chapter 8: linear programming

Chapter 8 Linear Programming Chapter 8 Linear Programming Nguyen Thi Minh Tam ntmtam vnuagmail com December 17, 2020 1 8 1 Graphical solution of linear programming problems 2 8 2 Application of linear programming Linear programming problems A linear programming problem is a problem of maximizing or minimizing a linear function subject to linear constraints The constraints may be equalities or inequalities When linear programming problems have exactly two variables, it is possible to solve them.

Chapter 8: Linear Programming

Nguyen Thi Minh Tam

ntmtam.vnua@gmail.com

December 17, 2020

1 8.1 Graphical solution of linear programming problems

2 8.2 Application of linear programming

Linear programming problems

A linear programming problemis a problem of maximizing or minimizing a linear function subject to linear constraints The constraints may be equalities or inequalities

When linear programming problems have exactly two

variables, it is possible to solve them graphically

Example 1

Minimize − 2x + y

subject to

x + 2y ≤ 12

− x + y ≤ 3

x ≥ 0, y ≥ 0 (non-negativity constraints)

How to sketch the solution set of a linear inequality in two

variables x , y

1 Sketch the straight line obtained by replacing the inequality sign with an equal sign

2 Choose a test point (a, b) not on the line and substitute

x = a, y = b into the given inequality

If the line does not pass through the origin, (0, 0) is a good choice.

If the line passes through the origin, a point on one of the axes would be a good choice.

3 If the inequality is satisfied, then the side containing the test point is the region of interest If not, then we choose the region on the other side of the line

Example 2 Sketch the solution set of the inequality

x + 2y ≤ 12

Feasible region

The set of points which satisfy all of the constraints in a linear programming problem is called thefeasible region

Example 3 Sketch the feasible region

x + 2y ≤ 10 3x + y ≤ 10

x ≥ 0

y ≥ 0

Example 4 Solve the linear programming problem

Maximise 3x + 5y subject to x + 2y ≤ 10

3x + y ≤ 10

x ≥ 0

y ≥ 0

Graphical method for solving linear programming problems in two variables

1 Sketch the feasible region

2 Identify the corners of the feasible region and find their coordinates

3 Evaluate the objective function at the corners and choose the one which has the maximum or minimum value

Example 5 Solve the linear programming problem

Minimise x + 2y subject to x + 3y ≥ 15

5x + y ≥ 20 2x + 3y ≥ 24

x ≥ 0

y ≥ 0

8.2 Application of linear programming

Linear programming is a vital tool for business managers who need

to allocate finite resources such as labour, machine time or raw materials to maximise profits or minimise costs

Example 6

A firm manufactures two products, X and Y To make 1 unit of product X requires 3 units of raw materials and 2 units of labour

To make 1 unit of product Y requires 5 units of raw materials and

2 units of labour The total number of units available for raw materials and labour is 31500 and 17000, respectively The firm makes a profit of $15 for making and selling product X The corresponding profit for Y is $20

a) Formulate a linear programming problem to maximise the firm’s profit

b) Solve the linear programming problem graphically

General strategy for problem formulation

1 Identify the decision variables and label them x and y

2 Write down an expression for the objective function in terms

of x and y , and decide whether it needs maximising or minimising

3 Write down all constraints on the variables x and y , including the non-negativity constraints

Shadow Prices

The shadow price is the change in the optimal value of the objective function due to a 1-unit increase in one of the available resources

The shadow price is the maximum price that a company is willing to pay for an extra unit of a given resource

Example 7 Find the shadow price of raw materials in Example 6 and give an interpretation of its value

Example 8 A farmer wishes to feed pigs with minimum cost but needs to ensure that each receives at least 1.6 kg of protein, at least 0.3 kg of amino acid and no more than 0.3 kg of calcium per day Foods available are fish meal and meat scraps, which contain protein, calcium and amino acid according to the following table:

Fish meal costs $0.65 per kg, whereas meat scraps cost $0.52 per

kg Determine a minimum-cost feeding programme

Exercise 8.2, 8.2*1-8 (page 612-613), 2, 4, 7 (page 614-616)