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Math 141H-copyright Joe Kahlig, 12A Page 1 Section 4.3: The Simplex Method: Non-Standard Problems A non-standard linear programming problem is one that can not be solve with one of the previous methods. Example: Can this problem be solved by the methods that we know from 4.1 and 4.2? A) maximize P = x + 2y 4x + 3y ≤ 18 −x + 3y ≥ 3 x, y ≥ 0 B) minimize C = 2x − 3y x+y ≤5 x + 3y ≥ 9 −2x + y ≤ 2 x, y ≥ 0 C) maximize P = x + 2y 2x + 3y ≤ 12 −x + 3y = 3 x, y ≥ 0 Solving a Non-Standard Linear Programming Problem: 1) If needed, rewrite the objective function as a maximize. 2) Rewrite all inequalities (except for x ≥ 0, y ≥ 0,...) as ≤ a constant. 3) Set up the simplex matrix. 4) To find the pivot column: Pick a row with a negative number in the answer column (last column). Then locate the most negative entry in that row (excluding the number in the answer column). The column that contains that number is the pivot column. Note: If that row only has a negative number in the answer column, then the problem does not have a solution. 5) To find the pivot row: Divide each number in the pivot column into the corresponding entry in the answer column. Choose the smallest positive quotient. That row will be the pivot row. 6) Pivot on the pivot element. If the last column contain a negative number, not counting the entry in the last row, then return to step 4. 7) Now apply the simplex method to solve the problem. Page 2 Math 141H-copyright Joe Kahlig, 12A Example: Solve the linear programming problem. maximize f = 2x − y + 2z x + 2y ≤ 40 x + 2y + 3z ≥ 40 y + z ≥ 30 x, y, z ≥ 0 1 −1 0 −2 2 −2 −1 1 0 −3 −1 −2 1 0 0 0 0 1 0 0 0 0 1 0 x + 2y + s1 = 40 −x − 2y − 3z + s2 = −40 −y − z + s3 = −30 −2x + y − 2z + f = 0 0 0 0 1 40 −40 −30 0 Math 141H-copyright Joe Kahlig, 12A Example: Solve the linear programming problem. maximize: f = x + 2y − z x + y + 2z ≤ 12 2x + y + z ≤ 16 x+y =3 x, y, z ≥ 0 Page 3 Math 141H-copyright Joe Kahlig, 12A Example: Solve the linear programming problem. maximize P = 4x + 5y + z x + 2y + 3z ≤ 6 x + y + 2z ≥ 3 3x + y ≥ 5 x, y, z ≥ 0 Page 4