### Linear Programming 1 Overview

Linear Programming is a mathematical technique for maximizing or minimizing a linear function of several variables, such as output or cost

Subjects > Mathematics > Linear Programming 1

Before moving direct into the problems of Linear Programming 1, let's see what we need to know about Linear Programming.

#### Meaning of Linear Programming

Linear programming sometimes called Linear Optimization (LP) may be defined in various ways as follows:

• Is a mathematical technique for maximizing or minimizing a linear function of several variables, such as output or cost
• Is the simplest technique where we depict complex relationships through linear functions and then find the optimum points.
• Is the method to achieve the best outcome (maximum or minimum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.
• Is the method to allocate scarce resources to competing activities in an optimal manner when the problem can be expressed using linear objective function and linear inequality constraints.

#### Common Terminologies used in Linear Programming

1. Decision variables:
Are the variables which will decide the output. They represent the ultimate solution.
2. Objective function:
It is defined as the objective of making decisions. For example "to make profit" is a objective function
3. Constraints:
Are restrictions or limitations on decision variables. They usually limit the value of the decision variables.
4. Non-negativity restrictions:
For all linear programs, the decision variables should always take non-negative values, which means the values for decision variables should be greater or equal to zero (0).

#### Process to formulate a Linear Programming problem

1. Identify the decision variables
2. Write the objective function
3. Mention the constraints
4. Explicitly state the non-negativity restrictions

For a problem to be a linear programming problem, the decision variables, objective function and constraints altogether have to be linear functions

#### Solutions of Linear Programming problem

Linear programming problems can be solved using different methods such as:

• Graphical method
• Using Open solver
• Simplex method
• Northwest Corner & least count method
The commonly used method is Graphical method, let's see how to solve using Graphical Method

#### How to solve LP problem using Graphical method ?

1. If a linear programming problem P has a solution, then it must occurs at a vertex, or corner points of feasible region (set), S associated with the problem.
2. Feasible region is the region where solutions are possible and likely to be achieved.
It is bounded by corner points

3. Suppose we are given a linear programming problem with a feasible set S and an objective function P = ax + by, then:
• If S (feasible region) is bounded then P has both maximum and minimum values on S.
• If S is unbounded and both a and b are non-negative, then P has a maximum value on S provided that the constraints defining S include x ≥ 0 and y ≥ 0.
4. Method of Corners:
• Graph the feasible set, S
• Find exact coordinates of all corner points
• Evaluate the objective function, P at each vertex.
• The maximum (if it exists) is the largest value of P at a vertex
• The minimum is the smallest value of P at a vertex.

Now, let's move in. Solved Linear Programming questions.

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Before moving direct into the problems of Linear Programming 1, let's see what we need to know about Linear Programming.

#### Meaning of Linear Programming

Linear programming sometimes called Linear Optimization (LP) may be defined in various ways as follows:

• Is a mathematical technique for maximizing or minimizing a linear function of several variables, such as output or cost
• Is the simplest technique where we depict complex relationships through linear functions and then find the optimum points.
• Is the method to achieve the best outcome (maximum or minimum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.
• Is the method to allocate scarce resources to competing activities in an optimal manner when the problem can be expressed using linear objective function and linear inequality constraints.

#### Common Terminologies used in Linear Programming

1. Decision variables:
Are the variables which will decide the output. They represent the ultimate solution.
2. Objective function:
It is defined as the objective of making decisions. For example "to make profit" is a objective function
3. Constraints:
Are restrictions or limitations on decision variables. They usually limit the value of the decision variables.
4. Non-negativity restrictions:
For all linear programs, the decision variables should always take non-negative values, which means the values for decision variables should be greater or equal to zero (0).

#### Process to formulate a Linear Programming problem

1. Identify the decision variables
2. Write the objective function
3. Mention the constraints
4. Explicitly state the non-negativity restrictions

For a problem to be a linear programming problem, the decision variables, objective function and constraints altogether have to be linear functions

#### Solutions of Linear Programming problem

Linear programming problems can be solved using different methods such as:

• Graphical method
• Using Open solver
• Simplex method
• Northwest Corner & least count method
The commonly used method is Graphical method, let's see how to solve using Graphical Method

#### How to solve LP problem using Graphical method ?

1. If a linear programming problem P has a solution, then it must occurs at a vertex, or corner points of feasible region (set), S associated with the problem.
2. Feasible region is the region where solutions are possible and likely to be achieved.
It is bounded by corner points

3. Suppose we are given a linear programming problem with a feasible set S and an objective function P = ax + by, then:
• If S (feasible region) is bounded then P has both maximum and minimum values on S.
• If S is unbounded and both a and b are non-negative, then P has a maximum value on S provided that the constraints defining S include x ≥ 0 and y ≥ 0.
4. Method of Corners:
• Graph the feasible set, S
• Find exact coordinates of all corner points
• Evaluate the objective function, P at each vertex.
• The maximum (if it exists) is the largest value of P at a vertex
• The minimum is the smallest value of P at a vertex.