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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.

simplex problem solving

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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Simplex Problem Solving Process

The Basadur Simplex Problem Solving Process - toolshero

Basadur Simplex Problem Solving Process: this article provides you with a practical explanation of the Basadur Simplex Problem Solving Process . After reading, you will understand the basics of this powerful and creative problem solving method .

What is the Basadur Simplex Problem Solving Process?

Problems come in all shapes and sizes. The important thing is that they can be solved. Before acting to solve problems, however, it is very important to first identify the problem in question. Still, the most important steps in the process of problem solving are often overlooked, meaning that good solutions are not found, or even that problems are not identified correctly.

The Basadur Simplex Problem Solving Process is a problem-solving method that is aware of that, and prevents such mistakes from being made. This model was developed by the American creativity guru Marino (Min) Sidney Basadur , who presented the method in his book ‘ the Power of Innovation ’.

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He was also the inventor of the patented Simplexity Thinking System. The Basadur Simplex Problem Solving Process works in steps during which problem-solving groups are given the opportunity to come up with creative solutions. By following the Basadur Simplex Problem Solving Process, they can diagnose the problem more easily and then suggest and implement solutions.

Basadur Simplex Problem Solving Process : three Phases

The Basadur Simplex Problem Solving Process consists of three phases, subdivided into eight steps:

Phase 1: Problem formulation

This first phase is about the first three steps of the Basadur Simplex Problem Solving Process, namely problem finding , fact finding , and problem definition . Only then can the problem be formulated correctly. When it’s clear what the problem is, you can look at where in the organisation this problem has its origins.

Phase 2: Solution formulation

After the focus on the description of the problem, the Basadur Simplex Problem Solving Process switches to the second phase of finding possible solutions. The following steps are involved; finding ideas that can lead to a solution and selecting and evaluation those ideas. This phase is about coming up with as many creative ideas as possible that might lead to solutions.

Phase 3: Implementing the solution

When the solution to a problem is close, the third and final phase follows. It contains the final three steps of the Basadur Simplex Problem Solving Process; planning all actions that need to be taken, creating support and engagement for the solution among all employees, and then take action and implement it . In this phase it’s important that the solution be prepared thoroughly. Implementing the solution should then be done in a professional manner. There also has to be room for flexibility, so the solution can be adapted if needed after implementation.

Simplex Process cyclus - toolshero

Basadur Simplex Problem Solving Process : eight Steps

Step 1: problem finding.

For this step it’s a good idea not to look for solutions right away; first the problem has to be made clear, as well as its symptoms. The cause is identified later. In this step, you look for the what customers and suppliers want and need, for instance, but employees too. Unlike many other problem-solving methods, in the Basadur Simplex Problem Solving Process the problem is not yet known when the first step is taken. Secondary questions can be used to accelerate the first step, for example:

  • What tips for improvement would our customers give us?
  • How would customers respond if we would communicate better?
  • What is not going well in customer service right now?

Step 2: Fact finding

This step involves gathering information related to the current situation and possibly the problem. Step 1 identified the symptoms of the problem. Based on that, facts have to be gathered. What does the organisation already know about the problem in question? The gathered facts have to be assessed and evaluated. The most relevant facts will play a role in the following steps. Again, secondary questions may be helpful, such as:

  • What are some of the complaints we received over the last year?
  • How do customers see the problem?
  • What solutions have already been attempted?
  • What proposals for improvement are there?

Step 3: Problem definition

The problem area is known, so now the specific problem definition can be made. It’s important to explain the problem in a way that’s not too broad, but not too narrow either. Many ‘why’ questions help to get an idea of the bigger picture. The facts chosen can help describe the problem. The following example shows how:

  • Question: ‘Why do we want to improve our customer service?’
  • Answer: ‘Because customers are currently being sent back and forth between different points of contact. They want a regular contact person who knows what’s going on.’

The problem definition is about the customer service department that does not connect customers to the right person in most cases.

Step 4: Solution finding

Within the Basadur Simplex Problem Solving Process, this step is about a wide range of ideas that might offer a solution. Departments and individual employees play an important role in this. It might be a good idea to set up a project group to come up with creative solutions.

Such brainstorming sessions aren’t intended to criticise each other’s ideas. That would hamper the creative process. This step will result in a number of potential options that might solve the problem.

Step 5: Selection and evaluation

The ideas that have been obtained are now examined more closely. The goal is to decide which idea is best for solving the problem. It’s a good idea to first generate a number of evaluation criteria, allowing for an objective evaluation. The idea that meets these criteria the most is usually the best solution to the problem.

Here secondary questions can also speed up the process, for instance:

  • What will be the impact of the chosen solution?
  • Are there any costs associated with the chosen solution?
  • How much time and effort will it take to implement the solution?

Step 6: Planning

Now that a choice has been made, it’s time for an action plan . In short that means making clear who does what, allowing everyone to work toward the solution. A small number of actions is usually enough to solve minor problems. Formal project management would be a better choice in the case of more significant problems.

Step 7: Generating engagement

It might seem superfluous, but it’s important to consider employee engagement throughout the entire process. Only when they have accepted the solution will they make an effort to bring everything to a good conclusion. Solutions to problems often involve changes. It’s usually difficult for employees to change their work habits. That’s why they have to be made aware of the negative impact of the problem and the value of the changes.

Step 8: Taking action

Only when sufficient support has been secured, can the action plan be implemented. This step deserves as much attention as the other ones; after all, if the plan is not followed, the problem won’t be solved. It’s also smart to evaluate along the way and check if the problem has become noticeably smaller.

The Basadur Simplex Problem Solving Process : cyclical process

The Basadur Simplex Problem Solving Process is a simple but powerful problem-solving model. However, it’s not a linear process; it’s a cyclical process. That means that there is room for continuous improvement.

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It’s Your Turn

What do you think? Do you recognize the explanation of the simplex process? Are you familiar with the different phases and steps of this model? How do you apply the Simplex process within your organization? Do you have any tips or additions?

Share your experience and knowledge in the comments box below.

More information

  • Basadur, M. (1995). The power of innovation: How to make innovation a way of life and put creative solutions to work . Financial Times Management.
  • Basadur, M. (1998). The Basadur Simplex creative problem-solving profile inventory: Development, reliability and validity .
  • Basadur, M. I. N., Runco, M. A., & VEGAxy, L. A. (2000). Understanding how creative thinking skills, attitudes and behaviors work together: A causal process model . The Journal of Creative Behavior, 34(2), 77-100.

How to cite this article: Mulder, P. (2019). Basadur Simplex Problem Solving Process . Retrieved [insert date] from toolshero: https://www.toolshero.com/problem-solving/basadur-simplex-problem-solving-process/

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Published on: 25/06/2019 | Last update: 04/03/2022

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Patty Mulder

Patty Mulder

Patty Mulder is an Dutch expert on Management Skills, Personal Effectiveness and Business Communication. She is also a Content writer, Business Coach and Company Trainer and lives in the Netherlands (Europe). Note: all her articles are written in Dutch and we translated her articles to English!

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What is the Simplex Process?

The Simplex Process is a Problem Solving Method that Proposes 8 Steps to Find Lasting Solutions to any Problem .

The 8 Steps Proposed by the Simplex Process are:

  • Problem Finding .
  • Fact Finding .
  • Problem Definition .
  • Idea Finding .
  • Selection .
  • Sell the Idea .

These Steps must be Repeated until the Problem is Resolved .

The 8 steps of the Simplex Process

1 . Problem Finding : Identify the Issue to be Solved.

  • What is causing Trouble.

2. Fact Finding : Collect as much data as you can, about the Problem.

  • How Often, When, How and under what Circumstances it occurs.

3. Problem Definition : Define The Root Cause of the Problem.

  • What’s Causing it to happen and Why.

4. Idea Finding : Explore Different Alternatives to Solve the Problem.

  • They should be Durable Solutions, rather than Temporary Patches.

5. Selection : Choose the Final Idea; the One that would Best Solve the Problem.

  • Maximizing results and Minimizing the necessary Resources.

6. Planning : Define How the Final Idea will be Implemented.

  • What Steps, Resources and Players will be Necessary.

7. Sell the Idea : Explain the Idea to those from whom you will need Help.

  • Especially, if you don’t have the Authority to implement it on your own.

8. Action : Implement the Final Idea.

  • Use Metrics to Track Progress with Objective data.

Repeat this Cycle until the Problem is Solved .

Now, let’s see the first example:

Simplex Process example

Do you think you always follow this Process when you have a Problem?

Does it seem obvious to you?

Think about your last Mobile Phone :

How do you decide that you need a new Mobile phone?

As soon as you see that it starts working slower, you probably think:

  • “ It has become old. I need a new one “.

You surely don’t think about all the Root Causes that may be slowing Down your Phone .

  • Too many Apps.
  • Too many Pictures.

What happens if, on the contrary, you use the Simplex Process?

  • You are forced to Study the Problem .
  • You are forced to Identify the Root Cause .
  • You have to Propose different Alternatives .
  • You have to Implement the Actions.
  • You have to Check if the Problem is Resolved.

You wouldn’t change your phone that often .

This is a simple example of How, even if this Method seems Obvious, we don’t use it .

People rarely think much about a Root Cause that they haven’t seen before .

Few People Really sit down and look for Unexpected Root Causes.

Few People really Wonder Why a Problem is Occurring.

People tend to settle for the First explanation found .

  • And that is why this method is important.

We’ll share a Practical example with you , so that you understand How it can be used.

Simplex Process examples

Let’s imagine you own a Restaurant with a Partner .

Recently, you have noticed that you have fewer Clients .

  • And you don’t know Why.

Therefore, you decide to use the Simplex Process to Find out what is happening.

Let’s begin:

Problem Finding - Simplex Process example

First, you decide to compare the number of Clients you have with those you had one and two years ago .

  • In case you have a Seasonal decrease.

You conclude that your Customer Decline is not Seasonal.

You are losing Clients.

Fact Finding - Simplex Process example

Once you have confirmed that you do indeed have a Decrease in Customers, you decide to collect more Data .

Then, You check What reviews your Clients have posted on the Internet .

  • An extremely valuable tool that can save you a lot of time.

Then, to be sure that there is a Correlation between what they say and their Consumption , you check:

  • What your Clients Order Now.
  • What they used to Order.
  • The evolution of their Reviews.

Problem Definition - Simplex Process example

You have found what appears to be the Root Cause of your Problem:

  • Your Clients think that your Menu is too ambiguous .

You have good Dishes , but you can’t offer High quality in everything  you offer .

  • One of the two options won’t be as good as the other.

You have checked the Orders and, they confirm the Reviews:

  • Ribs, Burgers and Baked Potatoes.

Idea Finding - Simplex Process example

After careful analysis, you come up with the following Alternatives :

  • Dividing the Restaurant into two separate Areas with different Specialties.
  • Hiring an Expert Chef to improve your less popular Dishes .
  • Focus on your best Dishes .

Selection - Simplex Process example

Finally, you decide that the best option you have is to Focus on your Best Dishes .

This Idea is:

  • The most Cost effective .
  • The Easiest to Implement.
  • Based on Real Results ; Your Clients actually love your Ribs and Burgers.

This way, you’ll be even more Focused on your Specialties and they will be even better .

You will Offer:

  • Different types of Ribs.
  • Different Sauces.
  • Burgers with different meats.

Your Diversity will be Focused on your Best Dishes .

  • Using the Simplex Process.

Planning - Simplex Process example

Then, you Plan what must be done :

  • Decide what Dishes will be offered.
  • Modify the Menu.
  • Create a Marketing Campaign to let People Know that the Restaurant has updated its Offering.
  • Modify the Interior of the Restaurant (decoration)

Sell Idea - Simplex Process example

Your Partner has to Agree in this Decision.

But, as this Idea involves practically no investment , your Partner agrees immediately .

Perhaps, in the future, if your Idea is Successful, you’ll need to Expand the Kitchen or add a Larger Barbecue.

You decide to communicate these changes to the Chefs and the Staff to hear their opinion .

  • The Old Menu was very Confusing for them.

Action - Simplex Process example

Finally, you Start Implementing the Idea .

In 2 weeks you have the Final menu and the Marketing Campaign working.

At the End of each Month , you’ll check your Progress in case you have to Repeat the Simplex Process Again .

  • Adding a New Item.
  • Improving the Quality of your Menu.
  • Updating the Prices.
  • Offering New Promotions.

Your main Metrics will be:

  • The amount of Clients you have (of course).
  • The Reviews on the Internet.
  • The Dishes they order.
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Learn about Basadur

The Basadur System of innovation and creative problem solving was developed after extensive academic research and experience in the field. It’s made up of a core problem solving process which puts people at its core throughout its implementation.

Basadur encompasses the entire process of innovation

Your Team + Basadur Process + Process Skills & Tools + Innovation profile = Innovative results for your organization

Organizations thrive when innovation processes are integrated into all aspects of their business

Basadur integrates the Simplexity process of problem solving into the fabric of an organization. They provide a clear process for problem solving, a metric to create balanced teams and they ensure that individuals understand their innovation styles and have the tools they need to tackle any kind of problem.

The Simplexity Thinking Process is applied across…

Individuals, teams are best equipped to problem solve when they have a diverse mix of innovation styles and understand how their colleagues think.

Are your teams are optimized for innovation success? A well-balanced team will feel empowered throughout the problem solving process. Understanding team composition will help you identify areas to focus on while problem solving. It will also make team members more respectful of each others’ innovation styles.

The Simplexity Thinking Process

Min Basadur realized that to effectively solve a problem, he first needed to properly identify and define the problem without existing bias. He developed the 8-step Simplexity Thinking Process as a result. This breaks down the process of solving a problem into 3 overarching steps and 8 detailed ones.

Problem Formulation | Solution Formulation | Solution Implementation

The Simplexity Thinking Process guides you through the 8 step process that asks “How might we” from problem finding to action.

Skills & Tools make the Basadur System accessible for everyone

Everyone can innovate with Basadur. No matter what your innovation style is, Skills and Tools help you get the most out of every stage of the process. As you move through the workshops, you’ll find that SImplexity’s systematic approach will make you an Innovation Master in no time!

Individuals can understand their innovation style and its corresponding creative potential

When you have the tools to problem solve you’ll be inspired to innovate. You can apply your innovation style effectively to the problem solving process. Most importantly, you can use tools to navigate the parts of the process that don’t suit your innovation style. You can be an Innovation Master

Applying the Simplexity Thinking Process to Design Thinking

The design thinking process and the basadur system are entirely complementary. many of the tools used in design thinking are easily integrated into the simplexity thinking process. the basadur system helps build cognitively diverse teams and is the backbone of design thinking., contact us  to find out how we can help your organization innovate today.

PM Calculator - Logo

Simplex Method Calculator – Two Phase Online 🥇

Simplex method calculator - free version, members-only content, do you already have a membership, get membership.

The free version of the calculator shows you each of the intermediate tableaus that are generated in each iteration of the simplex method, so you can check the results you obtained when solving the problem manually.

Advanced Functions of the simplex method online calculator – Two-Phase

Let's face it, the simplex method is characterized by being a meticulous and impractical procedure, because if you fail in an intermediate calculation you can compromise the final solution of the problem. In that sense, it is important for the student to know the step by step procedure to obtain each of the values in the iterations. Thus, in PM Calculators we have improved our application to include a complete step-by-step explanation of the calculations of the method. You can access this tool and others (such as the big m calculator and the graphical linear programming calculator ) by becoming a member of our membership .

Within the functionality that this application counts we have:

  • Ability to solve exercises with up to 20 variables and 50 constraints.
  • Explanation of how to determine the optimality condition.
  • Explanation of the criteria to establish the feasibility condition.
  • Detail of the calculations performed to obtain the vector of reduced costs, the pivot row and the other rows of the table.
  • For exercises with artificial variables it becomes a two-phase method calculator .
  • Explanation of the special cases such as unbounded and infeasible solutions.

You can find complete examples of how the application works in this link .

How to use the simplex method online calculator

To use our tool you must perform the following steps:

  • Enter the number of variables and constraints of the problem.
  • Select the type of problem: maximize or minimize .
  • Enter the coefficients in the objective function and the constraints. You can enter negative numbers, fractions, and decimals (with point).
  • Click on “Solve”.
  • The online software will adapt the entered values ​​to the standard form of the simplex algorithm and create the first tableau .
  • Depending on the sign of the constraints, the normal simplex algorithm or the two phase method is used.
  • We can see step by step the iterations and tableaus of the simplex method calculator.
  • In the last part will show the results of the problem.

We have considered for our application to solve problems with a maximum of 20 variables and 50 restrictions; this is because exercises with a greater number of variables would make it difficult to follow the steps using the simplex method. For problems with more variables, we recommend using other method.

Below we show some reference images of the step by step and the result of the following example:

The following problem is required to be maximized:

Objective Function Z = 3X 1 + 2X 2

Subject to the following restrictions

2X 1 + X 2 ≤ 18 2X 1 + 3X 2 ≤ 42 X 1 , X 2 ≥ 0

We enter the number of variables and constraints:

simplex method calculator online

Enter the coefficients of the equations / inequalities of the problem and click on Solve:

simplex calculator

Next you will see the step by step in obtaining the solution as well as the calculation of the vector of reduced costs:

two phase method calculator

The calculation of the values of the pivot row:

lpp calculator

Until the final result:

simplex algorithm calculator

Final reflection

Our free simplex minimizing and maximizing calculator is being used by thousands of students every month and has become one of the most popular online Simplex method calculators available. In addition, our full version has been helping hundreds of students study and do their homework faster and giving them more time to devote to their personal activities.

If you have questions about it or find an error in our application, we will appreciate if you can write to us on our contact page .

Simplex algorithm

Author: Guoqing Hu (SysEn 6800 Fall 2020)

  • 1 Introduction
  • 2 Algorithmic Discussion
  • 3 Numerical Example
  • 4.1 Mathematical Problem
  • 4.2 Industrial Application
  • 5 Conclusion
  • 6 References

Introduction

Simplex algorithm (or Simplex method) is a widely-used algorithm to solve the Linear Programming(LP) optimization problems. The simplex algorithm can be thought of as one of the elementary steps for solving the inequality problem, since many of those will be converted to LP and solved via Simplex algorithm. [1] Simplex algorithm has been proposed by George Dantzig , initiated from the idea of step by step downgrade to one of the vertices on the convex polyhedral. [2] "Simplex" could be possibly referred to as the top vertex on the simplicial cone which is the geometric illustration of the constraints within LP problems. [3]

Algorithmic Discussion

There are two theorems in LP:

  • The feasible region for an LP problem is a convex set (Every linear equation's second derivative is 0, implying the monotonicity of the trend). Therefore, if an LP has an optimal solution, there must be an extreme point of the feasible region that is optimal
  • For an LP optimization problem, there is only one extreme point of the LP's feasible region regarding every basic feasible solution. Plus, there will be a minimum of one basic feasible solution corresponding to every extreme point in the feasible region. [4]

simplex problem solving

Based on the two theorems above, the geometric illustration of the LP problem could be depicted. Each line of this polyhedral will be the boundary of the LP constraints, in which every vertex will be the extreme points according to the theorem. The simplex method is the way to adjust the nonbasic variables to travel to different vertex till the optimum solution is found. [5]

Consider the following expression as the general linear programming problem standard form:

{\displaystyle \max \sum _{i=1}^{n}c_{i}x_{i}}

With the following constraints:

{\displaystyle {\begin{aligned}s.t.\quad \sum _{j=1}^{n}a_{ij}x_{j}&\leq b_{i}\quad i=1,2,...,m\\x_{j}&\geq 0\quad j=1,2,...,n\end{aligned}}}

The first step of the simplex method is to add slack variables and symbols which represent the objective functions:

{\displaystyle {\begin{aligned}\phi &=\sum _{i=1}^{n}c_{i}x_{i}\\z_{i}&=b_{i}-\sum _{j=1}^{n}a_{ij}x_{j}\quad i=1,2,...,m\end{aligned}}}

With the progression of simplex method, the starting dictionary (which is the equations above) switches between the dictionaries in seeking for optimal values. Every dictionary will have m basic variables which form the feasible area, as well as n non-basic variables which compose the objective function. Afterward, the dictionary function will be written in the form of:

{\displaystyle {\begin{aligned}\phi &={\bar {\phi }}+\sum _{j=1}^{n}{\bar {c_{j}}}x_{j}\\x_{i}&={\bar {b_{i}}}-\sum _{j=1}^{n}{\bar {a_{ij}}}x_{ij}\quad i=1,2,...,n+m\end{aligned}}}

The leaving variables are defined as which go from basic to non-basic. The reason of their existence is to ensure the non-negativity of those basic variables. Once the entering variables are determined, the corresponding leaving variables will change accordingly from the equation below:

{\displaystyle x_{i}={\bar {b_{i}}}-{\bar {a_{ik}}}x_{k}\quad i\,\epsilon \,\{1,2,...,n+m\}}

Since the non-negativity of entering variables should be ensured, the following inequality can be derived:

{\displaystyle {\bar {b_{i}}}-{\bar {a_{i}}}x_{k}\geq 0\quad i\,\epsilon \,\{1,2,...,n+m\}}

Once the leaving-basic and entering-nonbasic variables are chosen, reasonable row operation should be conducted to switch from the current dictionary to the new dictionary, as this step is called pivot. [4]

As in the pivot process, the coefficient for the selected pivot element should be one, meaning the reciprocal of this coefficient should be multiplied to every element within this row. Afterward, multiplying this specific row with corresponding coefficients and adding this to different rows, one should get 0 values for all other entries in this pivot element's column.

If there are any negative variables after the pivot process, one should continue finding the pivot element by repeating the process above. At once there are no more negative values for basic and non-basic variables. The optimal solution is found. [6] [7]

Numerical Example

Considering the following numerical example to gain better understanding:

{\displaystyle \max {4x_{1}+x_{2}+4x_{3}}}

with the following constraints:

{\displaystyle {\begin{aligned}2x_{1}+x_{2}+x_{3}&\leq 2\\x_{1}+2x_{2}+3x_{3}&\leq 4\\2x_{1}+2x_{2}+x_{3}&\leq 8\\x_{1},x_{2},x_{3}&\geq 0\end{aligned}}}

With adding slack variables to get the following equations:

{\displaystyle {\begin{aligned}z-4x_{1}-x_{2}-4x_{3}&=0\\2x_{1}+x_{2}+x_{3}+s_{1}&=2\\x_{1}+2x_{2}+3x_{3}+s_{2}&=4\\2x_{1}+2x_{2}+x_{3}+s_{3}&=8\\x_{1},x_{2},x_{3},s_{1},s_{2},s_{3}&\geq 0\end{aligned}}}

The simplex tableau can be derived as following:

{\displaystyle {\begin{array}{c c c c c c c | r}x_{1}&x_{2}&x_{3}&s_{1}&s_{2}&s_{3}&z&b\\\hline 2&1&1&1&0&0&0&2\\1&2&3&0&1&0&0&4\\2&2&1&0&0&1&0&8\\\hline -4&-1&-4&0&0&0&1&0\end{array}}}

By performing the row operation still every other rows (other than first row) in column 1 are zeroes:

{\displaystyle {\begin{array}{c c c c c c c | r}x_{1}&x_{2}&x_{3}&s_{1}&s_{2}&s_{3}&z&b\\\hline 1&0.5&0.5&0.5&0&0&0&1\\0&1.5&2.5&-0.5&1&0&0&3\\0&1&0&-1&0&1&0&6\\\hline 0&1&-2&2&0&0&1&4\end{array}}}

By performing the row operation to make other columns 0's, the following could be derived

{\displaystyle {\begin{array}{c c c c c c c | r}x_{1}&x_{2}&x_{3}&s_{1}&s_{2}&s_{3}&z&b\\\hline 1&0.2&0&0.6&-0.2&0&0&0.4\\0&0.6&1&-0.2&0.4&0&0&1.2\\0&-0.1&0&0.2&0.6&-1&0&-4.2\\\hline 0&2.2&0&1.6&0.8&0&1&6.4\end{array}}}

Application

The simplex method can be used in many programming problems since those will be converted to LP (Linear Programming) and solved by the simplex method. Besides the mathematical application, much other industrial planning will use this method to maximize the profits or minimize the resources needed.

Mathematical Problem

The simplex method is commonly used in many programming problems. Due to the heavy load of computation on the non-linear problem, many non-linear programming(NLP) problems cannot be solved effectively. Consequently, many NLP will rely on the LP solver, namely the simplex method, to do some of the work in finding the solution (for instance, the upper or lower bound of the feasible solution), or in many cases, those NLP will be wholly linearized to LP and solved from the simplex method. [1] Other than solving the problems, simplex method can also be used reliably to support the LP's solution from other theorem, for instance the Farkas' theorem in which Simplex method proves the suggested feasible solutions. [1] Besides solving the problems, the Simplex method can also enlighten the scholars with the ways of solving other problems, for instance, Quadratic Programming (QP). [8] For some QP problems, they have linear constraints to the variables which can be solved analogous to the idea of the Simplex method.

Industrial Application

The industries from different fields will use the simplex method to plan under the constraints. With considering that it is usually the case that the constraints or tradeoffs and desired outcomes are linearly related to the controllable variables, many people will develop the models to solve the LP problem via the simplex method, for instance, the agricultural and economic problems

Farmers usually need to rationally allocate the existed resources to obtain the maximum profits. The potential constraints are raised from multiple perspectives including policy restriction, budget concerns as well as farmland area. Farmers may incline to use the simplex-method-based model to have a better plan, as those constraints may be constant in many scenarios and the profits are usually linearly related to the farm production, thereby forming the LP problem. Currently, there is an existing plant-model that can accept inputs such as price, farm production, and return the optimal plan to maximize the profits with given information. [9]

Besides agricultural purposes, the Simplex method can also be used by enterprises to make profits. The rational sale-strategy will be indispensable to the successful practice of marketing. Since there are so many enterprises international wide, the marketing strategy from enamelware is selected for illustration. After widely collecting the data of the quality of varied products manufactured, cost of each and popularity among the customers, the company may need to determine which kind of products well worth the investment and continue making profits as well as which won't. Considering the cost and profit factors are linearly dependent on the production, economists will suggest an LP model that can be solved via the simplex method. [10]

The above professional fields are only the tips of the iceberg to the simplex method application. Many other fields will use this method since the LP problem is gaining popularity in recent days and the simplex method plays a crucial role in solving those problems.

It is indisputable to acknowledge the influence of the Simplex method to programming, as this method won the 'National Medal of Science' to its inventor, George Dantzig. [11] Not only for its wide usage in the mathematic models and industrial manufacture, but the Simplex method also provides a new perspective in solving the inequality problems. As its contribution to the programming substantially boosts the advancement of the current technology and economy from making the optimal plan with the constraints. Nowadays, with the development of technology and economics, the Simplex method is substituted with some more advanced solvers which can solve the problems with faster speed and handle a larger amount of constraints and variables, but this innovative method marks the creativity at that age and continuously offer the inspiration to the upcoming challenges.

  • ↑ 1.0 1.1 Linear complementarity, linear and nonlinear programming Internet Edition .
  • ↑ Dantzig, G. B. (1987, May). Origins of the simplex method .
  • ↑ Strang, G. (1987). Karmarkar’s algorithm and its place in applied mathematics. The Mathematical Intelligencer, 9 (2), 4-10. doi:10.1007/bf03025891.
  • ↑ 4.0 4.1 Vanderbei, R. J. (2000). Linear programming: Foundations and extensions . Boston: Kluwer.
  • ↑ Sakarovitch M. (1983) Geometric Interpretation of the Simplex Method. In: Thomas J.B. (eds) Linear Programming. Springer Texts in Electrical Engineering. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4106-3_8
  • ↑ Evar D. Nering and Albert W. Tucker, 1993, Linear Programs and Related Problems , Academic Press. (elementary)
  • ↑ Robert J. Vanderbei, Linear Programming: Foundations and Extensions , 3rd ed., International Series in Operations Research & Management Science, Vol. 114, Springer Verlag, 2008. ISBN 978-0-387-74387-5.
  • ↑ Wolfe, P. (1959). The simplex method for quadratic programming. Econometrica, 27 (3), 382. doi:10.2307/1909468
  • ↑ Hua, W. (1998). Application of the revised simplex method to the farm planning model .
  • ↑ Nikitenko, A. V. (1996). Economic analysis of the potential use of a simplex method in designing the sales strategy of an enamelware enterprise. Glass and Ceramics, 53 (12), 367-369. doi:10.1007/bf01129674.
  • ↑ Cottle, R., Johnson, E. and Wets, R. (2007). George B. Dantzig (1914–2005). Notices Amer. Math. Soc. 54, 344–362.

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Mastering the Art of Problem Solving: A Comprehensive Guide to Using Min Basadur’s Simplex Model

Published by gautam ghosh on july 23, 2023 july 23, 2023.

simplex problem solving

Table of Contents

Introduction:

Problem Solving is an important skill for both personal and business life. Whether you face problems at work, at home, or anywhere else in your life, knowing how to solve them well can make a big difference. In this blog post, we’ll look at Min Basadur’s Simplex problem-solving model. It’s a powerful eight-step method for finding, analyzing, and fixing problems in a systematic way.

The Simplex Process: Understanding the Eight Steps

The Simplex process is a structured way to solve problems that makes sure choices are well-informed and work. Let’s look at each step of the Simplex model and the tools and techniques you can use for each one.

Step 1: Find the Problem

Problem-solving works best when problems are found ahead of time. Even if everything seems to be going well, looking for potential problems can help you escape emergencies and stay in charge when problems come up. Here are some ways to help you figure out what’s wrong:

PEST Analysis:

Use this tool to keep track of changes in your environment, your customers’ needs, the way the market works, and the trends in your business.

Risk analysis:

Figure out which business risks are the most important so you can deal with them before they become problems.

Failure Modes and Effects Analysis:

Pinpoint possible points of failure in your business processes to avoid future problems.

After Action Reviews:

Look at how things went in the past to see where they could be better in the future.

Prioritization and Pareto Analysis:

Put the problems in order of importance so that you can work on the most important ones first.

Step 2: Find the Facts

Once you know there might be a problem, it’s important to find out more about it. Learn about what caused the problem, who was involved, what was done in the past to solve it, and how others see it. When you do a lot of study, you don’t have to rely on assumptions and narrow points of view. Here’s how to find out what’s going on:

Research Thoroughly:

To get a good understanding of the problem, spend time gathering knowledge from many different sources. Use online study, talk to people, look at relevant data, and look at old records. When you’re trying to solve a problem, you’ll be able to make better choices if you know more about it.

Engage Stakeholders:

Talk to people who are directly touched by the problem and people who know a lot about it. Their ideas can give important background information and put light on possible solutions.

Examine Case Studies:

Find case studies or examples of how similar problems have been solved in the past. Analyzing these cases can teach you important lessons and give you ideas for new ways to solve your own problems.

Consider Multiple Perspectives:

Think about things from different points of view. Don’t just stick to one point of view. Involve different people or groups in your study to get new ideas and different ways to look at the problem.

Validate Information:

Make sure the information you gather is correct and reliable. If you don’t want to make choices based on bad information, check the data and look at it from more than one source.

Remember that the key to handling problems well is to fully understand the problem. In this step of the Simplex model, you do the study that will help you find good solutions in the next steps.

Step 3: Define the Problem

After you know what the problem is, you need to clearly and fully describe it. A well-defined problem helps set clear limits and stops the project’s goals from growing. Use the following tools to get a clear picture of the problem:

CATWOE is a useful tool that lets you look at a problem from six different points of view:

Customers: Think about how the problem affects the customers, clients, or end-users who are directly affected by it.

Figure out which people or groups in the business are part of the problem or have some control over it.

Transformation:

Look at the business tools or processes that are part of the problem.

Know how the problem fits into the bigger picture and what influences it from the outside.

Figure out who is in charge of the problem and has the power to fix it.

Environment:

Think about things outside of the problem that might affect it.

By looking at the problem from these six points of view, you can get a full picture of it and find hidden problems.

Cause and Effect Analysis:

This tool helps you figure out what causes the problem and tell the difference between the signs and the real causes. Make a cause-and-effect diagram, also called a “fishbone diagram” or a “Ishikawa diagram,” to map out clearly the possible causes and how they relate to each other. If you do this, you can focus on fixing the real problems instead of just treating the symptoms.

By using CATWOE and Cause and Effect Analysis to define the problem, the next steps of the Simplex model are more likely to lead to good answers. A clear description makes sure that your efforts to solve a problem are focused, efficient, and aimed at the real problems.

Step 4: Find Ideas

With a clear definition of the problem, you can start thinking of possible answers. Think about the problem from different points of view and use metaphors and comparisons. These methods can help you come up with creative solutions:

Brainstorming and Reverse Brainstorming:

In brainstorming, a group of people come up with a lot of ideas without judging any of them. Encourage people to easily share their ideas, even if they seem crazy or out of the ordinary. In reverse brainstorming, on the other hand, you look for ways to make the problem worse instead of finding ways to solve it. By looking at things from the other side, you can find new insights that lead to creative answers.

Metaphors and Analogies:

Use metaphors and analogies to compare the present problem to others or to other situations. This kind of creative thinking can help you come up with new ideas and see the problem from different points of view.

Mind mapping:

To make a mind map, write the main problem in the middle of a page and connect it to connected ideas and possible solutions with lines. This picture lets you look at how different ideas are linked and find links you didn’t expect.

SCAMPER Technique:

SCAMPER is an acronym that means for Substitute, Combine, Adapt, Modify, Put to Another Use, Eliminate, and Reverse. Use these questions to think of creative ways to change parts of the problem or to look at it from different angles, which will help you come up with new ideas.

Role Playing:

Have people play different roles to look at the problem from different points of view. This method can help find different solutions based on how each job sees the problem.

Remember that the goal of this step is to come up with a lot of different ideas. Think outside the box and include different points of view to look at all the options. In the next steps of the Simplex model, you’ll look at these ideas and choose the best ones to move forward with.

Step 5: Select and Evaluate

After coming up with ideas, you’ll have a number of options that need to be weighed. Set criteria for a good answer and use techniques like Paired Comparison Analysis, Decision Matrix Analysis, and Risk Analysis. Here’s what you should do:

Set Evaluation Criteria:

Before evaluating the possible options, set clear criteria for evaluation. Think about things like how easy it is to do, how much it will cost, how it will affect partners, and how well it fits with the organization’s goals. This step makes sure you have a good way to compare and rank the different choices.

Paired Comparison Analysis:

This method lets you compare each solution to all the others and give each one a relative worth or value based on the criteria you set up. By systematically judging each pair of answers, you can put them in order of how well they work overall.

Decision Matrix Analysis:

Make a decision matrix that shows all possible solutions and their criteria or attributes. Give each measure a weight based on how important it is. Then, give each answer a score based on these factors. The grid gives you a way to compare and rank the choices in a way that is based on numbers.

Risk Analysis:

Analyze the possible risks of each option with a risk analysis. Identify possible obstacles, uncertainties, and potential negative outcomes. By knowing the risks, you can make plans to reduce them and make better choices.

Final Choice:

Once you’ve used the chosen methods to evaluate the options, pick the most promising ones that best meet the set criteria. At this time, don’t judge ideas too soon or get rid of them too soon, because that can kill creativity and leave out good options.

By using these evaluation methods, you can look at possible solutions in a systematic way and make choices based on facts. Remember that choosing the best answer is an important part of solving a problem, and that it will help you get closer to a good solution in the next steps of the Simplex model.

Step 6: Plan

When an answer is chosen, the implementation phase can begin. Planning and getting ready are very important. For small projects, do a Risk Analysis and make Action Plans. For bigger projects, look into more complex methods. What you have to do is:

Risk Analysis and Impact Analysis:

Before you start putting the plan into action, you should do a full Risk Analysis to look for possible problems, uncertainties, and challenges. Also, do an Impact Analysis to see how the proposed solution will affect different parts of the company. When you know about possible risks and their effects, you can make backup plans and plan for possible roadblocks.

Force Field Analysis:

This method helps you figure out what forces are driving the answer and what forces are holding it back. Find the things that help the suggested solution and the things that hurt it. By looking at these forces, you can come up with plans to strengthen the ones that help and deal with the ones that hurt.

Go/No-Go Decision:

Make a well-informed Go/No-Go Decision before fully agreeing to the implementation. Check to see if the benefits of going with the chosen choice outweigh the risks and costs that come with it. This choice makes sure that you use your resources wisely to go after the best answer.

Action Plans and Project Management:

For smaller projects, make detailed Action Plans that list the tasks, who is responsible for them, when they need to be done, and what tools are needed. For bigger, more complicated projects, you might want to use project management methods to make sure everything goes smoothly and the project is done well.

Change Management:

Be aware that putting any solution into place may require changes to processes, procedures, and the culture of the company. Plan for good Change Management to help with a smooth transition and get people on board.

By carefully planning the implementation phase, you set the stage for putting the chosen answer into action in a way that works. Thorough risk analysis and a “Go” or “No-Go” decision give you the confidence you need to move forward. Detailed action plans and change management strategies pave the way for the solution to fit into the company without any problems.

Step 7: Sell the Idea

For your answer to work, you must get stakeholders to agree with it. Anticipate resistance and consider all repercussions. Listen to what people say and make the changes that are needed. Here’s how to get people to buy into your idea:

Communicate Effectively :

Effective communication is the key to getting people to agree with your suggested solution. Change your message so that it speaks to the needs and worries of different stakeholders. Explain how the solution helps the company reach its goals and what its benefits are. Be open to comments and willing to make changes based on what stakeholders tell you.

Empathize and Address Concerns:

Put yourself in the shoes of your audience and try to see things from their point of view and understand any concerns they might have. Address their worries ahead of time and show why the proposed answer is the best way to move forward.

Build a Compelling Value Proposition:

Build a compelling value proposition by emphasizing the solution’s worth and how it will help different stakeholders. Show how the solution solves specific problems and contributes to the success of the company as a whole.

Involve Key Stakeholders:

Include key players from the start of the decision-making process. Ask them what they think and what they have to say. When stakeholders feel like they are part of the solution, they are more likely to back its implementation.

Adapt Your Message:

Tailor your communication style and content to suit different groups. Focus on the high-level strategic benefits for the executives, and on the practical effects and application details for the operational teams.

Use Visual Aids:

Use visual aids like charts, graphs, and infographics to make complex material easy to understand and interesting. Visual tools can help people understand and remember what you say better.

Address Possible Risks:

Point out and talk about any possible risks that come with the answer. Be honest about problems and show how you plan to solve them to show that you have thought about every part of the application.

By convincing stakeholders of the value of your idea, you make it more likely that the suggested solution will be accepted and used. Open and honest communication builds trust and support, which makes execution easier and makes it more likely that the desired results will be reached.

Step 8: Act

Lastly, move forward with action if all stakeholders agree. This is the most satisfying part of problem-solving because it’s where your work makes a real difference. Once the execution is done, get ready for the next round of problem-solving. What you have to do is:

Watch and judge:

As you put the answer into place, keep a close eye on how it’s going and judge how well it’s working. Collect data and feedback on a regular basis to see how the answer affects the problem. Check to see if the expected results are being met and look for any changes from the plan. Use this knowledge to make changes and improvements at the right time.

Celebrate Success and Learn from Challenges:

Recognize and celebrate the accomplishments and wins that came from the implementation, and learn from the problems that came up. Recognize the work of the team and let people know about the good results. It’s just as important to learn from any problems or hurdles you face along the way. Accept them as chances to learn and grow, and try to figure out what you can use to help you solve problems in the future.

Reflect and try again:

Take the time to think about the whole process of fixing a problem. Check how well each part of the Simplex model works and look for ways to make it better. Involve your team in the process of thought to get a variety of views and ideas. Use this knowledge to improve how you solve problems and improve how you do things in the future.

Prepare for the Next Cycle:

Solving problems is a continuous process, and when one cycle ends, the next one starts. Adopt a mindset of continuous growth and be ready to use the Simplex model to solve new problems as they come up. Keep an open mind and be creative. Always look for better ways to solve problems and bring about good change.

By keeping an eye on the results of your implementation and evaluating them, you can make sure that the chosen answer is giving you the results you want. People are more likely to grow and get better when they celebrate their wins and learn from their failures. As you get ready for the next cycle of problem-solving, you take a dynamic and proactive approach to keep your company on a path of continuous improvement and innovation.

Conclusion:

Being able to solve problems well opens up a lot of doors to success. The Simplex problem-solving model by Min Basadur gives a structured and effective way to approach problems in a systematic way. By following these eight steps and using the right tools and methods, you can approach problem-solving with confidence, knowing that your choices are well-informed and effective.

Assessment Test: Problem-Solving Skills

Instructions:.

For each statement, choose the option that best reflects your typical behavior or response. Select only one option for each statement. Answer honestly based on your actual experiences and behavior. Choose from the following options:

Now, let’s begin:

1. When faced with a difficult problem, I avoid thinking about it.

Not at all                              Rarely                                   Sometimes

2. I can break down complex issues into smaller, manageable parts.

3. I seek feedback and opinions from others to gain different perspectives on a problem.

4. I get easily frustrated when a solution doesn’t come quickly.

5. I enjoy brainstorming and generating creative solutions.

6. I often use trial and error to find solutions to problems.

7. I take the time to research and gather relevant information before making a decision.

8. I am open to changing my approach if a better solution is presented.

9. I can identify patterns and trends to help solve problems more efficiently.

10. I remain calm and composed when facing urgent or high-pressure situations.

11. I ask clarifying questions to fully understand the nature of a problem.

12.I am good at identifying the root causes of a problem rather than just addressing the symptoms.

13. I am confident in my ability to find solutions to most challenges.

14. I am willing to take calculated risks in order to reach a solution.

15. I actively seek out new information and knowledge to enhance my problem-solving skills.

16. I can adapt and adjust my problem-solving approach based on the specific circumstances.

17. I involve others in the implementation of solutions to ensure success.

18. I evaluate the outcomes of my problem-solving efforts to learn from both successes and failures.

19. I set realistic and achievable goals to guide my problem-solving process.

20. I am proactive in anticipating potential problems and taking preventive actions.

Once you have completed the assessment, calculate your total score by adding the points from each question. Refer to the scoring interpretation provided earlier to assess your problem-solving skills. Remember that this is not an absolute measure of your abilities, but rather an indication of areas you might want to focus on for improvement.

Not at all: 0 points                            Rarely: 1 point                   Sometimes: 2 points

Interpretation:

0 to 20 points: Your problem-solving skills may need improvement, and you might find it challenging to tackle complex issues effectively.

21 to 20 points: Your problem-solving skills are moderately developed, but there is still room for improvement in certain areas.

41 to 60 points: You possess good problem-solving skills, and you often approach challenges with a thoughtful and analytical mindset.

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Simplex Method for Solution of L.P.P (With Examples) | Operation Research

simplex problem solving

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:

simplex problem solving

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.

simplex problem solving

Flow Chart of Simplex Method :

simplex problem solving

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4 Applying the Creative Problem Solving Process to Open Innovation

  • Published: June 2016
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Group idea generation is a notoriously difficult process to carry out successfully. Creative problem solving is a proven model for driving innovation when implemented as an organization-wide business process. Organizations with a culture of continuous problem finding, problem solving, and implementation—along with requisite attitudinal, behavioral, and cognitive skills—have the greatest long-term innovation success. This chapter examines a framework that was implemented successfully at Procter & Gamble. It examines tools and methods for defining problems, finding solutions, and creating plans to execute those solutions. The chapter pays careful attention to methods for developing a range of potential solutions at each stage of the idea-generation process as well as methods for allowing groups to converge on a solution.

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4.2.1: Maximization By The Simplex Method (Exercises)

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  • Page ID 37871

  • Rupinder Sekhon and Roberta Bloom
  • De Anza College

SECTION 4.2 PROBLEM SET: MAXIMIZATION BY THE SIMPLEX METHOD

Solve the following linear programming problems using the simplex method.

1) \[\begin{array}{ll} \text { Maximize } & \mathrm{z}=\mathrm{x}_{1}+2 \mathrm{x}_{2}+3 \mathrm{x}_{3} \\ \text { subject to } & \mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_3 \leq 12 \\ & 2 \mathrm{x}_{1}+\mathrm{x}_{2}+3 \mathrm{x}_{3} \leq 18 \\ & \mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3} \geq 0 \end{array} \nonumber \]

2) \[\begin{array}{ll} \text { Maximize } \quad z= & x_{1}+2 x_{2}+x_{3} \\ \text { subject to } & x_{1}+x_{2} \leq 3 \\ & x_{2}+x_{3} \leq 4 \\ & x_{1}+x_{3} \leq 5 \\ & x_{1}, x_{2}, x_{3} \geq 0 \end{array} \nonumber \]

3) A farmer has 100 acres of land on which she plans to grow wheat and corn. Each acre of wheat requires 4 hours of labor and $20 of capital, and each acre of corn requires 16 hours of labor and $40 of capital. The farmer has at most 800 hours of labor and $2400 of capital available. If the profit from an acre of wheat is $80 and from an acre of corn is $100, how many acres of each crop should she plant to maximize her profit?

4) A factory manufactures chairs, tables and bookcases each requiring the use of three operations: Cutting, Assembly, and Finishing. The first operation can be used at most 600 hours; the second at most 500 hours; and the third at most 300 hours. A chair requires 1 hour of cutting, 1 hour of assembly, and 1 hour of finishing; a table needs 1 hour of cutting, 2 hours of assembly, and 1 hour of finishing; and a bookcase requires 3 hours of cutting, 1 hour of assembly, and 1 hour of finishing. If the profit is $20 per unit for a chair, $30 for a table, and $25 for a bookcase, how many units of each should be manufactured to maximize profit?

5). The Acme Apple company sells its Pippin, Macintosh, and Fuji apples in mixes. Box I contains 4 apples of each kind; Box II contains 6 Pippin, 3 Macintosh, and 3 Fuji; and Box III contains no Pippin, 8 Macintosh and 4 Fuji apples. At the end of the season, the company has altogether 2800 Pippin, 2200 Macintosh, and 2300 Fuji apples left. Determine the maximum number of boxes that the company can make.

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9.3: Minimization By The Simplex Method

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  • Rupinder Sekhon and Roberta Bloom
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Learning Objectives

In this section, you will learn to solve linear programming minimization problems using the simplex method.

  • Identify and set up a linear program in standard minimization form
  • Formulate a dual problem in standard maximization form
  • Use the simplex method to solve the dual maximization problem
  • Identify the optimal solution to the original minimization problem from the optimal simplex tableau.

In this section, we will solve the standard linear programming minimization problems using the simplex method. Once again, we remind the reader that in the standard minimization problems all constraints are of the form \(ax + by ≥ c\).

The procedure to solve these problems was developed by Dr. John Von Neuman. It involves solving an associated problem called the dual problem . To every minimization problem there corresponds a dual problem. The solution of the dual problem is used to find the solution of the original problem. The dual problem is a maximization problem, which we learned to solve in the last section. We first solve the dual problem by the simplex method.

From the final simplex tableau, we then extract the solution to the original minimization problem.

Before we go any further, however, we first learn to convert a minimization problem into its corresponding maximization problem called its dual .

Example \(\PageIndex{1}\)

Convert the following minimization problem into its dual.

\[\begin{array}{ll} \textbf { Minimize } & \mathrm{Z}=12 \mathrm{x}_{1}+16 \mathrm{x}_{2} \\ \textbf { Subject to: } & \mathrm{x}_{1}+2 \mathrm{x}_{2} \geq 40 \\ & \mathrm{x}_{1}+\mathrm{x}_2 \geq 30 \\ & \mathrm{x}_{1} \geq 0 ; \mathrm{x}_{2} \geq 0 \end{array} \nonumber \]

To achieve our goal, we first express our problem as the following matrix.

\[\begin{array}{cc|c} 1 & 2 & 40 \\ 1 & 1 & 30 \\ \hline 12 & 16 & 0 \end{array} \nonumber \]

Observe that this table looks like an initial simplex tableau without the slack variables. Next, we write a matrix whose columns are the rows of this matrix, and the rows are the columns. Such a matrix is called a transpose of the original matrix. We get:

\[\begin{array}{cc|c} 1 & 1 & 12 \\ 2 & 1 & 16 \\ \hline 40 & 30 & 0 \end{array} \nonumber \]

The following maximization problem associated with the above matrix is called its dual.

\[\begin{array}{ll} \textbf { Maximize } & \mathrm{Z}=40 \mathrm{y}_{1}+30 \mathrm{y}_{2} \\ \textbf { Subject to: } & \mathrm{y}_{1}+\mathrm{y}_{2} \leq 12 \\ & 2 \mathrm{y}_1+\mathrm{y}_2 \leq 16 \\ & \mathrm{y}_{1} \geq 0 ; \mathrm{y}_{2} \geq 0 \end{array} \nonumber \]

Note that we have chosen the variables as y's, instead of x's, to distinguish the two problems.

Example \(\PageIndex{2}\)

Solve graphically both the minimization problem and its dual maximization problem.

Our minimization problem is as follows.

\[\begin{array}{ll} \textbf { Minimize } & \mathrm{Z}=12 \mathrm{x}_1+16 \mathrm{x}_2 \\ \textbf { Subject to: } & \mathrm{x}_{1}+2 \mathrm{x}_{2} \geq 40 \\ & \mathrm{x}_{1}+\mathrm{x}_{2} \geq 30 \\ & \mathrm{x}_{1} \geq 0 ; \mathrm{x}_{2} \geq 0 \end{array} \nonumber \]

We now graph the inequalities:

imageedit_3_7200556551.png

We have plotted the graph, shaded the feasibility region, and labeled the corner points. The corner point (20, 10) gives the lowest value for the objective function and that value is 400.

Now its dual is:

\[\begin{array}{ll} \textbf { Maximize } & \mathrm{Z}=40 \mathrm{y}_1+30 \mathrm{y}_{2} \\ \textbf { Subject to: } & \mathrm{y}_{1}+\mathrm{y}_{2} \leq 12 \\ & 2 \mathrm{y}_1+\mathrm{y} 2 \leq 16 \\ & \mathrm{y}_{1} \geq 0 ; \mathrm{y}_{2} \geq 0 \end{array}\nonumber \]

We graph the inequalities:

imageedit_6_8472935815.png

Again, we have plotted the graph, shaded the feasibility region, and labeled the corner points. The corner point (4, 8) gives the highest value for the objective function, with a value of 400.

The reader may recognize that Example \(\PageIndex{2}\) above is the same as Example 3.1.1, in section 3.1. It is also the same problem as Example 4.1.1 in section 4.1, where we solved it by the simplex method.

We observe that the minimum value of the minimization problem is the same as the maximum value of the maximization problem; in Example \(\PageIndex{2}\) the minimum and maximum are both 400. This is not a coincident. We state the duality principle.

The Duality Principle

The objective function of the minimization problem reaches its minimum if and only if the objective function of its dual reaches its maximum. And when they do, they are equal.

Our next goal is to extract the solution for our minimization problem from the corresponding dual. To do this, we solve the dual by the simplex method.

Example \(\PageIndex{3}\)

Find the solution to the minimization problem in Example \(\PageIndex{1}\) by solving its dual using the simplex method. We rewrite our problem.

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

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

Recall that we solved the above problem by the simplex method in Example 4.1.1, section 4.1. Therefore, we only show the initial and final simplex tableau.

The initial simplex tableau is

\[\begin{array}{ccccc|c} \mathrm{y}_1 & \mathrm{y}_2 & \mathrm{x}_{1} & \mathrm{x}_{2} & \mathrm{Z} & \mathrm{C} \\ 1 & 1 & 1 & 0 & 0 & 12 \\ 2 & 1 & 0 & 1 & 0 & 16 \\ \hline-40 & -30 & 0 & 0 & 1 & 0 \end{array}\nonumber \]

Observe an important change. Here our main variables are \(\mathrm{y}_1\) and \(\mathrm{y}_2\) and the slack variables are \(\mathrm{x}_1 and \mathrm{x}_2\).

The final simplex tableau reads as follows:

\[\begin{array}{ccccc|c} \mathrm{y}_1 & \mathrm{y}_2 & \mathrm{x}_{1} & \mathrm{x}_{2} & \mathrm{Z} & \\ 0 & 1 & 2 & -1 & 0 & 8 \\ 1 & 0 & -1 & 1 & 0 & 4 \\ \hline 0 & 0 & 20 & 10 & 1 & 400 \end{array} \nonumber \]

A closer look at this table reveals that the \(\mathrm{x}_1\) and \(\mathrm{x}_2\) values along with the minimum value for the minimization problem can be obtained from the last row of the final tableau. We have highlighted these values by the arrows.

\[\begin{array}{ccccc|c} \mathrm{y}_1 & \mathrm{y}_2 & \mathrm{x}_{1} & \mathrm{x}_{2} & \mathrm{Z} & \\ 0 & 1 & 2 & -1 & 0 & 8 \\ 1 & 0 & -1 & 1 & 0 & 4 \\ \hline 0 & 0 & 20 & 10 & 1 & 400 \\ & & \uparrow & \uparrow & & \uparrow \end{array} \nonumber \]

We restate the solution as follows:

The minimization problem has a minimum value of 400 at the corner point (20, 10)

We now summarize our discussion.

Minimization by the Simplex Method

  • Set up the problem.
  • Write a matrix whose rows represent each constraint with the objective function as its bottom row.
  • Write the transpose of this matrix by interchanging the rows and columns.
  • Now write the dual problem associated with the transpose.
  • Solve the dual problem by the simplex method learned in section 4.1.
  • The optimal solution is found in the bottom row of the final matrix in the columns corresponding to the slack variables, and the minimum value of the objective function is the same as the maximum value of the dual.

IMAGES

  1. Simplex Process explained with Examples

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  2. Steps to solve linear programming problems by using the simplex method

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  3. Simplex Problem-Solving Process PowerPoint Template

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  4. 10 Things You need to know about Simplex Method

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  5. Simplex Process: Un proceso sólido de resolución de problemas

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  1. OR1 Simplex Method Part3

  2. SIMPLEX PROBLEM 1 PART 2 2

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  6. CE422-Solving LP using Simplex Method-Part1 (in Arabic)

COMMENTS

  1. The Simplex Process

    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 ...

  2. Simplex Problem Solving Process

    The Basadur Simplex Problem Solving Process is a problem-solving method that is aware of that, and prevents such mistakes from being made. This model was developed by the American creativity guru Marino (Min) Sidney Basadur, who presented the method in his book ' the Power of Innovation '. He was also the inventor of the patented Simplexity ...

  3. 4.2: Maximization By The Simplex Method

    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.

  4. Simplex Process explained with Examples

    The Simplex Process is a Problem Solving Method that Proposes 8 Steps to Find Lasting Solutions to any Problem. The 8 Steps Proposed by the Simplex Process are: Problem Finding. Fact Finding. Problem Definition. Idea Finding. Selection. Planning. Sell the Idea. Action. These Steps must be Repeated until the Problem is Resolved.

  5. A Creative Problem Solving & Innovative Thinking Process

    Step 4:Idea Finding. This step is where ideas are created in order to solve the defined problem. Skilled idea finders are never content with a single good idea. They continue to look for more and better ideas, and are able to build on half-formed and other people's ideas. Seemingly radical, even 'impossible' ideas can be fine-tuned to ...

  6. 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.

  7. 3.4: Simplex Method

    Solving the Linear Programming Problem by Using the Initial Tableau. ... The entire process of solving using simplex method is: Simplex Method. Set up the problem. That is, write the objective function and the constraints. Convert the inequalities into equations. This is done by adding one slack variable for each inequality.

  8. Discover the processes and tools for innovation

    The Simplexity Thinking Process. Min Basadur realized that to effectively solve a problem, he first needed to properly identify and define the problem without existing bias. He developed the 8-step Simplexity Thinking Process as a result. This breaks down the process of solving a problem into 3 overarching steps and 8 detailed ones.

  9. 4: Linear Programming

    4.3: Minimization By The Simplex Method. In this section, we will solve the standard linear programming minimization problems using the simplex method. The procedure to solve these problems involves solving an associated problem called the dual problem. The solution of the dual problem is used to find the solution of the original problem.

  10. Simplex Method Calculator

    How to use the simplex method online calculator. Enter the number of variables and constraints of the problem. Select the type of problem: maximize or minimize. Enter the coefficients in the objective function and the constraints. You can enter negative numbers, fractions, and decimals (with point). Click on "Solve".

  11. Simplex algorithm

    In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming. ... and determining the number of iterations needed for solving a given problem, are both NP-hard problems.

  12. Simplex algorithm

    Simplex algorithm (or Simplex method) is a widely-used algorithm to solve the Linear Programming (LP) optimization problems. The simplex algorithm can be thought of as one of the elementary steps for solving the inequality problem, since many of those will be converted to LP and solved via Simplex algorithm. [1]

  13. Mastering Problem Solving: Min Basadur's Simplex Model

    The Simplex problem-solving model by Min Basadur gives a structured and effective way to approach problems in a systematic way. By following these eight steps and using the right tools and methods, you can approach problem-solving with confidence, knowing that your choices are well-informed and effective. Assessment Test: Problem-Solving Skills

  14. PDF Optimization Algorithms and the Simplex Method

    Simplex Method itself to solve the Phase I LP problem for which a starting BFS is known, and for which an optimal basic solution is a BFS for the original LP problem if it's feasible. For example, for the standard equality form with the right-hand-side nonnegative, the Phase-I problem is min z 1 +z 2+…+z m, s.t. Ax+z=b, (x,z) ≥0.

  15. 4.3: Minimization By The Simplex Method

    To do this, we solve the dual by the simplex method. Example 4.3.3. Find the solution to the minimization problem in Example 4.3.1 by solving its dual using the simplex method. We rewrite our problem. Minimize Z = 12x1 + 16x2 Subject to: x1 + 2x2 ≥ 40 x1 + x2 ≥ 30 x1 ≥ 0; x2 ≥ 0.

  16. Understanding Simplex method and implementation on Python

    Implementation of Simplex Algorithm — Solution by Hand. SOLUTION. STEP 1: Set the problem in standard form. For setting in standard form we need to do two things: Make the Objective function in ...

  17. 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 ...

  18. Simplex Creative Problem Solving

    This is the opposite approach to the staff in most public and private offices, factories and construction sites. Simplex creative problem solving process is a system developed by Min Basadur at McMaster University, Ontario that does allow time to spend finding the real problem. It then finds a solution and programmes the implementation.

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

    holds in Sec. 5.1.) This optimality test is the one used by the simplex method for deter-mining when an optimal solution has been reached. Now we are ready to apply the simplex method to the example. Solving the Example Here is an outline of what the simplex method does (from a geometric viewpoint) to solve the Wyndor Glass Co. problem.

  20. Applying the Creative Problem Solving Process to Open Innovation

    Creative problem solving is a proven model for driving innovation when implemented as an organization-wide business process (Basadur, 2001; Basadur & Gelade, 2003).Figure 4.1 is an adaptation of Basadur's Simplex model (Basadur, 2001) that has been used successfully in hundreds of innovation workshops at Procter & Gamble's innovation studio, the GYM.

  21. 4.2.1: Maximization By The Simplex Method (Exercises)

    SECTION 4.2 PROBLEM SET: MAXIMIZATION BY THE SIMPLEX METHOD. Solve the following linear programming problems using the simplex method. 4) A factory manufactures chairs, tables and bookcases each requiring the use of three operations: Cutting, Assembly, and Finishing. The first operation can be used at most 600 hours; the second at most 500 ...

  22. Linear Programming

    Finding the optimal solution to the linear programming problem by the simplex method. Complete, detailed, step-by-step description of solutions. Hungarian method, dual simplex, matrix games, potential method, traveling salesman problem, dynamic programming ... Solve linear programming tasks offline! The number of constraints: 4-----The Number ...

  23. 9.3: Minimization By The Simplex Method

    Solve the dual problem by the simplex method learned in section 4.1. The optimal solution is found in the bottom row of the final matrix in the columns corresponding to the slack variables, and the minimum value of the objective function is the same as the maximum value of the dual.