Question

Professor Proof is trying to arrange for the implementation of his latest operations research algorithm in a computer program. He can contract with any mix of three sources for help: unlimited hours from undergraduates at $4/hour; up to 500 hours of graduate students at the rate of $10/hour; or unlimited help from professional programmers at $25/hour. The full project would take a professional at least 1000 hours, but graduate students are only 0.3 as productive, and undergraduates are 0.2 as productive. Proof only has 164 hours of his own time to devote to the effort, and he knows from experience that undergraduate programmers require more supervision than graduate students, who, in turn, require more than professionals. In particular, he estimates that he will have to invest 0.2 hours of his own time per hour of undergraduate programming, 0.1 hour of his time per hour of graduate programming, and 0.05 hour of his time per hour of professional programming. a) Formulate the problem as an LP model and find the optimum solution. b) If the cost per hour of undergraduate, graduate, and professional were $6, $7, and $30 respectively, would the current solution still be optimal? c) Suppose that Professor Proof decides to allow unlimited hours of graduate student programming. Could this revision change the optimal solution? d) How much would the hourly rate of graduate student programmers have to be reduced before the professor might optimally hire some?

          Professor Proof is trying to arrange for the implementation of his latest operations research algorithm in a computer program. He can contract with any mix of three sources for help: unlimited hours from undergraduates at $4/hour; up to 500 hours of graduate students at the rate of $10/hour; or unlimited help from professional programmers at $25/hour. The full project would take a professional at least 1000 hours, but graduate students are only 0.3 as productive, and undergraduates are 0.2 as productive. Proof only has 164 hours of his own time to devote to the effort, and he knows from experience that undergraduate programmers require more supervision than graduate students, who, in turn, require more than professionals. In particular, he estimates that he will have to invest 0.2 hours of his own time per hour of undergraduate programming, 0.1 hour of his time per hour of graduate programming, and 0.05 hour of his time per hour of professional programming.

a) Formulate the problem as an LP model and find the optimum solution.
b) If the cost per hour of undergraduate, graduate, and professional were $6, $7, and $30 respectively, would the current solution still be optimal?
c) Suppose that Professor Proof decides to allow unlimited hours of graduate student programming. Could this revision change the optimal solution?
d) How much would the hourly rate of graduate student programmers have to be reduced before the professor might optimally hire some?

Added by Edward C.

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