solutionmidtermdecember11th2010

solutionmidtermdecember11th2010 - This exam consists of 3...

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Unformatted text preview: This exam consists of 3 pages Sabanci university Faculty of Engineering and Natural Sciences Industrial Engineering Solutions Midterm exam MS 301 December 11th 2010, 11.00 - 13.00 Remark: keep your answers as short as possible but of course complete. Also read very carefully the questions. The total number of points of this exam is 100. At the beginning of each part of the exercise the total number of points are listed. Please write your student number at each page and write clearly. To avoid too many disturbances for the other students you are only allowed to ask questions about the exam in the first hour. Within two weeks the results of the exam will be published on SUcourse. Question 1 . Consider the linear programming problem max- x 1- x 2 +2 x 3 5 x 1 +3 x 2 + x 3 ≥ 10 x 1 + x 2 +2 x 3 ≥ 1 x 2 ≤ 3 x 1 x 2 x 3 ≥ 1. ( 10 points ) After introducing the slack variables x 4 , x 5 and x 6 write down the initial start dictionaire with x 4 , x 5 and x 6 in the basis. Is this dictionaire primal feasible? Explain your answer! 2. ( 10 points ) Construct a primal feasible startdictionaire and write down the linear program you have to solve to obtain such a dictionaire. Show all the dictionaires and use in your calculations the smallest index rule. 3. ( 10 points ) Solve the original linear programming problem (show all the computed dictio- naires). Does this problem and the dual problem have an optimal solution?.Explain your answer! 1 4. ( 5 points) Explain why you could have seen the solution immediately without doing all the calculatioins. Solution 1. First write optimization problem in standard form max- x 1- x 2 +2 x 3- 5 x 1- 3 x 2- x 3 ≤ - 10- x 1- x 2- 2 x 3 ≤ - 1 x 2 ≤ 3 x 1 x 2 x 3 ≥ Starting dictionaire has the form x 4 =- 10 +5 x 1 +3 x 2 + x 3 x 5 =- 1 + x 1 + x 2 +2 x 3 x 6 = 3- x 2 z = 0- x 1- x 2 +2 x 3 This dictionaire is not primal feasible since the basic solution (=set nonbasic variables equal to ) is not primal feasible. This means geometrically that the vertex belonging to the above dictionaire does not belong to the feasible region. 2. We need to apply the two-phase simplex method. To determine a primal feasible start dic- tionaire representing a primal feasible vertex we need to solve the Help problem max- x- x- 5 x 1- 3 x 2- x 3 ≤ - 10- x- x 1- x 2- 2 x 3 ≤ - 1- x + x 2 ≤ 3 x x 1 x 2 x 3 ≥ 2 Introducing the slack variables...
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This note was uploaded on 03/17/2012 for the course MS 301 taught by Professor H during the Spring '12 term at Sabancı University.

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solutionmidtermdecember11th2010 - This exam consists of 3...

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