# A Fuzzy Logic Algorithm for Optimizing the Investment Decisions within Companies

^{1}

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^{*}

## Abstract

**:**

## 1. Introduction

- Ensures the ratio optimization between the cost of acquiring an asset and its economic performance;
- Ensures the comparability of the criteria for the acquisition of an asset even when these criteria have different measurement units or can be judged through ratings;
- Ensures the transition from crisp sets used for the quantification of asset acquisition criteria to fuzzy sets, characterized by membership degrees, so that each acquisition criterion $\left({C}_{i}\right)$ will have a membership degree $\mu \left({C}_{i}\right)\in \left[0;1\right]$;
- When there are multiple offers on the market for the acquisition of assets, the algorithm allows the selection of the offer that best combines the cost of the asset with its economic performance;
- Ensures the selection of an asset or portfolio of assets based on the company’s asset purchase policy, and the acquisition criteria considered to be a priority. This feature is provided by the fuzzy logic multi-criteria algorithm using the adjustment coefficients described in this paper.

- They are based on professional judgment;
- Does not ensure comparability between asset acquisition criteria;
- The determined score functions do not generate the best combination of the asset cost and its economic performance.

- The multitude and complexity of the selection criteria for assets to be purchased from the market;
- Lack of homogeneity in the content of the selection criteria;
- The possibility to provide metrics for measuring selection criteria in the form of linguistic variables;
- The existence of decision-shaping techniques based on the selection of market assets in line with company acquisition strategies.

## 2. State of the Art

- It is the first decision management tool in the literature to use artificial intelligence techniques over state-of-the-art methods/algorithms based on simple mathematical models. As we all know, these techniques are not always characterized by flexibility, adaptability or efficiency and are not well suited to the company’s needs;
- Modeling the asset selection decision is based on triangular fuzzy numbers. Considering the market selection criteria used in practice, it can be underlined that they are modeled by fuzzy logic, which allows for fuzzy modeling of those criteria that have the linguistic values as the sole alternative for quantification;
- It contains innovative components of the algorithm based on artificial intelligence, which refers to: the matrix of the degree of belonging, the global degree of belonging vector and the inference operator of the maximum of the global degree of belonging;
- It is a flexible managerial tool for companies that allows the selection of assets in portfolios, but also the adaptation of asset selection strategy to the real needs of the company;
- It effectively combines criteria for the selection of assets on the market with different content, such as asset-backed assets with selection criteria that take into account their economic performance;
- It solves extremely complex problems for substantiating the multi-criteria decisions underlying the selection of asset portfolios, regardless of the number and content of asset selection criteria.

## 3. The Development of the Fuzzy Logic Algorithm

#### 3.1. The Fuzzy Logic Multi-Criteria Algorithm Premises

**P1:**The company intends to purchase the assets A_{1}, A_{2}, .... A_{n}, with i = $\overline{1,\mathrm{n}}$ from the market, in order to improve its economic performance. These assets are required for investment activity and are purchased within the range of available funding sources.**P2:**For each asset A_{i}, i = $\overline{1,\mathrm{n}}$, the company sets acquisition criteria C_{j}with j = $\overline{1,\mathrm{m}},$ according to its purpose, so that each asset A_{i}, i = $\overline{1,\mathrm{n}}$, will have the associated criteria C_{j}, j = $\overline{1,\mathrm{m}},$ of the form:$$\begin{array}{ccc}{\mathrm{A}}_{1}=\mathrm{f}({\mathrm{C}}_{\mathrm{j}11}& {\mathrm{C}}_{\mathrm{j}12}\dots \dots \dots .& {\mathrm{C}}_{\mathrm{j}1\mathrm{m}})\\ {\mathrm{A}}_{2}=\mathrm{f}({\mathrm{C}}_{\mathrm{j}21}& {\mathrm{C}}_{\mathrm{j}22}\dots \dots \dots .& {\mathrm{C}}_{\mathrm{j}2\mathrm{m}})\\ \hfill \dots .& \hfill \dots .& \dots .\\ {\mathrm{A}}_{\mathrm{n}}=\mathrm{f}({\mathrm{C}}_{\mathrm{ji}1}& {\mathrm{C}}_{\mathrm{ji}2}\dots \dots \dots .& {\mathrm{C}}_{\mathrm{jim}})\end{array}$$**P3:**There are offers (${\mathrm{O}}_{\mathrm{ki}})$ on the market for acquiring the assets $({\mathrm{A}}_{\mathrm{i}})\text{}\mathrm{with}\text{}\mathrm{i}=\overline{1,\mathrm{n}\text{}}$ that meet the purchase criteria C_{j}, $\mathrm{j}=\overline{1,\mathrm{m}}$ of the form:$$\left|\begin{array}{c}{\mathrm{A}}_{1}\\ {\mathrm{A}}_{2}\\ \dots \\ {\mathrm{A}}_{\mathrm{n}}\end{array}\right|\to \left|\begin{array}{cccc}{\mathrm{c}}_{11}& {\mathrm{c}}_{12}& \dots & {\mathrm{c}}_{1\mathrm{m}}\\ {\mathrm{c}}_{21}& {\mathrm{c}}_{22}& \dots & {\mathrm{c}}_{2\mathrm{m}}\\ \dots & \dots & \dots & \dots \\ {\mathrm{c}}_{\mathrm{k}1}& {\mathrm{c}}_{\mathrm{k}2}& \dots & {\mathrm{c}}_{\mathrm{km}}\end{array}\right|\to \left|\begin{array}{cccc}{\mathrm{O}}_{11}& {\mathrm{O}}_{12}& \dots & {\mathrm{O}}_{1\mathrm{m}}\\ {\mathrm{O}}_{21}& {\mathrm{O}}_{22}& \dots & {\mathrm{O}}_{2\mathrm{m}}\\ \dots & \dots & \dots & \dots \\ {\mathrm{O}}_{\mathrm{k}1}& {\mathrm{O}}_{\mathrm{k}2}& \dots & {\mathrm{O}}_{\mathrm{km}}\end{array}\right|$$_{i}—assets; C_{j}—acquisition criteria; ${\mathrm{O}}_{\mathrm{ki}}$—the offers existing on the marketEach offer on the market meets the purchase criteria C_{j}, $\mathrm{j}=\overline{1,\mathrm{m}}$, otherwise the offer is rejected, namely:$$\begin{array}{l}{\mathrm{O}}_{11}\in \left\{{\mathrm{c}}_{11},{\mathrm{c}}_{12},\dots ,{\mathrm{c}}_{1\mathrm{n}}\right\};\\ {\mathrm{O}}_{22}\in \left\{{\mathrm{c}}_{21},{\mathrm{c}}_{22},\dots ,{\mathrm{c}}_{2\mathrm{n}}\right\};{\mathrm{O}}_{\mathrm{ki}}\in \left\{{\mathrm{c}}_{\mathrm{k}1},{\mathrm{c}}_{\mathrm{k}2},\dots ,{\mathrm{c}}_{\mathrm{kn}}\right\};\end{array}$$For each asset A_{i}, i = $\overline{1,\mathrm{n}}$ is satisfied the condition ${\mathrm{O}}_{\mathrm{ki}}$ ≥ 2: there are at least two eligible offers, on the contrary the existing of only an eligible offer on the market, would make the problem irrelevant.**P4:**For each acquisition criterion ${\mathrm{C}}_{\mathrm{j}}\left({\mathrm{A}}_{\mathrm{i}}\right)$ with $\mathrm{i},\mathrm{j}=\overline{1,\mathrm{n},\mathrm{m}}$, the decision maker sets a ${\mathrm{C}}_{\mathrm{jmax}}\left({\mathrm{A}}_{\mathrm{i}}\right)$ considered the maximum accepted level for each criterion, and a ${\mathrm{C}}_{\mathrm{jmin}}\left({\mathrm{A}}_{\mathrm{i}}\right)$ considered the level below which the company rejects any existing offer on the market. Any acquisition criterion respects the following inequality:$${\mathrm{C}}_{\mathrm{jmin}}\left({\mathrm{A}}_{\mathrm{i}}\right)\le {\mathrm{C}}_{\mathrm{j}}\left({\mathrm{A}}_{\mathrm{i}}\right)\text{}\le \text{}{\mathrm{C}}_{\mathrm{jmax}}\left({\mathrm{A}}_{\mathrm{i}}\right)$$**P5:**The global membership degree of an offer ${\mathsf{\mu}}_{\mathrm{g}}\left({\mathrm{O}}_{\mathrm{ji}}\right)$ substantiates the company’s investment decision. Any offer with a global membership degree ${\mathsf{\mu}}_{\mathrm{g}}\left({\mathrm{O}}_{\mathrm{k}}\left({\mathrm{A}}_{\mathrm{i}}\right)\right)$ below 0.5 is considered to be an offer that deviates from the company’s objectives, to identify an optimal ratio between the cost of asset acquisition and the economic performance.

#### 3.2. The Content of the Fuzzy Logic Multi-Criteria Algorithm

**Definition**

**1.**

**Definition**

**2.**

- For minimum criteria:$${\mathsf{\mu}}_{\mathrm{cjmin}}({\mathrm{O}}_{\mathrm{k}}\left({\mathrm{A}}_{\mathrm{i}}\right))=\{\begin{array}{c}1{\text{}\mathrm{if}\text{}\mathrm{V}}_{\mathrm{i}}\left({\mathrm{O}}_{\mathrm{k}}\left({\mathrm{A}}_{\mathrm{i}}\right)\right)={\mathrm{C}}_{\mathrm{jmax}}\\ 1-\text{}\frac{{\mathrm{V}}_{\mathrm{i}}\left({\mathrm{O}}_{\mathrm{k}}\left({\mathrm{A}}_{\mathrm{i}}\right)\right)-{\mathrm{C}}_{\mathrm{jmax}}}{{\mathrm{C}}_{\mathrm{jmin}}}{\text{}\mathrm{if}\text{}\mathrm{C}}_{\mathrm{jmax}}\le {\mathrm{V}}_{\mathrm{i}}\left({\mathrm{O}}_{\mathrm{k}}\left({\mathrm{A}}_{\mathrm{i}}\right)\right)\le {\mathrm{C}}_{\mathrm{jmax}\text{}}+{\mathrm{C}}_{\mathrm{jmin}}\\ 0{\text{}\mathrm{if}\text{}\mathrm{V}}_{\mathrm{i}}\left({\mathrm{Ok}}_{\mathrm{i}}\right)\ge {\mathrm{C}}_{\mathrm{jmax}\text{}}+{\mathrm{C}}_{\mathrm{jmin}}\end{array}$$
- For maximum criteria:$${\mathsf{\mu}}_{\mathrm{cjmax}}({\mathrm{O}}_{\mathrm{k}}\left({\mathrm{A}}_{\mathrm{i}}\right))=\{\begin{array}{c}1{\text{}\mathrm{if}\text{}\mathrm{V}}_{\mathrm{i}}\left({\mathrm{O}}_{\mathrm{k}}\left({\mathrm{A}}_{\mathrm{i}}\right)\right)={\mathrm{C}}_{\mathrm{jmax}}\\ 1-\text{}\frac{{\mathrm{C}}_{\mathrm{jmax}}-{\text{}\mathrm{V}}_{\mathrm{i}}\left({\mathrm{O}}_{\mathrm{k}}\left({\mathrm{A}}_{\mathrm{i}}\right)\right)}{{\mathrm{C}}_{\mathrm{jmin}}}{\text{}\mathrm{if}\text{}\mathrm{C}}_{\mathrm{jmax}\text{}}-{\mathrm{C}}_{\mathrm{jmin}\text{}}\le {\mathrm{V}}_{\mathrm{i}}\left({\mathrm{O}}_{\mathrm{k}}\left({\mathrm{A}}_{\mathrm{i}}\right)\right){\mathrm{C}}_{\mathrm{jmax}\text{}}\\ 0{\text{}\mathrm{if}\text{}\mathrm{V}}_{\mathrm{i}}\left({\mathrm{O}}_{\mathrm{k}}\left({\mathrm{A}}_{\mathrm{i}}\right)\right)\le {\mathrm{C}}_{\mathrm{jmax}\text{}}-{\mathrm{V}}_{\mathrm{i}}\left({\mathrm{O}}_{\mathrm{k}}\left({\mathrm{A}}_{\mathrm{i}}\right)\right)\end{array}$$

**Definition**

**3.**

**Proposition**

**1.**

- a)
- (${\mathrm{a}}_{\mathrm{i}}),\mathrm{i}=\overline{1,\mathrm{n}}$, cu ${\mathrm{a}}_{\mathrm{i}}\in \mathrm{R}$;
- b)
- ${{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{i}}=1$;
- c)
- ${\mathrm{a}}_{\mathrm{i}}:\mathrm{R}\to \left[0,1\right]$

## 4. Simulation of the Fuzzy Logic Multi-Criteria Algorithm

#### 4.1. First Scenario

_{1}) needed for the current activity of a company, for which the following information was known: six asset acquisition criteria, grouped into two categories, namely: economic criteria (asset price) and technical criteria (asset productivity, asset guarantee, operating costs, production capacity and quality of the products obtained), as well as five offers available on the market. Following the implementation of fuzzy logic multi-criteria algorithm, the following results were obtained:

_{3}(the operating cost), so that this offer has been removed from the algorithm. The first result of the algorithm is the matrix of the membership degree of the offers ${\mathrm{O}}_{\mathrm{k}}\left({\mathrm{A}}_{\mathrm{i}}\right)$ to the corresponding selection criterion ${\mathrm{C}}_{\mathrm{i}}\left({\mathrm{A}}_{\mathrm{i}}\right)$.

_{4}), for which ${\mathsf{\mu}}_{\mathrm{g}}{\mathrm{A}}_{1}({\mathrm{C}}_{2}\left({\mathrm{O}}_{\mathrm{k}}\right)=0.7$ has the higher value and has following features presented in Table 1.

_{4}, being the winning offer after applying the algorithm, falls within the category of offers that best fit the selection criteria formulated by the company. The global membership degree has the value of 0.7 for the $({\mathrm{O}}_{4}\left({\mathrm{A}}_{1}\right))$ and the highest value for each offer. The global membership degree provides information on how the offers best fit the selection criteria formulated by the company. A higher value of the degree indicates that all the technical and economic criteria of an offer fit “well” into the restrictions set for each offer.

_{1}), after applying the fuzzy logic multi-criteria algorithm, the following results are obtained:

_{3}the best combination between highest value of the global membership degree is recorded, respectively 0.025, which implies that the technical and economic parameters of this offer best fit in the company’s selection criteria and between the acquisition value of the asset.

#### 4.2. Second Scenario

_{1}, A

_{2}) needed for the current activity of a company, for which the following information is known: for asset (A

_{1}) are maintained the same data, while for the asset (A

_{2}), the information provided are related to: the asset purchase value (1000 * Euro), the asset productivity (pieces/hour), the operating cost (Euro/month) and the maintenance cost (Euro/month).

_{2}, the company obtained seven offers from the market, each of them with its technical-economic parameters. Following the implementation of the fuzzy logic algorithm the results that were obtained are:

_{i}(A

_{i}), related to A

_{1}and A

_{2}, (${\mathrm{A}}_{1},{\mathrm{A}}_{2}\in {\mathrm{m}}_{\mathrm{i}}\times {\mathrm{n}}_{\mathrm{i}})$ of each qualified offer.

_{1}), the adjustment coefficients for the asset acquisition policy remain the same as those presented under the first situation. These asset acquisition policies for (A

_{1}) and (A

_{2}), after applying the fuzzy logic algorithm lead to the following results:

_{i}(A

_{i}), (${\mathrm{A}}_{\mathrm{i}}\in {\mathrm{m}}_{\mathrm{i}}\times {\mathrm{n}}_{\mathrm{i}})$ for A

_{1}and A

_{2}is:

## 5. Conclusions

- The matrix of membership degrees $\left({\mu}_{{C}_{i}}\right)$ of the market offers to the asset selection criteria, which analyzes the extent to which the existing offers on the market correspond to the criteria for the selection of the assets. The fuzzy triangular numbers formed have their membership function and allow the appreciation of the selection criteria in linguistic terms. The higher the membership degree of an existing offer on the market to a criterion, the better the offer. The lower the membership degree of an offer to a criterion, the worse is the offer.
- The column vector of the global membership degree determines the hierarchy of existing offers on the market, according to the membership degree of the offers to all selection criteria. The higher the degree of compatibility of an offer with the selection criteria is, the more attractive the offer is for the company.
- Establishing the highest global membership degree used for identifying the best offer that suits to all criteria for the selection of assets on the market, including the situations where the company is about to acquire a chain of assets.

_{1}) the same criteria and data are maintained (six selection criteria and five offers), while for the asset (A

_{2}), the information provided are related to: the asset purchase value (1000 * Euro), the asset productivity (pieces/hour), the operating cost (Euro/month) and the maintenance cost (Euro/month): four selection criteria and seven offers. For each of these selection criteria a maximum and a minimum threshold were determined, respectively the matrix and the vector of the global membership degree. For each asset the algorithm steps were performed, so the validated offers were identified.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**The global membership degree of the offers to the selection criteria without adjustment coefficients for A

_{1}.

**Figure 4.**The global membership degree of the offers to the selection criteria with adjustment coefficients for A

_{1}.

**Figure 5.**The global membership degree of the offers to the selection criteria without adjustment coefficients for A

_{2}.

**Figure 6.**The global membership degree of the offers to the selection criteria with adjustment coefficients for A

_{2}.

**Table 1.**The technical-economic parameters of the offer O

_{4}, established as the winning offer after testing the fuzzy logic multi-criteria algorithm.

Name of Acquisition Criterion | Criterion | Unit of Measurement | Planned Indicator Value | Indicative Accepted Value | Winning Offer Features |
---|---|---|---|---|---|

Asset purchase value | ${\mathrm{C}}_{1}$ | 1000 * Euro | 20,000 | 15,000 | 16,000 |

Assets productivity | ${\mathrm{C}}_{2}$ | Pieces/hour | 25 | 15 | 22 |

Operating costs | ${\mathrm{C}}_{3}$ | Euro/month | 5000 | 3500 | 3500 |

Technical guarantee of the asset | ${\mathrm{C}}_{4}$ | Year | 3 | 2 | 3 |

Asset production capacity | ${\mathrm{C}}_{5}$ | Thousands pieces/month | 1000 | 500 | 900 |

Quality of products obtained | ${\mathrm{C}}_{6}$ | Qualitative | Very good | Average | Good |

**Table 2.**The technical-economic parameters of the offer O

_{1}, for asset A

_{2}established by applying the algorithm.

Name of Acquisition Criterion | Criterion | Unit of Measurement | Planned Indicator Value | Indicative Accepted Value | Winning Offer Features |
---|---|---|---|---|---|

Asset purchase value | ${C}_{1}$ | 1000 * Euro | 7500 | 5000 | 5400 |

Assets productivity | ${C}_{2}$ | Pieces/hour | 100 | 75 | 94 |

Operating costs | ${C}_{3}$ | Euro/month | 10,000 | 5000 | 6200 |

The maintenance cost | ${C}_{4}$ | Euro/month | 3000 | 1500 | 2600 |

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## Share and Cite

**MDPI and ACS Style**

Boloș, M.-I.; Bradea, I.-A.; Delcea, C.
A Fuzzy Logic Algorithm for Optimizing the Investment Decisions within Companies. *Symmetry* **2019**, *11*, 186.
https://doi.org/10.3390/sym11020186

**AMA Style**

Boloș M-I, Bradea I-A, Delcea C.
A Fuzzy Logic Algorithm for Optimizing the Investment Decisions within Companies. *Symmetry*. 2019; 11(2):186.
https://doi.org/10.3390/sym11020186

**Chicago/Turabian Style**

Boloș, Marcel-Ioan, Ioana-Alexandra Bradea, and Camelia Delcea.
2019. "A Fuzzy Logic Algorithm for Optimizing the Investment Decisions within Companies" *Symmetry* 11, no. 2: 186.
https://doi.org/10.3390/sym11020186