- Math Article

## Linear Programming

## Graphical Method

## What is Linear Programming?

- The number of constraints should be expressed in the quantitative terms
- The relationship between the constraints and the objective function should be linear
- The linear function (i.e., objective function) is to be optimised

## Components of Linear Programming

The basic components of the LP are as follows:

## Characteristics of Linear Programming

The following are the five characteristics of the linear programming problem:

Constraints – The limitations should be expressed in the mathematical form, regarding the resource.

Objective Function – In a problem, the objective function should be specified in a quantitative way.

Non-negativity – The variable value should be positive or zero. It should not be a negative value.

## Linear Programming Problems

## Methods to Solve Linear Programming Problems

## Linear Programming Simplex Method

Step 1 : Establish a given problem. (i.e.,) write the inequality constraints and objective function.

Step 6: Carry out pivoting to make all other entries in column is zero.

Step 8: Finally, determine the solution associated with the final simplex tableau.

Calculate the maximal and minimal value of z = 5x + 3y for the following constraints.

To begin with, first solve each inequality.

Here is the graph for the above equations.

Now pair the lines to form a system of linear equations to find the corner points.

Solving the above equations, we get the corner points as (2, 6)

Solving the above equations, we get the corner points as (6, 4)

Solving the above equations, we get the corner points as (-1, -3)

z = 5(2) + 3(6) = 10 + 18 = 28

z = 5(6) + 3(4) = 30 + 12 = 42

z = 5(-1) + 3(-3) = -5 -9 = -14

Hence, the maximum of z = 42 lies at (6, 4) and the minimum of z = -14 lies at (-1, -3)

## Linear Programming Applications

- Engineering – It solves design and manufacturing problems as it is helpful for doing shape optimisation
- Efficient Manufacturing – To maximise profit, companies use linear expressions
- Energy Industry – It provides methods to optimise the electric power system.
- Transportation Optimisation – For cost and time efficiency.

## Importance of Linear Programming

## Linear Programming Video Lesson

## Linear Programming Practice Problems

Solve the following linear programming problems:

- A doctor wishes to mix two types of foods in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 10 units of vitamin C. Food ‘I’ contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C. Food ‘II’ contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs 50 per kg to purchase Food ‘I’ and Rs 70 per kg to purchase Food ‘II’. Formulate this problem as a linear programming problem to minimise the cost of such a mixture
- One kind of cake requires 200g of flour and 25g of fat, and another kind of cake requires 100g of flour and 50g of fat. Formulate this problem as a linear programming problem to find the maximum number of cakes that can be made from 5kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.

## Frequently Asked Questions on Linear Programming

## Mention the different types of linear programming.

## What are the requirements of linear programming?

## Mention the advantages of Linear programming

## What is meant by linear programming problems?

Thank you so much for clearly explained notes. I benefited a lot from them

Thank you very much for this material.

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## Steps to Solve a Linear Programming Problem

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:

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

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

The primary parts of a linear programming problem 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

## 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

## 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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## linear programming

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- Mathematics LibreTexts Library - Linear Programming - The Simplex Method
- Story of Mathematics - Linear Programming – Explanation and Examples
- Wolfram MathWorld - Linear Programming

## Linear Programming

## What is Linear Programming?

## Linear Programming Definition

## Linear Programming Examples

## Linear Programming Formula

- Objective Function: Z = ax + by
- Constraints: cx + dy ≤ e, fx + gy ≤ h. The inequalities can also be "≥"
- Non-negative restrictions: x ≥ 0, y ≥ 0

## How to Solve Linear Programming Problems?

- Step 1: Identify the decision variables.
- Step 2: Formulate the objective function. Check whether the function needs to be minimized or maximized.
- Step 3: Write down the constraints.
- Step 4: Ensure that the decision variables are greater than or equal to 0. (Non-negative restraint)
- Step 5: Solve the linear programming problem using either the simplex or graphical method.

Let us study about these methods in detail in the following sections.

## Linear Programming Methods

## Linear Programming by Simplex Method

- 40\(x_{1}\) - 30\(x_{2}\) + Z = 0

\(x_{1}\) + \(x_{2}\) + \(y_{1}\) =12

2\(x_{1}\) + \(x_{2}\) + \(y_{2}\) =16

\(y_{1}\) and \(y_{2}\) are the slack variables.

Step 2: Construct the initial simplex matrix as follows:

Using the elementary operations divide row 2 by 2 (\(R_{2}\) / 2)

Now apply \(R_{1}\) = \(R_{1}\) - \(R_{2}\)

Finally \(R_{3}\) = \(R_{3}\) + 40\(R_{2}\) to get the required matrix.

Also, when \(x_{1}\) = 4 and \(x_{2}\) = 8 then value of Z = 400

## Linear Programming by Graphical Method

Suppose we have to maximize Z = 2x + 5y.

The constraints are x + 4y ≤ 24, 3x + y ≤ 21 and x + y ≤ 9

To solve this problem using the graphical method the steps are as follows.

Step 1: Write all inequality constraints in the form of equations.

Step 2: Plot these lines on a graph by identifying test points.

3x + y = 21 passes through (0, 21) and (7, 0).

x + y = 9 passes through (9, 0) and (0, 9).

Any point that lies on or below the line x + 4y = 24 will satisfy the constraint x + 4y ≤ 24.

Similarly, a point that lies on or below 3x + y = 21 satisfies 3x + y ≤ 21.

Also, a point lying on or below the line x + y = 9 satisfies x + y ≤ 9.

C = (4, 5) formed by the intersection of x + 4y = 24 and x + y = 9

33 is the maximum value of Z and it occurs at C. Thus, the solution is x = 4 and y = 5.

## Applications of Linear Programming

- Manufacturing companies make widespread use of linear programming to plan and schedule production.
- Delivery services use linear programming to decide the shortest route in order to minimize time and fuel consumption.
- Financial institutions use linear programming to determine the portfolio of financial products that can be offered to clients.

- Introduction to Graphing
- Linear Equations in Two Variables
- Solutions of a Linear Equation
- Mathematical Induction

Important Notes on Linear Programming

- Linear programming is a technique that is used to determine the optimal solution of a linear objective function.
- The simplex method in lpp and the graphical method can be used to solve a linear programming problem.
- In a linear programming problem, the variables will always be greater than or equal to 0.

go to slide go to slide go to slide

## Practice Questions on Linear Programming

## FAQs on Linear Programming

What is meant by linear programming.

## What is Linear Programming Formula?

The general formula for a linear programming problem is given as follows:

## What is the Objective Function in Linear Programming Problems?

## How to Formulate a Linear Programming Model?

The steps to formulate a linear programming model are given as follows:

- Identify the decision variables.
- Formulate the objective function.
- Identify the constraints.
- Solve the obtained model using the simplex or the graphical method.

## How to Find Optimal Solution in Linear Programming?

## How to Find Feasible Region in Linear Programming?

To find the feasible region in a linear programming problem the steps are as follows:

- Draw the straight lines of the linear inequalities of the constraints.
- Use the "≤" and "≥" signs to denote the feasible region of each constraint.
- The region common to all constraints will be the feasible region for the linear programming problem.

## What are Linear Programming Uses?

## What Is Linear Programming? Meaning, Methods, and Examples

## Table of Contents

What is linear programming, linear programming methods, examples of linear programming.

## Linear programming formula

The linear programming formula may be regarded as follows:

- The function of the formula: ax + by = Z
- The formula’s operating limitations: cx + dy ≤ e and fx + gy ≤ h
- Other, non-negative restrictions: x ≥ 0, y ≥ 0

- Determine the choice factors
- Develop the objective function
- Determine whether the function should be decreased or maximized
- Record the limitations
- Verify that decision variables are either larger than or equal to 0. (Non-negative inhibition)
- Utilize either the simplex or graphical method to resolve the linear programming issue

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## Why is linear programming necessary?

## Traits of a linear programming task

A problem being solved through linear programming will have the following traits or characteristics:

- Subject to constraints : Regarding the resource, one should represent the restrictions in mathematical form.
- Geared towards an objective function : The objective function of an issue should be described quantitatively.
- A linear relationship : The function’s connection across two or more independent variables must be linear. It indicates that the variable’s degree is one.
- Includes only finite numbers : There should be output and input numbers that are both finite and infinite. The optimum solution is not implementable if the function contains an unlimited number of elements.
- Does not include negative values : The variable’s value must be zero or positive. The value should not be negative.
- Hinges on decision variables : The result is determined by the decision variable. It provides the final solution to the issue. The first step in solving any issue is to determine the decision factors.

See More: What Is COBOL Programming Language? Definition, Examples, Uses, and Challenges

## 1. The simplex method

## 2. Solving linear programming problems using R

## 3. Graphical linear programming

## 4. Linear programming using OpenSolver

## 5. Mixed-integer linear programming

See More: Pivoting From Coder to Solution Architect: Four Skills and Certifications to Thrive

## Example 1: Optimizing dietary needs and cost constraints

## Example 2: Optimizing food ingredients and food volume

## Example 3: Optimizing goods transportation costs

## Example 4: Optimizing product sales to arrive at maximum profit

See More: Cobol Programmer: Job Description, Key Skills, and Salary in 2022

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## What is Linear Programming?

In particular, this topic will explain:

## How to Solve Linear Programming Problems

- Identify the variables and the constraints.
- Find the objective function.
- Graph the constraints and identify the vertices of the polygon.
- Test the values of the vertices in the objective function.

This will become clearer in context with the example problems.

We will come back to the objective function, but, for now, it is important to just identify it.

That is, the area inside the polygon contains all possible solutions to the problem.

Consider the geometric region shown in the graph.

- What are the inequalities that define this function?
- If the objective function is 3x+2y=P, what is the maximum value of P?
- If the objective function is 3x+2y=P, what is the minimum value of P

In summary, our system of linear inequalities is x ≤ 5 and y ≥ – 1 / 2 x+4 and y ≤ – 1 / 4 x+4.

(5, 2.75): P=3(5)+2(2.75)=20.5.

Therefore, the function P has a maximum at the point (5, 2.75).

Looking at part B, we see that this happens at the point (0, 4), with an output of 8.

## Constraints

Now, we know that the amount of time spent making a square box is 2x.

Thus, overall, we have the following constraints:

These constraints function line the boundaries in the graphical region from example 1.

## The Objective Function

Let’s plug all three values into the profit function and see what happens.

(15, 10): P=4(15)+5(10)=60+50=110.

(22.5, 5): P=4(22.5)+5(5)=90+25=115.

At (22, 5), P=4(22)+5(5)=88+25=113.

Thus, in summary, our constraints are:

This gives us the graph below.

Therefore, our four vertices are (10.5, 10.5), ( 40 / 3 , 40 / 3 ), (21, 0), and (40, 0).

## Finding the Maximum

Now, we test all four points in the function P=8x+20y.

( 40 / 3 , 40 / 3 )=1120/3 (or about 373.33)

Therefore, our system of linear inequalities is:

In this case, we are finding the overlap of 6 different functions!

When we create the polygonal shaded region, we find that it has 5 vertices, as shown below.

## The Vertices

Now, we need to consider all 5 vertices and test them in the original function.

Moving the x values to the left and the numbers without a coefficient to the right gives us

Putting the x-values on the left and numbers without a coefficient on the right gives us

## Finding the Minimum

( 10 / 9 , 112 / 9 ): 10 / 9 + 112 / 9 = 112 / 9 , which is about 13.5.

( 5 / 4 , 37 / 4 ): 5 / 4 + 37 / 4 , which is 42 / 4 =10.5.

( 20 / 9 , 92 / 9 ): 20 / 9 + 92 / 9 = 112 / 9 . This is about 12.4.

Then, we know x ≥ 5 and y ≥ 2.

Her total number of hours, however, cannot be more than 20. Therefore, x+y ≤ 20.

Since she wants to have at least as many library hours as tutoring hours, she wants x ≥ y.

In sum, then Amy’s constraints are

## The Feasible Region

This graph looks like the one below.

## Alternative Solution?

Now, her constraints are just x ≥ 5, y ≥ 2, y ≤ -x+20, and y ≥ – 3 / 4 x+18.

Then, we end up with this region.

## IMAGES

## VIDEO

## COMMENTS

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