Page 1 Revised Simplex Method 09/23/04 page 1 of 22 (includes both Phases I & II) © Dennis Bricker Dept of Mechanical & Industrial Engineering The University of Iowa Page 2 Revised Simplex Method 09/23/04 page 1 of 22 (includes both Phases I & II) © Dennis Bricker Dept of Mechanical & Industrial Engineering The University of Iowa Revised Simplex Method 09/23/04 page 2 of 22 1 2 34 56 Minimize z=3x 5 4 7 5 4 xx x x x ++ + + + subject to 12 4 6 13 4 5 6 23 4 5 2310 33212 42 3 15 xx x x xx x x x xx x x -+ + = ? ? +- + + = ? ? ++ + = ? and 0 1, 6 j x j =?= â€¦ Because of the lack of a slack variable in each constraint, we must use Phase I to find an initial feasible basis. Add variables X 9 , X 10 , X 11 (artificial variables), and a Phase I objective of minimizing the sum of these three variables. Page 3 Revised Simplex Method 09/23/04 page 1 of 22 (includes both Phases I & II) © Dennis Bricker Dept of Mechanical & Industrial Engineering The University of Iowa Revised Simplex Method 09/23/04 page 2 of 22 1 2 34 56 Minimize z=3x 5 4 7 5 4 xx x x x ++ + + + subject to 12 4 6 13 4 5 6 23 4 5 2310 33212 42 3 15 xx x x xx x x x xx x x -+ + = ? ? +- + + = ? ? ++ + = ? and 0 1, 6 j x j =?= â€¦ Because of the lack of a slack variable in each constraint, we must use Phase I to find an initial feasible basis. Add variables X 9 , X 10 , X 11 (artificial variables), and a Phase I objective of minimizing the sum of these three variables. Revised Simplex Method 09/23/04 page 3 of 22 Phase One 1 2 3 4 5 6 7 8 9 0 1 b 0 0 0 0 0 0 0 0 1 1 1 0 phase one objective 3 5 4 7 5 4 0 0 0 0 0 0 phase two objective 2 -1 0 1 0 3 0 0 1 0 0 10 1 0 3 -1 3 2 -1 0 0 1 0 12 0 4 2 3 1 0 0 -1 0 0 1 15 Values of basic (artificial) variables are: i Xi 9 10 10 12 11 15 Page 4 Revised Simplex Method 09/23/04 page 1 of 22 (includes both Phases I & II) © Dennis Bricker Dept of Mechanical & Industrial Engineering The University of Iowa Revised Simplex Method 09/23/04 page 2 of 22 1 2 34 56 Minimize z=3x 5 4 7 5 4 xx x x x ++ + + + subject to 12 4 6 13 4 5 6 23 4 5 2310 33212 42 3 15 xx x x xx x x x xx x x -+ + = ? ? +- + + = ? ? ++ + = ? and 0 1, 6 j x j =?= â€¦ Because of the lack of a slack variable in each constraint, we must use Phase I to find an initial feasible basis. Add variables X 9 , X 10 , X 11 (artificial variables), and a Phase I objective of minimizing the sum of these three variables. Revised Simplex Method 09/23/04 page 3 of 22 Phase One 1 2 3 4 5 6 7 8 9 0 1 b 0 0 0 0 0 0 0 0 1 1 1 0 phase one objective 3 5 4 7 5 4 0 0 0 0 0 0 phase two objective 2 -1 0 1 0 3 0 0 1 0 0 10 1 0 3 -1 3 2 -1 0 0 1 0 12 0 4 2 3 1 0 0 -1 0 0 1 15 Values of basic (artificial) variables are: i Xi 9 10 10 12 11 15 Revised Simplex Method 09/23/04 page 4 of 22 Page 5 Revised Simplex Method 09/23/04 page 1 of 22 (includes both Phases I & II) © Dennis Bricker Dept of Mechanical & Industrial Engineering The University of Iowa Revised Simplex Method 09/23/04 page 2 of 22 1 2 34 56 Minimize z=3x 5 4 7 5 4 xx x x x ++ + + + subject to 12 4 6 13 4 5 6 23 4 5 2310 33212 42 3 15 xx x x xx x x x xx x x -+ + = ? ? +- + + = ? ? ++ + = ? and 0 1, 6 j x j =?= â€¦ Because of the lack of a slack variable in each constraint, we must use Phase I to find an initial feasible basis. Add variables X 9 , X 10 , X 11 (artificial variables), and a Phase I objective of minimizing the sum of these three variables. Revised Simplex Method 09/23/04 page 3 of 22 Phase One 1 2 3 4 5 6 7 8 9 0 1 b 0 0 0 0 0 0 0 0 1 1 1 0 phase one objective 3 5 4 7 5 4 0 0 0 0 0 0 phase two objective 2 -1 0 1 0 3 0 0 1 0 0 10 1 0 3 -1 3 2 -1 0 0 1 0 12 0 4 2 3 1 0 0 -1 0 0 1 15 Values of basic (artificial) variables are: i Xi 9 10 10 12 11 15 Revised Simplex Method 09/23/04 page 4 of 22 Revised Simplex Method 09/23/04 page 5 of 22 Iteration 1 Current partition: (B = basis, N = non-basis) B= {9 10 11}, N= {1 2 3 4 5 6 7 8} Basis inverse is 1 0 0 0 1 0 0 0 1 Simplex multipliers (dual solution): i p 1 1 2 1 3 1 () [] 1 10 0 1,1,1 0 1 0 [ 1 ,1 ,1] 00 1 B B cA - ?? ?? ?p= = = ?? ?? ? ?Read More

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