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Unformatted text preview: UNIVERSITY OF WATERLOO MIDTERM EXAMINATION WINTER TERM 2011 Student Name (Print Legibly) ( family name ) ( given name ) Signature Student ID Number COURSE NUMBER CO 227 COURSE TITLE Introduction to Optimization COURSE SECTION 001 DATE OF EXAM Tuesday, March 1, 2011 TIME PERIOD 7:00  9:00 pm DURATION OF EXAM 2 hours NUMBER OF EXAM PAGES (Including cover page) 10 pages INSTRUCTORS P. Roh EXAM TYPE Closed Book ADDITIONAL MATERIALS ALLOWED Calculator (Nongraphing) Notes: 1. Fill in your name, ID number, and sign the paper. 2. Answer all questions in the space provided. Ask the proctor for ex tra blank pages if necessary. Show your work and give FORMAL solu tions. Leave your answers in exact form, such as 5 + √ 7, 4 1 / 3 sin( 3 10 ), etc. 3. Check that the examination has 10 pages. 4. Your grade will be influenced by how clearly you express your ideas, and how well you organize your solutions. Marking Scheme: Question Mark Out of 1 16 2 10 3 15 4 19 5 10 Total 70 DO NOT WRITE FORMULAS ON THE COVER PAGE. CO 227  Midterm Examination Winter 2011 Page 2 of 10 1. For these questions, write a mathematical program to solve the given problem and classify the type of mathematical program as linear, integer, or nonlinear. (Include an explanation of what each variable in your program represents.) (a) Coop Calculates makes the world’s best calculators. The workers work on a 3 day cycle in one of 2 shifts each day. The minimum number of workers needed for each shift is given by the following table: Shift \ Day 1 2 3 1 23 34 18 2 17 24 29 The contract with the workers union states that: 1. Workers work 2 shifts for each 3 day cycle and no two shifts are on the same day. 2. Workers are paid $2000 for each cycle if they work the same shift during the 3 day cycle and they are paid $2500 for each cycle if they work different shifts during the 3 day cycle. How should the workers be scheduled to minimize the labour cost during a 3 day cycle? Solution: Let x ij represent the number of workers who have day i off and always work shift j . Let y ij represent the number of workers who have day i off and work shift j on their first day of the 3 day cycle and another shift their second day. minimize 2000( x 11 + x 12 + x 21 + x 22 + x 31 + x 32 ) +2500( y 11 + y 12 + y 21 + y 22 + y 31 + y 32 ) subject to x 21 + x 31 + y 21 + y 31 ≥ 23 x 22 + x 32 + y 22 + y 32 ≥ 17 x 11 + x 31 + y 11 + y 32 ≥ 34 x 12 + x 32 + y 12 + y 31 ≥ 24 x 11 + x 21 + y 12 + y 22 ≥ 18 x 12 + x 22 + y 11 + y 21 ≥ 29 x ij ,y ij ≥ 1 ≤ i ≤ 3 , 1 ≤ j ≤ 2 x ij ,y ij integer 1 ≤ i ≤ 3 , 1 ≤ j ≤ 2 This is an integer program. CO 227  Midterm Examination Winter 2011 Page 3 of 10 (b) Coop Calculates sells very highend gold calculators with diamond buttons. The price of the calculator they sell is $50000/ √ x where x is the number of calculators they make per day. (This reflects that increased supply means lower prices are rethey make per day....
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 Fall '08
 Burbidge,John
 Optimization, Midterm Examination, Canonical form, Infeasibility

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