# 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 and 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.

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##### 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.
$\lambda=50$, $L=1$, $h=10$, $p=25$, $\alpha=0.90$
 $K$ $\lambda K$ $Q_d^*$ $Q^*$ $Q_{\alpha}^*$ $r_d^*$ $r^*$ $r_{\alpha}^*$ $c_d^*$ $c^*$ $c_{\alpha}^*$ $1$ $50$ $3.7$ $8$ $7$ $48.9$ $50$ $54$ $26.73$ $95.68$ $227.90$ $5$ $250$ $8.4$ $13$ $12$ $47.6$ $48$ $52$ $59.76$ $115.48$ $252.64$ $25$ $1250$ $18.7$ $24$ $20$ $44.7$ $44$ $50$ $133.63$ $171.49$ $318.37$ $100$ $5000$ $37.4$ $41$ $39$ $39.3$ $38$ $44$ $267.26$ $289.39$ $448.09$ $1000$ $50000$ $118.3$ $121$ $120$ $16.2$ $15$ $21$ $845.15$ $852.56$ $1023.23$
 $K$ $\lambda K$ $Q_d^*$ $Q^*$ $Q_{\alpha}^*$ $r_d^*$ $r^*$ $r_{\alpha}^*$ $c_d^*$ $c^*$ $c_{\alpha}^*$ $1$ $50$ $3.7$ $8$ $7$ $48.9$ $50$ $54$ $26.73$ $95.68$ $227.90$ $5$ $250$ $8.4$ $13$ $12$ $47.6$ $48$ $52$ $59.76$ $115.48$ $252.64$ $25$ $1250$ $18.7$ $24$ $20$ $44.7$ $44$ $50$ $133.63$ $171.49$ $318.37$ $100$ $5000$ $37.4$ $41$ $39$ $39.3$ $38$ $44$ $267.26$ $289.39$ $448.09$ $1000$ $50000$ $118.3$ $121$ $120$ $16.2$ $15$ $21$ $845.15$ $852.56$ $1023.23$
$\lambda=50$, $L=1$, $h=10$, $p=25$, $\alpha=0.965$
 $K$ $\lambda K$ $Q_d^*$ $Q^*$ $Q_{\alpha}^*$ $r_d^*$ $r^*$ $r_{\alpha}^*$ $c_d^*$ $c^*$ $c_{\alpha}^*$ $1$ $50$ $3.7$ $8$ $6$ $48.9$ $50$ $56$ $26.73$ $95.68$ $269.24$ $5$ $250$ $8.4$ $13$ $11$ $47.6$ $48$ $54$ $59.76$ $115.48$ $295.01$ $25$ $1250$ $18.7$ $24$ $21$ $44.7$ $44$ $51$ $133.63$ $171.49$ $362.73$ $100$ $5000$ $37.4$ $41$ $39$ $39.3$ $38$ $46$ $267.26$ $289.39$ $493.90$ $1000$ $50000$ $118.3$ $121$ $120$ $16.2$ $15$ $23$ $845.15$ $852.56$ $1068.81$
 $K$ $\lambda K$ $Q_d^*$ $Q^*$ $Q_{\alpha}^*$ $r_d^*$ $r^*$ $r_{\alpha}^*$ $c_d^*$ $c^*$ $c_{\alpha}^*$ $1$ $50$ $3.7$ $8$ $6$ $48.9$ $50$ $56$ $26.73$ $95.68$ $269.24$ $5$ $250$ $8.4$ $13$ $11$ $47.6$ $48$ $54$ $59.76$ $115.48$ $295.01$ $25$ $1250$ $18.7$ $24$ $21$ $44.7$ $44$ $51$ $133.63$ $171.49$ $362.73$ $100$ $5000$ $37.4$ $41$ $39$ $39.3$ $38$ $46$ $267.26$ $289.39$ $493.90$ $1000$ $50000$ $118.3$ $121$ $120$ $16.2$ $15$ $23$ $845.15$ $852.56$ $1068.81$
$\lambda=50$, $L=1$, $h=25$, $p=25$, $\alpha=0.90$
 $K$ $\lambda K$ $Q_d^*$ $Q^*$ $Q_{\alpha}^*$ $r_d^*$ $r^*$ $r_{\alpha}^*$ $c_d^*$ $c^*$ $c_{\alpha}^*$ $1$ $50$ $2.8$ $6$ $4$ $48.6$ $46$ $48$ $35.36$ $153.35$ $381.80$ $5$ $250$ $6.3$ $11$ $8$ $46.9$ $44$ $46$ $79.06$ $177.25$ $413.29$ $25$ $1250$ $14.1$ $19$ $15$ $43.0$ $40$ $42$ $176.78$ $245.58$ $499.99$ $100$ $5000$ $28.3$ $31$ $29$ $35.9$ $34$ $35$ $353.55$ $670.96$ $671.36$ $1000$ $50000$ $89.4$ $91$ $90$ $5.3$ $4$ $5$ $1118.03$ $1131.89$ $1431.33$
 $K$ $\lambda K$ $Q_d^*$ $Q^*$ $Q_{\alpha}^*$ $r_d^*$ $r^*$ $r_{\alpha}^*$ $c_d^*$ $c^*$ $c_{\alpha}^*$ $1$ $50$ $2.8$ $6$ $4$ $48.6$ $46$ $48$ $35.36$ $153.35$ $381.80$ $5$ $250$ $6.3$ $11$ $8$ $46.9$ $44$ $46$ $79.06$ $177.25$ $413.29$ $25$ $1250$ $14.1$ $19$ $15$ $43.0$ $40$ $42$ $176.78$ $245.58$ $499.99$ $100$ $5000$ $28.3$ $31$ $29$ $35.9$ $34$ $35$ $353.55$ $670.96$ $671.36$ $1000$ $50000$ $89.4$ $91$ $90$ $5.3$ $4$ $5$ $1118.03$ $1131.89$ $1431.33$
$\lambda=50$, $L=1$, $h=25$, $p=25$, $\alpha=0.965$
 $K$ $\lambda K$ $Q_d^*$ $Q^*$ $Q_{\alpha}^*$ $r_d^*$ $r^*$ $r_{\alpha}^*$ $c_d^*$ $c^*$ $c_{\alpha}^*$ $1$ $50$ $2.8$ $6$ $7$ $48.6$ $46$ $46$ $35.36$ $153.35$ $466.68$ $5$ $250$ $6.3$ $11$ $8$ $46.9$ $44$ $46$ $79.06$ $177.25$ $495.40$ $25$ $1250$ $14.1$ $19$ $15$ $43.0$ $40$ $43$ $176.78$ $245.58$ $580.36$ $100$ $5000$ $28.3$ $31$ $29$ $35.9$ $34$ $35$ $353.55$ $670.96$ $750.45$ $1000$ $50000$ $89.4$ $91$ $90$ $5.3$ $4$ $5$ $1118.03$ $1131.89$ $1510.09$
 $K$ $\lambda K$ $Q_d^*$ $Q^*$ $Q_{\alpha}^*$ $r_d^*$ $r^*$ $r_{\alpha}^*$ $c_d^*$ $c^*$ $c_{\alpha}^*$ $1$ $50$ $2.8$ $6$ $7$ $48.6$ $46$ $46$ $35.36$ $153.35$ $466.68$ $5$ $250$ $6.3$ $11$ $8$ $46.9$ $44$ $46$ $79.06$ $177.25$ $495.40$ $25$ $1250$ $14.1$ $19$ $15$ $43.0$ $40$ $43$ $176.78$ $245.58$ $580.36$ $100$ $5000$ $28.3$ $31$ $29$ $35.9$ $34$ $35$ $353.55$ $670.96$ $750.45$ $1000$ $50000$ $89.4$ $91$ $90$ $5.3$ $4$ $5$ $1118.03$ $1131.89$ $1510.09$
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