Explanation Consider a linear programming problem. Maximize p = 3 x + 2 y + z Subject to constraints, x + 3 y ≥ 5 − x + y ≤ 4 2 x + y ≤ 8 x ≥ 0 , y ≥ 0 Recall that to compute solution of general linear programming problems, use slack and surplus variables to covert inequalities into equations and star the rows for which the active variables have negative value. Subtract the surplus variable s and add the slack variables t and u . x + 3 y − s = 5 − x + y + t = 4 2 x + y + u = 8 Thus, the equations are x + 3 y − s = 5 − x + y + t = 4 2 x + y + u = 8 − 3 x − 2 y − z + p = 0 Write the equations in augmented matrix. x y z s t u p s 1 3 0 − 1 0 0 0 5 t − 1 1 0 0 1 0 0 4 u 2 1 0 0 0 1 0 8 p − 3 − 2 − 1 0 0 0 1 0 Check the values of the active variables s t and u . s = 5 − 1 = − 5 And, t = 4 1 = 4 And, u = 8 1 = 8 Since, the value of s is negative, star the s row. Thus, the tableau is x y z s t u p s * 1 3 0 − 1 0 0 0 5 t − 1 1 0 0 1 0 0 4 u 2 1 0 0 0 1 0 8 p − 3 − 2 − 1 0 0 0 1 0 Since, there are starred rows, use Phase I to proceed. Phase I: To remove the stars that is to make active variables have positive values

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Chapter 5.4, Problem 59E

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A linear programming problem in 3 variables that require one pivot in Phase I.

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A linear programming problem in 3 variables that require one pivot in Phase I.

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