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Unformatted text preview: CO 227  Homework assignment 4 Fall ‘10 Page 1 CO 227  Fall ‘10 Homework assignment #4: (Due at the beginning of the class, 10:30AM, on Wednesday, Nov 10th) Instructions: • Please show all your work and justify your answers. Answers without proper justification will not be considered. • Please list who you collaborated with. • Please staple your homework and write your name and student id in pen. • Total value = 100 points READ THIS: For all questions that you are asked to solve the LP, you must the TABLEAU simplex method. Also, you may have several choices for variables entering/leaving the basis. However, to make the correction easier, if there is a choice, always choose the lowest index variable. If you do not do that, you will be penalized. Question 1 Interpreting the tableau (21 points) Consider the following tableau for a given LP: T = 1 − 1 3 5 1 1 − 1 / 2 1 1 2 3 5 − 1 4 1 2 (a) (6 points) What is the LP that this tableau represents? Solution: The tableau represents the following LP. max z ( x ) = 5 + (0 , 1 , , − 3 , 0) x s.t. 1 1 − 1 / 2 1 2 3 − 1 4 1 x 1 x 2 x 3 x 4 x 5 = 1 5 2 x ≥ vector (b) (6 points) What is the current basis and basic feasible solution that it represents? Solution: The current basis is B = { 1 , 3 , 5 } and the basic feasible solution is ¯ x = (5 , , 1 , , 2) T . (3pts for basis, 3pts for basic feasible solution) (c) (3 points) What is the objective value of the corresponding basic feasible solution? Solution: The objective value of ¯ x is 5 + ¯ x 2 − 3¯ x 4 = 5. (d) (6 points) Is this tableau optimal? If not, then on which element should we pivot in the next step of the simplex algorithm? CO 227  Homework assignment 4 Fall ‘10 Page 2 Solution: This tableau is not optimal as − c = (0 , − 1 , , 3 , 0) T negationslash≥ 0. Since c 2 > 0, we select x 2 to be the entering variable. For each row i with T i, 2 > 0, we compute the ratio T i, 6 /T i, 2 . min { 1 / 1 , 5 / 2 , −} = 1 is attained by row 1, therefore we should pivot on the element (1 , 2). (2pts for figuring tableau is not optimal, 4pts for figuring the correct pivot) Question 2 Tableau and canonical form (31 points) Consider the following LP in standard equality form: max x 1 + x 2 s.t 2 x 1 + x 2 + x 3 = 4 x 1 +2 x 2 + x 4 = 3 x 1 , x 2 , x 3 , x 4 ≥ (1) The optimal tableau for the LP is: 1 1 / 3 1 / 3 7 / 3 1 − 1 / 3 2 / 3 2 / 3 1 2 / 3 − 1 / 3 5 / 3 (2) (a) (6 points) What is the current basis and basic feasible solution that it represents?...
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This note was uploaded on 01/18/2011 for the course CO 227 taught by Professor 1 during the Spring '10 term at Waterloo.
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