1. (a) In decision analysis models, what do the terms decision alternatives, states of nature, and payoff represent? Give a real world example and identify these terms in your example.
(b) What are the different types of integer programming problems? Briefly describe each type and give one real world example for each type.
(c) How is the simulation process used in Decision Sciences models? What are the advantages of using simulation? What are its limitations? How can a simulation model be verified? Give a real world example where using simulation is appropriate.
(d) Briefly describe a single-server waiting line model and a multiple-server waiting line model. How are the two models different? Give one real world example for each model and identify the difference(s).
2. The Charm City Construction Company is considering six projects. The projects, the number of supervisors and the number of workers required for each project, and the expected profits for each project are given below.
Project
1 2 3 4 5 6
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Supervisors Required 4 3 5 3 4 2
Workers Required 15 24 35 20 24 30
Profit (in thousands of dollars) 200 280 320 220 280 180
The objective is to maximize the company's total expected profit subject to the following constraints:
- Use no more than 15 supervisors
- Use no more than 100 workers
- If project 2 is done, then project 4 must be done and vice versa
- At least four projects are to be done.
Formulate a capital budgeting integer programming problem by defining
(a) The decision variables.
(b) The objective function. What does it represent?
(c) All the constraints. What does each constraint represent?
Note: Do NOT solve the problem after formulating.
3. The Charm City Consultants Inc. wants to build a new network of computers for its employees. The management of the company is considering three network sizes for the possible purchase: large, medium, or small. The management believes that the demand for their services will be either high level, medium level, or low level. The payoff (profit in dollars) table for the situation is given below:
Demand Level
Decision High Medium Low
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Large Size $150,000 $ 60,000 $20,000
Medium Size $100,000 $110,000 $50,000
Small Size $ 60,000 $ 70,000 $80,000
(a) What is the best decision using the maximax criterion? What is the payoff for it?
(b) What is the best decision using the maximin criterion? What is the payoff for it?
(c) What is the best decision using the minimax regret criterion? What is the payoff for it?
(d) What is the best decision using the Hurwicz's criterion if α = 0.4? What is the payoff for it?
4. For the problem given in Question 3, assume that the probability of high demand level is 0.3, the probability of medium demand level is 0.4, and the probability of low demand level is 0.3.
(a) Calculate the expected value of each decision alternative. What is your recommendation using the expected value criterion?
(b) Calculate the expected opportunity loss value of each decision alternative. What is your recommendation using the expected opportunity loss criterion?
(c) Calculate and interpret the value of perfect information.
5. Randy works at an ice cream counter of a food court. Customers arrive at a mean rate of 6 per hour. The mean service rate is 9.5 per hour. Assume it is a single-server waiting line model.
(a) Determine the mean arrival rate and the mean service rate.
(b) Determine the probability that a customer will have an empty queue.
(c) Determine the probability that 3 customers are in the queuing system.
(d) Determine the average number of customers in the queue and the average number of customers in the system.
(e) Determine the average waiting time in the queue and the average total time in the system for a customer.
(f) Find the utilization factor of the server.
6. In Question 5, suppose Randy can be replaced by another server, Jodi, who must be paid $12 per hour whereas Randy is paid $8 per hour. Jodi can serve 11 customers per hour. If a customer's time is considered to be worth $10 per hour, is it worth to replace Randy with Jodi?