American Institute of Mathematical Sciences

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, 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. [2] N. Altintas, F. Erhun and S. Tayur, Quantity discounts under demand uncertainty, Management Science, 54 (2008), 777-792. doi: 10.1287/mnsc.1070.0829. [3] L. C. Alwan, M. Xu, D. Q. Yao and X. Yue, The dynamic newsvendor model with correlated demand, Decision Sciences, 47 (2016), 11-30. [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. [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. [6] A. Burnetas, S. M. Gilbert and C. E. Smith, Quantity discounts in single-period supply contracts with asymmetric demand information, IIE Transactions, 39 (2007), 465-479. [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. [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. [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. [10] S. Ding and Y. Gao, The (σ, S) policy for uncertain multi-product newsboy problem, Expert Systems with Applications, 41 (2014), 3769-3776. [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. [12] G. A. Hanasusanto, D. Kuhn, S. 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. [13] D. Huang, H. 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. [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. [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. [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. [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. [18] P. Mileff and K. Nehéz, Solving capacity constraint problems in a multi-item, multi-period newsvendor model, Proc. of microCAD, (2007), 169-176. [19] R. Lotfi, M. Nayeri, S. 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. [20] R. Lotfi, A. Mostafaeipour, N. 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. [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. [22] B. Pal, S. 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. [23] W. L. Pearn, R. H. Su, M. 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. [24] G. Perakis and A. Sood, Competitive multi-period pricing with fixed inventories, (2004). [25] Y. Qin, R. Wang, A. J. Vakharia, Y. 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. [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. [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. [28] J. W. Tukey, Sufficiency, truncation and selection, The Annals of Mathematical Statistics, 20 (1949), 309-311. doi: 10.1214/aoms/1177730042. [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. [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. [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. [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.

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References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York, 1992. [2] N. Altintas, F. Erhun and S. Tayur, Quantity discounts under demand uncertainty, Management Science, 54 (2008), 777-792. doi: 10.1287/mnsc.1070.0829. [3] L. C. Alwan, M. Xu, D. Q. Yao and X. Yue, The dynamic newsvendor model with correlated demand, Decision Sciences, 47 (2016), 11-30. [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. [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. [6] A. Burnetas, S. M. Gilbert and C. E. Smith, Quantity discounts in single-period supply contracts with asymmetric demand information, IIE Transactions, 39 (2007), 465-479. [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. [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. [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. [10] S. Ding and Y. Gao, The (σ, S) policy for uncertain multi-product newsboy problem, Expert Systems with Applications, 41 (2014), 3769-3776. [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. [12] G. A. Hanasusanto, D. Kuhn, S. 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. [13] D. Huang, H. 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. [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. [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. [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. [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. [18] P. Mileff and K. Nehéz, Solving capacity constraint problems in a multi-item, multi-period newsvendor model, Proc. of microCAD, (2007), 169-176. [19] R. Lotfi, M. Nayeri, S. 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. [20] R. Lotfi, A. Mostafaeipour, N. 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. [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. [22] B. Pal, S. 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. [23] W. L. Pearn, R. H. Su, M. 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. [24] G. Perakis and A. Sood, Competitive multi-period pricing with fixed inventories, (2004). [25] Y. Qin, R. Wang, A. J. Vakharia, Y. 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. [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. [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. [28] J. W. Tukey, Sufficiency, truncation and selection, The Annals of Mathematical Statistics, 20 (1949), 309-311. doi: 10.1214/aoms/1177730042. [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. [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. [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. [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.
Process of implementation of the conceptual model
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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