Consider the production problem: One company wishes to produce two types of souvenirs: type A and type B. Each type-A souvenir will result in a profit of $1, and each type-B souvenir will result in a profit of $1.2. To manufacture a type-A souvenir requires 2 minutes on machine I and 1 minute on machine II. A type-B souvenir requires 1 minute on machine I and 3 minutes on machine II.There are 180 minutes available on machine I and 300 minutes available on machine II for processing the order.
Maximaze

Objective function
Subject to

Constrained 1

Constrained 2

Now suppose the time available on machine I is changed from 180 minutes to (180 + h) minutes, where h is a real number. Then the constraint on machine I is changed to

But the line with equation

is parallel to the line

associated with the original constraint 1.
As you can see from the figure , the result of adding the constant h to the right-hand side of constraint 1 is to shift the current optimal solution from the point C to the new optimal solution occurring at the point C. To find the coordinates of C, we observe that C is the point of intersection of the lines with equations

Thus, the coordinates of the point are found by solving the system of linear equations

The solutions are

and

The nonnegativity of x implies that

Next, the nonnegativity of y implies that

Thus, h must satisfy the inequalities

. Computations reveal that for a meaningful solution the time available for machine I must range between (180 - 80) and (180 + 420) minutes-that is, between 100 and 600 minutes.
According to the problem, optimal solution would be

and

The resulting profit is calculated as follows:

Upon setting h = 1, we find

Since the optimal profit for the original problem is $159.6, we see that the shadow price for the first resource is 149.16 - 148.80, or $.36.
Show that if the time available on machine II is changed from 300 min to (300 + k) min, with no change in the maximum capacity for machine I, then company profit is maximized by producing

type-A souvenirs and

type-B souvenirs, where

.
A) If the time available on machine II is changed from 300 min to (300 + k) min, with no change in the maximum capacity for machine I, the constraint on machine II is changed to

Optimal quantum of output is found by solving the system of linear equations

and

The solutions are

and

.
B) If the time available on machine I is changed from 180 min to (180 + h) min, with no change in the maximum capacity for machine I, the constraint on machine I is changed to

. Correspondingly constraint II is changed to

. In accordance with nonnegativity x and y (

and

) we could take x = 0 and y = 0 and substitute them to constraints:

. Finally optimal value of type-A is

and optimal value of type-B is

C) Suppose the time available on machine II is changed from 180 minutes to (180 + h) minutes, where h is a real number. Then the constraint on machine II is changed to

. New optimal solutions are found by solving the system of linear equations

and

. Finally,

and

Determine Graphically the Solution Set for the System of Inequalities

Multiple Choice

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