Applications

Example 1

A clothing manufacturer makes cotton and wool jackets. It costs $80 to make each cotton jacket and $160 to make each wool jacket. Each day the company can make at most a total of 1000 jackets. If it has a daily budget of $100,000 for jackets and it makes $20 on each cotton jacket and $30 on each wool jacket, how many of each should be made each day to maximize daily profit?

Solution .

Let x = the number of cotton jackets.

Let y = the number of wool jackets.

The information in this problem can be summarized in the following chart.

________cotton___wool_____notes_________________

cost/unit____80____160____total cost can't exceed 100000

number_____x______y_____total number can't exceed 1000

profit/unit___20_____30____maximize total profit

Maximize

subject to

( i ) (cost constraint)

( ii ) (capacity constraint)

( iii )

( iv )

While constraints ( iii ) and ( iv ) have not been explicitly stated in this problem, they are needed since the number of jackets cannot be negative.

Graphing the constraints we get the feasible region shown below.

There are four vertices. Two of them, (0, 0) and (1000, 0) are easy to identify on the graph. The one on the y-axis is the y-intercept of the line 80x + 160y = 100000. This is (0, 625). The fourth vertex is the intersection of the lines x + y = 1000 and 80x + 160y = 100000. Solving this system, give us (750, 250). Now find the value of P at each corner.

Vertex _______ value of P __________

(0, 0)_______20(0)+30(0) = 0

(1000, 0)____20(1000)+30(0) = 20000

(0, 625)_____20(0)+30(625) = 18750

(750, 250)___20(750) + 30(250) = 22500

Conclusion : The maximum daily profit is $22500. This is achieved by making 750 cotton coats and 250 wool coats.

Example 2

Each unit of food X contains 1 unit of vitamin A, 1 unit of vitamin B, 3 units of vitamin C, and 0.2 units of fat. Each unit of food Y contains 1 unit of vitamin A, 2 units of vitamin B, 1 unit of vitamin C, and 0.35 units of fat. If some one needs to have at least 100 units of vitamin A, 130 units of vitamin B, and 120 units of vitamin C while minimizing fat, how many units of food X and food Y should be consumed?

Solution .

Let x = the number of units of food X.

Let y = the number of units of food Y.

The information is summarized in the following chart.

_______food X___food Y____notes____________

vitamin A___1_______1_____need at least 100 units

vitamin B___1_______2_____need at least 130 units

vitamin C___3_______1_____need at least 120 units

fat_______0.2______0.35___minimize total fat

The linear programming problem is

Minimize

subject to

( i )

( ii )

( iii )

( iv )

( v )

Graphing the system of inequalities we get the feasible region pictured below.

The feasible region is unbounded with vertices P, Q, R, and S.

The coordinates of the vertices are gotten by noting which lines intersect at each point and then solving the corresponding systems of equations.

P = (0, 120)

P is obtained by solving the system x = 0 and 3x + y = 120.

Q = (10, 90)

Q is obtained by solving the system x + y = 100 and 3x + y =120.

R = (70, 30)

R is obtained by solving the system x + y = 100 and x + 2y =130

S = (130, 0)

S is obtained by solving the system y = 0 and x + 2y = 130.

To find the point meeting the restrictions but also providing the least amount of fat, we evaluate F at each corner

Vertex____value of F__________

(0, 120)___0.2(0) + 0.35(120) = 42

(10, 90)___0.2(10) + 0.35(90) = 33.5

(70, 30)___0.2(70) + 0.35(30) = 24.5

(130, 0)___0.2(130) + 0.35(0) = 26

Conclusion :

Use 70 units of food X and 30 units of food Y to keep fat a minimum and meet all other requirements.