CaliforniaStatePolytechnicUniversity, POMONA

Industrial and Manufacturing Engineering Department

IE 416 MIDTERM EXAM Date: 2-10-03

Last name: First name:

Instructions: Read all the questions carefully. Organize your work clearly, neatly, and legibly. Show all details of your work. Identify your answers. Do not write on the back of any page. This is a closed-book and closed-note exam. You may have one sheet of information. The total score is 35 points.

Good Luck

Dr. Parisay

1- What are the applications of slack variable, artificial variable, and excess variable from practical point of view and from simplex method point of view? Write your comment in the following table. (3 points)

Variable / Simplex Method / Practical
Slack
Excess
Artificial

2- Define “reduced cost” and provide one example for its application. (3 points)

3- Fruit Computer Company is ready to make its annual purchase of computer chips. Fruit can purchase chips from three suppliers. Each chip is rated as being excellent, good, or mediocre quality. The characteristics of the chips purchased (in lots of 100) from each supplier are shown in the following table. During the coming year, Fruit will need at least 4000 excellent chips, 5000 good chips, and 3000 mediocre chips. Each year, Fruit has budgeted $130,000 to spend on chips. Use preemptive goal programming to formulate the problem. Let the restriction on the number of Excellent chips have the higher priority. Then, followed by the budget constraint. (4 points)

. Characteristics of a lot of 100 chips .

.ExcellentGoodMediocrePrice per 100 chips

Supplier 1 50 15 35$1.5

Supplier 2 30 25 45$2.4

Supplier 3 65 20 15$3.1

4- Sailco Corporation must determine how many sailboats should be produced during each of the next three quarters (one quarter = three months). The demand during each of the next three quarters is as follows: first quarter, 50 sailboats: second quarter, 80 sailboats; third quarter, 70 sailboats. Sailco must meet demands on time. At the beginning of the first quarter, Sailco has an inventory of 30 sailboats. At the beginning of each quarter, Sailco must decide how many sailboats should be produced during that quarter. During each quarter, Sailco can produce up to 65 sailboats with regular-time labor at a total cost of $550 per sailboat. At the end of each quarter (after production has occurred and the current quarter’s demand has been satisfied), a carrying or holding cost of $15 per sailboat is incurred. Sailco would like to have at least 20 units of sailboats in inventory at the end of third quarter. (9 points)

a) Summarize the problem as a table as much as you can.

b) Define decision variables in detail.

c) Formulate the problem using linear programming to determine a production schedule to minimize the sum of production and inventory costs during the next four quarters.

5- Create the first simplex tableau for the following problem. (6 points)

Min Z = 10 X1 + 20 X2

S.T. X1 + X2 =< 15

2 X1 - X2 >= 22

6- Chandler Oil Company has 5000 barrels of oil 1 and 10,000 barrels of oil 2. The company sells two products: gasoline and heating oil. Both products are produced by combining oil 1 and oil 2. The quality level of each oil is as follows: oil 1, 10; oil 2, 5. Gasoline must have an average quality level of at least 8 and heating oil, at least 6. Demand for each product must be created by advertising. Each dollar spent advertising gasoline creates 5 barrels of demand and each spent on heating oil creates 10 barrel of demand. Gasoline is sold for $25 per barrel, heating oil for $20. Chandler would like to maximize profit. The problem formulation as an LP and its solution is attached. (10 points)

a) If Gasoline is sold for only $15 per barrel, how will the optimal solution and optimal objective function value change?

b) The company would like to consider producing more products. They plan to buy an extra 200 barrels of oil 1. However, the supplier is asking for $10 more than the basic price for each barrel of oil 1. Should the company accept this offer? Explain.

c) Draw a sensitivity graph to show the effect of changes in number of barrels of oil 2 available on the total profit.

d) Write a report of this solution for a hypothetical manager. Use a language understandable to most people. Do not use mathematical abbreviations. Also, explain any major findings from your sensitivity analysis.

Definition Variables:

Aj = dollars spend daily on advertising product j (where j = gasoline, heating oil) (Ag, Ah)

Xij = barrels of oil i used daily to produce product j (where i = 1,2 j = gasoline, heating oil) (i.e., X1G)

OF: Max Z = 25 X1G + 25X2G + 20X1H + 20X2H - 1Ag - 1Ah

ST:

X1G + X2G -5Ag = 0(gasoline demand created by ad)

X1H + X2H – 10 Ah = 0(heating oil demand created by ad)

X1G + X1H <= 5000(at most 5000 barrels of oil 1 available)

X2G + X2H <= 10000(at most 10000 barrels of oil 2 available)

2X1G - 3X2G >= 0(total quality in mixture/gasoline in mixture) >= 8

4X1H - 1X2H >= 0(total quality in mixture/heating oil in mixture) >= 6

Xij , Aj >= 0

Input Data:

Output Results: