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.

a. What is the manual operations queueing model (M/D/1 or M/M/1)?

b. What is the average time in the system in the manual operations (in minutes)?

c. What is the automatic operations queueing model (M/D/1 or M/M/1)?

d. What is the average time in the system in the automatic operations (in minutes)?

e. Calculate savings per product due to the automatic device?

f. Calculate savings for one year due to the automatic device?

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

Answer 1

(a)

M/M/1

(b)

L = average arrival rate = 50 per hour
M = average service rate = 60 per hour

Ws = average time in the system = 1 / (M - L) = 1/10 hours = 6 minutes

(c)

M/D/1

(d)

L = average arrival rate = 50 per hour
M = average service rate = 120 per hour

Ws = average time in the system = L/(2M*(M - L)) + 1/M = 50/(2*120*70) + 1/120 = 0.0113 hours = 0.679 minutes

(e)

Savings per product = (6 - 0.679) x $2 = $10.64

(f)

Savings for one year = 1500 x $10.64 = $15,964.3

(g)

Yes, because the savings exceeds the cost.