# ps1 - 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 1 Due February 15 th , 2007 4:30 pm in the 15.053 Box You will need 100 points out of 124 to receive a grade of 5. Problem 1: An optimized yarn (24 points, 4 points per part.) The purpose of this problem is to practice formulating optimization problems and to start thinking about the different properties they contain. Suppose we are given a piece of yarn that is 12m long. Part A: Our objective is to cut the yarn into four pieces and form a rectangle with maximum area. Formulate an optimization problem that when solved will give the lengths that maximize the area. Is your optimization problem a linear program? Part B: Solve the optimization problem from “part a” using results from single variable calculus. What is the optimum solution? Part C: Ollie the Owl, wise and curious, asks what is the largest area that can be formed if we cut the yarn into three pieces and form a triangle. Formulate an optimization problem whose solution will give you the optimal lengths. Be sure that your model eliminates spurious solutions such as 1, 1, 10. Use Heron’s Formula: The area of a triangle with sides w, x and y is s ( s w )( s x )( s y ) , where s = (w + x + y)/2. Part D: Solve the optimization problem in part c and comment on the solution. Page 1 of 10

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HINT 1: Suppose that f ( w , x , y ) > 0 for all w , x , y . A solution is maximum with respect to the objective function f ( w , x , y ) if and only if it is maximum with respect to the objective function f ( w , x , y ). HINT 2. Use results from single variable calculus to show that for any fixed value of y with 0 < y < 6, the optimal solution will have w = x. Does your proof also show that x = y in an optimal solution? Part E: Suppose you are asked to cut the yarn into four pieces to form a quadrilateral with maximum area. What do you think the optimal solution is? Formulate an optimization problem that when solved will give you the optimal solution. Hint: Go to mathworld.wolfram.com/BrahmaguptasFormula.html and use Brahmagupta’s Formula for your objective function, assuming that cos(.5(A+B)) = 0 Part F: (Bonus 2 Points): Solve the problem in part e, assuming that cos(.5(A+B)) = 0. Did you get the solution you were expecting? Problem 2: Beer Production (28 points, 4 points per part) The goal of this problem is to introduce you to a production problem and to practice adding additional constraints to a model as new restrictions become known. The key to this problem is to define the proper decision variables. One of the main uses of linear programming is to determine an optimal allocation of resources for a firm that produces multiple goods with limited resources. In this problem we develop a model for the Anteater-Bugs beer company. We start out with a basic core problem and then add additional constraints to further enhance the applicability of the model. The Anteater-Bugs corporation has two main brands of beer: Bugwheezer and Bug-Lite. Each
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ps1 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY 15.053...

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