A clever but ethically corrupted mathematics student used to sell assignment solutions to her lazy fellow students. The student, however, learned that she can make much more money by charging the students fees for doing the assignments with them. This semester the students need to complete three assignments. One in Statistical Models, one in Introduction to Partial Differential Equations and one very difficult one in Optimization and Financial Mathematics. It takes 2 hours to explain the assignment to one student for Statistical Models, 1/2 hours for Introduction to Partial Differential Equations and 5 hours for Optimization and Financial Mathematics. The student can charge 5$, 50 Cents and 10$ per hour for Statistical Models, Introduction to Partial Differential Equations and Optimization and Financial Mathematics, respectively. In Statistical Models there are not more than 10 students wanting her service. In Introduction to Partial Differential Equations there are more than 50 students wanting her service and she has obligations to take at least 50. In Optimization and Financial Mathematics,the course with the most diligent students, there are not more than 2 students wanting her service. How many students from each course will the clever student offer her service to maximize her profit, if she does not want to spend more than 50 hours on the assignments? (a) Formulate this as a linear programming problem (b) Solve the problem using the two-phase simplex algorithm.
A good way to start a problem such as this is to identify the variables, then the objective and constraints. What quantities does the question ask for? What objective is being maximized or minimized? Then see how you can express the objective and constraints in terms of these variables, and perhaps others.
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