Quantitative Method

Myrtle Air Express decided to offer direct service from Cleveland to Myrtle Beach. Management must decide between a full-price service using the company's new fleet of jet aircraft and a discount service using smaller capacity commuter planes. It is clear that the best choice depends on the market reaction to the service Myrtle Air offers. Management developed estimates of the contribution to profit for each type of service based upon two possible levels of demand for service to Myrtle Beach: strong and weak. The following table shows the estimated quarterly profits (in thousands of dollars).
  
Demand for Service

Service ------Strong--- Weak
Full Price-- $960------ (-$490)
Discount--- $670--------$320
What is the decision to be made, what is the chance event, and what is the consequence for this problem? How many decision alternatives  are there? How many outcomes are there for the chance event?
B. If nothing is known about the probabilities of the chance outcomes, determine the recommended decision using the optimistic, conservative, and minimax regret approaches.
C. Suppose that management of Myrtle Air Express believes that the probability of strong demand is 0.7 and the probability of weak demand is 0.3. Use the expected value approach to determine an optimal decision.
D. Suppose that the probability of strong demand is 0.8 and the probability of weak demand is 0.2. What is the optimal decision using the expected value approach?


Q. 25
George Johnson recently inherited a large sum of money; he wants to use a portion of this money to set up a trust fund for his two children. The trust fund has two investment options: (1) a bond fund and (2) a stock fund. The projected returns over the life of the investments are 6% for the bond fund and 10% for the stock fund. Whatever portion of the inheritance George finally decides to commit to the trust fund, he wants to invest at least 30% of that amount in the bond form. In addition, he wants to select a mix that will enable him to obtain a total return of at least 7.5%.
a.	Formulate a linear programming model that can be used to determine the percentage that should be allocated to each of the possible investment alternatives.
B. Solve the problem using the graphical solution procedure


Q. 25 Linear Programming Problem
Georgia Cabinets manufactures kitchen cabinets that are sold to local dealers throughout the Southeast. Because of a large backlog of orders for oak and cherry cabinets, the company decided to contract with three smaller cabinetmakers to do the final finishing operation. For the three cabinetmakers, the number of hours required to complete all the oak cabinets, the number of hours required to complete all the cherry cabinets, the number of hours available for the final finishing operation, and the cost per hour to perform the work are shown here. (see file attached).
For example, Cabinetmaker 1 estimates that it will take 50 hours to complete all the oak cabinets and 60 hours to complete all the cherry cabinets. However, Cabinetmaker 1 only has 40 hours available for the final finishing operation. Thus, Cabinetmaker 1 can only complete 40/50 = 0.80 or 80% of the oak cabinets if it worked only on oak cabinets. Similarly, Cabinetmaker 1 can only complete 40/60 = 0.67 or 67% of the cherry cabinets if it worked only on cherry cabinets.
Formulate a linear programming model that can be used to determine the percentage of the oak cabinets and the percentage of the cherry cabinets that should be given to each of the three cabinetmakers in order to minimize the total cost of completing both projects.
Solve the model formulated in part (a). What percentage of the oak cabinets and what percentage of the cherry cabinets should be assigned to each cabinetmaker? What is the total cost of completing both projects?
If Cabinetmaker 1 has additional hours available, would the optimal solution change? Explain.
If Cabinetmaker 2 has additional hours available, would the optimal solution change? Explain.
Suppose Cabinetmaker 2 reduced its cost to $38 per hour. What effect would this change have on the optimal solution? Explain.

Q.15
Doug Casey is in charge of planning and coordinating next spring- sales management training program for his company. Doug listed the following activity information for this project.

Activity Immediate Predecessor Optimistic Most probable Pesimistic
A Plan topic - 1.5 2.0 2.5
B Obtain speakers A 2.0 2.5 6.0
C List meeting location - 1.0 2.0 3.0
D Select location C 1.5 2.0 2.5
E Finalize speaker travel plan B,D 0.5 1.0 1.5
F Make final check with speakers E 1.0 2.0 3.0
G Prepare and mail brochure B,D 3.0 3.5 7.0
H Take reservations G 3.0 4.0 5.0
I Handle last minute details F,H 1.5 2.0 2.5


a. Draw a project network
b. Prepare an activity schedule.
c. What is the critical activities and what is the expected project completion time?
d.If Doug wants a 0.99 probability of completing the project on time, how far ahead of the scheduled meeting date should he begin working on the project?

Q.1.  
Willow Brook National Bank operates a drive-up teller window that allows customers to
complete bank transactions without getting out of their cars. On weekday mornings, arrivals to
the drive-up teller window occur at random, with an arrival rate of 24 customers per hour or 0.4
customers per minute.
a. What is the mean or expected number of customers that will arrive in a five-minute
period?
b. Assume that the Poisson probability distribution can be used to describe the arrival
process. Use the arrival rate in part (a) and compute the probabilities that exactly 0, 1,
2, and 3 customers will arrive during a five-minute period.
c. Delays are expected if more than three customers arrive during any five-minute period.
What is the probability that delays will occur?
Q.2. In the Willow Brook National Bank waiting line system (see Problem 1), assume that the service
times for the drive-up teller follow an exponential probability distribution with a service rate of
36 customers per hour, or 0.6 customer per minute. Use the exponential probability distribution
to answer the following questions:
a. What is the probability the service time is one minute or less?
b. What is the probability the service time is two minutes or less?
c. What is the probability the service time is more than two minutes?

Q3. 
Use the single-channel drive-up bank teller operation referred to in Problems 1 and 2 to
determine the following operating characteristics for the system:
a. The probability that no customers are in the system
b. The average number of customers waiting
c. The average number of customers in the system
d. The average time a customer spends waiting
e. The average time a customer spends in the system
f. The probability that arriving customers will have to wait for service