What is the average labor u$]$liza$]$on of the servers? Assume the process operates at its capacity.

Mr$.$ K$\text{'}$s Other Hair Salon $(20$ points$)$

Mr$.$ K also has a second hair salon. Similar to last week, this salon also offers high$-$quality

hair$-$styling and physical relaxation services at a reasonable price, so it always has unlimited

demand. The service process includes five activities that are conducted in the sequence

described below. $($The time required for each activity is shown in parenthesis$)$:

Activity $1$: Welcome a guest and offer a cup of coffee. $(5$ minutes$)$

Activity $2$: Neck, shoulder, and back stress release massage. $(15$ minutes$)$

Activity $3$: Design the hairstyle and do the hair. $(30$ minutes$)$

Activity $4$: Wash and condition hair. $(10$ minutes$)$

Activity $5$: Check out the guest. $(5$ minutes$)$

Three servers $($S$1,$ S$2,$ and S$3)$ offer the services in a worker$-$paced line. The assignment of tasks to servers is the following:

S$1$ does Activity $1$ and $5.$

S$2$ does Activities $2.$

S$3$ does Activities $3$ and $4.$

a$)$ Which server is the bottleneck of the process?

b$)$ What is the average labor utilization of the servers? Assume the process operates at its

capacity.

c$)$ Assume a wage rate of $$30$ per hour. What are the direct labor costs for one guest?

d$)$ To increase the service rate, $Mr.K$ is considering two alternatives:

Alternative I: To hire a new employee to help any one $($and only one$)$ of the servers

without changing the tasks performed by each server.

Alternative II: To redesign the assignment of tasks to servers. For this, Mr$.$ K is

evaluating reassigning Activity $4$ from S$3$ to S$1.$

What would be the direct labor costs of serving one guest under each of these two

alternatives? Assume that the system operates at its capacity.

e$)$ So far, we have assumed that demand is unlimited. Now assume variable demand that is$,$

on average, below bottleneck capacity and memoryless. Would customers in expectation

be required to wait before getting served?