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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.

Study Guides > Finite Math

Reading: solving standard maximization problems using the simplex method.

Standard Maximization Problem

• an objective function, and
• All of the a number represent real-numbered coefficients and
• the x number represent the corresponding variables.
• V is a non-negative (0 or larger) real number

Setting Up the Initial Simplex Tableau

Solving the Linear Programming Problem by Using the Initial Tableau

1. select a pivot column.

2. Select a Pivot Row

3. Using Gaussian Elimination, Eliminate Rows 1 and 3

4. Identify the Solution Set

• Pivot1, and

Calculator Clinic—Using the Simplex Program

• Read the solution by ignoring all inactive columns and using only those solutions that correspond to active columns. If a variable column is ever inactive, its value is set to 0. Reading from the matrix gives [latex-display]\displaystyle{\left[\matrix{{1}{x}=\frac{{6}}{{5}}={1.2}\{1}{y}=\frac{{6}}{{5}}={1.2}\{z}={22.8}}.}[/latex-display] With some minor rounding error, we confirm the solution triplet (1.2,1.2,22.8)

• Average price per flight is less than or equal to $200
• Average cost from airfare is no more than 10% of total
• Add prices and divide by 3 [latex]\displaystyle\frac{{{x}+{y}+{z}}}{{3}}le{200}[/latex]
• Revenue from San Diego tickets will total and 10% of this amount is estimated to be cost. That is Cost = .10(900 x ) = 90 x . Similarly, we have .12(700 y ) = 84 y and .14(1000 z ) = 140 z . We want the sum of these costs to be less than or equal to 10% of total revenue, which is .10(900 x + 700 y + 1000 z ) = 90 x + 70 y + 100 z . That is, 90 x + 84 y + 140 z le 90 x + 70 y + 100 z 0 x + 14 y + 40 z le 0

Practice Problems

• Maximize: P = x + y + 2 z , subject to x + 2 y – z le 100 – x – y + 10 z le 30 3 x + 4 y + 5 z le 200
• Maximize: P = .5 x + 1.2 y + .9 z , subject to 1.1 x + 2.1 y + .4 z le 9 0.7 x + .99 y + z le 12 – x – 4 y + 30 z le 400
• A gas station sells three types of energy drinks: High Power, Purple Elephant, and Creature. It earns $0.60, $0.76, and $0.99 in profit on each of the three drinks, respectively. It can stock no more than 400 cans in the store each week. Typically, at least twice as many Purple Elephants are sold as Creatures. Finally, the company never sells more than 100 High Powers in a week. If the company's goal is to maximize profit, how many of each energy drink should it stock?
• How much time should be allotted to each politician in order to get the maximum number of viewers?
• Under these circumstances, what will be the maximum number of viewers?
• What is the maximum revenue the company can generate, and how many of each breakfast will they need to sell?
• What is the maximum number of each breakfast the café can produce each day?
• Gives a possible reason for why we observe the same number of breakfasts in each of the two above scenarios. Your reasons should take into account what it is that is being maximized in each of the two cases, in addition to the constraints.

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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 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.

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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:

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 :

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

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The Simplex Process

A robust and creative problem-solving tool.

By the Mind Tools Content Team

Imagine that you and your team are tasked with eliminating bottlenecks in your organization's billing process. Suppliers are angry, managers are frustrated, and the problem is costing the company money.

But, try as you might, you just can't pinpoint what's wrong, and the fixes that you've tried so far haven't worked.

Here's where the Simplex Process, now known as Simplexity Thinking, could help. This powerful tool enables you to identify and deal with problems creatively and effectively. It takes you through an eight-step process, from identifying the problem to implementing a solution.

In this article, we'll explain what Simplexity Thinking is, and describe how to use each stage.

Click here to view a transcript of this video.

What Is Simplexity Thinking?

The Simplex Process was created by management and creativity specialist Min Basadur, and was popularized in his 1995 book, " The Power of Innovation ."

The process is made up of eight steps, grouped into three stages: Problem Formulation, Solution Formulation and Solution Implementation. It is a versatile tool that can be used in organizations of all sizes, and for almost any type of problem.

Basadur has developed and refined the Process since the original publication of his book. Figure 1, below explains the most recent version.

Figure 1. Follow eight steps to solve a problem by using Simplexity Thinking.

Reproduced with permission from Dr Min Basadur. See Basadur Applied Creativity for more information on Simplex and Simplexity Thinking. From " The Power of Innovation: How to Make Innovation a Way of Life & How to Put Creative Solutions to Work ," by Min Basadur. Copyright © 1995 and 2002.

How to Use the Process

Let's look at the eight steps in more detail, below.

1. Problem Finding

Often, the most difficult part of any problem-solving exercise is finding the right issue to tackle. So, this is the first step to carry out. Problems may be obvious but, if they're not, you can identify them by using “trigger questions” such as:

• What do our customers want us to improve? What are they complaining about?
• What could they be doing better if we were to help them?
• What small problems do we have that could grow into bigger ones?
• What slows down our work or makes it more difficult? How can we improve quality?
• What are our competitors doing that we could do?
• What is frustrating and irritating to our team?

You can also consider issues that may arise in the future.

For example, think about how you expect markets and customers to change over the next few years. There could be problems as your organization expands. Social, political or legal changes may affect it, too. See our article, PEST Analysis for more on this.

It's also worth exploring possible problems from the perspective of different "actors" in the situation. This is where techniques such as the CATWOE checklist are helpful.

You may not have enough information to define your problem precisely, even after asking plenty of questions. But don't worry about this until you reach Step 3!

2. Fact Finding

The next stage is to research the problem as fully as possible.

Start by analyzing the data you have to see whether the problem really does exist. Then, establish whether the benefits of solving the problem will be worth the effort and resources that you'll need to spend.

Be clear which processes, components, services or technologies you want to use, and explore any solutions that others have already tried.

Next, work out how different people perceive the situation, explore your customers' needs in more detail, and investigate your competitors' best ideas.

3. Problem Definition

Identify the problem at the right level. For example, if you ask questions about it in terms that are too broad, then you'll never have enough resources to answer them effectively. If, however, your questions are too narrow, you may end up fixing the symptoms of a problem, rather than the problem itself. Our article, The Problem Definition Process , explores this issue.

Min Basadur suggests asking "Why?" to broaden your definition of the problem, and "What's stopping you?" to narrow it.

Let's say your system has difficulty maintaining stock levels in your warehouse. Start by asking, "Why is the system not doing its job properly?" The answer might lead you to ask a broader question, such as, "Why are we asking the system to do something that it's not good at?"

A "What's stopping you?" question here could give you the answer, "We don't know enough about the capabilities of the system we're using." In this way you may realize that you're not actually looking to fix a malfunctioning part, but to get the warehouse to use the system correctly, or to introduce a new system that is a better fit.

Big problems are often made up of many smaller ones. In the Problem Definition stage you can use a technique like Drill Down to break the problem down to its component parts. You can also use the 5 Whys Technique , Cause and Effect Analysis and Root Cause Analysis to help you get to the root of a problem.

Negative thinking can affect the Problem Definition stage. You or your team might start using phrases such as "We can't," or "We don't," or "This costs too much." Shift the focus toward creating a solution by addressing objections with the phrase "How might we...?".

4. Idea Finding

Generate as many problem-solving ideas as possible.

Ways of doing this range from asking other people for their opinions, through programmed creativity tools such as Creative Problem Solving and lateral-thinking techniques, to brainstorming. You should also look at the problem from other perspectives .

Don't evaluate or criticize ideas during this stage. Instead, just concentrate on generating ideas. Remember, impractical ideas can often trigger good ones!

5. Evaluation and Selection

Once you have generated a number of possible solutions to your problem, you need to select the best one.

The best solution may be obvious. If it's not, then consider the criteria that you'll use to select the best idea. Our articles on Decision Making Techniques explore a wide range of methods for doing this.

Once you've selected an idea, develop it as far as possible . You then need to evaluate it. Common sense is more important than ego here: be objective, and consider each course of action on its merits.

If your idea doesn't offer a big enough benefit, either see whether you can generate more ideas, or restart the process. (You can waste years of your life developing creative ideas that no-one wants!)

6. Action Planning

When you've picked an idea, and you're confident that it's worthwhile, it's time to start planning its implementation.

Developing Action Plans is a good way to manage simple projects. Action plans lay out the who, what, when, where, why, and how of delivering the work.

For larger projects, it's worth using formal project management techniques . These enable you to deliver projects efficiently, successfully, and within a realistic timeframe.

7. Gaining Acceptance

Until this stage you may have been working on your own, or with just a small team. Now you have to sell your solution to the people you need support from. These people may include your boss, investors, and any other stakeholders involved with the project.

When you're selling your idea, you'll have to address not only the practicalities, but also other factors, such as internal politics and fear of change. Your goal should be to foster both a sense of ownership among the stakeholders, and an understanding of the benefits they will derive from what you're doing.

Also, think about change management in cases where implementation is likely to affect several people or groups of people. Understanding this will help you to make sure that your project gains support.

After the creativity and preparation comes action.

This is where your careful work and planning pays off. Again, if you're implementing a large-scale change or project, brushing up on your change-management skills can help you to implement the process smoothly.

When the action is underway, return to Stage 1, Problem Finding, to continue developing your idea. You can also adopt the principles of the Kaizen model of continuous improvement to refine your project.

Simplexity Thinking is a powerful approach to creative problem-solving. It is suitable for projects and organizations of almost any scale.

The process follows an eight-step cycle. When you've completed each step, you can start it again to find and solve another problem. This encourages a culture of continuous improvement.

The eight steps in the process are:

• Problem Finding.
• Fact Finding.
• Problem Definition.
• Idea Finding.
• Evaluation and Selection.
• Action Planning.
• Gaining Acceptance.

This process can foster intense creativity: by moving through these steps you give yourself the best chance of solving the most significant problems with the best solutions available.

Basadur, M. (2002). ' The Power of Innovation .' London: Pitman Publishing.

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