Explanation Given Information: Any linear programming problem. The coefficients of the objective function are represented in the bottom row and due to row reduction, the entries of the bottom row are added to a positive number, which is the pivot entry. Let all the pivots are 1s for simplicity. Suppose that the entry at the bottom of the pivot column is negative, the bottom row replaced by itself plus positive constant multiple of the pivot row. So, the value of objective function is replaced by itself plus a positive multiple of the nonnegative rightmost entry of the pivot row. Thus, it never decreases. Thus, the objective function cannot decrease in passing from one tableau to the next. For example- Maximize the function, p = x + y under subjected to 2 x + y ≤ 1 , x ≥ 0 , y ≥ 0 . Consider the constraints 2 x + y ≤ 1 x ≥ 0 , y ≥ 0 Step 1: Use slack variables and write the inequalities in equation form. Add the slack variable s . 2 x + y + s = 1 Step 2: Write the equations in an augmented matrix. The equations are 2 x + y + s = 1 − x − y + p = 0 Write the equations in augmented matrix. x y s p s 2 1 1 0 1 p − 1 − 1 0 1 0 Step 3: Choose the smallest negative number on the left side of the bottom row and pivot the column. Since, the smallest negative number on the left side of the bottom row is − 1 . Thus, pivot the x column. Step 4: Select the pivot which is the smallest of the test ratios a b , where a is the entry in the right most column and b is the corresponding entry in the pivot column. Since, the test ratios are neither 0 nor negative. Thus, the only test ratio is 2 1 = 2 . Thus, the pivot is 2 in column x . x y s p s 2 1 1 0 1 p − 1 − 1 0 1 0 Thus, the value of the objective function is 0 in the first tableau. Step 5: Clear the pivot column using the pivot and label the pivot row with the label of the pivot column. Since, the pivot is s in column x and replace s with x . Thus, x is the active variable. Use row operations to clear the pivot column of the augmented matrix

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Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 5.3, Problem 55E

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Whether the value of the objective function can decrease in passing from one tableau to the next.

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Whether the value of the objective function can decrease in passing from one tableau to the next.

Solution Summary: The author explains that the value of the objective function cannot decrease in passing from one tableau to the next.

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Whether the value of the objective function can decrease in passing from one tableau to the next.

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