Kalyan Singhal Corp. makes three products, and it has three machines available a

Question

Kalyan Singhal Corp. makes three products, and it has three machines available as resources as given in the following LP problem: Maximize contribution = 4x1 + 4x2 + 7x3 1x,+7x2+4X33 80 2X1+ 1X2+ 7X3 S 84 (C1: hours on machine 1) (C2: hours on machine 2) (C3: hours on machine 3) 8x, +4x2+1X3 5 80 8X1 +4X2 + 1X3 s 80 X1. X2.x3 20 X1, X2, X3 20 a) Using a computer software for solving LP, the optimal solution achieved is: X1 = 626 (round your response to two decimal places) X2= 5.12 (round your response to two decimal places). X3= 9.49 (round your response to two decimal places). Contribution (objective value)-111.86 (round your response to two decimal places). b) Machine 1 has 0.00 hours of unused time available at the optimal solution (round your response to two decimal places). Machine 2 has 0.00 hours of unused time available at the optimal solution (round your response to two decimal places). Machine 3 has 0.00 hours of unused time available at the optimal solution (round your response to two decimal places)

Explanation / Answer

The restrictions on the machine capacity are expressed in this manner:
To manufacture one standard unit requires two hours of grinding time; so that making x standard models uses 2x hours. Similarly, the production of y deluxe models uses 5y hours of grinding time. With 120 hours of grinding time available, the grinding capacity is written: 2x + 5y 120 hours of grinding capacity per week. The limitation on the polishing capacity is expressed: 4x + 2y 80 hours per week.

In summary, the relevant information is:

Grinding Time

Polishing Time

Contribution Margin

2 hours
5 hours
120 hours

4 hours
2 hours
80hours

$3
$4

When a linear programming problem involves only two variables, a two dimensional graph can be used to determine the optimal solution. In this example, the x-axis represents the number of standard models, and the y axis represents the number of deluxe models. The maximum number of each model that can be produced, given the constraints, are

Maximum Number of Models

Standard

Deluxe

(120 / 2) = 60

(120 / 5) = 24

(80 / 4) = 20

(80 / 2) = 40

The lowest number of each of the two columns measures the impact of the hours limitations. It appears that at best, the company can produce 20 standard models with a contribution margin of $60(20 × $3) or 24 deluxe models at a contribution margin of $96(24 × $4). However, producing a combination of standard and deluxe model may be a better solution.

To determine the combination of production levels in order to maximize the contribution margin, all the constraints are plotted on the graph. In this example, the polishing and grinding constraints are drawn by connecting the points that represent the extremes of production of each model. These points are:

When x = 0:    

When y = 0:    

Grinding Time

Polishing Time

Contribution Margin

Standard model
Deluxe model
Plant capacity

2 hours
5 hours
120 hours

4 hours
2 hours
80hours

$3
$4

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