Question

A manufacturer is studying a proposal to install an automatic device at one of its production operations. The device would perform the operation in exactly 0.5 minutes. At present, there is a (single server) manual operation with an average service rate of 60 per hour and exponential service times. The arrival rate is 50 products per hour and Poisson distributed. Each minute saved per product at the operation is worth $2. Assume that the total production for the year is 1,500.

What is the manual operations queueing model (M/D/1 or M/M/1; select from the list in the Answersheet)? (1 point)

What is the average time in the system in the manual operations (in minutes)? (2 points)

What is the automatic operations queueing model (M/D/1 or M/M/1; select from the list in the Answersheet)? (1 point)

What is the average time in the system in the automatic operations (in minutes)? (2 points)

Calculate savings per product due to the automatic device? (2 points)

Calculate savings for one year due to the automatic device? (1 point)

If the device costs $10,000, should the device be installed, i.e., would it offset its cost in one year (Yes/No)? (1 point)

Answer 1

1) The manual operation is a M/M/1 queueing model

Arrival rate, a = 50 per hour

Service rate s = 60 per hour

2) Average time in the system in manual operations (W) = 1/(s-a) = 1/(60-50) = 0.1 hour = 6 minutes

3) In the automatic operation, service time is deterministic. So, it is M/D/1 model.

Service rate = 1/(0.5/60) = 120 per hour

4) Average time in the system in automatic operations (W) = =1/s + a/(2s(s-a)) = 1/120 + 50/(2*120*(120-50)) = 0.0113 hour = 0.6786 minutes

5) Savings per product due to automatic device = (6-0.6786)*2 = $ 10.64

6) Savings per year = 1500*10.64 = $ 15,964

7) Yes, because the annual savings are $ 15,964, which will offset its cost within one year.