Explanation Given: The problem is given as, Minimize C = x + y + 3 z Subject to 2 x + y + 3 z ≥ 6 … ( 1 ) x + 2 y + 4 z ≥ 8 … ( 2 ) 3 x + y − 2 z ≥ 4 … ( 3 ) x ≥ 0 , y ≥ 0 , z ≥ 0 Approach: The Simplex Method for Solving Non-standard Problems is as follows: 1. If necessary, rewrite the problem as a maximization problem (recall that minimizing C is equivalent to maximizing − C ). 2. If necessary, rewrite all constraints (except x ≥ 0 , y ≥ 0 , ⋯ ) using less than or equal to ( ≤ ) inequalities. 3. Introduce slack variables and set up the initial simplex tableau. 4. Select the pivot column: locate the most negative entry to the left of the vertical line in the last row. The column containing this entry is called pivot column (if there is more than one such column choose any one). 5. Scan the upper part of the column of constants of the tableau for negative entries. a. If there are no negative entries, complete the solution using the simplex method for problems in standard form. b. If there are negative entries, proceed to Step 5. 6. a. Pick any negative entry in a row in which a negative entry in the column of constants occurs. The column containing this entry is the pivot column. b. Compute the positive ratios of the numbers in the column of constants to the corresponding numbers in the pivot column (above the last row). The pivot row corresponds to the smallest ratio. The intersection of the pivot column and the pivot row determines the pivot element. c. Pivot the tableau about the pivot element. Then return to Step 5. Calculation: Add surplus variables S 1 , S 2 , S 3 and artificial variables A 1 , A 2 , A 3 in the equations ( 1 ) , ( 2 ) , and ( 3 ) respectively. Introduce all these variables. Max C = x + y + 3 z + 0 S 1 + 0 S 2 + 0 S 3 + M A 1 + M A 2 + M A 3 Subject to 2 x + y + 3 z − S 1 + A 1 = 6 x + 2 y + 4 z − S 2 + A 2 = 8 3 x + y − 2 z − S 3 + A 3 = 4 and x , y , z , S 1 , S 2 , S 3 , A 1 , A 2 , A 3 ≥ 0 Use the simplex method. Iteration 1 C j 1 1 3 0 0 0 M M M B C B X B x y z S 1 S 2 S 3 A 1 A 2 A 3 Min Ratio X B x A 1 M 6 2 1 3 − 1 0 0 1 0 0 3 A 2 M 8 1 2 4 0 − 1 0 0 1 0 8 A 3 M 4 3 1 − 2 0 0 − 1 0 0 1 1.333 C = 18 M Z j 6 M 4 M 5 M − M − M − M M M M Z j − C j 6 M − 1 ↑ 4 M − 1 5 M − 3 − M − M − M 0 0 0 The positive maximum Z j − C j = 6 M − 1 and its column is first, the variable x will enter and minimum Ratio is 1.333 so the variable A 3 will depart. Therefore, the pivot element is 3 . Perform the transformations R 3 = R 3 3 . R 1 = R 1 − 2 R 3 R 2 = R 2 − R 3 Iteration 2 C j 1 1 3 0 0 0 M M B C B X B x y z S 1 S 2 S 3 A 1 A 2 Min Ratio X B x A 1 M 10 3 0 1 3 13 3 − 1 0 2 3 1 0 0.7692 A 2 M 20 3 0 5 3 14 3 0 − 1 1 3 0 1 1.4286 x 1 4 3 1 1 3 − 2 3 0 0 − 1 3 0 0 C = 10 M + 4 3 Z j 1 2 M + 1 3 9 M − 2 3 − M − M M − 1 3 M M Z j − C j 0 2 M − 2 3 9 M − 2 3 − M − M M − 1 3 0 0 The positive maximum Z j − C j = 9 M − 11 3 and its column is third, the variable z will enter and minimum Ratio is 0.7692 so the variable A 1 will depart. Therefore, the pivot element is 13 3 . Perform the transformations R 1 = R 1 ( 3 13 ) . R 2 = R 2 − ( 14 3 ) R 1 R 3 = R 3 + ( 2 3 ) R 1 Iteration 3 C j 1 1 3 0 0 0 M B C B X B x y z S 1 S 2 S 3 A 2 MinRatio X B x z 3 10 3 0 1 13 1 − 3 13 0 2 13 0 10 A 2 M 40 13 0 17 13 0 14 13 − 1 − 5 13 1 2

WebAssign Printed Access Card for Tan's Finite Mathematics for the Managerial, Life, and Social Sciences, 12th Edition, Single-Term

12th Edition

ISBN: 9781337652766

Author: Soo T. Tan

Publisher: Cengage Learning

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WebAssign Printed Access Card for Tan's Finite Mathematics for the Managerial, Life, and Social Sciences, 12th Edition, Single-Term

Ch. 4.1 - Give the three characteristics of a standard...Ch. 4.1 - a. When the initial simplex tableau is set up, how...Ch. 4.1 - In the simplex method, how is a pivoted column...Ch. 4.1 - In Exercises 1-6, a write the linear programming...Ch. 4.1 - In Exercises 1-6, a write the linear programming...Ch. 4.1 - In Exercises 1-6, a write the linear programming...Ch. 4.1 - In Exercises 1-6, a write the linear programming...Ch. 4.1 - In Exercises 1-6, a write the linear programming...Ch. 4.1 - Prob. 6E Ch. 4.1 - In Exercises 7-16, determine whether the given...

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Solution Summary: The author explains the simplex method for solving non-standard problems. The optimal solution is C=143.

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The solution of the linear programming problem.