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Federal Rent-a-Car is putting together a new fleet. It is considering package offers from three car manufacturers. Fred Motors is offering 5 small cars, 5 medium cars, and 10 large cars for $500,000. Admiral Motors is offering 5 small, 10 medium, and 5 large cars for $400,000. Chrysalis is offering 10 small, 5 medium, and 5 large cars for $300,000. Federal would like to buy at least 700 small cars, at least 600 medium cars, and at least 700 large cars. How many packages should it buy from each car maker to keep the total cost as small as possible?

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sqdancefan

Answer:

  • 40 packages from Fred Motors
  • 20 packages from Admiral Motors
  • 40 packages from Chrysalis

Step-by-step explanation:

I would formulate the problem like this. Let f, a, c represent the numbers of packages bought from Fred Motors, Admiral Motors, and Chrysalis, respectively. Then the function to minimize (in thousands) is …

  objective = 500f +400a +300c

The constraints on the numbers of cars purchased are …

  5f +5a +10c >= 700

  5f +10a +5c >= 600

  10f +5a +5c >= 700

Along with the usual f >=0, a>=0, c>=0. Of course, we want all these variables to be integers.

Any number of solvers are available in the Internet for systems like this. Shown in the attachments are the input and output of one of them.

The optimal purchase appears to be …

  • 40 packages from Fred Motors
  • 20 packages from Admiral Motors
  • 40 packages from Chrysalis

The total cost of these is $40 million.

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