Introduction to Linear Programming

It is an optimization method for a linear objective function and a system of linear inequalities or equations. The linear inequalities or equations are known as constraints. The quantity which needs to be maximized or minimized (optimized) is reflected by the objective function. The fundamental objective of the linear programming model is to look for the values of the variables that optimize (maximize or minimize) the objective function.

We know that in linear programming, we subject linear functions to multiple constraints. These constraints can be written in the form of linear inequality or linear equations. This method plays a fundamental role in finding optimal resource utilization. The word "linear" in linear programming depicts the relationship between different variables. It means that the variables have a linear relationship between them. The word "programming" in linear programming shows that the optimal solution is selected from different alternatives.

We assume the following things while solving the linear programming problems:

  • The constraints are expressed in the quantitative values
  • There is a linear relationship between the objective function and the constraints
  • The objective function which is also a linear function needs optimization

 

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Features of Linear Programming

The linear programming problem has the following five features:

Constraints

These are the limitations set on the main objective function. These limitations must be represented in the mathematical form.

Objective function

This function is expressed as a linear function and it describes the quantity that needs optimization.

Linearity

There is a linear relationship between the variables of the function.

Non-negativity

The value of the variable should be zero or non-negative.

Parts of Linear Programming

The primary parts of a linear programming problem are given below:

  • Objective function
  • Decision variables
  • Data
  • Constraints

Why we Need Linear Programming?

The applications of linear programming are widespread in many areas. It is especially used in mathematics, telecommunication, logistics, economics, business, and manufacturing fields. The main benefits of using linear programming are given below:

  • It provides valuable insights to the business problems as it helps in finding the optimal solution for any situation.
  • In engineering, it resolves design and manufacturing issues and facilitates in achieving optimization of shapes.
  • In manufacturing, it helps to maximize profits.
  • In the energy sector, it facilitates optimizing the electrical power system
  • In the transportation and logistics industries, it helps in achieving time and cost efficiency.

 

In the next section, we will discuss the steps involved in solving linear programming problems.

Steps to Solve a Linear Programming Problem

We should follow the following steps while solving a linear programming problem graphically.

Step 1 - Identify the decision variables

The first step is to discern the decision variables which control the behavior of the objective function. Objective function is a function that requires optimization.

 

Step 2 - Write the objective function

The decision variables that you have just selected should be employed to jot down an algebraic expression that shows the quantity we are trying to optimize. In other words, we can say that the objective function is a linear equation that is comprised of decision variables.

 

Step 3 - Identify Set of Constraints

Constraints are the limitations in the form of equations or inequalities on the decision variables. Remember that all the decision variables are non-negative; i.e. they are either positive or zero.

 

Step 4 - Choose the method for solving the linear programming problem

Multiple techniques can be used to solve a linear programming problem. These techniques include:

  • Simplex method
  • Solving the problem using R
  • Solving the problem by employing the graphical method
  • Solving the problem using an open solver

In this article, we will specifically discuss how to solve linear programming problems using a graphical method.

 

Step 5 - Construct the graph

After you have selected the graphical method for solving the linear programming problem, you should construct the graph and plot the constraints lines.

 

Step 6 - Identify the feasible region

This region of the graph satisfies all the constraints in the problem. Selecting any point in the feasible region yields a valid solution for the objective function.

 

Step 7 - Find the optimum point

Any point in the feasible region that gives the maximum or minimum value of the objective function represents the optimal solution.

 

Now, that you know what are the steps involved in solving a linear programming problem, we will proceed to solve an example using the steps above.

 

Example

A bakery manufacturers two kinds of cookies, chocolate chip, and caramel. The bakery forecasts the demand for at least 80 caramel and 120 chocolate chip cookies daily. Due to the limited ingredients and employees, the bakery can manufacture at most 120 caramel cookies and 140 chocolate chip cookies daily. To be profitable the bakery must sell at least 240 cookies daily.

Each chocolate chip cookie sold results in a profit of $0.75 and each caramel cookie produces $0.88 profit.

a) How many chocolate chip and caramel cookies should be made daily to maximize the profit?

b) Compute the maximum revenue that can be generated in a day?

 

Solution

Part a

Follow the following steps to solve the above problem.

Step 1 - Identify the decision variables

Suppose:

Number of caramel cookies sold daily = x

Number of chocolate chip cookies sold daily = y

 

Step 2 - Write the Objective Function

Since each chocolate chip cookie yields the profit of $0.75 and each caramel cookie produces a profit of $0.88, therefore we will write the objective function as:

P = 0.88 x + 0.75 y, where x and y are non negative

 

Step 3 - Identify Set of Constraints

It is mentioned in the problem that the demand forecast of caramel cookies is at least 80 and the bakery cannot produce more than 120 caramel cookies. Therefore, we will write this constraint as:

80 \leq x \leq 120

It is also mentioned that the expected demand for the chocolate chip cookies is at least 120 and the bakery can produce no more than 140 cookies. Therefore, the second constraint is:

120 \leq y \leq 140

 

 To be profitable the company must sell at least 240 cookies daily. Therefore, the third constraint is:
                                                                                        x + y \geq 240
Step 4 - Choose the method for solving the linear programming problem
We will find the solution of the above problem graphically.
Step 5 - Construct the graph
P = 0.88 x + 0.75 y is subjected to the following constraints:
                                                                                           80 \leq x \leq 120
                                                                                           120 \leq y \leq 140
                                                                                           x + y \geq 240
Step 6 - Identify the feasible region

Example graph

The green area of the graph is the feasibility region.

 

Step 7 - Find the Optimum point

 

Now, we will test the vertices of the feasibility region to determine the optimal solution. The vertices are:

(120, 120) , (100, 140), (120, 140)

(120, 120) P = 0.88 (120) + 0.75 (120) = $ 195.6

(100, 140) P = 0.88 (100) + 0.75 (140) = $ 193

(120, 140) P = 0.88 (120) + 0.75 (140) = $ 210.6

Hence, the bakery should manufacture 120 caramel cookies and 140 chocolate cookies daily to maximize the profit.

Now, we will proceed to solve the part b of the problem.

 

Part b

The answer to the part b is given in the previous section. The maximum profit that can be generated in a day is $210.6.

 

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.