Explanation Given: Find x 1 ≥ 0 and x 2 ≥ 0 such that x 1 + 2 x 2 ≤ 18 x 1 + 3 x 2 ≥ 12 2 x 1 + 2 x 2 ≤ 24 and z = 5 x 1 − 10 x 2 is maximized. Explanation: The following information is provided: Maximize z = 5 x 1 − 10 x 2 Subject to: x 1 + 2 x 2 ≤ 18 (I) x 1 + 3 x 2 ≥ 12 (II) 2 x 1 + 2 x 2 ≤ 24 (III) Step 1: First, add slack variable and subtract surplus variable in order to get equality constraints. Add slack variable (say s 1 ) to equation (I): x 1 + 2 x 2 + s 1 = 18 Subtract surplus variable (say s 2 ) from equation (II): x 1 + 3 x 2 − s 2 = 12 Add slack variable to equation (III): 2 x 1 + 2 x 2 + s 3 = 24 Step 2: Represent the system of linear equations obtained above in the simplex tableau. This is the initial tableau: [ x 1 x 2 s 1 s 2 s 3 z s 1 1 2 1 0 0 0 18 s 2 1 3 0 − 1 0 0 12 s 3 2 2 0 0 1 0 24 z − 5 10 0 0 0 1 0 ] _ The solution from the initial tableau is ( x 1 , x 2 , s 1 , s 2 , s 3 ) = ( 0 , 0 , 0 , 18 , − 12 , 24 ) . This is not a feasible solution because the value of surplus variable ( s 2 ) is negative. Step 3: If any basic variable has a negative value, identify the nonzero number in that variable’s column, and note that row. In the initial tableau, the basic variable s 2 is negative and the nonzero number in this variable column is − 1 , which is in row 2, that is, in R 2 . Step 4: Now, in the row located in step 3, find the most positive entry in the leftmost column. The most positive entry in column 1 is 2 . Step 5: From the column found in step 4, pick a pivot by the test ratio (or by quotients). Column 1 is the pivot column. Apply the test ratio: 18 1 = 18 , 12 1 = 12 , 24 2 = 12 Same ratio is obtained corresponding to row 1 and row 2. Hence, give priority to the element present at the intersection of 2nd row and 1st column. Therefore, 1 is the pivot element, which follows that s 2 is the depart variable and x 1 is the enter variable. Therefore, [ x 1 x 2 s 1 s 2 s 3 z s 1 1 2 1 0 0 0 18 x 1 1 3 0 − 1 0 0 12 s 3 2 2 0 0 1 0 24 z − 5 10 0 0 0 1 0 ] _ Step 6:

11th Edition

ISBN: 9780321979438

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Finite Mathematics (11th Edition)

Ch. 4.1 - Convert each inequality into an equation by adding...Ch. 4.1 - Prob. 2E Ch. 4.1 - Convert each inequality into an equation by adding...Ch. 4.1 - Prob. 4E Ch. 4.1 - For Exercises 5-8. (a) determine the number of...Ch. 4.1 - Prob. 6E Ch. 4.1 - For Exercises 5-8, (a) determine the number of...Ch. 4.1 - For Exercises 5-8, (a) determine the number of...Ch. 4.1 - Introduce slack variables as necessary, then...Ch. 4.1 - Introduce slack variables as necessary, then write...

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x 1 ≥ 0 , x 2 ≥ 0 , and value of z by using the simplex method.

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