To determine To calculate: The number of in interviews she should do in a week to minimize the time spent. Explanation Given: Time for political interview is 45 min and time for market interview is 55 min . And the additional information about interviews is shown in the below table. Interviews Earning (per week) in $ Bonus point (per week) Political 8 6 Market 10 5 Minimum 60 40 Formula used: Calculation: It is assumed that the time spends for political interview is y 1 and the time spends for market interview is y 2 . The total number of interviews in each week is. Then, The minimum cost function is: w = 45 y 1 + 55 y 2 Subjected to the constraints: 8 y 1 + 10 y 2 ≥ 60 6 y 1 + 5 y 2 ≥ 40 y 1 + y 2 ≥ 8 Where y 1 ≥ 0 , y 2 ≥ 0 . Now, find the dual problem for the above function then, Construct the argument matrix with coefficients of system of inequality and the coefficient of the minimization function. 8 10 60 6 5 40 1 1 8 45 55 0 The above argument matrix contains first three rows represent the coefficients of constraints and the last row shows the coefficient of the objective function. Now, find the transpose matrix of the argument matrix to find the dual problem. 8 6 1 45 10 5 1 55 60 40 8 0 Now, use ( x ) instead of ( y ) for dual problem, and the above transpose matrix shows that the first two rows represent the coefficients of constraints and the last row represents to coefficients of dual maximization problem. Then, The maximization function is z = 60 x 1 + 40 x 2 + 8 x 3 And inequalities equations are, 8 x 1 + 6 x 2 + x 3 ≤ 45 10 x 1 + 5 x 2 + x 3 ≤ 55 x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 Now, use simplex method to solve the problem. The maximization function is: z = 60 x 1 + 40 x 2 + 8 x 3 Subjected to the system of inequality or constraints. 8 x 1 + 6 x 2 + x 3 ≤ 45 10 x 1 + 5 x 2 + x 3 ≤ 55 x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 Now, use slack variables for converting the inequalities into equalities equations then the equations are, 8 x 1 + 6 x 2 + x 3 + s 1 = 45 10 x 1 + 5 x 2 + x 3 + s 2 = 55 Form the initial simplex table, x 1 x 2 x 3 s 1 s 2 8 6 1 1 0 45 10 5 1 0 1 55 − 60 − 40 − 8 0 0 0 The last row of the above table shows the coefficients of the objective function and the row-1, and row-2 show the coefficient of the constraints. Now, find the pivot column and pivot row of the table, Select most negative value column for the pivot column. From the above table, the most negative value is − 60 and it lies in the column-1. Now, find the quotient to select the pivot row. Then, Value of quotient for the row-1 is, 45 8 = 5.625 Value of quotient for the row-2 is, 55 10 = 5.5 Select the minimum value of the quotient to select the pivot row, Since, the minimum value of the quotient is ( 5.5 ) so the pivot row is row-2 and the pivot point is 10 as shown in the below table x 1 x 2 x 3 s 1 s 2 8 6 1 1 0 45 10 5 1 0 1 55 − 60 − 40 − 8 0 0 0 Now, the pivot point leads the table as follow then use the operation to find the pivot point as a unit. R 2 → 1 10 R 3 Then the table is, x 1 x 2 x 3 s 1 s 2 8 6 1 1 0 45 1 0.5 0.1 0 0.1 5.5 − 60 − 40 − 8 0 0 0 Now, the pivot point 1 lead the table as follow, Then, use the following operations to find the simplex table for the pivot point. Theoperations are used to reduce the negative sign from the last row for finding the feasible solution of the objective function. R 1 → R 1 − 8 R 2 , R 3 → R 3 + 60 R 2 Then the final table is: x 1 x 2 x 3 s 1 s 2 0 2 0.2 1 − 0.8 1 1 0.5 0.1 0 0.1 5.5 0 − 10 − 2 0 6 330 From the above simplex table, objective function row has the negative sign so the function is not having the feasible solution then use the second iteration to find the feasible solution for the function. Now, again find the pivot column and pivot row of the table, Select most negative value column for the pivot column. From the above table, the most negative value is − 10 and it lies in the column-2. Now, find the quotient to select the pivot row. Then, Value of quotient for the row-1 is, 1 2 = 0.5 Value of quotient for the row-2 is, 5.5 0.5 = 11 Select the minimum value of the quotient to select the pivot row, Since, the minimum value of the quotient is ( 0.5 ) so the pivot row is row-1 and the pivot point is 2 as shown in the below table. x 1 x 2 x 3 s 1 s 2 0 2 0.2 1 − 0.8 1 1 0.5 0.1 0 0.1 5.5 0 − 10 − 2 0 6 330 Now, the pivot point leads the table as follow then use the operation to find the pivot point as a unit

MATH W/APPLICAT.W/NOTES GDE +ACCESS CODE

11th Edition

ISBN: 9781323751671

Author: Lial

Publisher: PEARSON C

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Chapter 7.6, Problem 30E

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MATH W/APPLICAT.W/NOTES GDE +ACCESS CODE

Ch. 7.1 - Checkpoint 1 Graph the given...Ch. 7.1 - Prob. 2CP Ch. 7.1 - Prob. 3CP Ch. 7.1 - Prob. 4CP Ch. 7.1 - Prob. 5CP Ch. 7.1 - Checkpoint 6 Graph the feasible region of the...Ch. 7.1 - Prob. 7CP Ch. 7.1 - Match the inequality with its graph, which is one...Ch. 7.1 - Match the inequality with its graph, which is one...Ch. 7.1 - Match the inequality with its graph, which is one...

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