$3\mathrm{.}10($EOQ with Fixed Batch Sizes$)$ Suppose that in the EOQ model we can only order batches that are an integer multiple of some number $Q_B$; that is$,$ we can order a batch of size $Q_B,2Q_B,3Q_B,$ etc.

a$)$ Prove that, for the optimal order quantity hat$(Q)=m{Q}_B,$

$\sqrt[\phantom{2}]{\frac{m-1}{m}}\frac{Q_E}{(hat(Q))}\sqrt[\phantom{2}]{\frac{m+1}{m}}$

where $Q_E=\sqrt[\phantom{2}]{2K\frac{}{h}}$ is the optimal $($non$-$integer$-$multiple$)$ EOQ quantity.

b$)$ Suppose that $m2$ for hat$(Q).$ Using the result in part $($a$),$ prove that $g(hat(Q))1\mathrm{.}32g(Q_E),$ where $g(*)$ is the EOQ cost function.