A company produces two products, A and B, which have profits of $9 and $7, respe

Question

A company produces two products, A and B, which have profits of $9 and $7, respectively. Each unit of product must be processed on two assembly lines, where the required production times are as follows:

Product                Line 1               Line 2                        

A                             11                           5

B                             6                              9

Total Hours        65                           40

a. Formulate a linear programming model to determine the optimal product mix that will maximize profit. Show Sensitivity Range for the coefficient of # of units of product A in the objective function and sensitivity Range for the coefficient of # of units of product B in the objective function.                                                  

b. What are the sensitivity ranges for the objective function coefficients?

c. Determine the shadow prices for additional hours of production time on line 1 and line 2 and indicate whether the company would prefer additional line 1 or line 2 hours

Explanation / Answer

Lets the profit earned by company x and y for product A and B respectively,

accordingly,

Maximize         z=    9x +7y Subject to constraints, 11x+6y65 5x +9y 40 where x and y 0 By the addition of slack varibale S1, S2 the inequalties can be converted into equations. The problems thus become Z= 9x +7y + 0S1 + 0S2 subject to 11x +6y +S1 = 65 5x + 9y +S2 = 40 where x and y 0 Table1 9 7 0 0 profit qty x y S1 S2 0 S1 65 11 6 1 0 6 0 S2 40 5 9 0 1 8 0 0 0 0 Cj-Zj 9 7 0 0 Note : Cj-Zj is maximum at 9, therefore x shuld be introduced in place of S1 as it shows minimum cost, therefore new introduced rows can be computed as Formula is , New introduced row   old row - New row *key term First row Second row a b c= a/b d e f g= d- e*f 65 6 11 40 11 5 -15 11 6 2 5 2 5 -5 6 6 1 9 1 5 4 1 6 0 0 0 5 0 0 6 0 1 0 5 1 9 7 0 0 profit qty x y S1 S2 9 x 11 2 1 0 0 0 S2 -15 -5 4 0 1 99 18 9 0 0 Cj-Zj -9 -2 0 0 All the values of Cj-Zj are either zero or negative, therefore optimal program has been obtained here x = 11 therefore profit shall be maximixe when 11 units of x shall be produced that resultants to maiximize profit of 11*9 =99

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