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Unformatted text preview: UNIVERSITY OF WATERLOO FINAL 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 Saturday, April 9, 2011 TIME PERIOD 7:30  10:00 pm DURATION OF EXAM 2.5 hours NUMBER OF EXAM PAGES (Including cover page) 14 pages INSTRUCTOR 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 14 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 10 2 8 3 8 4 31 5 23 6 10 7 10 Total 100 DO NOT WRITE FORMULAS ON THE COVER PAGE. CO 227  Final Examination Winter 2011 Page 2 of 14 1. A company is auctioning off 3 different items owned by Justin Bieber. Since these items are considered so expensive, there are only 3 potential bidders for the items. A common way to force bidders to offer more money is to allow them to have only one item at the end of the auction. The bidders submit bids on each item and then the company determines who gets what based on the bids. The amount of money (in millions of dollars) the bidders are offering for each item is the following Bidder \ Item 1 2 3 1 22 33 18 2 16 24 28 3 19 37 15 Write a mathematical program to maximize the total sales for the company and classify the program as linear, integer, or nonlinear. (Make sure to explain what each variable and constraint does.) Solution: Let x ij = 1 Bidder i gets item j otherwise The objective function should be the total sales that we wish to maximize: 22 x 11 + 16 x 21 + 19 x 31 + 33 x 12 + 24 x 22 + 37 x 32 + 18 x 13 + 28 x 23 + 15 x 33 We have that x ij is either 0 or 1 so we need x ij ≥ 0 and x ij to be an integer. Each bidder gets exactly one item: x 11 + x 12 + x 13 = 1 x 21 + x 22 + x 23 = 1 x 31 + x 32 + x 33 = 1 Each item goes to only one bidder: x 11 + x 21 + x 31 = 1 x 12 + x 22 + x 32 = 1 x 13 + x 23 + x 33 = 1 The integer program to maximize total sales is: maximize 22 x 11 + 16 x 21 + 19 x 31 + 33 x 12 + 24 x 22 + 37 x 32 + 18 x 13 + 28 x 23 + 15 x 33 subject to x 11 + x 12 + x 13 = 1 x 21 + x 22 + x 23 = 1 x 31 + x 32 + x 33 = 1 x 11 + x 21 + x 31 = 1 x 12 + x 22 + x 32 = 1 x 13 + x 23 + x 33 = 1 x ≥ x integer CO 227  Final Examination Winter 2011 Page 3 of 14 2. For the following linear programs, convert them to standard equality form....
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This note was uploaded on 01/26/2012 for the course ECON 401 taught by Professor Burbidge,john during the Fall '08 term at Waterloo.
 Fall '08
 Burbidge,John

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