$)

$\delta $	$H^*$	$l^*_1$	$l^*_2$
0%	1532.1	271.1	240.88
20%	1532.5	274.55	240.88
40%	1533.1	271.1	240.88
60%	1534	265.4	240.88
80%	1535.9	254.17	240.88
100%	1541.3	220.98	240.88

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