# Retail System Scenario Modeling Using Fuzzy Cognitive Maps

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

## 3. Methods

#### 3.1. FCM as a Modeling Approach

#### 3.2. System Indicators

#### 3.3. Model Creation

- –
- “Employee loyalty”—“Staff turnover”
- –
- “Employee loyalty”—“Profit”
- –
- “Employee loyalty”—“Customer service level”

- –
- “Customer service level”—“Profit”
- –
- “Customer service level”—“Customer loyalty level”

- –
- “Staff turnover”—“Working conditions”
- –
- “Staff turnover”—“Cross department support”
- –
- “Staff turnover”—“Openness of communication with employees”
- –
- “Staff turnover”—“Profit”

- Publications on the company profit topic [39]:
- –
- “Profit”—“Bank loan”
- –
- “Profit”—“Fixed assets”
- –
- “Profit”—“Working capital”
- –
- “Profit”—“Stock prices”

#### 3.4. Software Implementation

## 4. Results

#### 4.1. Fuzzy Logic Operator Selection

#### 4.2. Structural Analysis of the FCM

- “Customer service level” (K31), “Company reputation” (K33), “Product quality” (K30), and “Technical level of equipment” (K1) form a cluster of customer security in the market.
- “Working capital” (K26), “Profit” (K19), “Sales revenue ” (K18), “Margin on goods” (K35), “Amount of taxes paid” (K17), “Total costs” (K20), and “Advertising costs” (K23) form a financial cluster.
- “IT infrastructure” (K4), “Integration of systems with suppliers ” (K44) form an IT cluster.
- “Customer income level” (K29), “Price segment of goods” (K36) are combined into a customer cost sensitivity cluster.
- “Production standards” (K2), “Labor productivity” (K7) constitute a production cluster.

#### 4.3. Scenario Modeling with FCM

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Fuzzy cognitive map of the retail system. Connections are only shown for absolute influences equal or greater than 0.4. Green arrows represent positive influences, while red arrows show negative influences. Thicker arrows imply stronger concept influences. Isolated concepts have absolute influences below 0.4.

**Figure 2.**Consonance of the overall impact of concepts using the operator combination T-norm Algebraic product and S-norm Maximum and relevancy threshold of 0.85.

**Figure 3.**Consonance of the overall influence of concepts when using the operator combination T-norm Minimum and S-norm Maximum and relevancy threshold of 0.85.

**Figure 4.**Number of unrelated concepts for the retail FCM when using different composition operators and relevancy thresholds.

**Figure 5.**Mutual influence of system elements when using the composition T-norm Algebraic product and S-norm Maximum, and relevancy threshold of 0.7.

**Table 1.**Commonly used T-norms and S-norms, where p and q are the state of the concepts j and i, respectively.

T-Norm | S-Norm |
---|---|

Minimum ${T}_{M}(p,q)=\mathrm{min}(p,q)$ | Maximum ${S}_{M}(p,q)=\mathrm{max}(p,q)$ |

Algebraic product ${T}_{P}(p,q)=pq$ | Algebraic sum ${S}_{P}(p,q)=p+q-pq$ |

Hamacher product ${T}_{H}(p,q)=\left\{\begin{array}{c}0,\phantom{\rule{3.33333pt}{0ex}}\mathrm{if}\phantom{\rule{3.33333pt}{0ex}}p=q=0\\ \frac{pq}{p+q-pq},\phantom{\rule{3.33333pt}{0ex}}\mathrm{otherwise}\end{array}\right.$ | Hamacher sum ${S}_{H}(p,q)=\left\{\begin{array}{c}0,\phantom{\rule{3.33333pt}{0ex}}\mathrm{if}\phantom{\rule{3.33333pt}{0ex}}p=q=0\\ \frac{p+q-2pq}{1-pq},\phantom{\rule{3.33333pt}{0ex}}\mathrm{otherwise}\end{array}\right.$ |

Einstein product ${T}_{E}(p,q)=\left\{\begin{array}{c}0,\phantom{\rule{3.33333pt}{0ex}}\mathrm{if}\phantom{\rule{3.33333pt}{0ex}}p=q=0\\ \frac{pq}{2-\left(p+q-pq\right)},\phantom{\rule{3.33333pt}{0ex}}\mathrm{otherwise}\end{array}\right.$ | Einstein sum ${S}_{E}(p,q)=\left\{\begin{array}{c}0,\phantom{\rule{3.33333pt}{0ex}}\mathrm{if}\phantom{\rule{3.33333pt}{0ex}}p\xb7q=-1\\ \frac{pq}{1+pq},\phantom{\rule{3.33333pt}{0ex}}\mathrm{otherwise}\end{array}\right.$ |

Drastic product ${T}_{D}(p,q)=\left\{\begin{array}{c}max(p,q),\phantom{\rule{3.33333pt}{0ex}}\mathrm{if}\phantom{\rule{3.33333pt}{0ex}}min(p,q)=0\\ 1,\phantom{\rule{3.33333pt}{0ex}}\mathrm{otherwise}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\end{array}\right.$ | Drastic sum ${S}_{D}(p,q)=\left\{\begin{array}{c}min(p,q),\phantom{\rule{3.33333pt}{0ex}}\mathrm{if}\phantom{\rule{3.33333pt}{0ex}}max(p,q)=1\\ 0,\phantom{\rule{3.33333pt}{0ex}}\mathrm{otherwise}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\end{array}\right.$ |

Nilpotent minimum ${T}_{N}(p,q)=\left\{\begin{array}{c}min(p,q),\mathrm{if}\phantom{\rule{3.33333pt}{0ex}}p+q\ge 1\\ 0,\mathrm{if}\phantom{\rule{3.33333pt}{0ex}}p+q<1\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\end{array}\right.$ | Nilpotent sum ${S}_{N}(p,q)=\left\{\begin{array}{c}max(p,q),\mathrm{if}\phantom{\rule{3.33333pt}{0ex}}p+q<1\\ 0,\phantom{\rule{3.33333pt}{0ex}}\mathrm{if}\phantom{\rule{3.33333pt}{0ex}}p+q\ge 1\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\end{array}\right.$ |

Lukasiewicz max ${T}_{L}(p,q)=max\left(p+q-1,0\right)$ | Lukasiewicz min ${S}_{L}(p,q)=min\left(p+q,1\right)$ |

Indicator | Formula |
---|---|

Concept’s consonance | ${c}_{ij}=\frac{\left|{v}_{ij}+{\overline{v}}_{ij}\right|}{\left|{v}_{ij}\right|+\left|{\overline{v}}_{ij}\right|}$ |

Concept’s dissonance | ${d}_{ij}=1-{c}_{ij}$ |

Concept’s impact on the system | ${p}_{ij}=sign({v}_{ij}+{\overline{v}}_{ij})max\left(\right|{v}_{ij},{\overline{v}}_{ij}\left|\right),$ ${v}_{ij}\ne -{\overline{v}}_{ij}$ |

Consonance of the i-th concept influence on the system | ${\overrightarrow{C}}_{i}=\frac{1}{n}{\displaystyle \sum _{j=1}^{n}}\phantom{\rule{0.166667em}{0ex}}{c}_{ij}$ |

Consonance of the system’s influence on the j-th concept | ${\overleftarrow{C}}_{j}=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}}\phantom{\rule{0.166667em}{0ex}}{c}_{ij}$ |

Dissonance of the i-th concept influence on the system | ${\overrightarrow{D}}_{i}=\frac{1}{n}{\displaystyle \sum _{j=1}^{n}}\phantom{\rule{0.166667em}{0ex}}{d}_{ij}$ |

Dissonance of the system’s influence on the j-th concept | ${\overleftarrow{D}}_{j}=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}}\phantom{\rule{0.166667em}{0ex}}{d}_{ij}$ |

Impact of the i-th concept on the system | ${\overrightarrow{P}}_{i}=\frac{1}{n}{\displaystyle \sum _{j=1}^{n}}\phantom{\rule{0.166667em}{0ex}}{p}_{ij}$ |

Impact of the system on the j-th concept | ${\overleftarrow{P}}_{j}=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}}\phantom{\rule{0.166667em}{0ex}}{p}_{ij}$ |

**Table 3.**Concepts and subsystems defined for the retail FCM. The K* column denotes concept identifiers.

Subsystems | Concepts | K* |
---|---|---|

Technology | Technical level of equipment | K1 |

Production standards | K2 | |

Speed of adoption of innovative technology | K3 | |

IT infrastructure | K4 | |

Employees | Number of trained staff | K5 |

Lost working time | K6 | |

Labor productivity | K7 | |

Working conditions | K8 | |

Employee loyalty | K9 | |

Staff turnover | K10 | |

Cross department support | K11 | |

Openness of communication with employees | K12 | |

Finance | Market competition level | K13 |

Interest rate on loans | K14 | |

Accounts payable | K15 | |

Bank loan | K16 | |

Amount of taxes paid | K17 | |

Sales revenue | K18 | |

Profit | K19 | |

Total costs | K20 | |

Fixed assets | K21 | |

Rent | K22 | |

Advertising costs | K23 | |

Currency exchange rate | K24 | |

Market share | K25 | |

Working capital | K26 | |

Stock prices | K27 | |

Customers | Customer demand | K28 |

Customer income level | K29 | |

Product quality | K30 | |

Customer service level | K31 | |

Customer loyalty level | K32 | |

Company reputation | K33 | |

Assortment of goods | K34 | |

Margin on goods | K35 | |

Price segment of goods | K36 | |

Share of the internal branded goods | K37 | |

External factors | Political stability | K38 |

Inflation expectations | K39 | |

Suppliers | Supplier’ purchase price | K40 |

Supplier’ purchase terms | K41 | |

Effectiveness of supplier selection | K42 | |

Supplier’ technical readiness | K43 | |

Integration of systems with suppliers | K44 | |

Investments | Domestic investments | K45 |

Capital investments | K46 | |

Foreign investment | K47 |

**Table 4.**Changes in the system concepts over five steps for the scenario of increasing “Product Quality” (K30).

Concept | Step 0 | Step 1 | Step 2 | Step 3 | Step 4 | Step 5 |
---|---|---|---|---|---|---|

Employee loyalty | 0 | 0.04 | 0.07 | 0.12 | 0.22 | 0.38 |

Staff turnover | 0 | 0 | −0.04 | −0.06 | −0.11 | −0.2 |

Customer demand | 0 | 0.04 | 0.04 | 0.04 | 0.05 | 0.07 |

Product quality | 0.1 | 0.1 | 0.12 | 0.14 | 0.18 | 0.25 |

Customer service level | 0 | 0 | 0.01 | 0.02 | 0.02 | 0.06 |

Customer loyalty level | 0 | 0.07 | 0.07 | 0.09 | 0.16 | 0.27 |

Company reputation | 0 | 0.07 | 0.09 | 0.18 | 0.3 | 0.47 |

Assortment of goods | 0 | 0 | 0 | 0.01 | 0.01 | 0.01 |

Margin on goods | 0 | 0 | 0 | 0 | 0 | 0 |

Profit | 0 | 0 | 0 | 0.15 | 0.08 | 0.16 |

**Table 5.**Changes in the system concepts over five steps for the scenario of increasing the concepts of “Product quality” (K30) and “Employee loyalty level” (K9).

Concept | Step 0 | Step 1 | Step 2 | Step 3 | Step 4 | Step 5 |
---|---|---|---|---|---|---|

Employee loyalty | 0.1 | 0.14 | 0.24 | 0.36 | 0.59 | 0.94 |

Staff turnover | 0 | −0.05 | −0.1 | −0.17 | −0.29 | −0.48 |

Customer demand | 0 | 0.04 | 0 | 0.04 | 0.05 | 0.08 |

Product quality | 0.1 | 0.1 | 0.14 | 0.2 | 0.28 | 0.43 |

Customer service level | 0 | 0.03 | 0.05 | 0.1 | 0.18 | 0.29 |

Customer loyalty level | 0 | 0.07 | 0.11 | 0.18 | 0.35 | 0.6 |

Company reputation | 0 | 0.12 | 0.16 | 0.35 | 0.61 | 0.99 |

Assortment of goods | 0 | 0 | 0 | 0.01 | 0.01 | 0.03 |

Margin on goods | 0 | 0.05 | 0.07 | 0.09 | 0.14 | 0.2 |

Profit | 0 | 0.01 | 0.02 | 0.21 | 0.18 | 0.35 |

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Petukhova, A.; Fachada, N. Retail System Scenario Modeling Using Fuzzy Cognitive Maps. *Information* **2022**, *13*, 251.
https://doi.org/10.3390/info13050251

**AMA Style**

Petukhova A, Fachada N. Retail System Scenario Modeling Using Fuzzy Cognitive Maps. *Information*. 2022; 13(5):251.
https://doi.org/10.3390/info13050251

**Chicago/Turabian Style**

Petukhova, Alina, and Nuno Fachada. 2022. "Retail System Scenario Modeling Using Fuzzy Cognitive Maps" *Information* 13, no. 5: 251.
https://doi.org/10.3390/info13050251