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M ASSACHUSETTS I NSTITUTE OF T ECHNOLOGY 15.053 – Optimization Methods in Management Science (Spring 2007) Problem Set 3 Due March 1 st , 2007 at 4:30 pm You will need 116 points out of 137 to receive a grade of 5. Problem 1: Britney’s New Life (30 Points) This problem allows you to practice the simplex algorithm and to review some concepts from geometry. Since splitting with Kevin, Britney is slowly learning the harsh realties of having to manage on her own. She no longer has Kevin to help her with taking care of the kids, cleaning the house, managing the garden, doing laundry, and most importantly solving her linear programs. Having not taken a linear programming class, she turns to “you” for help. Britney’s goal is to maximize her utility points. Currently Britney needs to divide her hours each day between activities including: partying with Paris and Lindsey, cutting her hair, and taking care of her children. According to her mother, each hour of partying is worth 5 utility points; each hour she spends on her hair is worth 3 utility points; and each hour taking care of the kids is worth 1 utility point. Due to the fact she has not toured or sold a CD in more then 5 years, Britney cannot spend more then $6 in a given day. Partying costs Britney $1 an hour. (Fortunately, Paris pays for all her drinks. Hair styling costs $1 an hour, and caring for the kids costs $3 an hour. (She actually watches while she pays a local teenager to care for them.) In a given day, Britney has at most 15 units of energy to spend. Partying for one hour takes up 6 energy units; hair styling for an hour takes up 3 energy units; and caring for the kids for one hour takes up 6 energy units. We assume that the total time spent on these activities can be at most 24 hours. Any amount of time not spend on these activities, she spends sleeping. Part A: Using Britney’s considerations above formulate a linear program that will optimally allocate the number of hours to partying, hair styling, and caring for the kids. (Hint: Your LP should have exactly three variables) Part B: Page 1 of 9
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Convert the LP in Part A to standard form by adding variables. For each variable you added write a sentence about what that variable represents. Part C: Is the constraint that models “the total amount of time spent in a day on the three activities must be 24 hours or less redundant”. If it is explain why and remove it, if not explain why and leave it in the formulation. Part D: Identify a feasible solution where all basic variables are slack variables. Part E: Using your solution in part D fill out the first Simplex Tableau. Part F: Solve the problem using the Simplex Method. For each iteration write down the starting Tableau and indicate the pivot element. Part G: Does Britney have multiple optimal ways to divide her time? Please explain your answer.
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