Problem $4$: $(25$ points$)$

The annual total inventory cost of the economic production quantity model is given by

$TC(Q)=\frac{SD}{Q}+\frac{H}{2}(1-\frac{D}{P})Q$

where $Q$ is the production quantity per cycle, $S$ is the setup cost, $H$ is the holding cost per unit per year, $D$ is the annual demand rate, and $P$ is the annual production rate.

a$-$ Show that the economic production quantity is given by $Q^{**}=\sqrt[\phantom{2}]{\frac{2SD}{H(1-\frac{D}{P})}}$

b$-$ Show that the minimum annual total cost is given by $TC(Q^{**})=\sqrt[\phantom{2}]{2SDH(1-\frac{D}{P})}$

c$-$ A product is demanded at a monthly rate of $450$ units. It costs AED $24$ to store one unit of the product for one year. It costs AED $120$ to setup the production facility in order to manufacture the product. The actual annual production rate is $7200$ units per year. Find the economic production quantity, the minimum total inventory costs, and the minimum number of setups per year.

d$-$ Due to some constraint's, the facility can only produce $400$ units of the product after each setup. What is the effects of this restriction on the total inventory cost?