# American Institute of Mathematical Sciences

June  2020, 13(6): 1637-1652. doi: 10.3934/dcdss.2020096

## Optimization model and solution method for dynamically correlated two-product newsvendor problems based on Copula

 1 School of Mathematics and Statistics, Central South University, Changsha 410083, China 2 School of Business, Central South University, Changsha 410083, China

* Corresponding author: Zhong Wan, wanmath@csu.edu.cn

Received  January 2018 Revised  September 2018 Published  September 2019

Fund Project: All the authors are supported by National Natural Science Foundation of China (Grant No. 71671190, 71631008)

In this paper, a two-product newsvendor problem is taken into consideration, where the demands of products are correlated random variables and the buyer is risk-averse. Some important qualitative properties of the constructed model are analyzed, particularly the gradient information of the model is obtained and incorporated into solution method of the model. Based on the theory of Copulas, an efficient algorithm, called the feasible-direction based BFGS algorithm, is developed for solution of the constrained optimization model. Case study shows the efficiency of model and algorithm, and numerical results demonstrate that compared with the situation of independent demands, the total order quantity reduces against the high dynamic interrelation coefficient of two demands with the same degree of risk-aversion, and the optimal order quantity decreases as the degree of risk-aversion becomes greater.

Citation: Songhai Deng, Zhong Wan, Yanjiu Zhou. Optimization model and solution method for dynamically correlated two-product newsvendor problems based on Copula. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1637-1652. doi: 10.3934/dcdss.2020096
##### References:
 [1] L. L. Abdel-Malek and M. Otegbeye, Separable programming/duality approach to solving the multi-product Newsboy/Gardener problem with linear constraints, Applied Mathematical Modelling, 37 (2013), 4497-4508.  doi: 10.1016/j.apm.2012.09.059. [2] L. L. Abdel-Malek and N. Areeratchakul, A quadratic programming approach to the multi-product newsvendor problem with side constraints, European Journal of Operational Research, 176 (2001), 1607-1619.  doi: 10.1016/j.ejor.2005.11.002. [3] L. L. Abdel-Malek, R. Montanari and R. D. Meneghetti, The capacitated newsboy problem with random yield: The Gardener problem, International Journal of Production Economics, 115 (2008), 113-127.  doi: 10.1016/j.ijpe.2008.05.003. [4] K. Arrow, T. Harris and J. Marshak, Optimal inventory policy, Econometrica, 19 (1951), 250-272.  doi: 10.2307/1906813. [5] P. $\acute{A}$vila-Torres, F. L$\acute{o}$pez-Irarragorri and R. Caballero, et al., The multimodal and multiperiod urban transportation integrated timetable construction problem with demand uncertainty, Journal of Industrial and Management Optimization, 14 (2018), 447-472. doi: 10.3934/jimo.2017055. [6] S. Choi and A. Ruszczy$\acute{n}$ski, A multi-product risk-averse newsvendor with exponential utility function, European Journal of Operational Research, 214 (2011), 78-84.  doi: 10.1016/j.ejor.2011.04.005. [7] S. H. Deng, Z. Wan and X. H. Chen, An improved spectral conjugate gradient algorithm for nonconvex unconstrained optimization problems, Journal of Optimization Theory and Applications, 157 (2013), 820-842.  doi: 10.1007/s10957-012-0239-7. [8] S. J. Erlebacher, Optimal and heuristic solutions for the multi-item newsvendor problem with a single capacity constraint, Production and Operations Management, 9 (2000), 303-318.  doi: 10.1111/j.1937-5956.2000.tb00139.x. [9] G. Hadley and T. M. Whitin, Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, New Jersey, 1963. [10] 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. [11] 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. [12] H. S. Lau and A. H. L. Lau, The multi-product multi-constraint newsboy problem: Applications formulation and solution, Journal of Operations Management, 13 (1995), 153-162.  doi: 10.1016/0272-6963(95)00019-O. [13] H. S. Lau and A. H. L. Lau, The newsstand problem: A capacitated multiple product single period inventory problem, European Journal of Operational Research, 94 (1996), 29-42.  doi: 10.1016/0377-2217(95)00192-1. [14] T. Li and Z. Wan, New adaptive Barzilar-Borwein step size and its application in solving large scale optimization problems, The ANZIAM Journal, 61 (2019), 76-98.  doi: 10.1017/S1446181118000263. [15] Y. Li, Z. Wan and J. Liu, Bi-level programming approach to optimal strategy for vendor-managed inventory problems under random demand, The ANZIAM Journal, 59 (2017), 247-270.  doi: 10.1017/S1446181117000384. [16] C. C. Murray, A. Gosavi and D. Talukdar, The multi-product price-setting newsvendor with resource capacity constraints, International Journal of Production Economics, 138 (2012), 148-158.  doi: 10.1016/j.ijpe.2012.03.014. [17] I. Polak and N. Privault, A stochastic newsvendor game with dynamic retail prices, Journal of Industrial and Management Optimization, 14 (2018), 731-742.  doi: 10.3934/jimo.2017072. [18] 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. [19] T. Schmidt, Coping with copulas, Copulas-From theory to application in finance, 2007, 3-34. [20] J. M. Shi, G. Q. Zhang and J. C. Sha, Jointly pricing and ordering for a multi-product multi-constraint newsvendor problem with supplier quantity discounts, Applied Mathematical Modelling, 35 (2011), 3001-3011.  doi: 10.1016/j.apm.2010.12.018. [21] E. A. Silver, D. F. Pyke and R. Peterson, Inventory Management and Production Planning and Scheduling, Wiley, New York, 1998. [22] Z. M. A. Strinka, H. E. Romeijn and J. Wu, Exact and heuristic methods for a class of selective newsvendor problems with normally distributed demands, Omega, 41 (2013), 250-258.  doi: 10.1016/j.omega.2012.05.004. [23] Z. Wan, H. Wu and L. Dai, A polymorphic uncertain equilibrium model and its deterministic equivalent formulation for decentralized supply chain management, Applied Mathematical Modelling, 58 (2018), 281-299.  doi: 10.1016/j.apm.2017.06.028. [24] X. B. Zhang, S. Huang and Z. Wan, Optimal pricing and ordering in global supply chain management with constraints under random demand, Applied Mathematical Modelling, 40 (2016), 10105-10130.  doi: 10.1016/j.apm.2016.06.054. [25] X. B. Zhang, S. Huang and Z. Wan, Stochastic programming approach to global supply chain management under random additive demand, Operational Research, 18 (2018), 389-420.  doi: 10.1007/s12351-016-0269-2. [26] B. Zhang, X. Xu and Z. Hua, A binary solution method for the multi-product newsvendor problem with budget constraint, International Journal of Production Economics, 117 (2009), 136-141. [27] G. Q. 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. [28] B. Zhang, Multi-tier binary solution method for multi-product newsvendor problem with multiple constraints, European Journal of Operational Research, 218 (2012), 426-434.  doi: 10.1016/j.ejor.2011.10.053. [29] B. Zhang and S. F. 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. [30] P. H. Zipkin, Foundations of Inventory Management, McGraw-Hill, Singapore, 2000. [31] Y. J. Zhou, Z. Shen, R. R. Ying and X. H. Xu, A loss-averse two-product ordering model with information updating in two-echelon inventory system, Journal of Industrial and Management Optimization, 14 (2018), 687-705.  doi: 10.3934/jimo.2017069.

show all references

##### References:
 [1] L. L. Abdel-Malek and M. Otegbeye, Separable programming/duality approach to solving the multi-product Newsboy/Gardener problem with linear constraints, Applied Mathematical Modelling, 37 (2013), 4497-4508.  doi: 10.1016/j.apm.2012.09.059. [2] L. L. Abdel-Malek and N. Areeratchakul, A quadratic programming approach to the multi-product newsvendor problem with side constraints, European Journal of Operational Research, 176 (2001), 1607-1619.  doi: 10.1016/j.ejor.2005.11.002. [3] L. L. Abdel-Malek, R. Montanari and R. D. Meneghetti, The capacitated newsboy problem with random yield: The Gardener problem, International Journal of Production Economics, 115 (2008), 113-127.  doi: 10.1016/j.ijpe.2008.05.003. [4] K. Arrow, T. Harris and J. Marshak, Optimal inventory policy, Econometrica, 19 (1951), 250-272.  doi: 10.2307/1906813. [5] P. $\acute{A}$vila-Torres, F. L$\acute{o}$pez-Irarragorri and R. Caballero, et al., The multimodal and multiperiod urban transportation integrated timetable construction problem with demand uncertainty, Journal of Industrial and Management Optimization, 14 (2018), 447-472. doi: 10.3934/jimo.2017055. [6] S. Choi and A. Ruszczy$\acute{n}$ski, A multi-product risk-averse newsvendor with exponential utility function, European Journal of Operational Research, 214 (2011), 78-84.  doi: 10.1016/j.ejor.2011.04.005. [7] S. H. Deng, Z. Wan and X. H. Chen, An improved spectral conjugate gradient algorithm for nonconvex unconstrained optimization problems, Journal of Optimization Theory and Applications, 157 (2013), 820-842.  doi: 10.1007/s10957-012-0239-7. [8] S. J. Erlebacher, Optimal and heuristic solutions for the multi-item newsvendor problem with a single capacity constraint, Production and Operations Management, 9 (2000), 303-318.  doi: 10.1111/j.1937-5956.2000.tb00139.x. [9] G. Hadley and T. M. Whitin, Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, New Jersey, 1963. [10] 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. [11] 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. [12] H. S. Lau and A. H. L. Lau, The multi-product multi-constraint newsboy problem: Applications formulation and solution, Journal of Operations Management, 13 (1995), 153-162.  doi: 10.1016/0272-6963(95)00019-O. [13] H. S. Lau and A. H. L. Lau, The newsstand problem: A capacitated multiple product single period inventory problem, European Journal of Operational Research, 94 (1996), 29-42.  doi: 10.1016/0377-2217(95)00192-1. [14] T. Li and Z. Wan, New adaptive Barzilar-Borwein step size and its application in solving large scale optimization problems, The ANZIAM Journal, 61 (2019), 76-98.  doi: 10.1017/S1446181118000263. [15] Y. Li, Z. Wan and J. Liu, Bi-level programming approach to optimal strategy for vendor-managed inventory problems under random demand, The ANZIAM Journal, 59 (2017), 247-270.  doi: 10.1017/S1446181117000384. [16] C. C. Murray, A. Gosavi and D. Talukdar, The multi-product price-setting newsvendor with resource capacity constraints, International Journal of Production Economics, 138 (2012), 148-158.  doi: 10.1016/j.ijpe.2012.03.014. [17] I. Polak and N. Privault, A stochastic newsvendor game with dynamic retail prices, Journal of Industrial and Management Optimization, 14 (2018), 731-742.  doi: 10.3934/jimo.2017072. [18] 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. [19] T. Schmidt, Coping with copulas, Copulas-From theory to application in finance, 2007, 3-34. [20] J. M. Shi, G. Q. Zhang and J. C. Sha, Jointly pricing and ordering for a multi-product multi-constraint newsvendor problem with supplier quantity discounts, Applied Mathematical Modelling, 35 (2011), 3001-3011.  doi: 10.1016/j.apm.2010.12.018. [21] E. A. Silver, D. F. Pyke and R. Peterson, Inventory Management and Production Planning and Scheduling, Wiley, New York, 1998. [22] Z. M. A. Strinka, H. E. Romeijn and J. Wu, Exact and heuristic methods for a class of selective newsvendor problems with normally distributed demands, Omega, 41 (2013), 250-258.  doi: 10.1016/j.omega.2012.05.004. [23] Z. Wan, H. Wu and L. Dai, A polymorphic uncertain equilibrium model and its deterministic equivalent formulation for decentralized supply chain management, Applied Mathematical Modelling, 58 (2018), 281-299.  doi: 10.1016/j.apm.2017.06.028. [24] X. B. Zhang, S. Huang and Z. Wan, Optimal pricing and ordering in global supply chain management with constraints under random demand, Applied Mathematical Modelling, 40 (2016), 10105-10130.  doi: 10.1016/j.apm.2016.06.054. [25] X. B. Zhang, S. Huang and Z. Wan, Stochastic programming approach to global supply chain management under random additive demand, Operational Research, 18 (2018), 389-420.  doi: 10.1007/s12351-016-0269-2. [26] B. Zhang, X. Xu and Z. Hua, A binary solution method for the multi-product newsvendor problem with budget constraint, International Journal of Production Economics, 117 (2009), 136-141. [27] G. Q. 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. [28] B. Zhang, Multi-tier binary solution method for multi-product newsvendor problem with multiple constraints, European Journal of Operational Research, 218 (2012), 426-434.  doi: 10.1016/j.ejor.2011.10.053. [29] B. Zhang and S. F. 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. [30] P. H. Zipkin, Foundations of Inventory Management, McGraw-Hill, Singapore, 2000. [31] Y. J. Zhou, Z. Shen, R. R. Ying and X. H. Xu, A loss-averse two-product ordering model with information updating in two-echelon inventory system, Journal of Industrial and Management Optimization, 14 (2018), 687-705.  doi: 10.3934/jimo.2017069.
The amount order quantity of products 1, 2 with different $\lambda$
The amount order quantity of different products with $\lambda = 0.005$
The amount order quantity of different products with $\lambda = 0.006$
The amount order quantity of different products with $\lambda = 0.01$
Degree of risk-aversion $\lambda = 0.005$
 (1, 2) (1, 3) (1, 4) $\tau$ $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT 0.0 141. 104. 0.4414820 200. 200. 0.0025293 140. 149. 0.0936952 0.2 149. 94. 0.4122465 200. 200. 0.0038017 142. 139. 0.0979983 0.5 170. 71. 0.3361799 200. 200. 0.0051565 146. 133. 0.0951468 0.8 196. 40. 0.2596440 200. 200. 0.0037477 153. 135. 0.0680937 (2, 3) (2, 4) (3, 4) $\tau$ $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT 0.0 106. 184. 0.0326635 104. 148. 1.9337362 190. 155. 0.0071639 0.2 90. 185. 0.0290358 101. 151. 1.9601832 182. 135. 0.0082129 0.5 63. 209. 0.0187051 93. 159. 1.9775577 177. 109. 0.0113830 0.8 47. 233. 0.0088329 92. 169. 1.7155343 194. 70. 0.0141149
 (1, 2) (1, 3) (1, 4) $\tau$ $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT 0.0 141. 104. 0.4414820 200. 200. 0.0025293 140. 149. 0.0936952 0.2 149. 94. 0.4122465 200. 200. 0.0038017 142. 139. 0.0979983 0.5 170. 71. 0.3361799 200. 200. 0.0051565 146. 133. 0.0951468 0.8 196. 40. 0.2596440 200. 200. 0.0037477 153. 135. 0.0680937 (2, 3) (2, 4) (3, 4) $\tau$ $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT 0.0 106. 184. 0.0326635 104. 148. 1.9337362 190. 155. 0.0071639 0.2 90. 185. 0.0290358 101. 151. 1.9601832 182. 135. 0.0082129 0.5 63. 209. 0.0187051 93. 159. 1.9775577 177. 109. 0.0113830 0.8 47. 233. 0.0088329 92. 169. 1.7155343 194. 70. 0.0141149
Degree of risk-aversion $\lambda = 0.006$
 (1, 2) (1, 3) (1, 4) $\tau$ $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT 0.0 132. 101. 0.6285704 200. 200. 0.0020828 131. 141. 0.0951745 0.2 140. 92. 0.5793933 200. 200. 0.0033609 132. 134. 0.1000532 0.5 162. 69. 0.4438878 200. 200. 0.0044802 140. 125. 0.0948748 0.8 189. 37. 0.3045889 200. 200. 0.0024427 147. 132. 0.0597808 (2, 3) (2, 4) (3, 4) $\tau$ $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT 0.0 102. 175. 0.0293271 101. 141. 3.0125868 183. 143. 0.0046145 0.2 88. 174. 0.0251259 98. 145. 3.1002305 177. 124. 0.0055739 0.5 65. 204. 0.0139032 90. 155. 3.1641422 172. 98. 0.0080783 0.8 66. 203. 0.0066768 87. 168. 2.5235262 186. 67. 0.0102763
 (1, 2) (1, 3) (1, 4) $\tau$ $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT 0.0 132. 101. 0.6285704 200. 200. 0.0020828 131. 141. 0.0951745 0.2 140. 92. 0.5793933 200. 200. 0.0033609 132. 134. 0.1000532 0.5 162. 69. 0.4438878 200. 200. 0.0044802 140. 125. 0.0948748 0.8 189. 37. 0.3045889 200. 200. 0.0024427 147. 132. 0.0597808 (2, 3) (2, 4) (3, 4) $\tau$ $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT 0.0 102. 175. 0.0293271 101. 141. 3.0125868 183. 143. 0.0046145 0.2 88. 174. 0.0251259 98. 145. 3.1002305 177. 124. 0.0055739 0.5 65. 204. 0.0139032 90. 155. 3.1641422 172. 98. 0.0080783 0.8 66. 203. 0.0066768 87. 168. 2.5235262 186. 67. 0.0102763
Degree of risk-aversion $\lambda = 0.01$
 (1, 2) (1, 3) (1, 4) $\tau$ $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT 0.0 113. 95. 4.3915092 200. 200. 0.0046826 114. 118. 0.1975858 0.2 123. 88. 3.9317405 175. 175. 0.0031067 117. 112. 0.2074626 0.5 145. 67. 2.3337793 175. 175. 0.0035576 126. 105. 0.1720581 0.8 177. 32. 0.8585376 200. 200. 0.0011583 133. 120. 0.0591331 (2, 3) (2, 4) (3, 4) $\tau$ $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT 0.0 95. 136. 0.0612705 95. 118. 34.7005708 162. 114. 0.0039200 0.2 83. 146. 0.0480698 93. 123. 39.6971756 138. 92. 0.0044741 0.5 55. 174. 0.0147401 85. 134. 43.1266161 161. 69. 0.0062166 0.8 67. 182. 0.0026076 76. 164. 21.1056347 168. 40. 0.0064492
 (1, 2) (1, 3) (1, 4) $\tau$ $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT 0.0 113. 95. 4.3915092 200. 200. 0.0046826 114. 118. 0.1975858 0.2 123. 88. 3.9317405 175. 175. 0.0031067 117. 112. 0.2074626 0.5 145. 67. 2.3337793 175. 175. 0.0035576 126. 105. 0.1720581 0.8 177. 32. 0.8585376 200. 200. 0.0011583 133. 120. 0.0591331 (2, 3) (2, 4) (3, 4) $\tau$ $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT $x_1^*$ $x_2^*$ CT 0.0 95. 136. 0.0612705 95. 118. 34.7005708 162. 114. 0.0039200 0.2 83. 146. 0.0480698 93. 123. 39.6971756 138. 92. 0.0044741 0.5 55. 174. 0.0147401 85. 134. 43.1266161 161. 69. 0.0062166 0.8 67. 182. 0.0026076 76. 164. 21.1056347 168. 40. 0.0064492
Values for test of algorithm
 product $p$ $c$ $s$ $v$ $\mu$ $\sigma$ 1 12 5 2 3 200 150 2 12 10 2 3 200 150 3 12 5 2 3 200 50 4 12 10 2 3 200 50
 product $p$ $c$ $s$ $v$ $\mu$ $\sigma$ 1 12 5 2 3 200 150 2 12 10 2 3 200 150 3 12 5 2 3 200 50 4 12 10 2 3 200 50
Optimal stocking with neutral risk and independent demands
 PC $x^*_1$ $x^*_2$ $\max\sum\limits_{i=1}^2E[\Pi_i]$ (1, 2) 345.97478848 170.36041138 929.83403949 (1, 3) 345.97505779 245.42380186 2224.05971701 (1, 4) 345.97908047 182.56556145 1259.63761786 (2, 3) 170.36075068 245.42456745 1079.73941595 (2, 4) 170.36011994 182.56530892 115.31731692 (3, 4) 245.42346457 182.56497486 1448.31275184
 PC $x^*_1$ $x^*_2$ $\max\sum\limits_{i=1}^2E[\Pi_i]$ (1, 2) 345.97478848 170.36041138 929.83403949 (1, 3) 345.97505779 245.42380186 2224.05971701 (1, 4) 345.97908047 182.56556145 1259.63761786 (2, 3) 170.36075068 245.42456745 1079.73941595 (2, 4) 170.36011994 182.56530892 115.31731692 (3, 4) 245.42346457 182.56497486 1448.31275184
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