Question 1
The Precision Engineering Service Company uses engineers and technicians to work on a design project for its customers. Engineers cost more per hour than technicians, but they also require less supervision and secretarial assistance than technicians. technicians. Engineers cost $100 per hour and technicians cost $50 per hour. Each hour of engineering time requires ¼ hour of supervision time and ¾ hour of secretarial assistance. assistance. Each hour of technician time requires r equires ½ hour of supervision time and 1 hour of secretarial assistance. The company company bid and won a large design project several months ago and now must determine how many engineering and technician hours to use on the project. It bid a total labor cost of $100,000, and management will not tolerate labor cost overruns because of a severe cash shortage. A maximum of 600 hours of supervision and 1,500 hours of secretarial assistance is available for this project. If Precision charges its customer a hourly rate when labor is used so that each hour of technician time yields $40 in profit and each hour of engineering time yields $30 in profit, how many engineering and technician hours, respectively, should be allocated to the project? a. In words, what is the objective function? Maximize the profit b. What are the decision parameters(Variables???)? parameters(Variables???)? X1 - Number of engineering hours ; X2 – Number of technician hours. c. Formulate the mathematical mathematical optimization problem based on the problem description. Max 30X1 + 40X2 S.t. ¼ X1 +1/2 X2 <= 600 (supervision-hours) 100 X1 + 50 X2 <= 100,000 (cash-$) ¾ X1 + X2 <=1500 (Secretarial-hours) X1, X2 >0 X1, X2 integer (optional)
This problem has been solved in Excel and the following output was received.
Answer Report:
Sensitivity Report:
d. How many hours of engineering and technician time should be allocated to the project? Engineering hours – 533, technician hours – 933 (except non-integer numbers as well) e. What is the maximum profit that can be achieved? 53,333.33 $ f.
One of the managers suggested that he can contribute 10 hours of his time (??paid or unpaid??) for supervision. How it will change the optimal solution (amount of hours and/or total profit)? It will increase the target function (profit) by 66.667 * 10 = 666.7$ (using the shadow price from sensitivity report)
It will change the final values of X1 and X2, but we can say by how much without re-running the excel. g. Another manager is willing to contribute 30 hours of his personal secretary to help the engineers and technicians working on the project. How it will change the optimal solution (amount of hours and/or total profit)? It will do noting as Secretarial-hours is not binding constrain. Question 2:
You are a senior business analyst in a chocolate making company. The company largely produces one particular type of crunchy chocolate bar. There are two machines in the plant that produce this chocolate bar. The maintenance costs per day incurred on these two machines are $100 and $120, respectively. The manufacturing cost per chocolate bar is $2.5 for Machine-1 and $2 for Machine-2. The maximum daily production capacity for Machine-1 and Machine-2 are 1100 and 1250, respectively, and the company must produce at least 1000 chocolate bars per day. A. Develop (give a mathematical representation of) an optimization programming model for minimizing the total cost of production. Answer: Let
C 1 =
the number of chocolate bars produced by Machine-1
C 2 =
the number of chocolate bars produced by Machine-2
Y 1 =
1 if Machine-1 produces chocolate bar; 0, otherwise
Y2 = 1 if Machine-2 produces chocolate bar; 0, otherwise
Min 2.5C 1 + 2C 2 + 100Y 1 + 120Y 2 s.t. C 1
≤ 1100 Y 1
C 2
≤ 1250 Y 2
C 1 + C 2
≥ 1000
C 1, C 2 ≥
0 and integer
Y 1, Y 2 =
0, 1
B. One of the junior business analysts has developed an optimization Excel file to solve this problem. Examine the print screens from his file, given below, and determine if there are any mistakes in the file.
Answer: 1. Forgot using the Y1 and Y2 2. Forgot C 1, C 2 ≥ 0 3. GRG nonlinear 4. Type in capacity of Machine-2 c. The company is now interesting in saving money by only operating one machine. What constraint should you add to the optimization problem in question a? Answer: Y 1 + Y 2 =
1
Question 3
An investor is to purchase one of three types of real estate . The investor must decide among an apartment building, an office building, and a warehouse. Two future states of nature determine how much profit the investor will make: state one is characterized by good economic conditions, and state two is characteriz ed by poor economic conditions. The profits that will result from each decision in the event of each state of nature are shown in the following payoff table:
A. Determine the best decision without probabilities based on the following criterion: 1. Maximax, 2. Maximin 3. Minimax Regret B. Let us suppose that, based on several economic forecasts, the investor is able to estimate a .60 probability that good economic conditions will prevail and a 0.40 probabilit y that poor economic conditions will prevail. 1. 2. 3. 4.
Determine the best decision using expected value Determine the best decision using expected opportunity loss Compute the expected value of perfect information Develop a decision tree with the expected value a t each of the nodes
C. Suppose that the investor has decided to hire a professional economic analyst who will provide additional information about the future economic conditions. Based on past history, the investor knows the prediction quality (conditional probabilities) of the economist as follows: P(Pg) = .80 and P(N p) = .90 Where g = good economic conditions p = poor economic conditions P = positive economic report (predict good) N = negative economic report (predict poor)
1. Determine posterior probabilities using Bayes’s rule (or Contingency Table)
2. Perform a decision tree analysis using the posteri or probability obtained in part 1.
