You are charged with designing a three$-$station flow line that must achieve a target throughput of $5$ jobs per hour and a total cycle time of $3$ hours or less. Each station must consist of a single machine purchased from a vendor who will construct it to your specifications, any speed you desire. However, the price depends on the speed as follows:

$K(i)=a(i)[\frac{1}{t_e(i)}{]}^{b(i)}$

where $K(i)$ is the $($total$)$ equipment cost at station $i$;$t_e(i)$ is the effective process time of the machine at station $i$; and $a(i)$ and $b(i)$ are constants. Assume that the arrival coefficient of variation $(CV)$ to the line is equal to one and that $c_e(i)=1$ for $i=1,2,3($i$.$e$.,$ the process $CV$ for all machines is equal to one, regardless of the speed$).$

$($a$)$ Suppose that $a(i)=$$$10,000$ and $b(i)=\frac{2}{3}$ for $i=1,2,3.$ Find the values of $t_e(i)$ for $i=1,2,3$ that achieve target throughput and cycle time with minimum total equipment cost. $($Hint: The Solver tool in Excel is very handy for this.$)$ Is the result a balanced line? Explain why or why not.

$($b$)$ Suppose that $a(1)=$$$1,000,a(2)=$$$100,000,a(3)=$$$10,000,$ and $b(i)=\frac{2}{3}$ for $i=1,2,3.$ Find the values of $t_e(i)$ for $i=1,2,3$ that achieve target throughput and cycle time with minimum total equipment cost. Is the result a balanced line? Explain why or why not.