Explanation Given Information: The LP problems can be solved by graphical method. Consider the graphical method. Since, the graphical method involves the linear programming problems with 2 unknowns. Thus, any problem with more than 2 unknowns cannot be solved by using the graphical method. Now, consider the simplex method. Since, there is no restriction as to the number of unknowns that can be solved using the simplex method. Thus, the simplex method can be used to solve any linear programming problem with 2 or more unknowns and is therefore, more efficient than the graphical method. For example. Consider the problem to calculate the maximum value of p = 6 x + y + 3 z subject to 3 x + y ≤ 15 2 x + 2 y + 2 z ≤ 20 x ≥ 0 , y ≥ 0 , z ≥ 0 Since, the equation 2 x + 2 y + 2 z ≤ 20 cannot be easily graphed as it involves 3 unknowns. Thus, the graphical method cannot be used to solve this problem. Consider the simplex method. Since, the simplex method can be used to solve the problems with any number of inequalities. Thus, the problem can be solved by simplex method. Consider the constraints 3 x + y ≤ 15 2 x + 2 y + 2 z ≤ 20 x ≥ 0 , y ≥ 0 , z ≥ 0 Step 1: Use slack variables and write the inequalities in equation form. Add the slack variables s and t . 3 x + y + s = 15 2 x + 2 y + 2 z + t = 20 Step 2: Write the equations in an augmented matrix. The equations are 3 x + y + s = 15 2 x + 2 y + 2 z + t = 20 − 6 x − y − 3 z + p = 0 Write the equations in augmented matrix. x y z s t p s 3 1 0 1 0 0 15 t 2 2 2 0 1 0 20 p − 6 − 1 − 3 0 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 − 6 . 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 test ratios are 15 3 = 5 and 20 2 = 10 . Thus, the pivot is 3 in column x . x y z s t p s 3 1 0 1 0 0 15 t 2 2 2 0 1 0 20 p − 6 − 1 − 3 0 0 1 0 Step 5: Clear the pivot column using the pivot and label the pivot row with the label of the pivot column

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Chapter 6.3, Problem 49E

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The reason that the simplex method is useful than graphical method.

Solution Summary: The author explains that the simplex method can be used to solve any type of maximization linear programming problem involving 2 or more unknowns.

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