Q $3.($Worker$-$Paced Line$)$

U$1$ U$2$ U$3$ U$4$ U$5$

The accompanying diagram depicts a five$-$step, worker$-$paced headphone manufacturing plant. The headphones are meant to be used with iPods and DVD players. Step $1$ involves a worker bending a metal strip into an arc shape. In step $2,$ the metal arc is fitted with a plastic sleeve. In step $3,$ the headphones are fitted at the end of the metal and plastic strips. In step $4,$ the wires are soldered into the headphones. Step $5$ involves a specially designed packaging unit. After the plant has been operational for a couple of hours, the manager inspects the plant. He is particularly interested in cutting labor costs. He observes the following. The process is capacity constrained and the entire process produces $36$ units in one hour. U$1$ through U$5$ denote the utilization at steps $1$ through $5$ respectively. Currently, there is a single worker at each step and the utilizations are as follows: U$1=4/30,$ U$2=4/15,$ U$3=4/5,$ U$4=1,$ U$5=2/5.$