INVENTORY The inventory model analyzed in Example $3\mathrm{.}5\mathrm{.}7$ is not the only such model possible. Suppose a company must supply $N$ units per time period at a uniform rate. Assume that the storage cost per unit is $D_1$ dollars per time period and that the setup $($ordering$)$ cost is $D_2$ dollars. If production is at a uniform rate of $m$ units per time period $($with no items in inventory at the end of each period$),$ it can be shown that the total storage cost is

$C_1=\frac{D_{1}x}{2}(1-\frac{N}{m})$

where $x$ is the number of items produced in each run.

a$.$ Show that the total average cost per period is

$C=\frac{D_{1}x}{2}(1-\frac{N}{m})+\frac{D_{2}N}{x}$

b$.$ Find an expression for the number of items that should be produced in each run in order to minimize the total average cost per time period.

c$.$ The optimum quantity found in the inventory problem in Example $3\mathrm{.}5\mathrm{.}7$ is sometimes called the economic order quantity $($EOQ$),$ while the optimum found in part $($b$)$ of this exercise is called the economic production quantity $($EPQ$).$ Modern inventory management goes far beyond the simple conditions in the EOQ and EPQ models, but elements of these models are still very important. For instance, the just$-$in$-$time inventory management described in Example $3\mathrm{.}5\mathrm{.}7$ fits well with the production philosophy of the Japanese. Read an article on Japanese production methods and write a paragraph on why the Japanese regard using space for the storage of materials as undesirable.$*$