Explanation Given: The problem Minimize C = 2 x + 4 y + z Subject to x + 2 y + 4 z ≥ 7 …… ( 1 ) 3 x + y − z ≥ 6 …… ( 2 ) x + 4 y + 2 z ≥ 24 …… ( 3 ) x ≥ 0 , y ≥ 0 , z ≥ 0 Approach: 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. After introducing all these variables Min C = 2 x + 4 y + z + 0 S 1 + 0 S 2 + 0 S 3 + M A 1 + M A 2 + M A 3 Subject to x + 2 y + 4 z − S 1 + A 1 = 7 3 x + y − z − S 2 + A 2 = 6 x + 4 y + 2 z − S 3 + A 3 = 24 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 2 4 1 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 7 1 2 4 − 1 0 0 1 0 0 3.5 A 2 M 6 3 1 − 1 0 − 1 0 0 1 0 6 A 3 M 24 1 4 2 0 0 − 1 0 0 1 6 C = 37 M Z j 5 M 7 M 5 M − M − M − M M M M Z j − C j 5 M − 2 7 M − 4 5 M − 1 − M − M − M 0 0 0 The positive maximum Z j − C j = 7 M − 4 and its column is second, the variable y will enter and minimum Ratio is 3.5 so the variable A 1 will depart from the basis. Therefore the pivot element is 2 Perform the transformations R 1 = R 1 2 . R 2 = R 2 − R 1 . R 3 = R 3 − 4 R 1 . Iteration 2 C j 2 4 1 0 0 0 − M − M B C B X B x y z S 1 S 2 S 3 A 2 A 3 Min Ratio X B x y 4 7 2 1 2 1 2 − 1 2 0 0 0 0 --- A 2 M 5 2 5 2 0 − 3 1 2 − 1 0 1 0 5 A 3 M 10 − 1 0 − 6 2 0 − 1 0 1 5 C = 25 M 2 + 14 Z j 3 M 2 + 2 4 − 9 M + 8 5 M 2 − 2 − M − M M M Z j − C j 3 M 2 0 − 9 M + 8 5 M 2 − 2 − M − M 0 0 The positive minimum Z j − C j = 5 M 2 − 2 and its column is fourth, the variable S 1 will enter and minimum Ratio is 5 so the variable A 2 will depart from the basis

11th Edition

ISBN: 8220100478185

Author: Tan

Publisher: CENGAGE L

See similar textbooks

Videos

Question

Chapter 4.2, Problem 2TE

To determine

To find:

The solution of the linear programming problem.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.2, Problem 1TE

Chapter 4.2, Problem 3TE

Students have asked these similar questions

Sox & Sin (px) dx 0

In Exercises 62-64, sketch a reasonable graph that models the given situation. The number of hours of daylight per day in your hometown over a two-year period The motion of a diving board vibrating 10 inches in each direction per second just after someone has dived off The distance of a rotating beam of light from a point on a wall

8 L 8 e ipx dx

Chapter 4 Solutions

EBK FINITE MATHEMATICS FOR THE MANAGERI

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...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The solution of the linear programming problem.

Solution Summary: The author explains the simplex method for solving non-standard problems.

Videos

To find: The solution of the linear programming problem.

Want to see the full answer?

Chapter 4 Solutions

To find:

The solution of the linear programming problem.