exam_1_s - ESE304 Introduction to Optimization(Exam#1 Fall...

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ESE304 - Introduction to Optimization (Exam #1) Fall Semester, 2010 M. Carchidi –––––––––––––––––––––––––––––––––––– Problem #1 (10 points) Let A , B and C be three fi xed n × n matrices and let S be the set of all n × n matrices X such that AX + XB C , i.e., S = { X | AX + XB C } . Prove that S is a convex set. –––––––––––––––––––––––––––––––––––– Problem #2 (30 points) Use the Big-M method to solve the following linear programming problem. min 3 x 1 + 5 x 2 + 7 x 3 (Objective Function) s.t. 2 x 1 + 4 x 2 + 5 x 3 30 (Constraint #1) 4 x 1 + 6 x 2 + 7 x 3 44 (Constraint #2) x 1 , x 2 , x 3 0 (Sign Restrictions) ––––––––––––––––––––––––––––––––––––
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–––––––––––––––––––––––––––––––––––– Problem #3 (30 points) Alyssa has a limited budget of $18 and can only buy two types of deserts. Desert 1 sells for $2 per piece and each piece of this desert contains 3 units of cake and 1 unit of chocolate, while Desert 2 sells for $1 per piece and each piece of this desert contains 1 unit of cake and 1 unit of chocolate. Her goals are to ingest at least 26 units of cake and 16 units of chocolate. a.) (6 points) Can Alyssa meet both of her goals with her present budget? Explain with graphs. b.) (8 points) By what minimum amount must her budget allocation for desert increase if she wants to meet both of her goals? c.) (8 points) By what minimum amount must her cake goal decrease if she wants to meet her chocolate goal and still stay within her budget? d.) (8 points) By what minimum amount must her chocolate goal decrease if she wants to meet her cake goal and still stay within her budget? –––––––––––––––––––––––––––––––––––– Problem #4 (30 points) Highland’s TV-Radio store must determine how many TVs and radios to keep in stock. A TV requires 12 square feet of fl oor space, whereas a radio requires 4 square feet, and a total of 260 square feet of fl oor space is available.
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