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## Table of contents

Decision algorithm: yes or no

1. Summary and structure of the present paper

2. Introduction

3. Fields of application

4. Structure of the algorithm

5. Conclusion and outlook

6. Bibliography

7. List of figures

## 1. Summary and structure of the present paper

In this paper an algorithm is presented which allows to make an accurate YES/NO decision. Such an algorithm can be used and applied in various fields of application. This paper was written as a by-product of my phd project (DBA studies) at Middlesex University in cooperation with the KMU in Austria. In the fight against COVID-19, such an algorithm is needed both in the production of medicine and in the detection of diseased people, which the present paper tries to illuminate. Furthermore, such an algorithm can be used in various applications, e.g. in the automotive industry, medical technology, aerospace technology and even in the defence industry. Wherever various factors have to be considered for a decision, such an algorithm can be used to put them in relation to each other so that a prognosis for the YES or NO decision can be calculated. The algorithm shall be called Brecht algorithm, after Bertolt Brecht, an influential German dramatist, librettist and poet of the 20th century. The YES and NO refer to the production "the yes and no speaker" but this time with an OR and semicolon at the end of the sentence. The structure of the present work is shown in figure 1.

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Figure 1: Structure of the present paper. Source: own representation

## 2. Introduction

The chain leading up to a decision always leads to a decision between YES (for something) or NO (against something). In most cases, no quantified data are available as a basis for decision-making, but rather qualitative phenomena or data that cannot be precisely measured. In addition, in many fields of application, several independent variables that influence a dependent variable play a role in decision-making. This often leads to the fact that an efficient and quick decision is difficult to make. The present paper aims to show a mathematical approach that can be used to make better and more accurate decisions.

## 3. Fields of application

There are many fields of application for such an algorithm: Wherever a decision between YES and NO is to be made, the algorithm can be used. In medical research and production "medical studies" the algorithm can be used for decisions about the production of medicine, and further it may be used in mobile applications for the detection of sick people. Furthermore, the algorithm can be used in medical technology (cf. Mach, 2019) for stimulation and shock delivery of a pacemaker or defibrillator in order to make a decision based on the available data. In the automotive industry (cf. Winkelhake, 2018), such a decision could become necessary, for example, to prevent accidents by means of the braking system (driver assistance system). In the automotive industry, such an algorithm can also be used in electric cars to predict the range or to charge the battery.

This algorithm belongs to the field of machine learning (cf. Otte, 2019) and should lead to more transparency and explainability in order to be able to use the advantages of deep learning algorithms in machine learning on this basis. The efficient implementation of such an algorithm using an 8-bit microcontroller offers a lot of leeway, which is reflected in cost minimization, simple implementation, fast decision making and independence of the microcontroller used

## 4. Structure of the algorithm

In physics, v = d/t (cf. Thomsen, 2018) is used to determine a relationship between distance and time as velocity. The distance d is then v * t. It is quite similar in the following considerations, where the distance to minimum and maximum is the object of consideration. On the basis of this analogy from physics, the risks/opportunities (YES/NO) for a decision should be determined in order to determine the relationship between opportunities (success factors - YES) and risks (failure factors - NO) through mathematics (cf. Papula, 2018) and statistics (cf. Rooch, 2014) and thus be able to make a better YES/NO decision. The following two formulae are used to establish a basic equation for risk and success, which makes use of the above-mentioned analogy. The core objectives of the consideration are to keep the risk to a minimum and the success to a maximum.

*Equation 1:*

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Where E is the probability of occurrence and S the severity of the risk.

*Equation 2:*

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Here E is the probability of occurrence and S the degree of importance of the success.

Since in practice many risk and success factors (YES/NO influencing factors) as independent variables influence the dependent variable "decision", another equation is to be used, namely the Taylor series (see Gleiser, 2016).

*Equation 3:*

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The fundamental equations 1 and 2 are to be inserted into the Taylor series as functions. The independent variables in both the profit area (YES area) and the risk area (NO area) are to be combined into categories so that their quantification can be mapped on the basis of E * S. The variables and the weighting of both equations are not known due to the diversity, as there are many possible experience-based categories formed by humans, which should be put into a ratio in the end. Both equations also result in zero at t = 0. Since the present equation and associated variables are not known at the beginning. The first derivative of F (X) or G (X) denotes the importance of the category. The further polynomials of the second and third derivation represent three further, less significant categories. If FIVE categories are selected for importance, the first derivative starts with **F(X) = ((E * S) * t)5** and **G(X) = ((E * S) * t)5**, then continues to the fifth derivative, **F(X) = ((E * S) * t)4 F(X) = ((E * S) * t)3 F(X) = ((E * S) * t)2** and **F(X) = ((E * S) * t)**.

If, as shown above using the example of risks, the categories and the corresponding importance have been estimated, then a result can be formed from the sum of the individual risk values (e.g. Y). This is done for both functions F(X) and G(X), whereby the core idea remains the same: The risk must be minimal.

Equation 4 shows this in a simplified way:

*Equation 4:*

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This results in equation 5 in the general form with Y, X and T:

*Equation 5:*

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Following the principle of maxima and minima in mathematics, the first derivative of X is set to zero. Similarly, the second derivative of X is set to zero to search for turning points. So if the second derivative was used to calculate the number of turning points (> zero àrisk-minimum), the values are entered into the equation (X * t)6 to calculate the absolute minimum. This minimum is the measure of risk. The difference between Y and this scale is an indication that the risks must be reduced. The same is done for the success measure. Figure 2 shows this relationship schematically.^{1}

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The same applies to success as to risk, but with the maximum as the measure and the second derivative < zero. The core idea remains the same: Success must be maximum.

*Equation 6:*

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This results in equation 7 in the general form Y, X and T:

*Equation 7:*

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Following the principle of maxima and minima in mathematics, the first derivative of X is set to zero. Similarly, the second derivative of X is set to zero to search for turning points. When the result of the second derivative, i.e. the number of turning points, has been calculated with < zeroà Success-Maximum, the values are entered into the equation (X * t)6 to calculate the absolute maximum. This maximum is the measure of success. The difference The distance between Y and this scale is an indicator of the success to be promoted. Figure 3 shows this relationship schematically.^{2}

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Figure 3: Schematic representation of the success equation. Source: own representation

In order to be able to calculate the two unknown variables of equations 5 and 7, a second equation is required, which on the one hand represents a normalization and on the other hand represents a closed loop. The formula of the unit circle is used for this purpose.

*Equation 8:*

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The complete equation contains two unknown, namely X1 (YES) and X2 (NO), which are to be related to their probabilities X is not known for the individual derivatives. By assuming the normally distributed categories, probabilities and predictions can be read so that a YES/NO decision can be made based on the ratios. Both the risks and the success factors play a major role in this process in order to be able to make a meaningful YES or NO decision. The risk/no prognosis therefore results from the probability that at F(X) the values lie in the normal distribution between C and the minimum, and the success/no prognosis results from the probability that at G(X) the values lie in the normal distribution between C and the maximum.^{3}

It should always be borne in mind that the risks (NO factors) can become success factors (YES factors) over time. This requires a corresponding re-evaluation (E * S * t; here time grows and the variables E * S must be re-evaluated). An objective overall evaluation is usually relative and always refers to a relationship between two standardized phenomena or circumstances such as success/failure, potential/threat, data protection/public, growth/resource consumption or YES/NO. The YES probability and NO probability can thus be calculated. It is sufficient to set the NO probability against the YES probability and to put them in relation to each other. The values are normally distributed under the assumption that 80% of the events are normally distributed.

Another step in the algorithm is that each category should be weighted differently. For this purpose, the need for consensus for decision making is important. All that is needed is to program a loop in the software implementation of the algorithm to give each category a different weighting. In the end, the basic idea decides whether the risk/NO is set to a minimum and, analogously, success/YES to a maximum. If one wants to be independent of the underlying probability distribution, the function (equation 3) should be integrated as a distribution function in order to be able to calculate the expected probability for each run from the area and the interval boundaries (between the calculated scale and zero). If there is an even number of categories, this results in 2n variants; if there is an odd number, one can add another zero category in order to make the whole mathematically calculable, since a summation of the categories is the basis.

To derive the function from the beginning to the end, instead of simply calculating the deltas, can be justified by the fact that when deriving, one does not only calculate the average and can calculate an instantaneous value of the course in each phase of the process. The above analogy from physics can be used to illustrate this reasoning: A function over time such as acceleration and the calculation of current velocity require the derivation of a function. In the case of the average, however, only the average speed over the entire distance and time could be calculated.

## 5. Conclusion and outlook

The present contribution serves as a concept to be able to implement such an algorithm by programming. The mathematical, pragmatic and statistical fundamentals are presented here logically in a closed loop. In the end it is hoped that the algorithm can lead to practically useful results, which will be investigated by a prototypical implementation and verification.

## 6. Bibliography

**Track**, J. (2016). *The Taylor series and the Taylor polynomial. Explanation of the procedure*. GRIN Publishing House

**Mach**, E. (2019). *Introduction to medical technology for healthcare professionals*. (2nd edition). Vienna: Facultas

**Otte**, R. (2019). *Artificial intelligence for dummies*. (1st edition). Wiley-VCH

**Papula**, L. (2018). *Mathematics for Engineers and Scientists Volume 1: A textbook and workbook for basic studies*. (15th edition). Wiesbaden: Springer Vieweg

**Rooch**, A. (2014). *Statistics for engineers: Probability calculation and data evaluation finally understandable*. (1st edition). Berlin: Springer Spectrum

**Thomsen**, C. (2018). *Physics for engineers for dummies*. (2nd edition). Weinheim: Wiley-VCH

**Winkelhake**, U. (2018). *The Digital Transformation of the Automotive Industry: Drivers - Roadmap - Practice*. (1st edition). Berlin: Springer Vieweg

## 7. List of figures

Figure 1: Structure of the present paper. Source: own representation

Figure 2: Schematic representation of the risk equation. Source: own representation

Figure 3: Schematic representation of the success equation. Source: own representation

**[...]**

^{1} C is the calculated value of equation 3 after finding the variable or unknown

^{2} C is the calculated value of equation 3 after finding the variable or unknown

^{3} F(X) is the YES factors of equation 3 G(X) is the NO factors of equation 3

- Quote paper
- Haider Karomi (Author), 2020, Decision Algorithm between Yes and No, Munich, GRIN Verlag, https://www.grin.com/document/908933

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