Article Contents
Article Contents

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

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.

Mathematics Subject Classification: Primary: 90B05.

 Citation:

• Table 1.  $\lambda=50$, $L=1$, $h=10$, $p=25$, $\alpha=0.90$

 $K$ $\lambda K$ $Q_d^*$ $Q^*$ $Q_{\alpha}^*$ $r_d^*$ $r^*$ $r_{\alpha}^*$ $c_d^*$ $c^*$ $c_{\alpha}^*$ $1$ $50$ $3.7$ $8$ $7$ $48.9$ $50$ $54$ $26.73$ $95.68$ $227.90$ $5$ $250$ $8.4$ $13$ $12$ $47.6$ $48$ $52$ $59.76$ $115.48$ $252.64$ $25$ $1250$ $18.7$ $24$ $20$ $44.7$ $44$ $50$ $133.63$ $171.49$ $318.37$ $100$ $5000$ $37.4$ $41$ $39$ $39.3$ $38$ $44$ $267.26$ $289.39$ $448.09$ $1000$ $50000$ $118.3$ $121$ $120$ $16.2$ $15$ $21$ $845.15$ $852.56$ $1023.23$

Table 2.  $\lambda=50$, $L=1$, $h=10$, $p=25$, $\alpha=0.965$

 $K$ $\lambda K$ $Q_d^*$ $Q^*$ $Q_{\alpha}^*$ $r_d^*$ $r^*$ $r_{\alpha}^*$ $c_d^*$ $c^*$ $c_{\alpha}^*$ $1$ $50$ $3.7$ $8$ $6$ $48.9$ $50$ $56$ $26.73$ $95.68$ $269.24$ $5$ $250$ $8.4$ $13$ $11$ $47.6$ $48$ $54$ $59.76$ $115.48$ $295.01$ $25$ $1250$ $18.7$ $24$ $21$ $44.7$ $44$ $51$ $133.63$ $171.49$ $362.73$ $100$ $5000$ $37.4$ $41$ $39$ $39.3$ $38$ $46$ $267.26$ $289.39$ $493.90$ $1000$ $50000$ $118.3$ $121$ $120$ $16.2$ $15$ $23$ $845.15$ $852.56$ $1068.81$

Table 3.  $\lambda=50$, $L=1$, $h=25$, $p=25$, $\alpha=0.90$

 $K$ $\lambda K$ $Q_d^*$ $Q^*$ $Q_{\alpha}^*$ $r_d^*$ $r^*$ $r_{\alpha}^*$ $c_d^*$ $c^*$ $c_{\alpha}^*$ $1$ $50$ $2.8$ $6$ $4$ $48.6$ $46$ $48$ $35.36$ $153.35$ $381.80$ $5$ $250$ $6.3$ $11$ $8$ $46.9$ $44$ $46$ $79.06$ $177.25$ $413.29$ $25$ $1250$ $14.1$ $19$ $15$ $43.0$ $40$ $42$ $176.78$ $245.58$ $499.99$ $100$ $5000$ $28.3$ $31$ $29$ $35.9$ $34$ $35$ $353.55$ $670.96$ $671.36$ $1000$ $50000$ $89.4$ $91$ $90$ $5.3$ $4$ $5$ $1118.03$ $1131.89$ $1431.33$

Table 4.  $\lambda=50$, $L=1$, $h=25$, $p=25$, $\alpha=0.965$

 $K$ $\lambda K$ $Q_d^*$ $Q^*$ $Q_{\alpha}^*$ $r_d^*$ $r^*$ $r_{\alpha}^*$ $c_d^*$ $c^*$ $c_{\alpha}^*$ $1$ $50$ $2.8$ $6$ $7$ $48.6$ $46$ $46$ $35.36$ $153.35$ $466.68$ $5$ $250$ $6.3$ $11$ $8$ $46.9$ $44$ $46$ $79.06$ $177.25$ $495.40$ $25$ $1250$ $14.1$ $19$ $15$ $43.0$ $40$ $43$ $176.78$ $245.58$ $580.36$ $100$ $5000$ $28.3$ $31$ $29$ $35.9$ $34$ $35$ $353.55$ $670.96$ $750.45$ $1000$ $50000$ $89.4$ $91$ $90$ $5.3$ $4$ $5$ $1118.03$ $1131.89$ $1510.09$
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