You are charged with designing a three$-$station flow line that must achieve a target throughput

$of$ five jobs per hour and a total cycle time $of$ three hours $or$ less. Each station must consist $of$

a single machine purchased from a vendor who will construct $itto$ 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.

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.

$c.$ Suppose that everything $is$ the same $asin$ part a except that now $t_e(i)$ can only $be$ chosen $in$

multiples $of0\mathrm{.}05$ hour $,$ etc.$t_e(i)$ 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.

$d.$ What implications $do$ the results $of$ this simplified model have for designing realistic flow

lines?