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

asked 2020-12-30

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

\(\begin{array}{|c|c|c|}\hline\text{Product}&\text{Line 1}&\text{Line 2}\\\hline A&12&4\\\hline\text{Total hours}&60&40\\\hline\end{array}\)

a. formulate a linear programming model to determine the optimal product mix that will maximize profit.

b. transform this model into standard form.

c. identify the amount of unused resources(i.e., slack)at each of the graphical extreme points.

d. what would be the effect on the optimal if the production time on line 1 were reduced to 40 hours.

c. What would be the effect on the optimal solution if the profit for product B were increased from $7 to $15? To$20?


Answers (1)


Plot the straight lines 12x+4y=60

Consider (shade) the region between this line and the origin of the graph paper, 4x+8y=40

consider (shade) the region between thisline and the origin of the graph paper

The region common for the two above is the critical region and one of its vertices give the maximum profit.The vertices of the critical region are:

(0, 0) with unused 60 hours ofline 1 and 40 hours of Line 2, NSK (0, 5), with unused 40 hoursof line 1 ,

(4, 3) with no unused resources. as (4, 3) represents 4 units of product A and 3 units of product B which utilizes 12*4+4*3=60 hours of Line 1 and 4*4+8*3=40 hours of Line 2.

and (5,0) with unused 20 hours of Line 2

The profit function is P=9x+7y

P is maximum at (4,3)

In the part (b) of the problem the line (1) will be 12x+4y=40

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