## Steps to Solve a Linear Programming Problem

Introduction to linear programming, features of linear programming, parts of linear programming, why we need 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

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.

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

Non-negativity

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

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

- Decision variables

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.

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

Step 3 - Identify Set of Constraints

## 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

## Step 5 - Construct the graph

## Step 6 - Identify the feasible region

## Step 7 - Find the optimum point

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?

Follow the following steps to solve the above problem.

Number of caramel cookies sold daily = x

Number of chocolate chip cookies sold daily = y

Step 2 - Write the Objective Function

The green area of the graph is the feasibility region.

Step 7 - Find the Optimum point

(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

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

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## Linear Programming Examples

Linear programming, linear programming problems and solutions, cancel reply.

I like it is easier to understand it the way you break it down

This is awesome and simple to digest. It indeed is user friendly 😊🙇🏽♀️

How many of each should be made to maximize profit?

1. The cost of capital applicable to both projects is 12%

2. Project A requires sh. 20,000 and Project B 10,000 initial investment.

3. The funds available are restricted as follows;

4. Funds not utilized one year will not be available in the subsequent years.

i. Formulate a linear programming model to solve the above problem.

ii. Solve the problem graphically and comment on the proportion of investing on the two projects.

have you got the answer for this problem sir

Please sir can you help me solve this

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## Ex 12.1, 1 - Chapter 12 Class 12 Linear Programming (Term 1)

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## What Is Linear Programming? Definition, Methods and Problems for Data Scientists

## Table of Contents

## Example of a Linear Programming Problem (LPP)

## Formulating a Problem

- Each unit of A requires 1 unit of Milk and 3 units of Choco
- Each unit of B requires 1 unit of Milk and 2 units of Choco

Let the total number of units produced by A be = X

Let the total number of units produced by B be = Y

Now, the total profit is represented by Z

which means we have to maximize Z.

Also, the values for units of A can only be integers.

So we have two more constraints, X ≥ 0 & Y ≥ 0

For the company to make maximum profit, the above inequalities have to be satisfied.

This is called formulating a real-world problem into a mathematical model.

Let us define some terminologies used in Linear Programming using the above example.

- Decision Variables: The decision variables are the variables that will decide my output. They represent my ultimate solution. To solve any problem, we first need to identify the decision variables. For the above example, the total number of units for A and B denoted by X & Y respectively are my decision variables.
- Objective Function: It is defined as the objective of making decisions. In the above example, the company wishes to increase the total profit represented by Z. So, profit is my objective function.
- Constraints: The constraints are the restrictions or limitations on the decision variables. They usually limit the value of the decision variables. In the above example, the limit on the availability of resources Milk and Choco are my constraints.
- Non-negativity Restriction: For all linear programs, the decision variables should always take non-negative values. This means the values for decision variables should be greater than or equal to 0.

## The Process of Formulating a Linear Programming Problem

Let us look at the steps of defining a Linear Programming problem generically:

- Identify the decision variables
- Write the objective function
- Mention the constraints
- Explicitly state the non-negativity restriction

If all the three conditions are satisfied, it is called a Linear Programming Problem .

Let’s understand this with the help of an example.

Solution: To solve this problem, first we gonna formulate our linear program.

## Formulation of a Linear Problem

Step 1: Identify the decision variables

The total area for growing Wheat = X (in hectares)

The total area for growing Barley = Y (in hectares)

X and Y are my decision variables.

Step 2: Write the objective function

Our objective function (given by Z) is, Max Z = 50X + 120Y Step 3: Writing the constraints

X + Y ≤ 110 Step 4: The non-negativity restriction

The values of X and Y will be greater than or equal to 0. This goes without saying.

We have formulated our linear program. It’s time to solve it.

## Solving an LP Through the Graphical Method

Since we know that X, Y ≥ 0. We will consider only the first quadrant.

To plot for the graph for the above equations, first I will simplify all the equations.

100X + 200Y ≤ 10,000 can be simplified to X + 2Y ≤ 100 by dividing by 100.

10X + 30Y ≤ 1200 can be simplified to X + 3Y ≤ 120 by dividing by 10.

The third equation is in its simplified form, X + Y ≤ 110.

Plot the first 2 lines on a graph in the first quadrant (like shown below)

The values for X and Y which gives the optimal solution is at (60,20).

The maximum profit the company will gain is,

Max Z = 50 * (60) + 120 * (20)

The objective function is: Max.Z=25x+20y

where x are the units of pipe A

Solution: First, I’m gonna formulate my linear program in a spreadsheet.

Solution: First I am going to formulate my problem for a clear understanding.

Step 1: Identify Decision Variables

The objective of the company is to maximize the audience. The objective function is given by:

Now, I will mention each constraint one by one.

So our equations are as follows:

## Northwest Corner Method

- The level of supply and demand at each source is given
- The unit transportation of a commodity from each source to each destination

Solution: Let’s understand what the above table explains.

The total cost of transportation is = 5*10+(2*10+7*5)+9*15+(20*5+18*10) = $520

## Least Cost Method

- Manufacturing industries use linear programming for analyzing their supply chain operations . Their motive is to maximize efficiency with minimum operation cost. As per the recommendations from the linear programming model, the manufacturer can reconfigure their storage layout, adjust their workforce and reduce the bottlenecks. Here is a small Warehouse case study of Cequent a US-based company, watch this video for a more clear understanding.
- Linear programming is also used in organized retail for shelf space optimization . Since the number of products in the market has increased in leaps and bounds, it is important to understand what does the customer want. Optimization is aggressively used in stores like Walmart, Hypercity, Reliance, Big Bazaar, etc. The products in the store are placed strategically keeping in mind the customer shopping pattern. The objective is to make it easy for a customer to locate & select the right products. This is subject to constraints like limited shelf space, a variety of products, etc.
- Optimization is also used for optimizing Delivery Routes . This is an extension of the popular traveling salesman problem. The service industry uses optimization for finding the best route for multiple salesmen traveling to multiple cities. With the help of clustering and greedy algorithm, the delivery routes are decided by companies like FedEx, Amazon, etc. The objective is to minimize the operation cost and time.
- Optimizations are also used in Machine Learning . Supervised Learning works on the fundamental of linear programming. A system is trained to fit on a mathematical model of a function from the labeled input data that can predict values from an unknown test data.

## Frequently Asked Questions

Q1. what is linear programming and why is it important.

## Q2. What is a linear programming problem in simple words?

## Q3. What is an objective function in LPP (linear programming problem)?

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