ESM 503 – Fall 2010 Assignment # 2 Dr. Ali Diabat i

Assignment # 2Due: October 7th, 2010

1. Ole Oil produces three products: heating oil, gasoline, and jet fuel. The averageoctane levels must be at least 4.5 for heating oil, 8.5 for gas, and 7.0 for jet fuel. Toproduce these products Ole purchases two types of crude: crude 1 (at $12 per barrel)and crude 2 (at $10 per barrel). Each day, at most 10,000 barrels of each type of oilcan be purchased.

Before crude can be used to produce products for sale, it must be distilled. Each day,at most 15,000 barrels of oil can be distilled. It costs 10 cents to distill a barrel ofoil. The result of distillation is as follows: (1) Each barrel of crude 1 yields 0.6 barrelof naptha, 0.3 barrel of distilled 1, and 0.1 barrel of distilled 2. (2) Each barrel ofcrude 2 yields 0.4 barrel of naptha, 0.2 barrel of distilled 1, and 0.4 barrel of distilled2. Distilled naptha can be used only to produce gasoline or jet fuel. Distilled oil canbe used to produce heating oil or it can be sent through the catalytic cracker (at acost of 15 cents per barrel). Each day, at most 5,000 barrels of distilled oil can besent through the cracker. Each barrel of distilled 1 sent through the cracker yields 0.8barrel of cracked 1 and 0.2 barrel of cracked 2. Each barrel of distilled 2 sent throughthe cracker yields 0.7 barrel of cracked 1 and 0.3 barrel of cracked 2. Cracked oil canbe used to produce gasoline and jet fuel but not to produce heating oil.

The octane level of each type of oil is as follows: naptha, 8; distilled 1, 4; distilled 2,5; cracked 1, 9; cracked 2, 6.

All heating oil produced can be sold at $14 per barrel; all gasoline produced, $18 perbarrel; and all jet fuel produced, $16 per barrel. Marketing considerations dictatethat at least 3,000 barrels of each product must be produced daily. Formulate an LPto maximize Ole’s daily profit.

2. Silicon Valley Corporation (Silvco) manufactures transistors. An important aspectof the manufacture of transistors is the melting of the element germanium (a majorcomponent of a transistor) in a furnace. Unfortunately, the melting process yieldsgermanium of highly variable quality.

Two methods can be used to melt germanium; method 1 costs $50 per transistor, andmethod 2 costs $70 per transistor. The qualities of germanium obtained by methods1 and 2 are shown in the table below.

Grade of Melted Percent Yielded by MeltingGermanium Method 1 Method 2Defective 30 201 30 202 20 253 15 204 5 15

Silvco can refire melted germanium in an attempt to improve its quality. It costs $25to refire the melted germanium for one transistor. The results of the refiring processare shown in the table below.

ESM 503 – Fall 2010 Assignment # 2 Dr. Ali Diabat ii

Refired Grade Percent Yielded by Refiringof Germanium Defective Grade 1 Grade 2 Grade 3Defective 30 0 0 01 25 30 0 02 15 30 40 03 20 20 30 504 10 20 30 50

Silvco has sufficient furnace capacity to melt or refire germanium for at most 20,000transistors per month. Silvco’s monthly demands are for 1,000 grade 4 transistors,2,000 grade 3 transistors, 3,000 grade 2 transistors, and 3,000 grade 1 transistors. Notethat grade 1 is poor, and grade 4 is excellent. The quality of the germanium dictatesthe quality of the manufactured transistor. Use linear programming to minimize thecost of producing the needed transistors.

3. An engineer is working on a design problem with n parameters to be chosen, andeach has 3 possible values. A highly valid IP model is available, and he is thinkingof choosing the best design simply by using the model to evaluate the effect of everycombination of design parameter values.

(a) Explain why this approach will require running the IP model on 3n combinationsof decision-variable values.

(b) For n = 10, 15, 20, and 30, compute the time a computer would require to checkall the combinations, assuming that it runs 24 hours per day, 365 days per year,and that it requires 1 second to apply the model for any particular choice ofdesign parameters.

(c) Comment on the practical limits of this “try all combinations” analysis strategyin the light of your results from part (b).

4. astinghouse sells air conditioners. The annual demand for air conditioners in eachregion of the country is as follows: East: 100,000; South, 150,000; Midwest, 110,000;West, 90,000. Eastinghouse is considering building the air conditioners in four dif-ferent cities: New York, Atlanta, Chicago, and Los Angeles. The cost of producingan air conditioner in a city and shipping it to a region of the country is given in thetable below.

Price By Region ($)City East South Midwest WestNew York 206 225 230 290Atlanta 225 206 221 270Chicago 230 221 208 262Los Angeles 290 270 262 215

Any factory can produce as many as 150,000 air conditioners per year. The annualfixed cost of operating a factory in each city is given in the table below.

City Annual Fixed Cost ($ Million)New York 6Atlanta 5.5Chicago 5.8Los Angeles 6.2

ESM 503 – Fall 2010 Assignment # 2 Dr. Ali Diabat iii

At least 50,000 units of the Midwest demand for air conditioners must come fromNew York, or at least 50,000 units of the Midwest demand must come from Atlanta.Formulate an IP whose solution will tell Eastinghouse how to minimize the annualcost of meeting demand for air conditioners.

5. At a machine tool plant, five jobs must be completed each day. The time it takesto do each job depends on the machine used to do the job. If a machine is used atall, there is a setup time required. The relevant times (in minutes) are given in thetable below. The company’s goal is to minimize the sum of the setup and machineoperation times needed to complete all jobs. Formulate and solve (with MicrosoftExcel Solver) an IP whose solution will do this.

Job Machine SetupMachine 1 2 3 4 5 Time (Minutes)

1 42 70 93 X X 302 X 85 45 X X 403 58 X X 37 X 504 58 X 55 X 38 605 X 60 X 54 X 20