# American Institute of Mathematical Sciences

January  2020, 16(1): 117-140. doi: 10.3934/jimo.2018143

## 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

Received  March 2017 Revised  May 2018 Published  September 2018

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.

Citation: Reza Lotfi, Gerhard-Wilhelm Weber, S. Mehdi Sajadifar, Nooshin Mardani. Interdependent demand in the two-period newsvendor problem. Journal of Industrial & Management Optimization, 2020, 16 (1) : 117-140. doi: 10.3934/jimo.2018143
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Process of implementation of the conceptual model
]">Figure 2.  Chart of Differences between the proposed model and Matsuyama [17]
Chart of the ratio ($\alpha$)
Chart of the ratio ($\beta$)
Chart of the ratio ($\delta$)
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
 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
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 $l_{j+1}-\alpha (l_j -x_j )$ $l_{j+1}+\beta ({x_j-l}_j)$
 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 $l_{j+1}-\alpha (l_j -x_j )$ $l_{j+1}+\beta ({x_j-l}_j)$
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%
 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%
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  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
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
 $\alpha$ $H^*$ $l^*_1$ $l^*_2$ 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
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
 $\beta$ $H^*$ $l^*_1$ $l^*_2$ 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
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
 $\delta$ $H^*$ $l^*_1$ $l^*_2$ 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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