Systems of Linear Inequalities

Systems of Linear Inequalities and Introduction to Linear Programming

Fundamental theorem of linear programming

Over the closed convex polygons, the maximum and minimum values of any linear expansion occur at corner points or vertices of the region or at all points along one side.

The inequalities that determine the solution region form the set of feasible solutions. Any solution satisfies the constraints or restrictions placed on the variables of the given problem. An optimal feasible solution is any feasible solution which optimizes the subject function under consideration.

APPLICATIONS

In real situations, x and y often represent quantities or amounts, which cannot be negative.In this case our graphs are restricted to the first quadrant, where x and y are both nonnegative.

Example 1:

An animal feed is to be a mixture of foodstuffs, each unit of which contains protein, fat, and carbohydrates in the number of grams:

Each bag of the resulting mixture is to contain at least 400g of protein, 21g of fat, and at most 1,400g of carbohydrates.Find the mixtures that meet these requirements. If each gram of foodstuff A costs P100 and each gram of foodstuff B costs P200, how much of each foodstuff should be used to get the least cost?

Foodstuff

Nutrients

A

B

Protein

100

50

Fats

1

9

Carbohydrates

100

300

Example 1:

Solution:Let x be the number of units of foodstuff ALet y be the number of units of foodstuff B

The following are the constraints, demands, or restrictions placed on the variables of our problem:Protein: at least 400g100x + 50 y 400

Foodstuff

Nutrients

A

B

Protein

100

50

Fats

1

9

Carbohydrates

100

300

Example 1:

Solution:The following are the constraints, demands, or restrictions placed on the variables of our problem:Fats: at least 21gx + 9y 21Carbohydrates: at most 1,400g100x + 300y 1400Nonnegativity constraintx 0y 0

Foodstuff

Nutrients

A

B

Protein

100

50

Fats

1

9

Carbohydrates

100

300

6

Example 1:

Solution:The cost for each combination could be represented byC = 100x + 200y in pesos.

Objective: minimize cost, C = 100x +200y.

7

Example 1:

Solution:Graph the system of inequalities.

100x + 50 y 400x + 9y 21100x + 300y 1400x 0y 0

8

Example 1:

Example 1:

Identify the vertices of the polygonal region.Point A:100x + 50y = 400x + 9y = 21A(3, 2)Point B:100x + 50y = 400100x + 300y = 1400B(2, 4)Point C:100x + 300y = 1400x + 9y = 21C(21/2, 7/6)

Example 1:

Evaluate the value of C at each vertex to find the minimum cost.C = 100x + 200y (Php)

A(3, 2)C = 100(3) + 200(2) C = P700

B(2, 4)C = 100(2) + 200(4) C = P1,000

C(21/2, 7/6)C = 100(21/2) + 200(7/6) C = P1,283

The least cost is P700, that is when 3 units of foodstuff A and 2 units of foodstuff B are taken.

Example 2:

A plastic toy factory manufactures two types of toy cars type A and type B. Regular orders require a daily output of at least 40 pieces of type A and 30 pieces of type B. It takes 1.5 man hours to produce a type A car and 2 man hours to produce a type B car . The staff can work no more than 204 man hours per day.A. Represent the given information in the form of inequalities and graph them to show the possible numbers of each type that can be produced each day.B. If the profit on each type A car is P30 and on each type B car is P20, find the number of each type that should be produced to maximize profit granting that all toy cars can be sold.

Example 2:

A plastic toy factory manufactures two types of toy cars type A and type B. Regular orders require a daily output of at least 40 pieces of type A and 30 pieces of type B. It takes 1.5 man hours to produce a type A car and 2 man hours to produce a type B car . The staff can work no more than 204 man hours per day.A. Represent the given information in the form of inequalities and graph them to show the possible numbers of each type that can be produced each day.Solution:Let x be the number of type A toy cars produced per day.Let y be the number of type B toy cars produced per day.

Example 2:

A plastic toy factory manufactures two types of toy cars type A and type B. Regular orders require a daily output of at least 40 pieces of type A and 30 pieces of type B. It takes 1.5 man hours to produce a type A car and 2 man hours to produce a type B car. The staff can work no more than 204 man hours per day.Solution:Let x be the number of type A toy cars produced per day.Let y be the number of type B toy cars produced per day.Constraints:x 40y 301.5x + 2y 204 x, y 0

Example 2:

Graph the constraints to show the possible numbers of each type that can be produced each day. :x 40y 301.5x + 2y 204

Example 2:

Graph the constraints to show the possible numbers of each type that can be produced each day. :x 40y 301.5x + 2y 204

Example 2:

Identify the vertices of the polygonal region.Point A:x = 40y = 30A(40, 30) Point B:x = 401.5x + 2y = 204B(40, 72)Point C:y = 301.5x + 2y = 204C(96, 30)

Example 2:

B. If the profit on each type A car is P30 and on each type B car is P20, find the number of each type that should be produced to maximize profit granting that all toy cars can be sold.

Objective: maximize profit, PP = 30x +20y ( in pesos)

Evaluate the value of P at each vertex to find the maximum profitP = 30x + 20y (Php)

Example 2:

Evaluate the value of P at each vertex to find the maximum profit.P = 30x + 20y (Php)

A(40, 30)P = 30(40) + 20(30) P = P1,800

96 type A and 30 type B toy cars must be produced per day to maximize profit.

B(40, 72)P = 30(40) + 20(72) P = P2,640

C(96, 30)P = 30(96) + 20(30) P = P3, 480

HOMEWORK #5:

On a particular airline flight a passenger is allowed to check a piece of luggage, without additional cost, provided the sum of the 3 dimensions (length, width, height) does not exceed 62 in. Suppose a piece of luggage has a height of 20in., and the length is greater than the width, but not more than twice its width. If the length is x inches and the width I y inches, what is the system of linear inequalities involving x and y?Graph the region of permissible values of x and y.

HOMEWORK #5:

Suppose, on the flight in #1, that a passenger has 2 pieces of luggage and that she is allowed to check the two pieces without additional cost provided their total dimension do not exceed 106 in. The sum of the 3 dimensions of the larger piece is 58 in. Furthermore, the smaller piece has a height of 12in. And the length greater than the width but not more than 10 in. greater.If the smaller piece of luggage has a length of x inches and a width of y inches, what is the system of linear inequalities involving x and y?Graph the region of permissible values of x and y.

An electronic company manufactures 2 radio models, each on a separate production line. The daily capacity of the first line is 60 radios and that of the second is 75 radios. Each unit of the first model uses 10 pieces of a certain electronic component, whereas each unit of the second model requires 8 pieces of the same component. The maximum daily availability of the special component is 800 pieces. The profit per unit of models 1 and 2 is $30 and $20, respectively. Determine the optimum daily production of each model.

HOMEWORK #5:

A company can advertise its product by using local radio and TV stations. Its budget limits the advertisement expenditures to $1000 a month. Each minute of radio advertisement costs $5 and each minute of TV advertisement costs $100. The company would like to use the radio at least twice as much as the TV. Past experience shows that each minute of TV advertisement will usually generate 25 times as many sales as each minute of radio advertisement. Determine the optimum allocation of the monthly budget to radio and TV advertisements.

HOMEWORK #5:

A company produces tow types of cowboy hats. Each hat of the first type requires twice as much labor time as does each hat the second type. If all hats are of the second type only, the company can produce a total of 500hats a day. The market limits daily sales of the first and second types to 150 and 200 hats. Assume that the profit per hat is $8 for type 1 and $5 for type 2. Determine the number of hats of each type to produce in order to maximize profit.

HOMEWORK #5:

Reference:

http://www.quickmath.com/webMathematica3/quickmath/graphs/inequalities/advanced.jsp