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.

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

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

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

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

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

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

7. 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

Given

Arrival rate of products, - 50 per hour, Poisson distribution

Manual, Service time distribution – exponential distribution

Automatic, Service time distribution – constant

Number of servers- single for both manual and automatic

1. What is the manual operations queuing model (M/D/1 or M/M/1)? (1 point)

Given - for manual

Arrival rate of products, - 50 per hour, Poisson distribution

Manual, Service time distribution – exponential distribution

Automatic, Service time distribution – constant

Number of server- single

M/M/1 –criteria

• Arrival of consumers- Poisson distribution
• Service time distribution – exponential distribution
• Single server

M/D/1 -criteria

• Single server
• Arrival of consumers- Poisson distribution
• Service time distribution – constant

Manual operations has the same as M/M/1 –criteria

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

Arrival rate of products,- 50 per hour, Poisson distribution

Service rate, µ = 60 per hour

Average time spent in the system (waiting line + being served),W

=1/ (µ -) =1/(60-50) =0.1 hour or 6 minutes

3. What is the automatic operations queuing model (M/D/1 or M/M/1) ? (1 point)

Given- for automatic

Arrival rate of products, - 50 per hour, Poisson distribution

Automatic, Service time distribution – constant

Number of servers- single

M/M/1 –criteria

• Arrival of consumers- Poisson distribution
• Service time distribution – exponential distribution
• Single server

M/D/1 - Single server model with constant service time -criteria

• Single server
• Arrival of consumers- Poisson distribution
• Service time distribution – constant

Automatic operation has the same as M/D/1 –criteria

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

Arrival rate of products, - 50 per hour, Poisson distribution

Service time = 0.5 minutes = 0.5/60 hour

Service rate per hour, µ= 1/service time = 1/( 0.5/60 ) =120 products per hour

LQ = = 502 / 2.120(120-50) =0.1488 product

WQ = LQ / =0.1488/50 =0.0030 hour

Average time spent in the system (waiting line + being served), W

= WQ + 1/µ

=0.0030 + 1/120

=0.0030 + 0.0833

=0.0113 hour or

0.6786 minutes

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

Manual, Average time spent in the system for a product, W= 6 minutes

Automatic, Average time spent in the system for a product, W =0.6786 minutes

Time saved in automatic per product = 6-0.6786 =5.3214 minutes

Savings per product = 5.3214 minutes * $2 savings per minute

=$ 10.64

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

Annual production =1500 products

Savings per product = $ 10.64

Savings for one year = 1500 *$ 10.64=$15964.29

7. 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)

If the device is installed, Savings for one year = $15964.29.

Hence the device should be installed and the device would offset its cost in one year