Example $2\mathrm{.}7$

Analyzing the assignment of operators to multiple machines

For the situation illustrated in Figure $2\mathrm{.}18,$ it takes $0\mathrm{.}5$ minute to travel between machines,

$1$ minute to load a machine, $1$ minute to unload a machine, $6$ minutes of automatic machine

time, and $0\mathrm{.}5$ minute to inspect and pack a finished part. As shown, the analysis begins

with each machine empty and the operator standing in front of Machine $1($M$-1).$ The oper$-$

ator loads M$-1,$ walks to M$-2,$ loads M$-2,$ walks to M$-3,$ loads M$-3,$ walks to M$-1,$ unloads

M$-1,$ loads M$-1,$ inspects and packs the part removed from M$-1,$ travels to M$-2,$ and so forth.

As shown in Figure $2\mathrm{.}18,$ it takes $12$ minutes for the operator and three machines to achieve

a steady$-$state condition; thereafter, a repeating cycle of $9$ minutes in duration occurs. $($In

other words, if nothing interrupts the activities of the operator and the three machines, the

$9-$minute cycle will repeat indefinitely.$)$ In this example, cost per operator$-$hour is $$15$ and cost per machine$-$hour is $$50.$ In this case two

machines should be assigned to an operator. Now, suppose the cost per machine hour is unknown. For

what range of values of $$ will the optimum assignment remain the same? Justify your answer