# American Institute of Mathematical Sciences

January  2017, 13(1): 135-146. doi: 10.3934/jimo.2016008

## $(Q,r)$ Model with $CVaR_α$ of costs minimization

 1 School of Engineering, University of Medellín, Medellín 3300, Colombia 2 Basic Sciences Department, EAFIT University, Medellín 3300, Colombia 3 Risk Engineering, Empresas Públicas de Medellín, Medellín 3300, Colombia

Received  February 2013 Revised  April 2015 Published  March 2016

Fund Project: The first and third authors are supported by Medellín University project SIDI 489. The second and forth authors are supported by EAFIT University project SIDI 220-000001 .

In the classical stochastic continuous review, $(Q,r)$ model [18, 19], the inventory cost $c(Q,r)$ has an averaging term which is given as an integral of the expected costs over the different inventory positions during the lead time on any given cycle. The main objective of the article is to study risk averse optimization in the classical $(Q,r)$ model using $CVaR_{α}$ as a coherent risk measure with respect to the probability distribution of the demand $D$ on inventory position costs (the sum of the inventory holding and backorder penality cost), for any given (generic) confidence level $α∈[0,1)$.

We show that the objective function is jointly convex in $(Q,r)$. We also compare the risk averse solution and some other solutions in both analytical and computational ways. Additionally, some general and useful results are obtained.

Citation: María Andrea Arias Serna, María Eugenia Puerta Yepes, César Edinson Escalante Coterio, Gerardo Arango Ospina. $(Q,r)$ Model with $CVaR_α$ of costs minimization. Journal of Industrial & Management Optimization, 2017, 13 (1) : 135-146. doi: 10.3934/jimo.2016008
##### References:
 [1] S. Ahmed, U. Cakmak and A. Shapiro, Coherent risk measures in inventory problems, European Journal of Operational Research, 1 (2007), 226-238. doi: 10.1016/j.ejor.2006.07.016. [2] P. Artzner, F. Delbaen, J. Eber and D. Heath, Coherent measure of risk, Mathematical Finance, 9 (1999), 203-227. doi: 10.1111/1467-9965.00068. [3] X. Chen, M. Sim, D. Simchi-Levi and P. Sun, Risk aversion in inventory management, Operations Research, 55 (2007), 828-842. doi: 10.1287/opre.1070.0429. [4] L. Cheng and Z. Wana, Bilevel newsvendor models considering retailer with CVaR objective, Computers Industrial Engineering, 57 (2009), 310-318. doi: 10.1016/j.cie.2008.12.002. [5] A. Federgruen and Y. S. Zheng, A simple and efficient algorithm for computing optimal (r, Q) Policies in continuous-review stochastic inventory systems, Operations Research, 40 (1992), 808-813. doi: 10.1287/opre.40.2.384. [6] J. Gotoh and Y. Takano, Newsvendor solutions via conditional value-at-risk minimization, EuropeanJournal of Operational Research, 179 (2007), 80-96. doi: 10.1016/j.ejor.2006.03.022. [7] G. Hadley and M. Whittin, Analysis of Inventory Systems, 2 edition, Prentice-Hall, New York, 1963. [8] W. J. Hopp and M. L. Spearman, Factory Physics, 2 edition, McGraw-Hill, New York, 2001. [9] S. Moosa, A. Mohammed and S. S. Yadavalli, A note on evaluating the risk in continuous review inventory systems, International Journal of Production Research, 47 (2009), 5543-5558. [10] J. G. Murillo, M. A. Arias and L. C. Franco, Riesgo Operativo: Técnicas de modelación cuantitativa, 1st Sello Editorial Universidad de Medellín, Colombia, 2014. [11] G. Pflug, Some remarks on the value-at-risk and the conditional value-at-risk, in Probabilistic Constrained Optimization, Nonconvex Optim. Appl., 49, Kluwer Acad. Publ., Dordrecht, 2000,272-281. doi: 10.1007/978-1-4757-3150-7_15. [12] D. E. Platt, L. W. Robinson and R. B. Freund, Tractable (Q, R) heuristic models for constrained service levels, Management Science, 43 (1997), 951-965. doi: 10.1287/mnsc.43.7.951. [13] M. E. Puerta, M. A. Arias and J. I. Londoño, Matemáticas Aplicadas: Optimización de Inventarios Aleatorios, 1st Sello Editorial Universidad de Medellín, Colombia, 2011. [14] R. T. Rockafellar and S. P. Uryasev, Conditional Value-at-Risk for general loss distributions, Journal of Banking and Finance, 23 (2002), 1443-1471. [15] H. N. Shi, D. Li and Ch. Gu, The Schur-convexity of the mean of a convex function, Applied Mathematics Letters, 22 (2009), 932-937. doi: 10.1016/j.aml.2008.04.017. [16] R. Vinod, S. Amitabh and J. B. Raturi, On incorporating business risk into continuous review inventory models, European Journal of Operational Research, 75 (1994), 136-150. [17] X. M. Zhang and Y. M. Chu, Convexity of the integral arithmetic mean of a convex function, Rocky Mountain Journal of Mathematics, 40 (2010), 1061-1068. doi: 10.1216/RMJ-2010-40-3-1061. [18] Y. Zheng, On properties of stochastic inventory systems, Rocky Mountain Journal of Mathematics, 38 (1992), 87-101. doi: 10.1287/mnsc.38.1.87. [19] P. H. Zipkin, Foundations of Inventory Management, 2 edition, McGraw-Hill, New York, 2000.

show all references

##### References:
 [1] S. Ahmed, U. Cakmak and A. Shapiro, Coherent risk measures in inventory problems, European Journal of Operational Research, 1 (2007), 226-238. doi: 10.1016/j.ejor.2006.07.016. [2] P. Artzner, F. Delbaen, J. Eber and D. Heath, Coherent measure of risk, Mathematical Finance, 9 (1999), 203-227. doi: 10.1111/1467-9965.00068. [3] X. Chen, M. Sim, D. Simchi-Levi and P. Sun, Risk aversion in inventory management, Operations Research, 55 (2007), 828-842. doi: 10.1287/opre.1070.0429. [4] L. Cheng and Z. Wana, Bilevel newsvendor models considering retailer with CVaR objective, Computers Industrial Engineering, 57 (2009), 310-318. doi: 10.1016/j.cie.2008.12.002. [5] A. Federgruen and Y. S. Zheng, A simple and efficient algorithm for computing optimal (r, Q) Policies in continuous-review stochastic inventory systems, Operations Research, 40 (1992), 808-813. doi: 10.1287/opre.40.2.384. [6] J. Gotoh and Y. Takano, Newsvendor solutions via conditional value-at-risk minimization, EuropeanJournal of Operational Research, 179 (2007), 80-96. doi: 10.1016/j.ejor.2006.03.022. [7] G. Hadley and M. Whittin, Analysis of Inventory Systems, 2 edition, Prentice-Hall, New York, 1963. [8] W. J. Hopp and M. L. Spearman, Factory Physics, 2 edition, McGraw-Hill, New York, 2001. [9] S. Moosa, A. Mohammed and S. S. Yadavalli, A note on evaluating the risk in continuous review inventory systems, International Journal of Production Research, 47 (2009), 5543-5558. [10] J. G. Murillo, M. A. Arias and L. C. Franco, Riesgo Operativo: Técnicas de modelación cuantitativa, 1st Sello Editorial Universidad de Medellín, Colombia, 2014. [11] G. Pflug, Some remarks on the value-at-risk and the conditional value-at-risk, in Probabilistic Constrained Optimization, Nonconvex Optim. Appl., 49, Kluwer Acad. Publ., Dordrecht, 2000,272-281. doi: 10.1007/978-1-4757-3150-7_15. [12] D. E. Platt, L. W. Robinson and R. B. Freund, Tractable (Q, R) heuristic models for constrained service levels, Management Science, 43 (1997), 951-965. doi: 10.1287/mnsc.43.7.951. [13] M. E. Puerta, M. A. Arias and J. I. Londoño, Matemáticas Aplicadas: Optimización de Inventarios Aleatorios, 1st Sello Editorial Universidad de Medellín, Colombia, 2011. [14] R. T. Rockafellar and S. P. Uryasev, Conditional Value-at-Risk for general loss distributions, Journal of Banking and Finance, 23 (2002), 1443-1471. [15] H. N. Shi, D. Li and Ch. Gu, The Schur-convexity of the mean of a convex function, Applied Mathematics Letters, 22 (2009), 932-937. doi: 10.1016/j.aml.2008.04.017. [16] R. Vinod, S. Amitabh and J. B. Raturi, On incorporating business risk into continuous review inventory models, European Journal of Operational Research, 75 (1994), 136-150. [17] X. M. Zhang and Y. M. Chu, Convexity of the integral arithmetic mean of a convex function, Rocky Mountain Journal of Mathematics, 40 (2010), 1061-1068. doi: 10.1216/RMJ-2010-40-3-1061. [18] Y. Zheng, On properties of stochastic inventory systems, Rocky Mountain Journal of Mathematics, 38 (1992), 87-101. doi: 10.1287/mnsc.38.1.87. [19] P. H. Zipkin, Foundations of Inventory Management, 2 edition, McGraw-Hill, New York, 2000.
$\lambda=50$, $L=1$, $h=10$, $p=25$, $\alpha=0.90$
 $K$ $\lambda K$ $Q_d^*$ $Q^*$ $Q_{\alpha}^*$ $r_d^*$ $r^*$ $r_{\alpha}^*$ $c_d^*$ $c^*$ $c_{\alpha}^*$ $1$ $50$ $3.7$ $8$ $7$ $48.9$ $50$ $54$ $26.73$ $95.68$ $227.90$ $5$ $250$ $8.4$ $13$ $12$ $47.6$ $48$ $52$ $59.76$ $115.48$ $252.64$ $25$ $1250$ $18.7$ $24$ $20$ $44.7$ $44$ $50$ $133.63$ $171.49$ $318.37$ $100$ $5000$ $37.4$ $41$ $39$ $39.3$ $38$ $44$ $267.26$ $289.39$ $448.09$ $1000$ $50000$ $118.3$ $121$ $120$ $16.2$ $15$ $21$ $845.15$ $852.56$ $1023.23$
 $K$ $\lambda K$ $Q_d^*$ $Q^*$ $Q_{\alpha}^*$ $r_d^*$ $r^*$ $r_{\alpha}^*$ $c_d^*$ $c^*$ $c_{\alpha}^*$ $1$ $50$ $3.7$ $8$ $7$ $48.9$ $50$ $54$ $26.73$ $95.68$ $227.90$ $5$ $250$ $8.4$ $13$ $12$ $47.6$ $48$ $52$ $59.76$ $115.48$ $252.64$ $25$ $1250$ $18.7$ $24$ $20$ $44.7$ $44$ $50$ $133.63$ $171.49$ $318.37$ $100$ $5000$ $37.4$ $41$ $39$ $39.3$ $38$ $44$ $267.26$ $289.39$ $448.09$ $1000$ $50000$ $118.3$ $121$ $120$ $16.2$ $15$ $21$ $845.15$ $852.56$ $1023.23$
$\lambda=50$, $L=1$, $h=10$, $p=25$, $\alpha=0.965$
 $K$ $\lambda K$ $Q_d^*$ $Q^*$ $Q_{\alpha}^*$ $r_d^*$ $r^*$ $r_{\alpha}^*$ $c_d^*$ $c^*$ $c_{\alpha}^*$ $1$ $50$ $3.7$ $8$ $6$ $48.9$ $50$ $56$ $26.73$ $95.68$ $269.24$ $5$ $250$ $8.4$ $13$ $11$ $47.6$ $48$ $54$ $59.76$ $115.48$ $295.01$ $25$ $1250$ $18.7$ $24$ $21$ $44.7$ $44$ $51$ $133.63$ $171.49$ $362.73$ $100$ $5000$ $37.4$ $41$ $39$ $39.3$ $38$ $46$ $267.26$ $289.39$ $493.90$ $1000$ $50000$ $118.3$ $121$ $120$ $16.2$ $15$ $23$ $845.15$ $852.56$ $1068.81$
 $K$ $\lambda K$ $Q_d^*$ $Q^*$ $Q_{\alpha}^*$ $r_d^*$ $r^*$ $r_{\alpha}^*$ $c_d^*$ $c^*$ $c_{\alpha}^*$ $1$ $50$ $3.7$ $8$ $6$ $48.9$ $50$ $56$ $26.73$ $95.68$ $269.24$ $5$ $250$ $8.4$ $13$ $11$ $47.6$ $48$ $54$ $59.76$ $115.48$ $295.01$ $25$ $1250$ $18.7$ $24$ $21$ $44.7$ $44$ $51$ $133.63$ $171.49$ $362.73$ $100$ $5000$ $37.4$ $41$ $39$ $39.3$ $38$ $46$ $267.26$ $289.39$ $493.90$ $1000$ $50000$ $118.3$ $121$ $120$ $16.2$ $15$ $23$ $845.15$ $852.56$ $1068.81$
$\lambda=50$, $L=1$, $h=25$, $p=25$, $\alpha=0.90$
 $K$ $\lambda K$ $Q_d^*$ $Q^*$ $Q_{\alpha}^*$ $r_d^*$ $r^*$ $r_{\alpha}^*$ $c_d^*$ $c^*$ $c_{\alpha}^*$ $1$ $50$ $2.8$ $6$ $4$ $48.6$ $46$ $48$ $35.36$ $153.35$ $381.80$ $5$ $250$ $6.3$ $11$ $8$ $46.9$ $44$ $46$ $79.06$ $177.25$ $413.29$ $25$ $1250$ $14.1$ $19$ $15$ $43.0$ $40$ $42$ $176.78$ $245.58$ $499.99$ $100$ $5000$ $28.3$ $31$ $29$ $35.9$ $34$ $35$ $353.55$ $670.96$ $671.36$ $1000$ $50000$ $89.4$ $91$ $90$ $5.3$ $4$ $5$ $1118.03$ $1131.89$ $1431.33$
 $K$ $\lambda K$ $Q_d^*$ $Q^*$ $Q_{\alpha}^*$ $r_d^*$ $r^*$ $r_{\alpha}^*$ $c_d^*$ $c^*$ $c_{\alpha}^*$ $1$ $50$ $2.8$ $6$ $4$ $48.6$ $46$ $48$ $35.36$ $153.35$ $381.80$ $5$ $250$ $6.3$ $11$ $8$ $46.9$ $44$ $46$ $79.06$ $177.25$ $413.29$ $25$ $1250$ $14.1$ $19$ $15$ $43.0$ $40$ $42$ $176.78$ $245.58$ $499.99$ $100$ $5000$ $28.3$ $31$ $29$ $35.9$ $34$ $35$ $353.55$ $670.96$ $671.36$ $1000$ $50000$ $89.4$ $91$ $90$ $5.3$ $4$ $5$ $1118.03$ $1131.89$ $1431.33$
$\lambda=50$, $L=1$, $h=25$, $p=25$, $\alpha=0.965$
 $K$ $\lambda K$ $Q_d^*$ $Q^*$ $Q_{\alpha}^*$ $r_d^*$ $r^*$ $r_{\alpha}^*$ $c_d^*$ $c^*$ $c_{\alpha}^*$ $1$ $50$ $2.8$ $6$ $7$ $48.6$ $46$ $46$ $35.36$ $153.35$ $466.68$ $5$ $250$ $6.3$ $11$ $8$ $46.9$ $44$ $46$ $79.06$ $177.25$ $495.40$ $25$ $1250$ $14.1$ $19$ $15$ $43.0$ $40$ $43$ $176.78$ $245.58$ $580.36$ $100$ $5000$ $28.3$ $31$ $29$ $35.9$ $34$ $35$ $353.55$ $670.96$ $750.45$ $1000$ $50000$ $89.4$ $91$ $90$ $5.3$ $4$ $5$ $1118.03$ $1131.89$ $1510.09$
 $K$ $\lambda K$ $Q_d^*$ $Q^*$ $Q_{\alpha}^*$ $r_d^*$ $r^*$ $r_{\alpha}^*$ $c_d^*$ $c^*$ $c_{\alpha}^*$ $1$ $50$ $2.8$ $6$ $7$ $48.6$ $46$ $46$ $35.36$ $153.35$ $466.68$ $5$ $250$ $6.3$ $11$ $8$ $46.9$ $44$ $46$ $79.06$ $177.25$ $495.40$ $25$ $1250$ $14.1$ $19$ $15$ $43.0$ $40$ $43$ $176.78$ $245.58$ $580.36$ $100$ $5000$ $28.3$ $31$ $29$ $35.9$ $34$ $35$ $353.55$ $670.96$ $750.45$ $1000$ $50000$ $89.4$ $91$ $90$ $5.3$ $4$ $5$ $1118.03$ $1131.89$ $1510.09$
 [1] Xi Chen, Zongrun Wang, Songhai Deng, Yong Fang. Risk measure optimization: Perceived risk and overconfidence of structured product investors. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1473-1492. doi: 10.3934/jimo.2018105 [2] Yufei Sun, Grace Aw, Kok Lay Teo, Guanglu Zhou. Portfolio optimization using a new probabilistic risk measure. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1275-1283. doi: 10.3934/jimo.2015.11.1275 [3] Kegui Chen, Xinyu Wang, Min Huang, Wai-Ki Ching. Compensation plan, pricing and production decisions with inventory-dependent salvage value, and asymmetric risk-averse sales agent. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1397-1422. doi: 10.3934/jimo.2018013 [4] Zvi Drezner, Carlton Scott. Approximate and exact formulas for the $(Q,r)$ inventory model. Journal of Industrial & Management Optimization, 2015, 11 (1) : 135-144. doi: 10.3934/jimo.2015.11.135 [5] Yuwei Shen, Jinxing Xie, Tingting Li. The risk-averse newsvendor game with competition on demand. Journal of Industrial & Management Optimization, 2016, 12 (3) : 931-947. doi: 10.3934/jimo.2016.12.931 [6] Xiulan Wang, Yanfei Lan, Wansheng Tang. An uncertain wage contract model for risk-averse worker under bilateral moral hazard. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1815-1840. doi: 10.3934/jimo.2017020 [7] Helmut Mausser, Oleksandr Romanko. CVaR proxies for minimizing scenario-based Value-at-Risk. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1109-1127. doi: 10.3934/jimo.2014.10.1109 [8] Prasenjit Pramanik, Sarama Malik Das, Manas Kumar Maiti. Note on : Supply chain inventory model for deteriorating items with maximum lifetime and partial trade credit to credit risk customers. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1289-1315. doi: 10.3934/jimo.2018096 [9] Yinghui Dong, Guojing Wang. Ruin probability for renewal risk model with negative risk sums. Journal of Industrial & Management Optimization, 2006, 2 (2) : 229-236. doi: 10.3934/jimo.2006.2.229 [10] Sanjoy Kumar Paul, Ruhul Sarker, Daryl Essam. Managing risk and disruption in production-inventory and supply chain systems: A review. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1009-1029. doi: 10.3934/jimo.2016.12.1009 [11] Hao-Zhe Tay, Kok-Haur Ng, You-Beng Koh, Kooi-Huat Ng. Model selection based on value-at-risk backtesting approach for GARCH-Type models. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-20. doi: 10.3934/jimo.2019021 [12] Zhimin Zhang. On a risk model with randomized dividend-decision times. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1041-1058. doi: 10.3934/jimo.2014.10.1041 [13] Jesús Fabián López Pérez, Tahir Ekin, Jesus A. Jimenez, Francis A. Méndez Mediavilla. Risk-balanced territory design optimization for a Micro finance institution. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-18. doi: 10.3934/jimo.2018176 [14] Abhyudai Singh, Roger M. Nisbet. Variation in risk in single-species discrete-time models. Mathematical Biosciences & Engineering, 2008, 5 (4) : 859-875. doi: 10.3934/mbe.2008.5.859 [15] K. F. Cedric Yiu, S. Y. Wang, K. L. Mak. Optimal portfolios under a value-at-risk constraint with applications to inventory control in supply chains. Journal of Industrial & Management Optimization, 2008, 4 (1) : 81-94. doi: 10.3934/jimo.2008.4.81 [16] Yanju Zhou, Zhen Shen, Renren Ying, Xuanhua Xu. A loss-averse two-product ordering model with information updating in two-echelon inventory system. Journal of Industrial & Management Optimization, 2018, 14 (2) : 687-705. doi: 10.3934/jimo.2017069 [17] Min Zhu, Xiaofei Guo, Zhigui Lin. The risk index for an SIR epidemic model and spatial spreading of the infectious disease. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1565-1583. doi: 10.3934/mbe.2017081 [18] Emilija Bernackaitė, Jonas Šiaulys. The finite-time ruin probability for an inhomogeneous renewal risk model. Journal of Industrial & Management Optimization, 2017, 13 (1) : 207-222. doi: 10.3934/jimo.2016012 [19] Zhimin Zhang, Eric C. K. Cheung. A note on a Lévy insurance risk model under periodic dividend decisions. Journal of Industrial & Management Optimization, 2018, 14 (1) : 35-63. doi: 10.3934/jimo.2017036 [20] Yang Yang, Kaiyong Wang, Jiajun Liu, Zhimin Zhang. Asymptotics for a bidimensional risk model with two geometric Lévy price processes. Journal of Industrial & Management Optimization, 2019, 15 (2) : 481-505. doi: 10.3934/jimo.2018053

2017 Impact Factor: 0.994