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The Derivation for the Cost based optimization of inventory level

The derivation of the Cost based optimization of inventory level formula: $K\left(q\right)={{c}_{f}}+{{c}_{v}}q+pE\left[\max \left\{D-q,0\right\}\right]+hE\left[\max \left\{q-D,0\right\}\right]$

note that $\max \left\{D-q,0\right\}$ is the same as $D-\min \left\{D,q\right\}$ and $\max \left\{q-D,0\right\}$ is the same as $q-\min \left\{D,q\right\}$

so we can reuse the $\min \left\{D,q\right\}$ from the main article and do the following transforms:
${\begin{aligned}K\left(q\right)&={{c}_{f}}+{{c}_{v}}q+pE\left[\max \left\{D-q,0\right\}\right]+hE\left[\max \left\{q-D,0\right\}\right]=\\&={{c}_{f}}+{{c}_{v}}q+pE\left[D-\min \left\{q,D\right\}\right]+hE\left[q-\min \left\{q,D\right\}\right]=\\&={{c}_{f}}+{{c}_{v}}q+pE\left[D\right]-pE\left[\min \left\{q,D\right\}\right]+hq-hE\left[\min \left\{q,D\right\}\right]=\\&={{c}_{f}}+\left({{c}_{v}}+h\right)q+pE\left[D\right]-\left(p+h\right)E\left[\min \left\{q,D\right\}\right]=\\&={{c}_{f}}+\left({{c}_{v}}+h\right)q+pE\left[D\right]-\left(p+h\right)\left(\int \limits _{x\leq q}{xf\left(x\right)dx}+q\left[1-F\left(q\right)\right]\right)\end{aligned}}$

and take its partial derivative with respect to $q$:
${\begin{aligned}{\frac {\partial }{\partial q}}K\left(q\right)&=\left({{c}_{v}}+h\right)-\left(p+h\right)\left(qf\left(q\right)+1-F\left(q\right)-qf\left(q\right)\right)=\\&=\left({{c}_{v}}+h\right)-\left(p+h\right)\left(1-F\left(q\right)\right)\end{aligned}}$

for the optimal solution $q^{*}$ the above should be equal to zero:
${\begin{aligned}0&=\left({{c}_{v}}+h\right)-\left(p+h\right)\left(1-F\left({{q}^{*}}\right)\right)\\F\left({{q}^{*}}\right)&=1-{\frac {{{c}_{v}}+h}{p+h}}\\{{q}^{*}}&={{F}^{-1}}\left(1-{\frac {{{c}_{v}}+h}{p+h}}\right)={{F}^{-1}}\left({\frac {p-{{c}_{v}}}{p+h}}\right)\end{aligned}}$ Val.khokhlov (talk) 13:49, 19 December 2024 (UTC)