INDIVIDUAL ASSIGNMENT #$3$

Question$-1.(40$ points$)$

The process of fulfilling orders at the warehouse of an online retailer involves three steps:

Order taking, picking, and packing. The warehouse operates $24$ hours a day. Currently, the

processing times to take, pick, and pack an order are $3$ minutes, $9$ minutes, and $5$ minutes per

worker per order. There are $2$ workers assigned to order taking, $8$ workers assigned to picking

an order, and $4$ workers assigned to packing an order. Answer the questions below.

a$)(4$ points$)$ What is the total capacity of order taking $($in orders per hour$)?$

b$)(4$ points$)$ What is the total capacity of order picking $($in orders per hour$)?$

c$)(4$ points$)$ What is the total capacity of order packing $($in orders per hour$)?$

d$)(4$ points$)$ Which resources $($or task$)$ is the bottleneck of the process and why?

e$)(12$ points$)$ Suppose the demand rate is $60$ orders per hour. Answer the following

questions using this information:

i$.(3$ points$)$ What is the flow rate of the process?

ii$.(3$ points$)$ Is the process demand$-$constrained or capacity$-$constrained?

iii. $(3$ points$)$ What is the process cycle time?

iv$.(1$ point$)$ What is the utilization rate of the workers at order taking?

v$.(1$ point$)$ What is the utilization rate of the workers at order picking?

vi$.(1$ point$)$ What is the utilization rate of the workers at order packing?f$)(12$ points$)$ Suppose the demand rate is $35$ orders per hour. Answer the following

questions using this information:

i$.(3$ points$)$ What is the flow rate of the process?

ii$.(3$ points$)$ Is the process demand$-$constrained or capacity$-$constrained?

iii. $(3$ points$)$ What is the process cycle time?

iv$.(1$ point$)$ What is the utilization rate of the workers at order taking?

v$.(1$ point$)$ What is the utilization rate of the workers at order picking?

vi$.(1$ point$)$ What is the utilization rate of the workers at order packing?

Question $2.(60$ points$)$

Consider a call center that responds to calls from the customers and passengers of a large

airline. Suppose both the interarrival times and service times of the calls are variable. The

average arrival rate is $12\mathrm{.}5$ calls per hour and the coefficient of variation of time between

arrivals is $0\mathrm{.}5$ minutes. The average service time is $8$ minutes per call and the standard deviation

of the service time is $2$ minutes. There are currently $2$ operators taking the calls. There is only

one queue. All calls are answered in the order they are received $($first come first serve$).$ The

following is the average time spent in the system per call, measured in minutes.

No$.$ of servers Avg. Wait time in the system $($in minutes$)$

$1$ INFINITY

$2.2\mathrm{.}879114->($crossed out$)26\mathrm{.}426$

$3.0\mathrm{.}320055->($crossed out$)10\mathrm{.}048$

$4.0\mathrm{.}080688->($crossed out$)8\mathrm{.}5164$

NOTE: SHOW ALL YOUR WORK. YOU MUST WRITE DOWN ALL YOUR FORMULA AND

CALCULATIONS. IT IS NOT ENOUGH TO PROVIDE NUMBER AS YOUR ANSWER.

a$)(3$ points$)$ What is the average time between arrivals $($in minutes$)?$

b$)(3$ points$)$ What is the average arrival rate $($per hour$)?$

c$)(3$ points$)$ What is the average service time $($in minutes$)?$ d$)(3$ points$)$ What is the average service rate $($per hour$)?$

e$)(3$ points$)$ What is the coefficient of variation of service time?

f$)(3$ points$)$ If a call arrives at $10$:$00$:$00$am$,$ what is the expected arrival time of the next

call?

g$)(3$ points$)$ If a call arrives at $10$:$00$:$00$am$,$ what is the expected time the call will be picked

up by an operator after waiting in the queue?

h$)(3$ points$)$ If a call arrives at $10$:$00$:$00$am$,$ what is the expected time for the call to end

after getting served?

i$)(5$ points$)$ What is the average utilization rate of an operator?

j$)(4$ points$)$ What is the average number of calls in the system at any time?

k$)(4$ points$)$ What is the average number of calls getting serviced at any time?

l$)(4$ points$)$ What is the average number of calls waiting in the queue at any time?

m$)(4$ points$)$ If the average service time increases to $15$ minutes per call but the coefficient

of variation of service times remains the same as before, what is the minimum number of

operators needed for the queueing system to be stable $($i$.$e$.$ average utilization to be below

$100\%)?$

n$)(15$ points$)$ If the cost of delays $($associated with the queueing time$)$ is $$30$ per hour per

call and the labor cost is $$15$ per hour per operator, is it better to employ $2$ or $3$ operators?

Calculate the total relevant costs for each option. Show all your work.

Total relevant costs $($including labor and delay costs$)$ with one operator: $\_\_\_\_\_\_\_\_$

Total relevant costs $($including labor and delay costs$)$ with two operators: $\_\_\_\_\_\_\_\_$

Which one is better? Explain clearly why.