# ps4 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY 15.053...

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M ASSACHUSETTS I NSTITUTE OF T ECHNOLOGY 15.053 – Optimization Methods in Management Science (Spring 2007) Problem Set 4 Due March 8 th , 2007 at 4:30 pm. You will need 162 points out of 189 to receive a grade of 5. Problem 1: Phase 1 & Phase 2 (30 Points) The purpose of this problem is to give you practice solving a linear program completely. Donald Trump is too busy getting ready for Wrestlemania and worrying that he may soon look like Britney. http://abcnews.go.com/Entertainment/wireStory?id=2887820 Due to this he has no time to solve a linear program to decide how much to invest in three Projects: Project A is to determine how much to sell his casino for in Las Vegas. That is why it is non positive) Project B is a machine to create his new wives each time one ages past 30. Project C is a media campaign to destroy Rosie O’Donnell and “The View.” The variables are a, b, and c. Don’t be concerned if the model doesn’t make much sense. In order to determine how much he should invest in these projects he has formulated the following linear program not in standard form. min: zabc =++ St : a 2 b + 3 c = 3 −+ a + 2 b + 6 c = 5 a 0 , 0 bc He hires you, a bright 15.053 student, (Kevin Federline was busy watching “The View”) as his apprentice to solve the problem. He gives you the following set of instructions. Don’t mess up or else you will be fired. Part A: Page 1 of 12

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Convert the problem from a min to a max problem Part B: Convert the problem to standard form. Be sure to show all of your steps. Part C: Give an example of a constraint that if added would make the problem infeasible. Part D: Form the auxiliary Phase 1 Problem from the linear program you formulated in part B. Part E: Write out the initial phase one tableau and indicate the initial BFS. Part F: Using the phase one method find an initial BFS of the standard form problem. For each iteration write out the tableau and indicate the pivot element. Hint: You will need three pivots Part F: What is the optimal solution to the phase 1 problem? Part G: Write out the initial tableau for the Phase 2 problem. What is the initial BFS? Part H: Solve the Phase 2 part of the problem using the simplex method. Be sure to write out each of your tableaus and indicate the pivot element on each iteration. Part I: Are there multiple optimal solutions to this problem? Problem 2: Alex and Paris in Paris Texas (28 Points) The purpose of this problem is to get you to think about the geometric and algebraic interplay of the Simplex Method. Page 2 of 12
Paris Hilton and Alex Trebeck were having lunch at a café in Paris, Texas. (These were two of your top celebrity choices). After talking about their respective run-ins with the law, they decide to so something more fun. They solve a maximization problem using the simplex method. However halfway through, Paris spills her Jack and Coke on the sheet of paper they were using, blurring some of the entries (these unknown entries are shown with letters in the tableau). Their tableau is shown below: z x 1 x 2 x 3 x 4 x 5 x 6 x 7 RHS 1 0 0 0 C I D E F 0 0 1 0

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## This note was uploaded on 01/17/2012 for the course MGMT 15.053 taught by Professor Jamesorli during the Spring '07 term at MIT.

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ps4 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY 15.053...

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