Problem Set 8

B.1, B.5, B.7, D.1, and D.9 (parts a through e; assume µ = 15)

B.1 Solve the following linear programming problem graphically:

B.5 Solve the following LP problem graphically:

B.7 The Attaran Corporation manufactures two electrical products: portable air conditioners and portable heaters. The assembly process for each is similar in that both require a certain amount of wiring and drilling. Each air conditioner takes 3 hours of wiring and 2 hours of drilling. Each heater must go through 2 hours of wiring and 1 hour of drilling. During the next production period, 240 hours of wiring time are available and up to 140 hours of drilling time may be used. Each air conditioner sold yields a profit of $25. Each heater assembled may be sold for a $15 profit. Formulate and solve this LP production-mix situation, and find the best combination of air conditioners and heaters that yields the highest profit.

D.1 Customers arrive at Rich Dunn’s Styling Shop at a rate of 3 per hour, distributed in a Poisson fashion. Rich can perform haircuts at a rate of 5 per hour, distributed exponentially.

a) Find the average number of customers waiting for haircuts.

b) Find the average number of customers in the shop.

c) Find the average time a customer waits until it is his or her turn.

d) Find the average time a customer spends in the shop.

e) Find the percentage of time that Rich is busy.

D.9 Neve Commercial Bank is the only bank in the town of York, Pennsylvania. On a typical Friday, an average of 10 customers per hour arrive at the bank to transact business. There is currently one teller at the bank, and the average time required to transact business is 4 minutes. It is assumed that service times may be described by the exponential distribution. If a single teller is used, find:

arrival time 6

processing time 4

exponential distribution 1

CV factor 1

Utilization 0.67

Utlization factor 2

a) The average time in the line.

b) The average number in the line.

c) The average time in the system.

d) The average number in the system.

e) The probability that the bank is empty

CEO Benjamin Neve is considering adding a second teller (who would work at the same rate as the first) to reduce the waiting time for customers. A single line would be used, and the customer at the front of the line would go to the first available bank teller. He assumes that this will cut the waiting time in half. If a second teller is added, find the new answers to parts (a) to (e).

With one more teller added

utilization

utilization factor

a) The average time in the line.

b) The average number in the line.

c) The average time in the system.

d) The average number in the system.

e) The probability that the bank is empty