CO227mid_soln-W2011

CO227mid_soln-W2011 - UNIVERSITY OF WATERLOO MIDTERM...

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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 (Non-graphing) 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 non-linear. (Include an explanation of what each variable in your program represents.) (a) Co-op Calculates makes the worlds 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) Co-op Calculates sells very high-end 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 re-they make per day....
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CO227mid_soln-W2011 - UNIVERSITY OF WATERLOO MIDTERM...

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