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
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York, 1992.  Google Scholar

[2]

N. AltintasF. Erhun and S. Tayur, Quantity discounts under demand uncertainty, Management Science, 54 (2008), 777-792.  doi: 10.1287/mnsc.1070.0829.  Google Scholar

[3]

L. C. AlwanM. XuD. Q. Yao and X. Yue, The dynamic newsvendor model with correlated demand, Decision Sciences, 47 (2016), 11-30.   Google Scholar

[4]

H. Behret and C. Kahraman, A multi-period newsvendor problem with pre-season extension under fuzzy demand, Journal of Business Economics and Management, 11 (2010), 613-629.  doi: 10.3846/jbem.2010.30.  Google Scholar

[5]

M. Bouakiz and M. J. Sobel, Inventory control with an exponential utility criterion, Operations Research, 40 (1992), 603-608.  doi: 10.1287/opre.40.3.603.  Google Scholar

[6]

A. BurnetasS. M. Gilbert and C. E. Smith, Quantity discounts in single-period supply contracts with asymmetric demand information, IIE Transactions, 39 (2007), 465-479.   Google Scholar

[7]

J. M. Chen and H. L. Cheng, Effect of the price-dependent revenue-sharing mechanism in a decentralized supply chain, Central European Journal of Operations Research, 20 (2012), 299-317.  doi: 10.1007/s10100-010-0182-3.  Google Scholar

[8]

S. P. Chen and Y. H. Ho, Analysis of the newsboy problem with fuzzy demands and incremental discounts, International Journal of Production Economics, 129 (2011), 169-177.  doi: 10.1016/j.ijpe.2010.09.014.  Google Scholar

[9]

S. P. Chen and Y. H. Ho, Optimal inventory policy for the fuzzy newsboy problem with quantity discounts, Information Sciences, 228 (2013), 75-89.  doi: 10.1016/j.ins.2012.12.015.  Google Scholar

[10]

S. Ding and Y. Gao, The (σ, S) policy for uncertain multi-product newsboy problem, Expert Systems with Applications, 41 (2014), 3769-3776.   Google Scholar

[11]

H. Gaspars-Wieloch, Newsvendor problem under complete uncertainty: A case of innovative products, Central European Journal of Operations Research, 25 (2017), 561-585.  doi: 10.1007/s10100-016-0458-3.  Google Scholar

[12]

G. A. HanasusantoD. KuhnS. W. Wallace and S. Zymler, Distributionally robust multi-item newsvendor problems with multimodal demand distributions, Mathematical Programming, 152 (2015), 1-32.  doi: 10.1007/s10107-014-0776-y.  Google Scholar

[13]

D. HuangH. Zhou and Q. H. Zhao, A competitive multiple-product newsboy problem with partial product substitution, Omega, 39 (2011), 302-312.  doi: 10.1016/j.omega.2010.07.008.  Google Scholar

[14]

J. Kamburowski, The distribution-free newsboy problem under the worst-case and best-case scenarios, European Journal of Operational Research, 237 (2014), 106-112.  doi: 10.1016/j.ejor.2014.01.066.  Google Scholar

[15]

J. Kamburowski, The distribution-free newsboy problem and the demand skew, International Transactions in Operational Research, 22 (2015), 929-946.  doi: 10.1111/itor.12139.  Google Scholar

[16]

M. Khouja, The single-period (news-vendor) problem: literature review and suggestions for future research, Omega, 27 (1999), 537-553.  doi: 10.1016/S0305-0483(99)00017-1.  Google Scholar

[17]

K. Matsuyama, The multi-period newsboy problem, European Journal of Operational Research, 171 (2006), 170-188.  doi: 10.1016/j.ejor.2004.08.030.  Google Scholar

[18]

P. Mileff and K. Nehéz, Solving capacity constraint problems in a multi-item, multi-period newsvendor model, Proc. of microCAD, (2007), 169-176.   Google Scholar

[19]

R. LotfiM. NayeriS. Sajadifar and N. Mardani, Determination of start times and ordering plans for two-period projects with interdependent demand in project-oriented organizations: A case study on molding industry, Journal of Project Management, 2 (2018a), 119-142.  doi: 10.5267/j.jpm.2017.9.001.  Google Scholar

[20]

R. LotfiA. MostafaeipourN. Mardani and S. Mardani, Investigation of wind farm location planning by considering budget constraints, International Journal of Sustainable Energy, 37 (2018), 799-817.  doi: 10.1080/14786451.2018.1437160.  Google Scholar

[21]

M. Fakhrzad and R. Lotfi, Green vendor managed inventory with backorder in two echelon supply chain with Epsilon-Constraint and NSGA-Ⅱ approach, Journal of Industrial Engineering Research in Production Systems, 5 (2018), 193-209.  doi: 10.22084/ier.2017.11270.1509.  Google Scholar

[22]

B. PalS. S Sana and K. Chaudhuri, A distribution-free newsvendor problem with nonlinear holding cost, International Journal of Systems Science, 46 (2015), 1269-1277.  doi: 10.1080/00207721.2013.815828.  Google Scholar

[23]

W. L. PearnR. H. SuM. W. Weng and C. H. Hsu, Optimal production run time for two-stage production system with imperfect processes and allowable shortages, Central European Journal of Operations Research, 19 (2011), 533-545.  doi: 10.1007/s10100-010-0143-x.  Google Scholar

[24]

G. Perakis and A. Sood, Competitive multi-period pricing with fixed inventories, (2004). Google Scholar

[25]

Y. QinR. WangA. J. VakhariaY. Chen and M. M. 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.  Google Scholar

[26]

P. Ray and M. Jenamani, Sourcing decision under disruption risk with supply and demand uncertainty: A newsvendor approach, Annals of Operations Research, 237 (2016), 237-262.  doi: 10.1007/s10479-014-1649-8.  Google Scholar

[27]

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.  Google Scholar

[28]

J. W. Tukey, Sufficiency, truncation and selection, The Annals of Mathematical Statistics, 20 (1949), 309-311.  doi: 10.1214/aoms/1177730042.  Google Scholar

[29]

C. X. Wang and S. Webster, The loss-averse newsvendor problem, Omega, 37 (2009), 93-105.  doi: 10.1016/j.omega.2006.08.003.  Google Scholar

[30]

B. Zhang and S. Du, Multi-product newsboy problem with limited capacity and outsourcing, European Journal of Operational Research, 202 (2010), 107-113.  doi: 10.1016/j.ejor.2009.04.017.  Google Scholar

[31]

B. Zhang and Z. Hua, A portfolio approach to multi-product newsboy problem with budget constraint, Computers & Industrial Engineering, 58 (2010), 759-765.  doi: 10.1016/j.cie.2010.02.007.  Google Scholar

[32]

G. Zhang, The multi-product newsboy problem with supplier quantity discounts and a budget constraint, European Journal of Operational Research, 206 (2010), 350-360.  doi: 10.1016/j.ejor.2010.02.038.  Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York, 1992.  Google Scholar

[2]

N. AltintasF. Erhun and S. Tayur, Quantity discounts under demand uncertainty, Management Science, 54 (2008), 777-792.  doi: 10.1287/mnsc.1070.0829.  Google Scholar

[3]

L. C. AlwanM. XuD. Q. Yao and X. Yue, The dynamic newsvendor model with correlated demand, Decision Sciences, 47 (2016), 11-30.   Google Scholar

[4]

H. Behret and C. Kahraman, A multi-period newsvendor problem with pre-season extension under fuzzy demand, Journal of Business Economics and Management, 11 (2010), 613-629.  doi: 10.3846/jbem.2010.30.  Google Scholar

[5]

M. Bouakiz and M. J. Sobel, Inventory control with an exponential utility criterion, Operations Research, 40 (1992), 603-608.  doi: 10.1287/opre.40.3.603.  Google Scholar

[6]

A. BurnetasS. M. Gilbert and C. E. Smith, Quantity discounts in single-period supply contracts with asymmetric demand information, IIE Transactions, 39 (2007), 465-479.   Google Scholar

[7]

J. M. Chen and H. L. Cheng, Effect of the price-dependent revenue-sharing mechanism in a decentralized supply chain, Central European Journal of Operations Research, 20 (2012), 299-317.  doi: 10.1007/s10100-010-0182-3.  Google Scholar

[8]

S. P. Chen and Y. H. Ho, Analysis of the newsboy problem with fuzzy demands and incremental discounts, International Journal of Production Economics, 129 (2011), 169-177.  doi: 10.1016/j.ijpe.2010.09.014.  Google Scholar

[9]

S. P. Chen and Y. H. Ho, Optimal inventory policy for the fuzzy newsboy problem with quantity discounts, Information Sciences, 228 (2013), 75-89.  doi: 10.1016/j.ins.2012.12.015.  Google Scholar

[10]

S. Ding and Y. Gao, The (σ, S) policy for uncertain multi-product newsboy problem, Expert Systems with Applications, 41 (2014), 3769-3776.   Google Scholar

[11]

H. Gaspars-Wieloch, Newsvendor problem under complete uncertainty: A case of innovative products, Central European Journal of Operations Research, 25 (2017), 561-585.  doi: 10.1007/s10100-016-0458-3.  Google Scholar

[12]

G. A. HanasusantoD. KuhnS. W. Wallace and S. Zymler, Distributionally robust multi-item newsvendor problems with multimodal demand distributions, Mathematical Programming, 152 (2015), 1-32.  doi: 10.1007/s10107-014-0776-y.  Google Scholar

[13]

D. HuangH. Zhou and Q. H. Zhao, A competitive multiple-product newsboy problem with partial product substitution, Omega, 39 (2011), 302-312.  doi: 10.1016/j.omega.2010.07.008.  Google Scholar

[14]

J. Kamburowski, The distribution-free newsboy problem under the worst-case and best-case scenarios, European Journal of Operational Research, 237 (2014), 106-112.  doi: 10.1016/j.ejor.2014.01.066.  Google Scholar

[15]

J. Kamburowski, The distribution-free newsboy problem and the demand skew, International Transactions in Operational Research, 22 (2015), 929-946.  doi: 10.1111/itor.12139.  Google Scholar

[16]

M. Khouja, The single-period (news-vendor) problem: literature review and suggestions for future research, Omega, 27 (1999), 537-553.  doi: 10.1016/S0305-0483(99)00017-1.  Google Scholar

[17]

K. Matsuyama, The multi-period newsboy problem, European Journal of Operational Research, 171 (2006), 170-188.  doi: 10.1016/j.ejor.2004.08.030.  Google Scholar

[18]

P. Mileff and K. Nehéz, Solving capacity constraint problems in a multi-item, multi-period newsvendor model, Proc. of microCAD, (2007), 169-176.   Google Scholar

[19]

R. LotfiM. NayeriS. Sajadifar and N. Mardani, Determination of start times and ordering plans for two-period projects with interdependent demand in project-oriented organizations: A case study on molding industry, Journal of Project Management, 2 (2018a), 119-142.  doi: 10.5267/j.jpm.2017.9.001.  Google Scholar

[20]

R. LotfiA. MostafaeipourN. Mardani and S. Mardani, Investigation of wind farm location planning by considering budget constraints, International Journal of Sustainable Energy, 37 (2018), 799-817.  doi: 10.1080/14786451.2018.1437160.  Google Scholar

[21]

M. Fakhrzad and R. Lotfi, Green vendor managed inventory with backorder in two echelon supply chain with Epsilon-Constraint and NSGA-Ⅱ approach, Journal of Industrial Engineering Research in Production Systems, 5 (2018), 193-209.  doi: 10.22084/ier.2017.11270.1509.  Google Scholar

[22]

B. PalS. S Sana and K. Chaudhuri, A distribution-free newsvendor problem with nonlinear holding cost, International Journal of Systems Science, 46 (2015), 1269-1277.  doi: 10.1080/00207721.2013.815828.  Google Scholar

[23]

W. L. PearnR. H. SuM. W. Weng and C. H. Hsu, Optimal production run time for two-stage production system with imperfect processes and allowable shortages, Central European Journal of Operations Research, 19 (2011), 533-545.  doi: 10.1007/s10100-010-0143-x.  Google Scholar

[24]

G. Perakis and A. Sood, Competitive multi-period pricing with fixed inventories, (2004). Google Scholar

[25]

Y. QinR. WangA. J. VakhariaY. Chen and M. M. 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.  Google Scholar

[26]

P. Ray and M. Jenamani, Sourcing decision under disruption risk with supply and demand uncertainty: A newsvendor approach, Annals of Operations Research, 237 (2016), 237-262.  doi: 10.1007/s10479-014-1649-8.  Google Scholar

[27]

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.  Google Scholar

[28]

J. W. Tukey, Sufficiency, truncation and selection, The Annals of Mathematical Statistics, 20 (1949), 309-311.  doi: 10.1214/aoms/1177730042.  Google Scholar

[29]

C. X. Wang and S. Webster, The loss-averse newsvendor problem, Omega, 37 (2009), 93-105.  doi: 10.1016/j.omega.2006.08.003.  Google Scholar

[30]

B. Zhang and S. Du, Multi-product newsboy problem with limited capacity and outsourcing, European Journal of Operational Research, 202 (2010), 107-113.  doi: 10.1016/j.ejor.2009.04.017.  Google Scholar

[31]

B. Zhang and Z. Hua, A portfolio approach to multi-product newsboy problem with budget constraint, Computers & Industrial Engineering, 58 (2010), 759-765.  doi: 10.1016/j.cie.2010.02.007.  Google Scholar

[32]

G. Zhang, The multi-product newsboy problem with supplier quantity discounts and a budget constraint, European Journal of Operational Research, 206 (2010), 350-360.  doi: 10.1016/j.ejor.2010.02.038.  Google Scholar

Figure 1.  Process of implementation of the conceptual model
17]">Figure 2.  Chart of Differences between the proposed model and Matsuyama [17]
Figure 3.  Chart of the ratio ($\alpha $)
Figure 4.  Chart of the ratio ($\beta $)
Figure 5.  Chart of the ratio ($\delta $)
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
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
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 $l_{j+1}-\alpha (l_j -x_j )$ $l_{j+1}+\beta ({x_j-l}_j)$
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 $l_{j+1}-\alpha (l_j -x_j )$ $l_{j+1}+\beta ({x_j-l}_j)$
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%
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%
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
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
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
$\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
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
$\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
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
$\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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Caichun Chai, Tiaojun Xiao, Zhangwei Feng. Evolution of revenue preference for competing firms with nonlinear inverse demand. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021071

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Ru Li, Guolin Yu. Strict efficiency of a multi-product supply-demand network equilibrium model. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2203-2215. doi: 10.3934/jimo.2020065

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