Suppose you are the manager of a coffee shop with three servers, who each take a

Question

Suppose you are the manager of a coffee shop with three servers, who each take an average of 1.6 minutes to serve a customer. You are concerned about your service during peak hours, and are considering making an investment to improve service. During peak hours, you have 1.2 customers per minute arriving on average. You have two options to ensure faster service: hire a fourth server at an annual cost of $39,000, or rent faster dispensing machines at an annual cost of $27,000, which would reduce service time to 1.25 minutes, on average. You would like to find the cheapest solution that will allow you to serve the customers adequately - and you decide this means that you don't want to have a more than 10% chance of more customers arriving than you can serve. For instance, with your current operation, you can serve three customers in 1.6 minutes, so you don't want the chance of more than three customers within 1.6 minutes to be greater than 10%. What should you do - continue the current operation, hire a fourth server, or rent faster dispensing machines?

Explanation / Answer

Solution

Back-up Theory

If X = number of customer arrivals per minute, Y = number of customer arrivals per 1.6 minutes, Z = number of customer arrivals per 1.25 minutes, and X ~ Poisson (), then

Y ~ Poisson (1.6)…………………………………………………………………(1)

Z ~ Poisson (1.25)……………………………………………………………..…(2)

Now, to work out the solution,

Given = 1.2

Decision Option 1: Continue Current Operation

3 servers service 3 customers in 1.6 minutes and we want P[Number of arrivals > 3] 0.1 or

1 - P[Number of arrivals 3] 0.1 or P[Number of arrivals 3] 0.9.

Here the mean arrival rate = 1.6 = 1.92 [vide (1) under Back-up Theory].

Using Excel Function, P[Number of arrivals 3] = 0.8713 which is not greater than 0.9. So, the current operation does not meet the requirement ………………………………. (3)

Decision Option 2: Hire a 4th server

4 servers service 4 customers in 1.6 minutes and we want P[Number of arrivals > 4] 0.1 or

1 - P[Number of arrivals 4] 0.1 or P[Number of arrivals 4] 0.9.

Here the mean arrival rate = 1.6 = 1.92.[vide (1) under Back-up Theory]

Using Excel Function, P[Number of arrivals 4] = 0.9543 which is greater than 0.9. So, hiring 4th server does meet the requirement, but will cost $39000 …………………. (4)

Decision Option 3: Rent Faster Dispensing Machines

3 servers service 3 customers in 1.25 minutes and we want P[Number of arrivals > 3] 0.1 or 1 - P[Number of arrivals 3] 0.1 or P[Number of arrivals 3] 0.9.

Here the mean arrival rate = 1.25 = 1.5[vide (2) under Back-up Theory].

Using Excel Function, P[Number of arrivals 3] = 0.9344 which is greater than 0.9. So, Rent Faster Dispensing Machines

does meet the requirement, but will cost $27000 ……………….…………………. (5)

Comparing (3), (4) and (5), the optimum decision is:

Rent Faster Dispensing Machines ANSWER

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.