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Mathematics LibreTexts

4.2: Maximization By The Simplex Method

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  • Page ID 37869

  • Rupinder Sekhon and Roberta Bloom
  • De Anza College

Learning Objectives

In this section, you will learn to solve linear programming maximization problems using the Simplex Method:

  • Identify and set up a linear program in standard maximization form
  • Convert inequality constraints to equations using slack variables
  • Set up the initial simplex tableau using the objective function and slack equations
  • Find the optimal simplex tableau by performing pivoting operations.
  • Identify the optimal solution from the optimal simplex tableau.

In the last chapter, we used the geometrical method to solve linear programming problems, but the geometrical approach will not work for problems that have more than two variables. In real life situations, linear programming problems consist of literally thousands of variables and are solved by computers. We can solve these problems algebraically, but that will not be very efficient. Suppose we were given a problem with, say, 5 variables and 10 constraints. By choosing all combinations of five equations with five unknowns, we could find all the corner points, test them for feasibility, and come up with the solution, if it exists. But the trouble is that even for a problem with so few variables, we will get more than 250 corner points, and testing each point will be very tedious. So we need a method that has a systematic algorithm and can be programmed for a computer. The method has to be efficient enough so we wouldn't have to evaluate the objective function at each corner point. We have just such a method, and it is called the simplex method .

The simplex method was developed during the Second World War by Dr. George Dantzig. His linear programming models helped the Allied forces with transportation and scheduling problems. In 1979, a Soviet scientist named Leonid Khachian developed a method called the ellipsoid algorithm which was supposed to be revolutionary, but as it turned out it is not any better than the simplex method. In 1984, Narendra Karmarkar, a research scientist at AT&T Bell Laboratories developed Karmarkar's algorithm which has been proven to be four times faster than the simplex method for certain problems. But the simplex method still works the best for most problems.

The simplex method uses an approach that is very efficient. It does not compute the value of the objective function at every point; instead, it begins with a corner point of the feasibility region where all the main variables are zero and then systematically moves from corner point to corner point, while improving the value of the objective function at each stage. The process continues until the optimal solution is found.

To learn the simplex method, we try a rather unconventional approach. We first list the algorithm, and then work a problem. We justify the reasoning behind each step during the process. A thorough justification is beyond the scope of this course.

We start out with an example we solved in the last chapter by the graphical method. This will provide us with some insight into the simplex method and at the same time give us the chance to compare a few of the feasible solutions we obtained previously by the graphical method. But first, we list the algorithm for the simplex method.

THE SIMPLEX METHOD

  • Set up the problem. That is, write the objective function and the inequality constraints.
  • Convert the inequalities into equations. This is done by adding one slack variable for each inequality.
  • Construct the initial simplex tableau. Write the objective function as the bottom row.
  • The most negative entry in the bottom row identifies the pivot column.
  • Calculate the quotients. The smallest quotient identifies a row. The element in the intersection of the column identified in step 4 and the row identified in this step is identified as the pivot element. The quotients are computed by dividing the far right column by the identified column in step 4. A quotient that is a zero, or a negative number, or that has a zero in the denominator, is ignored.
  • Perform pivoting to make all other entries in this column zero. This is done the same way as we did with the Gauss-Jordan method.
  • When there are no more negative entries in the bottom row, we are finished; otherwise, we start again from step 4.
  • Read off your answers. Get the variables using the columns with 1 and 0s. All other variables are zero. The maximum value you are looking for appears in the bottom right hand corner.

Now, we use the simplex method to solve Example 3.1.1 solved geometrically in section 3.1.

Example \(\PageIndex{1}\)

Niki holds two part-time jobs, Job I and Job II. She never wants to work more than a total of 12 hours a week. She has determined that for every hour she works at Job I, she needs 2 hours of preparation time, and for every hour she works at Job II, she needs one hour of preparation time, and she cannot spend more than 16 hours for preparation. If she makes $40 an hour at Job I, and $30 an hour at Job II, how many hours should she work per week at each job to maximize her income?

In solving this problem, we will follow the algorithm listed above.

STEP 1. Set up the problem. Write the objective function and the constraints.

Since the simplex method is used for problems that consist of many variables, it is not practical to use the variables \(x\), \(y\), \(z\) etc. We use symbols \(x_1\), \(x_2\), \(x_3\), and so on.

  • \(x_1\) = The number of hours per week Niki will work at Job I and
  • \(x_2\) = The number of hours per week Niki will work at Job II.

It is customary to choose the variable that is to be maximized as \(Z\).

The problem is formulated the same way as we did in the last chapter.

\[\begin{array}{ll} \textbf { Maximize } & \mathrm{Z}=40 \mathrm{x}_{1}+30 \mathrm{x}_{2} \\ \textbf { Subject to: } & \mathrm{x}_{1}+\mathrm{x}_{2} \leq 12 \\ & 2 \mathrm{x}_{1}+\mathrm{x}_{2} \leq 16 \\ & \mathrm{x}_{1} \geq 0 ; \mathrm{x}_{2} \geq 0 \end{array}\nonumber \]

STEP 2. Convert the inequalities into equations. This is done by adding one slack variable for each inequality.

For example to convert the inequality \(x_1 + x_2 ≤ 12\) into an equation, we add a non-negative variable \(y_1\), and we get

\[x_1 + x_2 + y_1 = 12 \nonumber \]

Here the variable \(y_1\) picks up the slack, and it represents the amount by which \(x_1 + x_2\) falls short of 12. In this problem, if Niki works fewer than 12 hours, say 10, then \(y_1\) is 2. Later when we read off the final solution from the simplex table, the values of the slack variables will identify the unused amounts.

We rewrite the objective function \(Z = 40x_1 + 30x_2\) as \(- 40x_1 - 30x_2 + Z = 0\).

After adding the slack variables, our problem reads

\[\begin{array}{ll} \text { Objective function } & - 40x_1 - 30x_2 + Z = 0 \\ \text { Subject to constraints: } & x_1 + x_2 + y_1 = 12 \\ & 2x_1 + x_2 + y_2 = 16 \\ & x1 ≥ 0; x2 ≥ 0 \end{array} \nonumber \]

STEP 3. Construct the initial simplex tableau . Each inequality constraint appears in its own row. (The non-negativity constraints do not appear as rows in the simplex tableau.) Write the objective function as the bottom row.

Now that the inequalities are converted into equations, we can represent the problem into an augmented matrix called the initial simplex tableau as follows.

Example4.2.1a.png

Here the vertical line separates the left hand side of the equations from the right side. The horizontal line separates the constraints from the objective function. The right side of the equation is represented by the column C.

The reader needs to observe that the last four columns of this matrix look like the final matrix for the solution of a system of equations. If we arbitrarily choose \(x_1 = 0\) and \(x_2 = 0\), we get

\[\left[\begin{array}{ccccc} y_{1} & y_{2} & Z & | & C \\ 1 & 0 & 0 & | & 12 \\ 0 & 1 & 0 & | & 16 \\ 0 & 0 & 1 & | & 0 \end{array}\right]\nonumber \]

which reads

\[y_1 = 12 \quad y_2 = 16 \quad Z = 0 \nonumber \]

The solution obtained by arbitrarily assigning values to some variables and then solving for the remaining variables is called the basic solution associated with the tableau. So the above solution is the basic solution associated with the initial simplex tableau. We can label the basic solution variable in the right of the last column as shown in the table below.

Example4.2.1b.png

STEP 4. The most negative entry in the bottom row identifies the pivot column.

The most negative entry in the bottom row is -40; therefore the column 1 is identified.

Example4.2.1c.png

Question Why do we choose the most negative entry in the bottom row?

Answer The most negative entry in the bottom row represents the largest coefficient in the objective function - the coefficient whose entry will increase the value of the objective function the quickest.

The simplex method begins at a corner point where all the main variables, the variables that have symbols such as \(x_1\), \(x_2\), \(x_3\) etc., are zero. It then moves from a corner point to the adjacent corner point always increasing the value of the objective function. In the case of the objective function \(Z = 40x_1+ 30x_­2\), it will make more sense to increase the value of \(x_1\) rather than \(x_2\). The variable \(x_1\) represents the number of hours per week Niki works at Job I. Since Job I pays $40 per hour as opposed to Job II which pays only $30, the variable \(x_1\) will increase the objective function by $40 for a unit of increase in the variable \(x_1\).

STEP 5. Calculate the quotients. The smallest quotient identifies a row. The element in the intersection of the column identified in step 4 and the row identified in this step is identified as the pivot element.

Following the algorithm, in order to calculate the quotient, we divide the entries in the far right column by the entries in column 1, excluding the entry in the bottom row.

Example4.2.1e.png

The smallest of the two quotients, 12 and 8, is 8. Therefore row 2 is identified. The intersection of column 1 and row 2 is the entry 2, which has been highlighted. This is our pivot element.

Question Why do we find quotients, and why does the smallest quotient identify a row?

Answer When we choose the most negative entry in the bottom row, we are trying to increase the value of the objective function by bringing in the variable \(x_1\). But we cannot choose any value for \(x_1\). Can we let \(x_1 = 100\)? Definitely not! That is because Niki never wants to work for more than 12 hours at both jobs combined: \(x_1 + x_2 ≤ 12\). Can we let \(x_1 = 12\)? Again, the answer is no because the preparation time for Job I is two times the time spent on the job. Since Niki never wants to spend more than 16 hours for preparation, the maximum time she can work is 16 ÷ 2 = 8.

Now you see the purpose of computing the quotients; using the quotients to identify the pivot element guarantees that we do not violate the constraints.

Question Why do we identify the pivot element?

Answer As we have mentioned earlier, the simplex method begins with a corner point and then moves to the next corner point always improving the value of the objective function. The value of the objective function is improved by changing the number of units of the variables. We may add the number of units of one variable, while throwing away the units of another. Pivoting allows us to do just that.

The variable whose units are being added is called the entering variable , and the variable whose units are being replaced is called the departing variable . The entering variable in the above table is \(x_1\), and it was identified by the most negative entry in the bottom row. The departing variable \(y_2\) was identified by the lowest of all quotients.

STEP 6. Perform pivoting to make all other entries in this column zero.

In chapter 2, we used pivoting to obtain the row echelon form of an augmented matrix. Pivoting is a process of obtaining a 1 in the location of the pivot element, and then making all other entries zeros in that column. So now our job is to make our pivot element a 1 by dividing the entire second row by 2. The result follows.

Example4.2.1f.png

To obtain a zero in the entry first above the pivot element, we multiply the second row by -1 and add it to row 1. We get

Example4.2.1g.png

To obtain a zero in the element below the pivot, we multiply the second row by 40 and add it to the last row.

Example4.2.1h.png

We now determine the basic solution associated with this tableau. By arbitrarily choosing \(x_2 = 0\) and \(y_2 = 0\), we obtain \(x_1 = 8\), \(y_1 = 4\), and \(z = 320\). If we write the augmented matrix, whose left side is a matrix with columns that have one 1 and all other entries zeros, we get the following matrix stating the same thing.

\[\left[\begin{array}{ccccc} \mathrm{x}_{1} & \mathrm{y}_1 & \mathrm{Z} & | & \mathrm{C} \\ 0 & 1 & 0 & | & 4 \\ 1 & 0 & 0 & | & 8 \\ 0 & 0 & 1 & | & 320 \end{array}\right] \nonumber \]

We can restate the solution associated with this matrix as \(x_1 = 8\), \(x_2 = 0\), \(y_1 = 4\), \(y_2 = 0\) and \(z = 320\). At this stage of the game, it reads that if Niki works 8 hours at Job I, and no hours at Job II, her profit Z will be $320. Recall from Example 3.1.1 in section 3.1 that (8, 0) was one of our corner points. Here \(y_1 = 4\) and \(y_2 = 0\) mean that she will be left with 4 hours of working time and no preparation time.

STEP 7. When there are no more negative entries in the bottom row, we are finished; otherwise, we start again from step 4.

Since there is still a negative entry, -10, in the bottom row, we need to begin, again, from step 4. This time we will not repeat the details of every step, instead, we will identify the column and row that give us the pivot element, and highlight the pivot element. The result is as follows.

Example4.2.1i.png

We make the pivot element 1 by multiplying row 1 by 2, and we get

Example4.2.1j.png

Now to make all other entries as zeros in this column, we first multiply row 1 by -1/2 and add it to row 2, and then multiply row 1 by 10 and add it to the bottom row.

Example4.2.1k.png

We no longer have negative entries in the bottom row, therefore we are finished.

Question Why are we finished when there are no negative entries in the bottom row?

Answer The answer lies in the bottom row. The bottom row corresponds to the equation:

\[\begin{array}{l} 0 x_{1}+0 x_{2}+20 y_{1}+10 y_{2}+Z=400 \quad \text { or } \\ z=400-20 y 1-10 y 2 \end{array}\nonumber \]

Since all variables are non-negative, the highest value \(Z\) can ever achieve is 400, and that will happen only when \(y_1\) and \(y_2\) are zero.

STEP 8. Read off your answers.

We now read off our answers, that is, we determine the basic solution associated with the final simplex tableau. Again, we look at the columns that have a 1 and all other entries zeros. Since the columns labeled \(y_1\) and \(y_2\) are not such columns, we arbitrarily choose \(y_1 = 0\), and \(y_2 = 0\), and we get

\[\left[\begin{array}{ccccc} \mathrm{x}_{1} & \mathrm{x}_{2} & \mathrm{Z} & | & \mathrm{C} \\ 0 & 1 & 0 & | & 8 \\ 1 & 0 & 0 & | & 4 \\ 0 & 0 & 1 & | & 400 \end{array}\right] \nonumber \]

The matrix reads \(x_1 = 4\), \(x_2= 8\) and \(z = 400\).

The final solution says that if Niki works 4 hours at Job I and 8 hours at Job II, she will maximize her income to $400. Since both slack variables are zero, it means that she would have used up all the working time, as well as the preparation time, and none will be left.

LP Ch.5: Linear Programming with the Simplex Method

Chapter 5: Linear Programming with the Simplex Method

how to solve simplex method linear programming problem

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

how to solve simplex method linear programming problem

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.

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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:

how to solve simplex method linear programming problem

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.

how to solve simplex method linear programming problem

Flow Chart of Simplex Method :

how to solve simplex method linear programming problem

  • max Z = 3x1 + 5x2 + 4x3 subject to 2x1 + 3x2 2x2 + 5x3 3x1 + 2x2 + 4x3 and x1,x2,x3 >= 0
  • max Z = 5x1 + 10x2 + 8x3 subject to 3x1 + 5x2 + 2x3 4x1 + 4x2 + 4x3 2x1 + 4x2 + 5x3 and x1,x2,x3 >= 0
  • max Z = 4x1 + 3x2 subject to 2x1 + x2 x1 + x2 x1 x2 and x1,x2 >= 0
  • =,>=`4,7`');">min Z = x1 + x2 subject to 2x1 + 4x2 >= 4 x1 + 7x2 >= 7 and x1,x2 >= 0
  • =,>=`80,60`');">min Z = 600x1 + 500x2 subject to 2x1 + x2 >= 80 x1 + 2x2 >= 60 and x1,x2 >= 0
  • =`12,10,10`');">min Z = 5x1 + 3x2 subject to 2x1 + 4x2 2x1 + 2x2 = 10 5x1 + 2x2 >= 10 and x1,x2 >= 0
  • max Z = x1 + 2x2 + 3x3 - x4 subject to x1 + 2x2 + 3x3 = 15 2x1 + x2 + 5x3 = 20 x1 + 2x2 + x3 + x4 = 10 and x1,x2,x3,x4 >= 0
  • max Z = 3x1 + 9x2 subject to x1 + 4x2 x1 + 2x2 and x1,x2 >= 0
  • max Z = 3x1 + 2x2 + x3 subject to 2x1 + 5x2 + x3 = 12 3x1 + 4x2 = 11 and x2,x3 >= 0 and x1 unrestricted in sign
  • max Z = 3x1 + 3x2 + 2x3 + x4 subject to 2x1 + 2x2 + 5x3 + x4 = 12 3x1 + 3x2 + 4x3 = 11 and x1,x2,x3,x4 >= 0
  • =`30,24,3`');">max Z = 6x1 + 4x2 subject to 2x1 + 3x2 3x1 + 2x2 x1 + x2 >= 3 and x1,x2 >= 0
  • =`6,10,1`');">max Z = 3x1 + 5x2 subject to x1 - 2x2 x1 x2 >= 1 and x1,x2 >= 0
  • =`5,8`');">max Z = 6x1 + 4x2 subject to x1 + x2 x2 >= 8 and x1,x2 >= 0
  • =,>=`-5,8`');">max Z = 6x1 + 4x2 subject to -x1 - x2 >= -5 x2 >= 8 and x1,x2 >= 0

how to solve simplex method linear programming problem

Geektonight

Linear Programming: Simplex Method

  • Post last modified: 22 July 2022
  • Reading time: 6 mins read
  • Post category: Operations Research

What is Simplex Method Linear Programming?

The simplex method is an algorithm used to calculate the optimal solution to an LP problem. It is a systematically performed iterative procedure to identify the optimal solution from the set of feasible solutions. You might remember that in the graphical solution, the unique optimal solution to the LP problem occurred at a corner point or vertex of the feasible region.

The simplex algorithm also starts at one corner point of the feasible region and at each iteration moves to an adjacent vertex in sequence, until the corner point corresponding to the optimal solution is reached.

Table of Content

  • 1 What is Simplex Method Linear Programming?
  • 2 Introduction LP Simplex Method
  • 3 Important Terms of Linear Programming for Simplex Method
  • 4 Steps for Solving Linear Programming using Simplex Method

Introduction LP Simplex Method

In the previous article, you studied how to solve linear programming problems graphically . You also studied some special cases in the previous chapter.

The graphical approach is not applicable to problems with more than two variables are involved. The simplex method is more suitable for solving LP problems in three or more variables, or problems involving many constraints. The simplex method is a mathematical solution technique where the model is formulated as a tableau on which a series of repetitive mathematical steps are performed to reach the optimal solution.

The simplex method was developed in 1947 by George B. Dantzig. He put forward the simplex method for obtaining an optimal solution to a linear programming problem, i.e., for obtaining a non-negative solution of a system of m linear equations in n variables which maximises a given linear functional of the vector of variables.

It is one of the most universally applied mathematical techniques, the popularity of the simplex method comes from the fact that it can indicate at each phase if the solution is optimal and if the solution can be improved and what that improved solution would be.

All LP problems can be solved using the simplex method. It is much more adaptable to computers than the graphical method, therefore, it is more suited for complex problems despite being mathematically more complex. Using the simplex method, a decision maker can also identify degeneracy, unbounded solutions, alternate solutions, and infeasible solutions along with redundant constraints.

Important Terms of Linear Programming for Simplex Method

  • Pivot column : In a row-echelon matrix, the first non-zero entry of each row is called a pivot, and the columns where pivots occur are called pivot columns or key columns. This is the column with the most negative index number, and it shows the entering variable in the basis.
  • Pivot row : It is the row which contains the smallest non-negative ratio is called the pivot row or key row. This row has the smallest quotient obtained after dividing the values of quantity column by key column for each row. It shows the exiting variable from the basis.
  • Pivot element/ number : The pivot element of a matrix is selected first by an algorithm to do certain computations. The pivot element is at the intersection of the pivot column with pivot row.
  • Simplex tableau : The simplex tableau organises the model into a form that simplifies the application of the mathematical steps. An LP problem in standard form can be represented as a tableau of the form given below:
  • Basis : It is the set of variables not constrained to equal zero in the current basic solution. Basic variables are those variables which make up the basis.
  • Non-basic variables : These are all variables other than basic variables.
  • Iteration: This refers to the steps performed to progress from one feasible solution to another in simplex method.
  • Cj Row : The coefficients of the variables in the objective function occur in this row.
  • Zj Row : The Zj row element shows the increase or decrease in objective function value when one unit of that variable is brought into the solution.
  • Zj – Cj Row : It is also called the index row; the elements of this row depict net contribution/ loss per unit when one unit of that vari- able is brought into the solution.

Steps for Solving Linear Programming using Simplex Method

  • To apply the simplex method to solve an LP problem, the problem first needs to be put into the standard form. For this, the inequalities in constraints must be replaced by equalities by adding slack variables.
  • Now, organise a simplex tableau using slack variables.
  • Select a pivot column i.e., the column that has the smallest number in the last row.
  • Divide each element in the right most columns with the corresponding element in the pivot column. The row with the smallest non-negative quotient is the pivot row.
  • Locate the pivot element/number at the intersection of the pivot row and pivot column.
  • Calculate new values for the pivot row by dividing every number in the pivot row by the pivot element.
  • Calculate new values for each remaining row using the formula: (New row number) = (no in old row) – (no in old row above or below pivot number) x (corresponding no in the new row)
  • The goal is to have no negative indicators in the first row. The simplex method is iterative, i.e., we repeat steps 5, 6 and 7 until all numbers on the first row are positive.

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Simplex Algorithm – Tabular Method

Max/Min Z = c^tX s.t. AX \leq b X \geq 0

  • Build your matrix A. A will contain the coefficients of the constraints.
  • Matrix b will contain the amount of resources.
  • And matrix c will contain the coefficients of objective function or cost.

For the above problem – Matrix A – At Iteration 0

how to solve simplex method linear programming problem

At Iteration 0

Explanation of table- B : Basis and contains the basic variables. Simplex algorithm starts with those variables which form an identity matrix. In the above eg x4 and x3 forms a 2×2 identity matrix. CB : Its the coefficients of the basic variables in the objective function. The objective functions doesn’t contain x4 and x3, so these are 0. XB : The number of resources or we can say the RHS of the constraints. yi : The complete Matrix A.

min ratio test:  XBr/y_{rk} = min\{XB_i/y_{ik}\}

Table at Iteration 1

how to solve simplex method linear programming problem

Table at iteration 1

min ratio test: XBr/y_{rk} = min\{8/1, 10/2\}

Table at iteration 2

Relative profits = 0, 1/2, -1/2, 0 Pivot index = [1, 5] Pivot element = 1/2 Perform necessary row operations. See next table

how to solve simplex method linear programming problem

Table At iteration 3

Relative profits = 0, 0, 0, -1 Since all relative profits are less than or equal to 0. So optimality is reached. This will be the final simplex table and the optimal one. Value of Z at optimality = 6*1 + 2*1 = 8 Following cases can occur while performing this algorithm.

  • Case 1 – Unbounded Solution If the column corresponding to the max relative profit contains only non-positive real numbers then we won’t be able to perform the min ratio test. Therefore it is reported as unbounded solution.
  • Case 2 – Alternate Solution If at any iteration any one of the non-basic variable’s relative profit comes out to be 0, then it contains alternate solutions. Many optimal solutions will exist.

MAX 2x_1 + 5x_2 s.t. x_1 + x_2 \leq 6 x_2 \leq 3 x_1 + 2x_2 \leq 9

Table at iteration 0

Now continuing as the previous example. Table at iteration 1

how to solve simplex method linear programming problem

Relative profits = 2, 5, 0, 0, 0 Pivot Index = [2, 5] Pivot element = 1 Table at Iteration 2

how to solve simplex method linear programming problem

Relative Profits = 2, 0, 0, -5, 0 Pivot Index = [1, 4] Pivot Element = 1 Table at iteration 3

how to solve simplex method linear programming problem

Table at iteration 3

Relative profits = 0, 0, 0, -2, -3, 0 Since all relative profits are less than equal to 0. Optimality is reached. This is the final simplex table and the optimal one. Value of Z at optimality = 3*2 + 3*5 + 0*0 = 21 Code Implementation of Simplex Algorithm  

For the above just plug in the required values and you will get a detailed step by step solution of your LPP by the simplex algorithm.

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Linear Programming

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:

  • Characteristics
  • Linear programming Problems
  • Simplex Method

Graphical Method

  • Applications
  • 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
  • Constraints
  • 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.

Linear Programming - Graph

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.

how to solve simplex method 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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COMMENTS

  1. 4.2: Maximization By The Simplex Method

    STEP 1. Set up the problem. Write the objective function and the constraints. Since the simplex method is used for problems that consist of many variables, it is not practical to use the variables x x, y y, z z etc. We use symbols x1 x 1, x2 x 2, x3 x 3, and so on. Let.

  2. PDF 4 Solving Linear Programming Problems: The Simplex Method

    Solving Linear Programming Problems: The Simplex Method We now are ready to begin studying the simplex method,a general procedure for solving linear programming problems. Developed by George Dantzig in 1947, it has proved to be a remarkably efficient method that is used routinely to solve huge problems on today's computers.

  3. PDF Linear Programming: Chapter 2 The Simplex Method

    Linear Programming: Chapter 2 The Simplex Method Robert J. Vanderbei October 17, 2007 ... This is how we detect unboundedness with the simplex method. Initialization Consider the following problem: maximize 3x 1 + 4x 2 subject to 4x 1 2x 2 8 2x 1 2 3x 1 + 2x 2 10 x 1 + 3x 2 1 3x 2 2 x 1;x

  4. A Beginner's Guide to Linear Programming and the Simplex Algorithm

    The simplex algorithm is a widely used method for solving linear programming problems. It was developed by George Dantzig in 1947. The intuition behind the algorithm is to 'walk' from corner to corner in the feasible region space in a systematic way. When the optimal solution is found, the process stops.

  5. Simplex method

    simplex method, standard technique in linear programming for solving an optimization problem, typically one involving a function and several constraints expressed as inequalities. The inequalities define a polygonal region, and the solution is typically at one of the vertices. The simplex method is a systematic procedure for testing the vertices as possible solutions.

  6. PDF Simplex method

    The simplex method 7 §Two important characteristics of the simplex method: •The method is robust. §It solves any linear program; §It detects redundant constraints in the problem formulation; §It identifies instances when the objective value is unbounded over the feasible region; and §It solves problems with one or more optimal solutions.

  7. Chapter 5: Linear Programming with the Simplex Method

    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 ...

  8. PDF Lecture 6 Simplex method for linear programming

    Examples and standard form Fundamental theorem Simplex algorithm Simplex method I Simplex method is first proposed by G.B. Dantzig in 1947. I Simply searching for all of the basic solution is not applicable because the whole number is Cm n. I Basic idea of simplex: Give a rule to transfer from one extreme point to another such that the objective function is decreased.

  9. Simplex Method for Solution of L.P.P (With Examples)

    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 ...

  10. Simplex Method Problem 1- Linear Programming Problems (LPP ...

    Subject - Engineering Mathematics - 4Video Name -Simplex Method Problem 1Chapter - Linear Programming Problems (LPP)Faculty - Prof. Farhan MeerUpskill and ge...

  11. How to Solve a Linear Programming Problem using the Simplex Method

    In this listen we first learn the concept of slack variables and then we learn how to solve a linear programming problem using the simplex method.

  12. Part 1

    This video is the 1st part of a video that demonstrates how to solve a standard maximization problem using the simplex method. References to using the TI-84 ...

  13. Simplex method calculator

    Simplex method calculator - Solve the Linear programming problem using Simplex method, step-by-step online We use cookies to improve your experience on our site and to show you relevant advertising. By browsing this website, you agree to our use of cookies.

  14. Linear Programming: Simplex Method

    The simplex method was developed in 1947 by George B. Dantzig. He put forward the simplex method for obtaining an optimal solution to a linear programming problem, i.e., for obtaining a non-negative solution of a system of m linear equations in n variables which maximises a given linear functional of the vector of variables.

  15. Explanation of Simplex Method for Minimization.

    Simplex method is an approach to solving linear programming models by hand using slack variables, tableaus, and pivot variables as a means to finding the optimal solution of an optimization problem.

  16. Simplex Algorithm

    Simplex Algorithm is a well-known optimization technique in Linear Programming. The general form of an LPP (Linear Programming Problem) is Example: Let's consider the following maximization problem. Initial construction steps : Build your matrix A. A will contain the coefficients of the constraints.

  17. Linear Programming (Definition, Methods & Examples)

    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.

  18. Lec 5: Solving Maximization Problems using the Simplex Method in Linear

    In this operations research tutorial, learn how to solve maximization problems using the simplex method in linear programming. The simplex method is a powerf...

  19. Simplex Method for Linear Programming

    The steps of the simplex algorithm is: Set the problem in standard (correct) format. Set the objective function as maximum problem (if you have minimum problem multiply the objective function by ...

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  21. Answered: Use the simplex method to solve the…

    Use the simplex method to solve the linear programming problem. OA. The maximum is Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. when X₁ = (Simplify our answers.) Maximize z = 2x₁ + 3x2 subject to: 5x₁ + xX2 ≤60 3x₁ + 2x₂ ≤80 x₁ + x2 ≤70 X1, X₂ 20. B.

  22. Solve the linear programming problem using the

    Question: Solve the linear programming problem using the simplex method.Maximize P=-x1+2x2subject to -x1+x2≤2-x1+3x2≤14x1-4x2≤8x1,x2≥0Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.A. The maximum value of P is P= when x1= and x2=(Simplify your answers:)B.