uring has a division that model A and model B. To 3 lb of cast Iron and 6 min of

Question

uring has a division that model A and model B. To 3 lb of cast Iron and 6 min of labor. To hibachi requires 4 lb of cast iron and 3 min of labor. The profit for the production of hibachis each week, how many hibachis of each for each model A hibachi is $7, and the profit for each model B hibachl is $6.50. If 1000 of cast iron and 2o are available model should the division produce each week to maximize Kane's profie? X hibachis hlbachis model B 333 vlb labor o labor-hr plans to Iivest up to s whereas Project 8 yields a return of 15% on the reject a s nekier than th investment in Project A, the nnancer has dec ded that the nvestment in Project a gold not exceed 40% of the total investment. How much should she invest in each project to maximize the return on her investment? Project A $496000 Project B 330000

Explanation / Answer

1. Let x = quantity of model A hibachis produced

Let y = quantity of model B hibachis produced

Our objective function to maximize is profit: P = 7x1 + 6.5x2

We have 1000 lb of cast iron available

The amount of iron used in making model A’s = 3x1

The amount of iron used in making model B’s = 4x2

So constraint 1 is:

Con 1: 3x1 + 4x2 <= 1000

We have 20 labor hours = 1200 minutes available

The amount of time (in minutes) spent creating model A’s = 6x1

The amount of time (in minutes) spent creating model B’s = 3x2

So constraint 2 is: Con 2: 6x + 3y <= 1200

Thus we are to:

Maximize P = 7x1 + 6.5x2  

subject to 3x1 + 4x2 <= 1000

6x1 + 3x2 <= 1200

The problem is converted to canonical form by adding slack, surplus and artificial variables as appropiate

1. As the constraint 1 is of type '' we should add slack variable S1

2. As the constraint 2 is of type '' we should add slack variable S2

After introducing slack variables

Positive maximum Cj-Zj is 7 and its column index is 1. So, the entering variable is x1.

Minimum ratio is 200 and its row index is 2. So, the leaving basis variable is S2.

The pivot element is 6.

Entering =x1, Departing =S2, Key Element =6

R2(new)=R2(old)÷6

R1(new)=R1(old)-3R2(new)

Positive maximum Cj-Zj is 3 and its column index is 2. So, the entering variable is x2.

Minimum ratio is 160 and its row index is 1. So, the leaving basis variable is S1.

The pivot element is 52.

Entering =x2, Departing =S1, Key Element =52

R1(new)=R1(old)×25

R2(new)=R2(old)-12R1(new)

Since all Cj-Zj0

Hence, optimal solution is arrived with value of variables as :
x1=120,x2=160

Max Z=1880

We conclude that making 120 model A hibachis and 160 model B hibachis will maximize the profit to be $480. We note that both slack variables are zero so there will be no resources left over.

Max Z = 7 x1 + 6.5 x2 + 0 S1 + 0 S2 subject to 3 x1 + 4 x2 + S1 = 1000 6 x1 + 3 x2 + S2 = 1200 and x1,x2,S1,S20

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