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Running Head: SIMPLEX METHOD 1

Application of the simplex method in solving a linear programming optimization

problem

Name: Hanyi Wu

Date: April 11, 2021

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Introduction

Background information about the simplex method

Linear programming is a powerful technique for obtaining optimal solutions to the

problem, expressed using linear equations and inequalities. When a problem can be represented

accurately by mathematical equations in a linear program, the method will find the best

solution.

The simplex method is a typical linear programming approach which is used when

determining the point to operate on, basically in production, to get the highest output

(optimization point) while operating to the lowest cost as much as possible to produce the same

results. George Dantzig gave the method in 1945 to solve linear programming model by hand

using slack variables, tableaus, and pivot variables to find the optimal solution of an

optimization problem. Simplex tableau is used to perform row operations on the linear

programming modal and check its optimality.

This methodology is applied in mathematics and another related field that applies linear

optimization in finding a point in the feasible region of operation that will give us the mentioned

results. Over time, different fields such as management and accounting have been using the

optimization process as a tool when making optimal decisions or trying to have the highest

output using the lowest cost possible. There are various limiting factors or constraints that must

be expressed through this approach to have such results. The problem is that most of the

formulae used are complicated, making the process much more difficult. With the use of

optimization, the problem of complexity is solved because it is more straightforward and

simpler than the logarithmic formula of solving a programming problem.

For instance, a company doing popcorn and potato crisps production majorly use salt

denoted as T and cooking oil denoted as K. In the production of X1 amount of popcorn; the

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company needs T1 grams of salt and K1 litters of cooking oil. In the production of X2 amount

of crisps, the company needs T2 grams of salt and K2 litters of cooking oil. Assuming that X1

amount of popcorn trades at price D1 and X2 of crisps trades at D2, and the optimal output of

the company is C, then we can have the following equation:

D1 . X1 + D2 . X2 (Maximize revenue)

X1 + X2 ≤ C (optimal output limit)

T1 . X1 + T2 . X2≤ T (salt limit)

K1 . X1 + K2 . X2≤ K (cooking oil limit)

X1≥0, X2≥0

(these are the non-negative variables since it is impossible to have a negative

production).

After the above step, the matrix is then formed.

In the world, most problems consist of multiple variables due to the complexity of

nature. Hence these variables form many equations which are slow to solve unless a high-

speed processor is used. The simplex method enables one to program a simplified version of

the solving process of these variables. In most cases, the variables obtained from equations or

functions when solved each time hence becoming complex. The simplex method works by

beginning from the corners of the model while progressing to other model corners. For every

progression, the value of each primary function becomes more accurate. As what said at the

beginning, this procedure is repetitive and stops when ideal values are obtained.

Without proper calculation, it is hard for the management, especially in the business

field, to make better decisions considering that their application revolves around optimization

strategy. Within the business organization, the primary objective is to maximize the profit and

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to be able to achieve that objective; there is a need for the organization to determine the optimal

reduction units that the organizations need to produce to maximize its profits.

The knowledge of optimization is also used when maximization consumer satisfaction,

which is also the business organization's objective. It's the duty of the organization to consider

the interest of the customers as a way of expanding its market share. Therefore, the knowledge

of the simplex method of solving a linear programming method is necessary because it solves

optimization problems facing different mathematicians and other fields.

Statement problem

Many business organizations face an optimization challenge when dealing with various

management decisions to maximize profits and consumer satisfaction. To come up with an

exact unit of production that will maximize the profit, there is a need to calculate linear

programming to identify the unknown values that will represent the units to be produced. One

of the formulae that helps the management come up with optimization problems is the simplex

method. It helps the management understand the linear programming method when calculating

facts about optimization problems. The existence of multiple formulas for solving a linear

programming optimization problem makes it hard to determine the exact formula that can

quickly solve the optimization problem. This paper addresses how easy and compelling is a

simple method in solving a linear programming optimization problem.

The objective of the study

• To determine the effectiveness of the simplex method on solving an

optimization problem.

• To determine the ease of solving linear programming using the simplex method.

Significance of the study

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The study is significant by providing solutions to most organizations applying

optimization knowledge, such as business organizations, to allow for easy calculation of linear

programming equations.

Limitations and scope of the study

The study only covers the linear programming method to solve the optimization

problem but not the non-linear programming method. The study will be limited only to business

organizations that face optimization problems during the business's daily running.

Literature review

The simplex method will get into a loop in two scenarios. First, a variable is repeated

in the iterative process due to many variables that are solved first. This occurs mainly in real-

life variables (Sarode, 2017). The second situation is when parameters used as the primary

function do not change due to a complex process. The simplex method application helps in

solving complex mathematics expressions, hence offering practical solutions to many

mathematical problems.

In the simplex method, there is a function in the form of an equation and a constraint

that must be considered, limiting the operations and must be considered to come up with the

accurate value of the units (Lee, Resiga, & Wern, 2020). Entering variables in the simplex

method involves using the current non-basic value to improve the objective by increasing the

value from zero.

Considering the complexity of linear programming, it is better to implement the more

straightforward formula of calculating so that those involved can be in a better position of

getting the required solution to the optimization problem they are facing (Jing, Wang, Liang,

Wang, Li, Shah, & Zhao, 2018). The ease of calculating is determined by the time taken when

calculating values to get optimization solutions. When the method is complex to apply, it is

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estimated that it will take a longer time for users to get the correct answer, unlike when it is

easier where it takes minimum time before arriving at the accurate answer.

The mathematical formula's effectiveness depends on the accuracy of the answer

arrived at within the shortest time possible. Indeed, it is expected that when the simplex method

is more straightforward, it allows for the users to take little time to arrive at an seemingly

correct answer that will help in solving optimization problems (Zhou, Zhou, Luo, & Abdel-

Basset, 2017). It is believed that the effectiveness of the method is correlated with the simplicity

of its application. Therefore, the simplex method is considered the best alternative method to

using in organizations facing optimization problems to come up with accurate findings or

answers for decision-making.

Methodology of the study

Research design

The study will use an experimental research design with two categories to be studied:

the control group and the experimental group. The experimental group is the group that uses

the simplex method as a way of solving the optimization challenge, and the record results are

on time taken during calculations and the accuracy of the answer. The second category, the

control group, consists of individuals who use other methods to solve an optimization problem

using linear programming.

Target population

The study targets business organizations where the rate of the application of the

optimization problem is high. The target population provides the required information about

the simplex method's ease and effectiveness in solving linear programming.

Sample size

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The study covers 300 respondents that were involved in the experimental research. The

number was selected because with more respondents, it is easier to generalize the findings to

the general population, which assuring the results' validity.

Selection of the sample size

The study respondents were selected using the probability random sampling technique,

which gives each respondent's equal chance of being selected. In scientific research, the validity

of the findings is critical, and the respondents' selection provides a better opportunity to select

respondents without biasness.

Data collection technique

Primary data was used to collect the respondents' information through the use of an

online questionnaire to faster administration of the questions to get accurate responses from

the respondents.

Results and discussions

The study's findings indicated a positive relationship between using the simplex method

in linear programming with the effectiveness of and ease of getting an accurate answer. Over

70% of the respondents who used the simplex method when calculating for the optimization

problem got accurate answers within the shorter time possible, unlike respondents in the control

group who used other techniques in solving an optimization problem, who took more time to

arrive at accurate answers.

The ease of calculating mathematical problems depends on the simplicity of methods

and procedures involved. With the simplex method, it is easier to get answers considering that

it has few steps that are apparent to users when following, making it a faster method due to the

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lack of complexity of the application. The ease of the formula is evident in most business , and

many departments uses the method to solve the optimization problem.

In the world, most problems consist of multiple variables due to the complexity of

nature. Hence these variables form many equations which are slow to solve unless a high-

speed processor is used. The simplex method enables one to program a simplified version of

the solving process of these variables.

In most cases, the variables obtained from equations or functions when solved each

time hence becoming complex. The simplex method works by beginning from the corners of

the model while progressing to other model corners. For every progression, the value of each

primary function becomes more accurate. This procedure is repetitive and stops when ideal

values are obtained.

Addressing the complexity of linear programming, it is better to implement the more

straightforward formula of calculating. Those involved can be better positioned to get the

required solution to the optimization problem they are facing. The ease of calculating is

determined by the time taken when calculating values to get optimization solutions. When the

method is complex to apply, it is estimated that it will take a longer time for users to get the

correct answer, unlike when it is easier where it takes minimum time before arriving at the

accurate answers.

On the effectiveness of the simplex method of calculating linear programming, the

findings indicate that the method provides accurate answers, making it an effective method of

solving optimization problems. From the respondents used in the analysis, 80% got accurate

answers, unlike those in the control group, where only 30% got accurate answers, citing the

complexity of the nature of other methods used to solve the optimization problem.

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Conclusion

Accuracy and ease of applying the mathematical formula are used as a baseline to

determine the method's effectiveness. Different organizations are encouraged to adopt formulas

that are easy to apply and give accurate results. With the simplex method, it is easier to get

accurate results within the shortest time possible because of its ability to use programming

technique that has fewer but simple steps.

With the simplex method, it is easier to get answers considering that it has few steps

that are apparent to users when following, making it a faster method due to the lack of

complexity of the application. The ease of the formula is evident in most business

organizations; most departments are using the method to solve the optimization problem.

Addressing the complexity of linear programming, it is better to implement the more

straightforward formula of calculating. Those involved can be better positioned to get the

required solution to the optimization problem they are facing. The ease of calculating is

determined by the time taken when calculating values to get optimization solutions. When the

method is complex to apply, it is estimated that it will take a longer time for users to get the

correct answer, unlike when it is easier where it takes minimum time before arriving at the

accurate answers.

Recommendations

It is recommended that business organizations facing optimization problems in

applying management techniques adopt the simplex method when solving for optimization

using linear programming technique. The technique will help the organization save time

because it is easier and only require little time, giving the management adequate time to

concentrate on the production sector. The method also guarantees management accuracy

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because of the validity of the results, which is the main component and advantage of the

simplex method of solving an optimization problem.

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References

Jing, R., Wang, M., Liang, H., Wang, X., Li, N., Shah, N., & Zhao, Y. (2018). Multi-objective

optimization of a neighbourhood-level urban energy network: Considering Game-theory

inspired multi-benefit allocation constraints. Applied Energy, 231, 534-548.

Lee, Y., Resiga, A., Yi, S., & Wern, C. (2020). The Optimization of Machining Parameters for

Milling Operations by Using the Nelder–Mead Simplex Method. Journal of

Manufacturing and Materials Processing, 4(3), 66.

Sarode, M. V. (2017). Application of a Simplex Method to Find the Optimal Solution.

International Journal of Innovations of Engineering and Science, 2(2), 21-24.

Zhou, Y., Zhou, Y., Luo, Q., & Abdel-Basset, M. (2017). A simplex method-based social

spider optimization algorithm for clustering analysis. Engineering Applications of Artificial

Intelligence, 64, 67-82.

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