# Linear Programming Calculator

Linear Programming Calculator is a free online tool that displays the best optimal solution for a given constraint. STUDYQUERIES’ linear programming calculator tool works more efficiently and displays the best optimal solution in just a fraction of a second for a given objective function and linear constraint system.

## How to Use Linear Programming Calculator?

To use the linear programming calculator, follow these steps:

• Step 1: Enter the objective function and constraints in the appropriate input fields
• Step 2: Now click “Submit” to obtain the best result
• Step 3: The best optimal solution and the graph will be displayed in the new window

## What is Linear Programming?

Linear programming (LP, also known as linear optimization) is a mathematical method to achieve the best outcome (such as maximum profit or least cost) within a mathematical model whose requirements are represented by linear relationships. Mathematics programming (also referred to as mathematical optimization) is a type of linear programming.

Essentially, linear programming is a technique for optimizing a linear objective function, subject to linear equality and linear inequality constraints. This set consists of a convex polytope, where a convex polytope is defined as the intersection of a finitely many half-spaces, where each half-space is defined by a linear inequality. A real-valued affine (linear) function defined on this polyhedron defines its objective function.

If a point of the polytope has this function at its smallest (or largest) value, then linear programming finds it.

Linear programs are problems that can be expressed in canonical form as

Find a vector $$X$$

that maximizes $$C^T\times X$$

subject to $$AX \le B$$

and $$X \geq 0.$$

Components of x are the variables to be determined, c and b are given vectors (with $$C^T$$ indicating that the coefficients of c are used as a single-row matrix for the purpose of forming the matrix product), and A is a given matrix. The function whose value is to be maximized or minimized ($$X →C^T\times X$$ in this case) is called the objective function.

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The inequalities $$Ax ≤ B$$ and $$x ≥ 0$$ are the constraints that specify a convex polytope over which the objective function is to be optimized. When two vectors have the same dimensions, they are comparable. Every entry in the first vector is less-than or equal to the corresponding entry in the second vector, so the first vector is less-than or equal to the second vector.

Various fields of study can benefit from linear programming. Mathematicians and business people utilize it extensively, while some engineers use it for some problems as well. Manufacturing, transportation, energy, and telecommunications are among the industries that utilize linear programming models. It can be used to model diverse types of problems in planning, routing, scheduling, assignment, and design.

Example: Consider a chocolate manufacturing company that produces only two types of chocolate – A and B. Both the chocolates require Milk and Choco only. To manufacture each unit of A and B, the following quantities are required:

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

The company kitchen has a total of 5 units of Milk and 12 units of Choco. On each sale, the company makes a profit of

Rs 6 per unit A sold

Rs 5 per unit B sold.

Now, the company wishes to maximize its profit. How many units of A and B should it produce respectively?

### Common terminologies used in Linear Programming

We can examine some of the terms used in Linear Programming as

• Decision Variables: My output will be determined by the decision variables. These variables represent the ultimate outcome. Identifying the decision variables is the first step to solving any problem. My decision variables are the total number of units for A and B, denoted by X and Y respectively.
• Objective Function: It is defined as the purpose of making a decision. As shown in the example above, the company wishes to increase the total profit represented by Z. The optimization objective is the increase in profit.
• Constraints: Constrained variables are those that limit or restrict the variables of decision-making. Usually, they limit the value of the decision variables. My constraints in the above example are the limited availability of the resources Milk and Choco.
• Non-negativity restriction: For all linear programs, decision variables should always have positive values. This means the values for decision variables should be greater than or equal to 0.

### The Process to Identify a Linear Programming Problem

Defining a Linear Programming problem involves the following steps:

• Decide which variables need to be taken into account
• You need to write an objective function
• Describe the constraints
• The restriction on negativity should be stated explicitly

Linear programming is a problem whose decision variables, objective functions, and constraints are all linear functions.

It is called a Linear Programming Problem when all three conditions are met.

## Methods to Solve Linear Programming Problems

Linear programming can be solved in a variety of ways by using tools like R, the open solver, and other tools, including the graphical method and the simplex method. We will discuss in detail the simplex method and the graphical method, which are two of the most important methods.

### Linear Programming Simplex Method

One of the most popular methods to solve linear programming problems is the simplex method. The solution is iterated a number of times until it meets all the requirements. Using this method, the value of the basic variable is transformed to obtain the maximum value for the objective function. A simplex algorithm for linear programming is presented below.

• Step 1: Establish a given problem. (For example, write inequalities and objective functions.)
• Step 2: To solve the given inequalities, add the slack variable to each inequality expression.
• Step 3: Create an initial simplex tableau. Write the objective function at the bottom. Every inequality constraint appears in its own row. Now we can represent the problem as an augmented matrix, also known as an initial simplex tableau.
• Step 4: Identify the largest negative entry in the bottom row, which will be used to identify the pivot column. The largest negative entry in the bottom row represents the largest coefficient in the objective function, allowing us to increase its value as quickly as possible.
• Step 5: Calculate the quotients. We need to divide the entries in the far right column by the entries in the first column, excluding the bottom row, in order to calculate the quotient. A row is identified by the smallest quotient. The pivot point will be taken from the row identified in this step, as well as the element identified in this step.
• Step 6: Perform pivoting so that all other entries in the column are zero.
• Step 7: If no negative entries appear in the bottom row, stop the process. Otherwise, continue from step 4.
• Step 8: Determine the solution associated with the final simplex tableau.

### Graphical Method

Optimization of two-variable linear programming is done using the graphical method. A graphical method is the best way to find the optimal solution to a problem with two decision variables. The set of inequalities is subjected to constraints in this method. Afterward, the inequalities are plotted in the XY plane.

Hence, the intersection region will be considered to help decide the feasible region after all inequalities have been plotted in the YX graph. As well as describing what values our model can hold, the feasible region provides the optimal solution. Here is an example to help you understand the concept of linear programming better.

With the help of an example, let’s better understand this.

Example: A farmer has recently acquired a 110 hectares piece of land. He plans to grow wheat and barley on that land. With the quality of the sun and the region’s excellent climate, the entire production of Wheat and Barley can be sold. According to the data below, he wants to know how to plant each variety in the 110 hectares, given the costs, net profits, and labor requirements:

During the planning horizon, the farmer has a budget of US$10,000 and is available for 1,200 man-days. Ensure that the farmer gets the most value for his money. Solution: To solve this problem, we will first formulate our linear program. Formulation of Linear Problem Step 1: Determine the decision variables The total area for growing $$Wheat = X (hectares)$$ The total area for growing $$Barley = Y (hectares)$$ My decision variables are X and Y. Step 2: Formulate the objective function All the land’s production can be sold in the market. The farmer would want to maximize the profits from his harvest. Barley and wheat have respective net profits. Wheat earns the farmer US$50 per hectare and barley earns US$120 per hectare. Our objective function (given by Z) is, $$Max Z = 50X + 120Y$$ Step 3: Writing the constraints • According to the farmer’s budget, he has US$10,000. As well as the cost of producing wheat and barley per hectare, we receive this information as well. Farmers are limited in the total amount they can spend. So our equation becomes: $$100X + 200Y ≤ 10,000$$
• The next constraint is the upper limit of the number of man-days available for the planning horizon. There are 1200 man-days available. We are given the number of man-days per hectare for Wheat and Barley in the table. $$10X + 30Y ≤ 1200$$
• The third constraint involves the area available for plantations. There are 110 hectares available for plantations. So the equation becomes, $$X + Y ≤ 110$$

Step 4: Non-negativity restriction

X and Y will be greater or equal to 0. This is obvious.

$$X ≥ 0, Y ≥ 0$$

We have developed our linear program. Let’s solve it now.

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

I will simplify all the equations before plotting the graph for the above 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, $$X + Y ≤ 110.$$

Put the first two lines on a graph in the first quadrant (as shown below)

When budget and man-day constraints are active, the optimal feasible solution is obtained. This means the point at which the equations $$X + 2Y ≤ 100$$ and $$X + 3Y ≤ 120$$ intersect gives us the optimal solution.

The optimal values for X and Y are (60,20).

In order to maximize profit, the farmer should plant Wheat and Barley on 60 hectares of land and 20 hectares of land, respectively.

It is anticipated that the company will earn a maximum profit of,

$$Max Z = 50 \times 60 + 120 \times 20$$

= US\$5400

Everything taught here has also been taught in a free online course- Linear Programming for Data Science Professionals.

## Applications of Linear Programming

Optimization and linear programming are widely used in various industries. Manufacturing and service industries use linear programming regularly. We are going to examine the various applications of linear programming in this section.

• Linear programming is used by manufacturing industries to analyze their supply chains. Manufacturing industries strive to make operations as efficient as possible. A linear programming model can recommend changes to a manufacturer’s storage layout, workforce, and cutting production bottlenecks. Watch this video for a more in-depth look at a Warehouse case study from Cequent, a US-based company.
• In organized retail, linear programming is also used to optimize shelf space. There are so many products on the market today that it is imperative to understand what customers want. Big Bazaar, Walmart, Hypercity, Reliance, and others use optimization aggressively. The store’s products are strategically placed according to how customers shop. Customers should be able to locate & select the appropriate products easily. Due to limitations like limited shelf space and a wide range of products, this is not possible.
• In addition to optimizing delivery routes, optimization is also used to optimize cost. That is an extension of the popular traveling salesman problem. For an industry that involves multiple salesmen visiting multiple cities, optimization is used to find the most efficient route. FedEx, Amazon, etc. determine the delivery routes by using a clustering and greedy algorithm. Our goal is to reduce operation costs and times.
• In Machine Learning, optimizations are also used. Linear programming is the foundation for supervised learning. The system is trained to determine the values from the unknown input data by fitting a mathematical model based on the labeled input data.
• Well, linear programming has many other applications as well. In the real world, linear programming is applied by Shareholders, Sports, Stock Markets, etc. Explore further.

## Importance of Linear Programming

The field of optimization broadly applies linear programming for a variety of reasons. The concept of linear programming has application in many aspects of operations analysis. Linear programming problems such as network flow queries and multi-commodity flow queries are considered important enough to have generated extensive research on functional algorithms to solve them.

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## Linear Programming Calculator Ti 84:

The “Linear Programming Calculator Ti 84” refers to using the Texas Instruments TI-84 graphing calculator to solve linear programming problems. The TI-84 calculator is a popular tool used by students and professionals for various mathematical calculations and graphing functions.

Example: Suppose you have a linear programming problem where you want to maximize the profit given certain constraints. You can input the objective function, constraints, and variables into the TI-84 calculator and use its linear programming capabilities to find the optimal solution.

Solution: To solve a linear programming problem using the TI-84 calculator, you would typically follow these steps:
1. Enter the objective function and constraints into the calculator.
2. Use the calculator’s linear programming function to find the optimal solution.
3. The calculator will provide the optimal values for the variables and the maximum (or minimum) value of the objective function.

## Linear Programming Problem Calculator:

A “Linear Programming Problem Calculator” is a tool or software that helps users solve linear programming problems. These calculators are designed to handle complex calculations and provide solutions to linear programming problems efficiently.

Example: Let’s say you have a manufacturing company that produces two products, A and B. You want to determine the optimal production quantities that maximize profit while satisfying certain constraints, such as limited resources and demand. By using a linear programming problem calculator, you can input the relevant data, constraints, and objective function to find the optimal solution.

Solution: The linear programming problem calculator will take the input data, constraints, and objective function and apply mathematical algorithms, such as the simplex method, to determine the optimal solution. It will provide the values of decision variables that maximize (or minimize) the objective function while satisfying all constraints.

## Maximize Linear Programming Calculator:

A “Maximize Linear Programming Calculator” is a specific type of calculator or software designed to solve linear programming problems where the objective is to maximize a given function.

Example: Consider a company that produces two products, X and Y, and wants to maximize their total revenue. The company has limited resources and production capacities for both products. Using a maximize linear programming calculator, you can input the production constraints, resource availability, and revenue functions to find the production quantities that maximize the total revenue.

Solution: The maximize linear programming calculator will use mathematical optimization techniques to determine the optimal values for decision variables that maximize the given objective function. It will consider constraints and other relevant factors to provide the solution that yields the maximum possible value for the objective function.

## Linear Programming Graphical Method Calculator:

A “Linear Programming Graphical Method Calculator” is a tool that helps users solve linear programming problems using graphical methods. It allows users to visualize the problem’s constraints and objective function on a graph for easier interpretation and solution finding.

Example: Suppose you have a linear programming problem with two variables and two constraints. Using a linear programming graphical method calculator, you can plot the constraints as lines on a graph and identify the feasible region. By visually analyzing the feasible region and the objective function, you can determine the optimal solution.

Solution: The linear programming graphical method calculator will plot the constraints and objective function on a graph and shade the feasible region. It allows users to identify the corner points of the feasible region and evaluate the objective function at each corner point to find the optimal solution.

## Linear Programming Calculator With Steps:

A “Linear Programming Calculator With Steps” is a tool or software that provides a detailed step-by-step solution to linear programming problems. It not only gives the final result but also shows the intermediate calculations and the logic behind each step.

Example: Consider a linear programming problem with multiple constraints and variables. By using a linear programming calculator with steps, you can input the problem’s data and constraints. The calculator will then provide a complete solution, showing each step in detail, including how it formulates the problem, applies optimization algorithms, and finds the optimal solution.

Solution: A linear programming calculator with steps provides a comprehensive solution to linear programming problems, showing the mathematical transformations, optimization algorithms, and calculations involved in solving the problem. It helps users understand the entire process and verify the correctness of the solution.

## Simplex Method Calculator:

A “Simplex Method Calculator” is a specific type of calculator or software that solves linear programming problems using the simplex method. The simplex method is a widely used algorithm for solving linear programming problems.

Example: Suppose you have a linear programming problem with multiple variables and constraints. By using a simplex method calculator, you can input the problem’s data and constraints. The calculator will apply the simplex method algorithm to iteratively improve the objective function’s value until an optimal solution is reached.

Solution: A simplex method calculator uses the simplex algorithm to solve linear programming problems. It performs matrix operations, pivoting, and iteration to identify the optimal solution. The calculator provides the values of the decision variables and the maximum or minimum value of the objective function based on the given constraints.

## Nonlinear Programming Calculator:

A “Nonlinear Programming Calculator” is a tool or software designed to solve nonlinear programming problems. Nonlinear programming involves optimizing a function that is not necessarily linear, introducing complexities compared to linear programming problems.

Example: Let’s say you have a nonlinear programming problem where the objective function and constraints involve nonlinear equations or functions. Using a nonlinear programming calculator, you can input the relevant data and equations. The calculator will then apply appropriate algorithms, such as gradient-based methods or evolutionary algorithms, to find the optimal solution.

Solution: A nonlinear programming calculator utilizes advanced algorithms to solve nonlinear programming problems. It may involve techniques such as gradient descent, genetic algorithms, or constrained optimization methods to find the optimal solution. The calculator will provide the optimal values for decision variables and the maximum or minimum value of the objective function, subject to the given constraints.

## FAQs

What is linear programming used for?

With linear programming, you can find the optimal solution to a problem given a set of constraints. By formulating our real-life problem into a mathematical model, we are able to solve it. Linear inequalities and constraint-constraint objective functions are involved.

What are linear programming examples?

The most classic example of a linear programming problem is when a company must allocate its resources to create two different products. Different products require different amounts of time and money, which are typically limited resources, and they sell for different prices.

What is linear programming formula?

It can be defined as a problem in which a linear function is maximized or minimized under linear constraints. In mathematics, f(x,y) is called the objective function , where ax + by + c represents that function.

How do you solve a linear programming problem?

Steps to Solve a Linear Programming Problem

• 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.
• Step 5: Construct the graph.
• Step 6: Identify the feasible region.

What are the types of linear programming?

The different types of linear programming are:

• Solving linear programming by Simplex method.
• Solving linear programming using R.
• Solving linear programming by graphical method.
• Solving linear programming with the use of an open solver.

What is the basic concept of linear programming?

Linear programming is a mathematical technique for finding optimal solutions to problems that can be expressed using linear equations and inequalities. If a real-world problem can be represented accurately by the mathematical equations of a linear program, the method will find the best solution to the problem.

Why is it called linear programming?

One of the areas of mathematics that has extensive use in combinatorial optimization is called linear programming (LP). It derives its name from the fact that the LP problem is an optimization problem in which the objective function and all the constraints are linear.

Which Scientific Calculator Can You Solve Linear Programming?

Linear programming problems can be solved using various scientific calculators that have advanced mathematical capabilities. Some popular scientific calculators that can handle linear programming include the Texas Instruments TI-84, TI-89, Casio fx-9860GII, and HP Prime. These calculators often have built-in functions or programming capabilities specifically designed for linear programming problem-solving.

How To Put A Linear Programming Equation In Standard Form Calculator?

To put a linear programming equation in standard form using a calculator, you would typically follow these steps:

1. Identify the decision variables: Determine the variables that represent the quantities you want to optimize in the linear programming problem.

2. Formulate the objective function: Write the objective function that you want to maximize or minimize using the decision variables.

3. Set up the constraints: Write down the constraints that limit the values the decision variables can take. Ensure that each constraint is expressed as an inequality or an equality.

4. Convert inequalities to equalities: If any of the constraints are in the form of inequalities (e.g., “>=” or “<=”), convert them to equalities by introducing slack or surplus variables.

5. Add non-negativity constraints: Include non-negativity constraints for each decision variable by setting them to be greater than or equal to zero.

By following these steps, you can put a linear programming equation in standard form using a calculator, which allows you to input the problem accurately for further analysis and solution finding.

How To Do Linear Programming On Calculator?

To perform linear programming on a calculator, you would typically need a calculator with linear programming capabilities or a calculator that supports programming and advanced mathematical functions. Here is a general approach to doing linear programming on a calculator:

1. Formulate the linear programming problem: Write down the objective function and constraints of the problem, specifying the decision variables and their constraints.

2. Input the problem into the calculator: Depending on the calculator’s capabilities, you may have specific functions or programming features designed for linear programming. Use these features to input the problem accurately, including the objective function and constraints.

3. Apply the linear programming algorithm: Depending on the calculator, it may utilize various algorithms such as the simplex method or interior point methods to solve the linear programming problem. Follow the instructions provided by the calculator or programming language to apply the appropriate algorithm.

4. Obtain the optimal solution: After running the linear programming algorithm, the calculator will provide the optimal solution, including the values of the decision variables that optimize the objective function and satisfy the given constraints.

By following these steps and utilizing the calculator’s specific features, you can perform linear programming on a calculator efficiently.

How To Maximize Linear Programming Calculator?

To maximize a linear programming problem using a calculator, you would typically follow these steps:

1. Formulate the objective function: Write down the objective function that you want to maximize using the decision variables.

2. Set up the constraints: Write down the constraints that limit the values the decision variables can take.

3. Input the problem into the calculator: Use the calculator’s linear programming capabilities or programming features to input the objective function and constraints accurately.

4. Apply the appropriate optimization algorithm: Depending on the calculator, it may utilize algorithms such as the simplex method or interior point methods to maximize the objective function. Follow the calculator’s instructions or programming language to apply the appropriate algorithm.

5. Obtain the optimal solution: After running the optimization algorithm, the calculator will provide the optimal solution, including the values of the decision variables that maximize the objective function while satisfying the given constraints.

By following these steps and utilizing the calculator’s capabilities, you can use a linear programming calculator to maximize a linear programming problem effectively.

How To Find Optimal Solution In Linear Programming Calculator?

To find the optimal solution in a linear programming calculator, you would typically follow these steps:

1. Formulate the linear programming problem: Write down the objective function and constraints of the problem, specifying the decision variables and their constraints.

2. Input the problem into the calculator: Use the calculator’s linear programming capabilities or programming features to input the objective function and constraints accurately.

3. Apply the appropriate optimization algorithm: Depending on the calculator, it may use algorithms such as the simplex method or interior point methods to find the optimal solution. Follow the calculator’s instructions or programming language to apply the appropriate algorithm.

4. Interpret the results: After running the optimization algorithm, the calculator will provide the optimal solution. Interpret the results to obtain the values of the decision variables that optimize the objective function and satisfy the given constraints.

The specific steps may vary depending on the calculator and its features, but generally, by following these steps and utilizing the calculator’s capabilities, you can find the optimal solution to a linear programming problem.