Chapter 5: Linear Programming with the Simplex Method
One of the most significant advancements in linear programming is the simplex method, developed by George Dantzig. This algorithm provides a systematic approach to finding the optimal solution to linear programming problems. In this article, we will explore the simplex method, its key concepts, and how it is applied to solve linear programming problems.
Overview of the Linear Programming with the Simplex Method
The simplex method is a systematic approach to traverse the vertices of the polyhedron containing feasible solutions in a linear programming problem. It aims to find the optimal solution by iteratively improving the objective function value. This method is considered one of the greatest inventions of modern times due to its broad applicability in solving business-related problems.
Formulating the Original Linear Programming Problem
To illustrate the simplex method, let’s consider a furniture production problem. We want to maximize the revenue generated by producing chairs and tables, subject to constraints on the availability of mahogany and labor. The original problem can be formulated as follows:
- Maximize Revenue = 45×1 + 80×2
- 5×1 + 20×2 ≤ 400 (Mahogany constraint)
- 10×1 + 15×2 ≤ 450 (Labor constraint)
- x1, x2 ≥ 0 (Non-negativity constraint)
In this formulation, x1 represents the number of chairs produced, x2 represents the number of tables produced, and the objective function maximizes the total revenue. The constraints limit the consumption of mahogany and labor within the available capacities.
Transforming into Standard Form
To apply the simplex method, we transform the original problem into the standard form, which involves converting the inequalities into equalities. We introduce slack variables to represent the surplus capacity of each constraint. In this case, we add h1 and h2 as slack variables for the mahogany and labor constraints, respectively. The problem in standard form becomes:
- Maximize Revenue = 45×1 + 80×2 + 0h1 + 0h2
- 5×1 + 20×2 + h1 = 400 (Mahogany constraint)
- 10×1 + 15×2 + h2 = 450 (Labor constraint)
- x1, x2, h1, h2 ≥ 0 (Non-negativity constraint)
The slack variables h1 and h2 represent the unused capacities of mahogany and labor, respectively. We still aim to maximize the revenue while satisfying these transformed equalities.
Basic Feasible Solutions and Canonical Form
A basic feasible solution is an initial production plan that satisfies all the constraints, with some variables set to zero. In our case, the initial solution where no chairs or tables are produced (x1=0, x2=0) represents a basic feasible solution. This solution generates zero revenue, as expected.
The non-basic variables in a basic feasible solution are set to zero, while the basic variables take positive values. In our initial solution, h1 and h2 are the basic variables, representing the unused capacities of mahogany and labor. The non-basic variables are x1 and x2, which are set to zero. This configuration is called a basic feasible solution and is an important concept in linear programming.
To express the problem in canonical form, we represent the basic variables (h1 and h2) in terms of the non-basic variables (x1 and x2). Similarly, we substitute the non-basic variables in the objective function. This process is known as pivoting. The canonical form of the problem becomes:
- Maximize z = 45×1 + 80×2 + 0h1 + 0h2
- x1 = 0 + (1/5)h1 – (4/5)h2 (Transformed mahogany constraint)
- x2 = 0 – (2/5)h1 + (3/5)h2 (Transformed labor constraint)
- x1, x2, h1, h2 ≥ 0
In the canonical form, the objective function and constraints are expressed with respect to the basic variables x1 and x2. The reduced costs associated with the non-basic variables x1 and x2 (coefficients in the objective function) indicate the potential improvement in the objective function value if these variables enter the basis.
Iteration and Optimal Solution
In each iteration of the simplex method, we analyze the non-basic variables and their coefficients. If all the non-basic variables have coefficients ≤ 0, the current solution is optimal. Negative reduced costs associated with the non-basic variables indicate that increasing these variables would decrease the objective function value.
If there is a non-basic variable with a positive reduced cost, we choose the one with the largest coefficient to enter the basis. To determine the maximum value for this variable, we perform the minimum ratio test using the transformed equations. The minimum ratio identifies the constraint that limits the increase of the non-basic variable while staying feasible.
After selecting the entering variable, we perform pivoting to express the problem in canonical form with respect to the new basic variables. This process continues iteratively until we reach an optimal solution or determine that the problem is unbounded.
The simplex method provides a systematic approach to solving linear programming problems by iteratively improving the objective function value. By transforming the problem into the standard form and expressing it in canonical form, we can identify basic feasible solutions and optimize the objective function. The simplex method is a fundamental tool in linear programming, enabling efficient optimization in various industries and applications.
Download the complete Linear Programming Tutorial Series slide deck .
View the entire series:
- Welcome: Linear Programming Tutorial
- Chapter 1: Mathematical Programming
- Chapter 2: Introduction to Linear Programming
- Chapter 3: Mixed Integer Linear Programming Problems
- Chapter 4: Furniture Factory Problem
- Chapter 5: Simplex Method
- Chapter 6: Modeling and Solving Linear Programming Problems
- Chapter 7: Sensitivity Analysis of Linear Programming Problems
- Chapter 8: Multiple Optimal Solutions
- Chapter 9: Unbounded Linear Programming Problems
- Chapter 10: Infeasible Linear Programming Problems
- Chapter 11: Linear Programming Overview – Further Considerations
- Chapter 12: Duality in Linear Programming
- Chapter 13: Optimality Conditions
- Chapter 14: Dual Simplex Method
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Simplex Method for Solution of L.P.P (With Examples) | Operation Research
After reading this article you will learn about:- 1. Introduction to the Simplex Method 2. Principle of Simplex Method 3. Computational Procedure 4. Flow Chart.
Introduction to the Simplex Method :
Simplex method also called simplex technique or simplex algorithm was developed by G.B. Dantzeg, An American mathematician. Simplex method is suitable for solving linear programming problems with a large number of variable. The method through an iterative process progressively approaches and ultimately reaches to the maximum or minimum values of the objective function.
Principle of Simplex Method :
It has not been possible to obtain the graphical solution to the LP problem of more than two variables. For these reasons mathematical iterative procedure known as ‘Simplex Method’ was developed. The simplex method is applicable to any problem that can be formulated in-terms of linear objective function subject to a set of linear constraints.
The simplex method provides an algorithm which is based on the fundamental theorem of linear programming. This states that “the optimal solution to a linear programming problem if it exists, always occurs at one of the corner points of the feasible solution space.”
The simplex method provides a systematic algorithm which consist of moving from one basic feasible solution to another in a prescribed manner such that the value of the objective function is improved. The procedure of jumping from vertex to the vertex is repeated. The simplex algorithm is an iterative procedure for solving LP problems.
It consists of:
(i) Having a trial basic feasible solution to constraints equation,
(ii) Testing whether it is an optimal solution,
(iii) Improving the first trial solution by repeating the process till an optimal solution is obtained.
Computational Procedure of Simplex Method :
The computational aspect of the simplex procedure is best explained by a simple example.
Consider the linear programming problem:
Maximize z = 3x 1 + 2x 2
Subject to x 1 + x 2 , ≤ 4
x 1 – x 2 , ≤ 2
x 1 , x 2 , ≥ 4
< 2 x v x 2 > 0
The steps in simplex algorithm are as follows:
Formulation of the mathematical model:
(i) Formulate the mathematical model of given LPP.
(ii) If objective function is of minimisation type then convert it into one of maximisation by following relationship
Minimise Z = – Maximise Z*
When Z* = -Z
(iii) Ensure all b i values [all the right side constants of constraints] are positive. If not, it can be changed into positive value on multiplying both side of the constraints by-1.
In this example, all the b i (height side constants) are already positive.
(iv) Next convert the inequality constraints to equation by introducing the non-negative slack or surplus variable. The coefficients of slack or surplus variables are zero in the objective function.
In this example, the inequality constraints being ‘≤’ only slack variables s 1 and s 2 are needed.
Therefore given problem now becomes:
The first row in table indicates the coefficient c j of variables in objective function, which remain same in successive tables. These values represent cost or profit per unit of objective function of each of the variables.
The second row gives major column headings for the simple table. Column C B gives the coefficients of the current basic variables in the objective function. Column x B gives the current values of the corresponding variables in the basic.
Number a ij represent the rate at which resource (i- 1, 2- m) is consumed by each unit of an activity j (j = 1,2 … n).
The values z j represents the amount by which the value of objective function Z would be decreased or increased if one unit of given variable is added to the new solution.
It should be remembered that values of non-basic variables are always zero at each iteration.
So x 1 = x 2 = 0 here, column x B gives the values of basic variables in the first column.
So 5, = 4, s 2 = 2, here; The complete starting feasible solution can be immediately read from table 2 as s 1 = 4, s 2 , x, = 0, x 2 = 0 and the value of the objective function is zero.
Flow Chart of Simplex Method :
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- Math Article
In Mathematics, linear programming is a method of optimising operations with some constraints. The main objective of linear programming is to maximize or minimize the numerical value. It consists of linear functions which are subjected to the constraints in the form of linear equations or in the form of inequalities. Linear programming is considered an important technique that is used to find the optimum resource utilisation. The term “linear programming” consists of two words as linear and programming. The word “linear” defines the relationship between multiple variables with degree one. The word “programming” defines the process of selecting the best solution from various alternatives.
Linear Programming is widely used in Mathematics and some other fields such as economics, business, telecommunication, and manufacturing fields. In this article, let us discuss the definition of linear programming, its components, and different methods to solve linear programming problems.
Table of Contents:
- Linear programming Problems
- Simplex Method
- Practice Problems
What is Linear Programming?
Linear programming (LP) or Linear Optimisation may be defined as the problem of maximizing or minimizing a linear function that is subjected to linear constraints. The constraints may be equalities or inequalities. The optimisation problems involve the calculation of profit and loss. Linear programming problems are an important class of optimisation problems, that helps to find the feasible region and optimise the solution in order to have the highest or lowest value of the function.
In other words, linear programming is considered as an optimization method to maximize or minimize the objective function of the given mathematical model with the set of some requirements which are represented in the linear relationship. The main aim of the linear programming problem is to find the optimal solution.
Linear programming is the method of considering different inequalities relevant to a situation and calculating the best value that is required to be obtained in those conditions. Some of the assumptions taken while working with linear programming are:
- 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:
- Decision Variables
- Objective Functions
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.
Linearity – The relationship between two or more variables in the function must be linear. It means that the degree of the variable is one.
Finiteness – There should be finite and infinite input and output numbers. In case, if the function has infinite factors, the optimal solution is not feasible.
Non-negativity – The variable value should be positive or zero. It should not be a negative value.
Decision Variables – The decision variable will decide the output. It gives the ultimate solution of the problem. For any problem, the first step is to identify the decision variables.
Linear Programming Problems
The Linear Programming Problems (LPP) is a problem that is concerned with finding the optimal value of the given linear function. The optimal value can be either maximum value or minimum value. Here, the given linear function is considered an objective function. The objective function can contain several variables, which are subjected to the conditions and it has to satisfy the set of linear inequalities called linear constraints. The linear programming problems can be used to get the optimal solution for the following scenarios, such as manufacturing problems, diet problems, transportation problems, allocation problems and so on.
Methods to Solve Linear Programming Problems
The linear programming problem can be solved using different methods, such as the graphical method, simplex method, or by using tools such as R, open solver etc. Here, we will discuss the two most important techniques called the simplex method and graphical method in detail.
Linear Programming Simplex Method
The simplex method is one of the most popular methods to solve linear programming problems. It is an iterative process to get the feasible optimal solution. In this method, the value of the basic variable keeps transforming to obtain the maximum value for the objective function. The algorithm for linear programming simplex method is provided below:
Step 1 : Establish a given problem. (i.e.,) write the inequality constraints and objective function.
Step 2: Convert the given inequalities to equations by adding the slack variable to each inequality expression.
Step 3 : Create the initial simplex tableau. Write the objective function at the bottom row. Here, each inequality constraint appears in its own row. Now, we can represent the problem in the form of an augmented matrix, which is called the initial simplex tableau.
Step 4 : Identify the greatest negative entry in the bottom row, which helps to identify the pivot column. The greatest negative entry in the bottom row defines the largest coefficient in the objective function, which will help us to increase the value of the objective function as fastest as possible.
Step 5 : Compute the quotients. To calculate the quotient, we need to divide the entries in the far right column by the entries in the first column, excluding the bottom row. The smallest quotient identifies the row. The row identified in this step and the element identified in the step will be taken as the pivot element.
Step 6: Carry out pivoting to make all other entries in column is zero.
Step 7: If there are no negative entries in the bottom row, end the process. Otherwise, start from step 4.
Step 8: Finally, determine the solution associated with the final simplex tableau.
The graphical method is used to optimize the two-variable linear programming. If the problem has two decision variables, a graphical method is the best method to find the optimal solution. In this method, the set of inequalities are subjected to constraints. Then the inequalities are plotted in the XY plane. Once, all the inequalities are plotted in the XY graph, the intersecting region will help to decide the feasible region. The feasible region will provide the optimal solution as well as explains what all values our model can take. Let us see an example here and understand the concept of linear programming in a better way.
Calculate the maximal and minimal value of z = 5x + 3y for the following constraints.
x + 2y ≤ 14
3x – y ≥ 0
x – y ≤ 2
The three inequalities indicate the constraints. The area of the plane that will be marked is the feasible region.
The optimisation equation (z) = 5x + 3y. You have to find the (x,y) corner points that give the largest and smallest values of z.
To begin with, first solve each inequality.
x + 2y ≤ 14 ⇒ y ≤ -(1/2)x + 7
3x – y ≥ 0 ⇒ y ≤ 3x
x – y ≤ 2 ⇒ y ≥ x – 2
Here is the graph for the above equations.
Now pair the lines to form a system of linear equations to find the corner points.
y = -(½) x + 7
Solving the above equations, we get the corner points as (2, 6)
y = -1/2 x + 7
y = x – 2
Solving the above equations, we get the corner points as (6, 4)
Solving the above equations, we get the corner points as (-1, -3)
For linear systems, the maximum and minimum values of the optimisation equation lie on the corners of the feasibility region. Therefore, to find the optimum solution, you only need to plug these three points in z = 3x + 4y
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
A real-time example would be considering the limitations of labours and materials and finding the best production levels for maximum profit in particular circumstances. It is part of a vital area of mathematics known as optimisation techniques. The applications of LP in some other fields are
- 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 is broadly applied in the field of optimisation for many reasons. Many functional problems in operations analysis can be represented as linear programming problems. Some special problems of linear programming are such as network flow queries and multi-commodity flow queries are deemed to be important to have produced much research on functional algorithms for their solution.
Linear Programming Video Lesson
Linear programming problem.
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
Linear programming is a process of optimising the problems which are subjected to certain constraints. It means that it is the process of maximising or minimizing the linear functions under linear inequality constraints. The problem of solving linear programs is considered as the easiest one.
Mention the different 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 are the requirements of linear programming?
The five basic requirements of linear programming are: Objective function Constraints Linearity Non-negativity Finiteness
Mention the advantages of Linear programming
The advantages of linear programming are: Linear programming provides insights to the business problems It helps to solve multi-dimensional problems According to the condition change, LP helps in making the adjustments By calculating the cost and profit of various things, LP helps to take the best optimal solution
What is meant by linear programming problems?
The linear programming problems (LPP) helps to find the best optimal solution of a linear function (also, known as the objective function) which are placed under certain constraints (set of linear inequality constraints)
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CSCA 5424: Approximation Algorithms and Linear Programming
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Solving Linear Programming Problems Using Simplex Method
To understand this subject matter first lets have a look at linear programming problems using the simplex algorithm. For those who don’t know what the simplex algorithm is let me give you a quick background. This simplex algorithm is a way of solving linear programming problems by taking a set of inputs and transforming them into another set of outputs. The main motivation behind the use of the simplex algorithm is that it makes the solving of linear problems much easier because it essentially simplifies everything.
One of the problems with linear programming is that there are usually too many factors to take into account when solving it. So when solving linear programming problems often we have to take into account such things as; time, input variables, output variables, average values and so on. In order to solve these problems effectively we need to keep track of the results of each step we make. To do this manually would be very tedious and probably impossible. Luckily we can solve linear programming problems using simplex programming assignment help.
We can solve linear programming problems by using the simplex algorithm but how can we get these kind of answers? Well if we want to use the simplex algorithm correctly, we first need to learn how to use the simplex software. The software is available online and is written in Java, which is one of the simplest languages to use and understand. The basic usage of the simplex software is to solve linear equations and the assignment to help guide us through the whole process.
Before using the software we will need to know a few details about linear equations. The first thing we should know is that every linear equation has an associated function called the basis function, and depending on the type of equation there will also be a range of unknowns called the range. The function and the range are using to solve the equation so that we can get the best possible answer. The next thing we need to do is to define a few variables for our linear programming problem and then we can start solving.
Let us start with solving linear equations. We can use the Java Simplex language so that the equation will be written in Java. We should include a main variable called a solution variable so that we can get the initial value before our solving the linear function. Next we should create two arrays that will hold the starting value of the x and the y coordinates of the point and the final value that will be the end result after the function is done. Then we can solve for both the x and y by using the formula Solving Linear Programming Problems Using Simplex Method. We can use one or two arrays for both the x and y.
Another important thing we should learn is how to solve quadratic equations. Quadratic function of a polynomial equation can be solved using the same method as we used for the linear function. We should define a few variables for our problem and then we can solve using the following code:
Simplex method was created to make it easier for people to solve almost any problems. You can find more information on solving linear and quadratic equations using this method on the internet. If you do not have the time to read books or tutorials, you can use the software that will solve these problems for you in minutes. Once you are done with the problems, you will be able to solve your problems easily using the software.