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