# Fuzzy Model Identification Using Monolithic and Structured Approaches in Decision Problems with Partially Incomplete Data

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Fuzzy Set Theory

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

#### 2.2. The COMET Method

#### 2.3. Correlation Coefficients

#### 2.3.1. The Sample Pearson Correlation Coefficient

#### 2.3.2. Weighted Spearman’s Rank Correlation Coefficient

#### 2.3.3. Rank Similarity Coefficient

## 3. Empirical Study Case

#### 3.1. Material

#### 3.2. Methods

## 4. Results and Discussion

#### 4.1. Comparison of the Monolithic and Structured Approaches

#### 4.2. Significance Analysis of Criteria

#### 4.3. Incomplete Data

## 5. Conclusions

- research of using other number generalizations instead of interval numbers to solve problems with partially incomplete data,
- more extensive research on the accuracy of monolithic and structured approaches using computer simulations, and
- developing a new method to the identification of criteria significance.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MCDA | Multicriteria Decision Analysis |

MCDM | Multicriteria Decision-Making |

COMET | Characteristic Object METhod |

MEJ | Matrix of Expert Judgment |

CO | Characteristic Object |

TFN | Triangular Fuzzy Number |

TOPSIS | Technique for Order of Preference by Similarity to Ideal Solution |

PROMETHEE | Preference Ranking Organization METHod for Enrichment of Evaluation |

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**Figure 3.**Triangular fuzzy numbers (TFNs) generated for the criteria ${C}_{1}$–${C}_{9}$ and the preferences ${P}_{1}$–${P}_{3}$, where colors mean the following, blue-low, orange-medium, and green-high.

**Figure 5.**Matrix of Expert Judgment (MEJ) matrices for performance (

**left**), battery (

**center**), and engine (

**right**) models, where green 1.0, blue 0.5, and red 0.0 points.

**Figure 7.**Visualization of Pearson’s correlation coefficients between inputs values (${C}_{1}$–${C}_{9}$, ${P}_{1}$–${P}_{3}$) and the final preferences (${P}_{str}$, ${P}_{mon}$).

**Figure 8.**Visualization of changes taking place in the initial ranking when eliminating three criterion from the set of criteria.

**Figure 9.**Visualization of changes taking place in the initial ranking when eliminating one criterion from the set of criteria.

**Figure 10.**Visualization of changes taking place in the initial ranking when eliminating two criterion from the set of criteria.

**Figure 11.**Visualization of preferences for alternatives ${A}_{1}$–${A}_{3}$ expressed as intervals, where (

**a**) monolithic approach and (

**b**) structured approach.

Group of Criteria | ${\mathit{C}}_{\mathit{i}}$ | Criterion Name | Units | Direction |
---|---|---|---|---|

${P}_{1}$ Performance | ${C}_{1}$ | Carrying capacity | [kg] | max |

${C}_{2}$ | Max velocity | [km/h] | max | |

${C}_{3}$ | Travel range | [km] | max | |

${P}_{2}$ Engine | ${C}_{4}$ | Engine power | [kW] | max |

${C}_{5}$ | Engine torque | [Nm] | max | |

${P}_{3}$ Battery | ${C}_{6}$ | Battery charging time 100% | [h] | min |

${C}_{7}$ | Battery charging time 80% | [min] | min | |

${C}_{8}$ | Battery capacity | [kWh] | max | |

Price | ${C}_{9}$ | Price | [thous. USD] | min |

${\mathit{A}}_{\mathit{i}}$ | Name | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ${\mathit{C}}_{5}$ | ${\mathit{C}}_{6}$ | ${\mathit{C}}_{7}$ | ${\mathit{C}}_{8}$ | ${\mathit{C}}_{9}$ |
---|---|---|---|---|---|---|---|---|---|---|

${A}_{1}$ | EVI MD | 3000 | 96 | 145 | 200 | 610 | 10.0 | 120 | 99.0 | 120.0 |

${A}_{2}$ | EVI Walk-In Van | 2000 | 100 | 145 | 200 | 610 | 10.0 | 120 | 99.0 | 90.0 |

${A}_{3}$ | e-NV200+ | 705 | 120 | 170 | 80 | 270 | 4.0 | 30 | 24.0 | 25.0 |

${A}_{4}$ | e-Wolf Omega 0.7 | 613 | 140 | 180 | 140 | 400 | 8.0 | 40 | 24.2 | 50.0 |

${A}_{5}$ | Minicab-MiEV Truck | 350 | 100 | 110 | 30 | 196 | 4.5 | 15 | 10.5 | 12.9 |

${A}_{6}$ | Mitsubishi Minicab-MiEV (10.5 kWh) | 350 | 100 | 100 | 30 | 196 | 4.5 | 15 | 10.5 | 15.5 |

${A}_{7}$ | Mitsubishi Minicab-MiEV (16 kWh) | 350 | 100 | 150 | 30 | 196 | 7.0 | 35 | 16.0 | 18.7 |

${A}_{8}$ | Partner Panel Van | 635 | 110 | 170 | 49 | 200 | 8.0 | 35 | 22.5 | 31.5 |

${A}_{9}$ | Phoenix Motorcars SUV | 340 | 150 | 160 | 110 | 500 | 6.0 | 10 | 35.0 | 45.0 |

${A}_{10}$ | Piaggio Porter electric-power | 750 | 57 | 110 | 10 | 80 | 8.0 | 120 | 35.0 | 24.4 |

${\mathit{C}}_{\mathit{i}}$ | Name | Min | Mean | Max |
---|---|---|---|---|

${C}_{1}$ | Carrying capacity | 340 | 909.3 | 3000 |

${C}_{2}$ | Max velocity | 57 | 107.3 | 150 |

${C}_{3}$ | Travel range | 100 | 144 | 180 |

${C}_{4}$ | Engine power | 10 | 87.9 | 200 |

${C}_{5}$ | Engine torque | 80 | 325.8 | 610 |

${C}_{6}$ | Battery charging time 100% | 4 | 7 | 10 |

${C}_{7}$ | Battery charging time 80% | 10 | 54 | 120 |

${C}_{8}$ | Battery capacity | 10.5 | 37.57 | 99 |

${C}_{9}$ | Price | 12.9 | 43.3 | 120 |

${\mathit{A}}_{\mathit{i}}$ | ${\mathit{P}}_{1}$ | ${\mathit{P}}_{2}$ | ${\mathit{P}}_{3}$ | ${\mathit{P}}_{\mathit{struct}}$ |
---|---|---|---|---|

${A}_{1}$ | 0.7851 | 0.3684 | 1.0000 | 0.5732 |

${A}_{2}$ | 0.6136 | 0.3684 | 1.0000 | 0.5972 |

${A}_{3}$ | 0.5950 | 0.7151 | 0.3492 | 0.7718 |

${A}_{4}$ | 0.7187 | 0.3927 | 0.6850 | 0.6996 |

${A}_{5}$ | 0.1378 | 0.6439 | 0.1715 | 0.4911 |

${A}_{6}$ | 0.0917 | 0.6439 | 0.1715 | 0.4619 |

${A}_{7}$ | 0.3298 | 0.4342 | 0.1715 | 0.4595 |

${A}_{8}$ | 0.5354 | 0.4062 | 0.2110 | 0.5138 |

${A}_{9}$ | 0.6491 | 0.6901 | 0.6709 | 0.8374 |

${A}_{10}$ | 0.0824 | 0.1496 | 0.0000 | 0.1512 |

**Table 5.**Preference values and rankings for structured and monolithic approaches, where $Ref$ means a place in the reference ranking from [41]; ${P}_{str}$ and ${P}_{mon}$ are preference values for the structured and monolithic approaches, respectively; and r(·) means ranking.

Alt | Ref | ${\mathit{P}}_{\mathit{str}}$ | $\mathit{r}\left({\mathit{P}}_{\mathit{str}}\right)$ | ${\mathit{P}}_{\mathit{mon}}$ | $\mathit{r}\left({\mathit{P}}_{\mathit{mon}}\right)$ |
---|---|---|---|---|---|

${A}_{1}$ | 5 | 0.5732 | 5 | 0.5974 | 4 |

${A}_{2}$ | 4 | 0.5972 | 4 | 0.5913 | 5 |

${A}_{3}$ | 2 | 0.7718 | 2 | 0.6643 | 2 |

${A}_{4}$ | 3 | 0.6996 | 3 | 0.6410 | 3 |

${A}_{5}$ | 7 | 0.4911 | 7 | 0.4019 | 7 |

${A}_{6}$ | 9 | 0.4619 | 8 | 0.3774 | 8 |

${A}_{7}$ | 8 | 0.4595 | 9 | 0.3647 | 9 |

${A}_{8}$ | 6 | 0.5138 | 6 | 0.4321 | 6 |

${A}_{9}$ | 1 | 0.8374 | 1 | 0.7379 | 1 |

${A}_{10}$ | 10 | 0.1512 | 10 | 0.0835 | 10 |

**Table 6.**Comparison of the values of ${r}_{w}$ and $WS$ coefficients of the reference ranking with the ranking achieved by using a structural and monolithic approach.

Coefficient | Structured Approach | Monolithic Aproach |
---|---|---|

${r}_{w}$ | 0.9945 | 0.9802 |

$WS$ | 0.9992 | 0.9825 |

Excluding | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{2}$ | ${\mathit{r}}_{\mathit{w}}$ | $\mathit{WS}$ | Distance |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

none | 4 | 5 | 2 | 3 | 7 | 8 | 9 | 6 | 1 | 10 | 1.0000 | 1.0000 | 0.0000 |

${C}_{1}$ | 5 | 4 | 2 | 3 | 7 | 9 | 8 | 6 | 1 | 10 | 0.9802 | 0.9825 | 0.2048 |

${C}_{2}$ | 3 | 4 | 2 | 5 | 7 | 8 | 9 | 6 | 1 | 10 | 0.9537 | 0.9476 | 0.1083 |

${C}_{3}$ | 3 | 4 | 2 | 5 | 6 | 7 | 9 | 8 | 1 | 10 | 0.9273 | 0.9395 | 0.1716 |

${C}_{4}$ | 4 | 5 | 2 | 3 | 7 | 8 | 9 | 6 | 1 | 10 | 1.0000 | 1.0000 | 0.1732 |

${C}_{5}$ | 4 | 5 | 2 | 3 | 7 | 8 | 9 | 6 | 1 | 10 | 1.0000 | 1.0000 | 0.1600 |

${C}_{6}$ | 2 | 3 | 5 | 4 | 8 | 9 | 7 | 6 | 1 | 10 | 0.8314 | 0.8527 | 0.2140 |

${C}_{7}$ | 1 | 2 | 4 | 5 | 7 | 9 | 8 | 6 | 3 | 10 | 0.7300 | 0.7399 | 0.2284 |

${C}_{8}$ | 4 | 5 | 2 | 3 | 7 | 8 | 9 | 6 | 1 | 10 | 1.0000 | 1.0000 | 0.2093 |

${C}_{9}$ | 2 | 3 | 5 | 4 | 7 | 8 | 9 | 6 | 1 | 10 | 0.8512 | 0.8551 | 0.2134 |

${C}_{4}$, ${C}_{5}$ | 8 | 9 | 1 | 3 | 5 | 7 | 6 | 4 | 2 | 10 | 0.7333 | 0.8364 | 0.3659 |

${C}_{4}$, ${C}_{8}$ | 8 | 9 | 1 | 3 | 5 | 6 | 7 | 4 | 2 | 10 | 0.7410 | 0.8361 | 0.3958 |

${C}_{5}$, ${C}_{8}$ | 8 | 9 | 2 | 3 | 5 | 6 | 7 | 4 | 1 | 10 | 0.7620 | 0.9229 | 0.4139 |

${C}_{4}$, ${C}_{5}$, ${C}_{8}$ | 8 | 9 | 1 | 3 | 5 | 7 | 6 | 4 | 2 | 10 | 0.7333 | 0.8364 | 0.6527 |

${\mathit{A}}_{\mathit{i}}$ | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ${\mathit{C}}_{5}$ | ${\mathit{C}}_{6}$ | ${\mathit{C}}_{7}$ | ${\mathit{C}}_{8}$ | ${\mathit{C}}_{9}$ | ${\mathit{P}}_{\mathit{monolith}}$ | ${\mathit{P}}_{\mathit{structured}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

${A}_{1}$ | 695 | 110 | 170 | 49 | 200 | 7.5 | 30 | 22.5 | [12.9, 120.0] | [0.3064, 0.4971] | [0.2460, 0.6037] |

${A}_{2}$ | 650 | 130 | 170 | 44 | 226 | 8 | [10, 120] | 22 | 22.0 | [0.3590, 0.5396] | [0.0000, 0.6536] |

${A}_{3}$ | 2500 | 80 | 150 | 90 | [80, 610] | 7 | 120 | 40 | 81.0 | [0.2863, 0.4799] | [0.2881, 0.4500] |

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Shekhovtsov, A.; Kołodziejczyk, J.; Sałabun, W.
Fuzzy Model Identification Using Monolithic and Structured Approaches in Decision Problems with Partially Incomplete Data. *Symmetry* **2020**, *12*, 1541.
https://doi.org/10.3390/sym12091541

**AMA Style**

Shekhovtsov A, Kołodziejczyk J, Sałabun W.
Fuzzy Model Identification Using Monolithic and Structured Approaches in Decision Problems with Partially Incomplete Data. *Symmetry*. 2020; 12(9):1541.
https://doi.org/10.3390/sym12091541

**Chicago/Turabian Style**

Shekhovtsov, Andrii, Joanna Kołodziejczyk, and Wojciech Sałabun.
2020. "Fuzzy Model Identification Using Monolithic and Structured Approaches in Decision Problems with Partially Incomplete Data" *Symmetry* 12, no. 9: 1541.
https://doi.org/10.3390/sym12091541