Problem Set (Integer Programming: Formulations) 1. The board of directors of General Wheels Co. is considering six large capital investments. Each investment can be made only once. These investments differ in the estimated long-run profit (NPV) that they will generate as well as in the amount of capital required, as shown by the following table (in units of millions of dollars): Investment Opportunity Estimated 1 2 3 4 5 6 15 12 16 18 9 11 38 33 39 45 23 27 Profit Capital Required The total amount of capital available for these investments is $100 million. Investment opportunities 1 and 2 are mutually exclusive, and so are 3 and 4. Furthermore, neither 3 nor 4 can be undertaken unless one of the first two opportunities is undertaken. There are no such restrictions on investment opportunities 5 and 6. The objective is to select the combination of capital investments that will maximize the total estimated long-run profit (NPV). Formulate a BIP model for this problem. 2. DAC is considering manufacturing of three types of cars: compact, sedan, and SUV. The resources required for, and the profits yielded by, each type of car are shown in the following table. Currently, 6000 tons of steel and 60000 hours of labor are available. For production of a type of car to be economically feasible, at least 1000 cars of that type must be produced. Formulate an IP to maximize DAC’s profit. Car Type Resources Compact Sedan SUV Steel required (in tons) 1.5 3 5 Labor required (in hours) 30 25 40 Profit yielded ($) 2000 3000 4000 3. There are six cities (Cities 1-6) in Joka County. The county must determine where to build fire stations. The county wants to build the minimum number of fire stations needed to ensure that at least one fire station is within 15 minutes (driving time) of each city. The times (in minutes) required to drive between the cities in Joka County are shown in the table below. Formulate an IP that will tell Joka County how many fire stations should be built and where they should be located. To From City 1 City 2 City 3 City 4 City 5 City 6 City 1 0 10 20 30 30 20 City 2 10 0 25 35 20 10 City 3 20 25 0 15 30 20 City 4 30 35 15 0 15 25 City 5 30 20 30 15 0 14 City 6 20 10 20 25 14 0 4. EGas produces two types of gasoline (Gas 1 and Gas 2) from two types of oil (Oil 1 and Oil 2). Each gallon of Gas 1 must contain at least 50% Oil 1 and each gallon of Gas 2 must contain at least 60% Oil 1. Each gallon of Gas 1 can be sold for $12 and each gallon of Gas 2 can be sold for $14. Currently, 500 gallons of Oil 1 and 1000 gallons of Oil 2 are available. As many as 1500 more gallons of Oil 1 can be purchased at the following prices: first 5000 gallons at $25 per gallon; next 500 gallons at $20 per gallon; next 500 gallons at $15 per gallon. Formulate an IP that will maximize EGas’s profit. 5. TSire has a $20000 advertising budget. Tsire can purchase full-page ads in two magazines: Business Tomorrow (BT) and Family Square (FS). An exposure occurs when a person reads a TSire ad for the first time. The number of exposures generated by each ad in BT is as follows: ads 1-6 give 10000 exposures each; ads 7-10 give 3000 exposures each; ads 11-15 give 2500 exposures each; while ads 16 onwards give zero exposures each. For example, 8 ads in BT would generate 6*10000 + 2*3000 = 66000 exposures. The number of exposures generated by each ad in FS is as follows: ads 1-4, 8000 exposures each; ads 5-12, 6000 exposures each; ads 13-15, 2000 exposures; ads 16+, 0 exposures each. Thus, 13 ads in FS would generate 4*8000 + 8*6000 + 1*2000 = 82000 exposures. Each full page ad in either magazine costs $1000. Assume there is no overlap in the readership of the two magazines. Formulate an IP to maximize the number of exposures that TSire can obtain with the limited advertising budget.