Explanation Given: Find y 1 ≥ 0 , y 2 ≥ 0 , and y 3 ≥ 0 such that 5 y 1 + 3 y 2 + 3 y 3 ≤ 150 5 y 1 + 10 y 2 + y 3 ≥ 90 and w = 10 y 1 + 12 y 2 + 10 y 3 is minimized. Explanation: Minimize w = 10 y 1 + 12 y 2 + 10 y 3 subject to: 5 y 1 + 3 y 2 + 3 y 3 ≤ 150 (1) 5 y 1 + 10 y 2 + y 3 ≥ 90 (2) Step 1: First, if a minimization problem is given, change it to a maximization problem; that is, take objective function of maximization problem z = − w with the same constraints of minimization problem. This leads to z = − w = − ( 10 y 1 + 5 y 2 + 10 y 3 ) = − 10 y 1 − 5 y 2 − 10 y 3 Then, add slack variable and subtract surplus variable in order to get equality constraints. Add slackvariable (say s 1 ) to equation (1) to get 5 y 1 + 3 y 2 + 3 y 3 + s 1 = 150 Subtractsurplusvariable (say s 2 ) from equation (2) to get 5 y 1 + 10 y 2 + y 3 − s 2 = 90 Step 2: Represent the system of linear equations obtained above in simplex tableau. The initial tableau is shown below. [ y 1 y 2 y 3 s 1 s 2 z s 1 5 3 3 1 0 0 150 s 2 5 10 3 0 − 1 0 90 z 10 12 10 0 0 1 0 ] _ The solution from initial tableau is ( y 1 , y 2 , y 3 s 1 , s 2 ) = ( 0 , 0 , 150 , − 90 ) . 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 initial tableau, basic variable s 1 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 5 . Step 5: From the column found in step 4, pick a pivot by test ratio (or by quotients). Column 1 is a pivot column. Apply test ratio: 150 5 = 30 , 90 5 = 18 Therefore, 5 is the pivot because smallest positive ratio is obtained by the element of R 2 , which shows that s 2 is depart variable and y 1 is enter variable. Therefore, [ y 1 y 2 y 3 s 1 s 2 z s 1 5 3 3 1 0 0 150 y 1 5 10 3 0 − 1 0 90 z 10 12 10 0 0 1 0 ] _ Since pivot is not 1 , make it 1 by R 2 → R 2 5

Finite Mathematics, Books a la Carte Plus MyLab Math Access Card Package (11th Edition)

11th Edition

ISBN: 9780133886818

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

Publisher: PEARSON

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Chapter 4.4, Problem 15E

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Finite Mathematics, Books a la Carte Plus MyLab Math Access Card Package (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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