# A Parameterized Intuitionistic Type-2 Fuzzy Inference System with Particle Swarm Optimization

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## Abstract

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## 1. Introduction

_{2.5}, PM

_{10}, and total suspended particles (TSP), as well as the health-risk level, as the output variable, were fuzzed by using a fuzzy inference system. This method could be used effectively in other workplaces, such as hospitals and health-care facilities. Wang et al. [9] developed the Genetic Algorithm and Rough Set Incorporated Neural Fuzzy Inference System (GARSINFIS), which is an integrated autonomous computational model for underpricing forecasting in initial public offerings. The experimental results showed a higher yield of initial returns for initial public offerings, by following the advice provided by GARSINFIS in comparison to any other benchmarking model. Hence, the GARSINFIS model was capable of offering investors highly interpretable and reliable decision support to gain the money-left-on-the-table in initial public offerings. Maciel and Ballini [10] proposed interval-valued fuzzy inference system (iFIS) modeling to predict interval-valued time series. The fuzzy c-means clustering algorithm was used in interval-valued data with adaptive distances for antecedent identification. In order to fit a linear regression model to symbolize the interval data, the center-range methodology estimated the parameters of the linear consequents. The results indicated that iFIS could obtain a better performance than traditional alternative approach methods.

- (1)
- A novel Type-2 fuzzy inference system (IT-2 FIS) for enhancing traditional FIS in uncertain environments is developed. The proposed IT-2 FIS adopts the parameterized Yager-generating function to determine the degrees of hesitation in Type-2 fuzzy set, and optimal target values based on particle swarm optimization.
- (2)
- The proposed IT-2 FIS is capable of dealing with complex capacity loading and medical diagnosis problems in which various uncertain variables and incomplete knowledge are involved. It is more suitable for revealing expert knowledge and constructing fuzzy models in a human tractable form.

## 2. IT-2 FIS with a Novel Intuitionistic Type-2 Fuzzy Set

#### 2.1. Intuitionistic Type-2 Fuzzy Sets with Yager-Generating Functions (Fuzzy Input)

_{x}⊆ [0, 1] are calculated as follows:

#### 2.2. Fuzzy IF-THEN Rules

^{l}: IF x

_{1}is ${F}_{1}^{l}$ and x

_{2}is ${F}_{2}^{l}$ and … and x

_{q}is ${F}_{q}^{l}$, THEN y is G

^{l}”, where x

_{i}represents the inputs, ${F}_{i}^{l}$ represents the antecedent sets (i = 1, …, q), y is the map of input x

_{i}, and G

^{l}is the linguistic variable. The lth rule can also take the form “R

^{l}: IF x

_{1}is $\tilde{{F}_{1}^{l}}$ and x

_{2}is $\tilde{{F}_{2}^{l}}$ and … and x

_{q}is $\tilde{{F}_{q}^{l}}$, THEN y is $\tilde{{G}^{l}}$”. The input and output sets can be replaced by intuitionistic Type-2 sets. The structure of this intuitionistic fuzzy Type-2 rule is exactly the same as that of a fuzzy Type-1 rule; the only difference is in the nature of the membership functions using the upper and lower membership functions of intuitionistic Type-2 fuzzy sets.

#### 2.3. Type Reduction and Defuzzification

_{cos}is an interval set determined by two end points, y

_{l}and y

_{r}; f

^{i}∈ = [$\underset{\_}{{f}^{M}},\overline{{f}^{M}}$]; y

^{i}∈ Y

^{i}= [${y}_{l}^{i},{y}_{r}^{i}$]; Y

^{i}is the centroid of the intuitionistic fuzzy Type-2 interval set of $\tilde{{G}^{i}}$; and, i = 1, …, M. Due to Y

_{cos}being an interval set, it can be defuzzified by using the average of y

_{l}and y

_{r}. The defuzzified output of intuitionistic fuzzy interval Type-2 is as follows:

#### 2.4. Particle Swarm Optimization in the Proposed FIS

Population_size | is initial population size; |

p_{best} | is the best movement; |

g_{best} | is the best position movement; |

v_{id} | is a modification of velocity; |

x_{id} | is the position of the ith particle; |

rand(.) | represents random variables with a uniform distribution |

C_{1} and C_{2} | are two acceleration constants that regulate the velocities to the best global and local positions; |

K | is the current generation number; |

w^{k} | is the inertia weight; |

w_{max} | is the initial weight; |

w_{min} | is the final weight; |

k_{max} | is the maximum number of generations; |

$\overline{d}$ | is the spread of membership functions which can associate with the upper bound; |

$\underset{\_}{d}$ | is the spread of membership functions which can associate with the lower bound; |

α | is tuned parameter of Yager-generating functions. |

Algorithm 1 Particle swarm optimization in an IT-2 FIS | |

1. | Initial populations (Randomly) |

2. | Iteration = 0 |

3. | Setting Population_size, x = ($\overline{d},\underset{\_}{d}$, α) |

4. | Setting C_{1}, C_{2}, w_{max}, w_{min} |

5. | While (Iteration < Maximum number of iterations) do |

6. | If f(x_{i}) < p_{best} then |

7. | p_{best} = x_{i} |

8. | end if |

9. | If p_{bes} < g_{best} then |

10. | g_{best} = p_{best} |

11. | end if |

12. | Calculating the modification of velocity and position of the ith particle |

13. | ${v}_{id}^{k}={w}^{k}{v}_{id}^{k-1}+{C}_{1}rand(.)({p}_{best}-{x}_{id}^{k-1})+{C}_{2}rand(.)({g}_{best}-{x}_{id}^{k-1})$ |

14. | Calculating inertia weight ${w}^{k}={w}_{\mathrm{max}}-\frac{{w}_{\mathrm{max}}-{w}_{\mathrm{min}}}{{k}_{\mathrm{max}}}\times k$ |

15. | Calculating new position of the particle ${x}_{id}^{k}={x}_{id}^{k-1}+{v}_{id}^{k}$ |

16. | Iteration++ |

17. | End while |

18. | Return$\overline{d},\underset{\_}{d}$, and α |

## 3. Numerical Examples

#### 3.1. Proposed FIS in Capacity-Planning Problems

_{1}) fuzzy sets are shown in Table 3. The output of this stage is the TCR of each period, and it shows the approaching values under FIS, Type-2 FIS, and the proposed FIS. The UMF and LMF of the CR for the four periods are presented in Table 3, and the TCR is shown in Table 3, after the exploration of BOM.

#### 3.2. Proposed FIS in a Medical Diagnosis Problem

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**The compatibilities of the fuzzy total capacity requirements (TCR) and the fuzzy capacity requirements (CR).

**Figure 6.**Bill of material [23].

Author(s) | Year | Method | Applied Field |
---|---|---|---|

Olvera-García et al. | 2016 | FIS + AHP | Air quality index |

Blanes-Vidal et al. | 2017 | NFIS | Air pollution exposures |

Kang et al. | 2017 | FIS | Diagnosis of feedwater heater performance degradation |

Milan et al. | 2018 | FIS | Determine the groundwater withdrawal |

Štěpnička and Mandal | 2018 | FIS with the satisfaction of Moser–Navara axioms | None |

Toseef and Khan | 2018 | FIS | Diagnosis of crop diseases in Pakistan |

Jamshidi et al. | 2018 | FIS | Estimating health risk of suspended dust |

Wang et al. | 2018 | GARSINFIS | Predictions of underpricing in initial public offerings |

Maciel and Ballini | 2019 | iFIS | Simulated interval-valued time series |

Parameters | Values |
---|---|

Population_size | 20 |

C_{1} | 1.4 |

C_{2} | 1.4 |

w_{max} | 0.9 |

w_{min} | 0.4 |

Maximum number of iterations | 500 |

Period | 1 | 2 | 3 | 4 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

UMF | LMF | α_{1} | UMF | LMF | α_{1} | UMF | LMF | α_{1} | UMF | LMF | α_{1} | ||

Fuzzy D | 64 | 50 | (75, 85, 95) | (60, 70, 80) | |||||||||

Fuzzy FR (%) | 1.5 | 0.8 | (1, 2, 3) | (1.5, 2, 2.5) | |||||||||

TCR | FIS | 67 | − | 52 | − | 92 | − | 72 | − | ||||

Type-2 FIS | 66 | 65 | − | 53 | 52 | 0.9 | 93 | 87 | − | 73 | 67 | − | |

Proposed IT-2 FIS | 69 | 67 | 0.7 | 52 | 52 | 0.9 | 78 | 78 | 0.3 | 78 | 68 | 0.5 |

Period | 1 | 2 | 3 | 4 |
---|---|---|---|---|

TCR | (477, 502, 527) | (362, 387, 412) | (658, 683, 708) | (507, 557, 607) |

CR | (500, 550, 600) | (500, 550, 600) | (500, 550, 600) | (500, 550, 600) |

CL | −46 | −162 | 127 | 7 |

Period | 1 | 2 | ||||||

UMF | LMF | UMF | LMF | |||||

TCR | (465, 490, 515) | (471, 496, 521) | (368, 393, 418) | (362, 387, 412) | ||||

CC | (500, 550, 600) | (500, 550, 600) | ||||||

CL | UMF | LMF | UMF | LMF | UMF | LMF | UMF | LMF |

−56 | −67 | −52 | −57 | −148 | −145 | −149 | −148 | |

Defuzzification | −58 | −147.5 | ||||||

Period | 3 | 4 | ||||||

UMF | LMF | UMF | LMF | |||||

TCR | (622, 647, 672) | (658, 683, 708) | (513, 563, 613) | (477, 527, 577) | ||||

CC | (500, 550, 600) | (500, 550, 600) | ||||||

CL | UMF | LMF | UMF | LMF | UMF | LMF | UMF | LMF |

98 | 95 | 138 | 125 | 20 | 5 | −24 | −21 | |

Defuzzification | 114 | −5 |

Period | 1 | 2 | ||||||

UMF | LMF | UMF | LMF | |||||

TCR | (477, 502, 527) | (489, 514, 539) | (387, 412, 437) | (387, 412, 437) | ||||

CC | (500, 550, 600) | (500, 550, 600) | ||||||

α_{2} | 0.7 | 0.5 | 0.8 | 0.8 | ||||

CL | UMF | LMF | UMF | LMF | UMF | LMF | UMF | LMF |

−42 | −50 | −49 | −50 | −159 | −165 | −148 | −176 | |

Defuzzification | −47.75 | −162 | ||||||

Period | 3 | 4 | ||||||

UMF | LMF | UMF | LMF | |||||

TCR | (543, 568, 593) | (543, 568, 593) | (483, 508, 533) | (543, 568, 593) | ||||

CC | (500, 550, 600) | (500, 550, 600) | ||||||

α_{2} | 0.5 | 0.5 | 0.4 | 0.06 | ||||

CL | UMF | LMF | UMF | LMF | UMF | LMF | UMF | LMF |

125 | 125 | 125 | 125 | 125 | 125 | 125 | 125 | |

Defuzzification | 125 | 9.75 |

Period | 1 | 2 | 3 | 4 | T-Test Estimate for Difference (p-Value) |
---|---|---|---|---|---|

FIS to Type-2 FIS | 12 | 14.5 | 13 | 12 | −11.250 (0.000)* |

FIS to proposed IT-2 FIS | 1.75 | 0 | 2 | 2.75 |

Variables | Decision | |||||
---|---|---|---|---|---|---|

Micro-calcification clusters | H | Calcification density | H | Calcification abnormal shape | H | 4A |

M | 4A | |||||

L | 3 | |||||

M | H | 4A | ||||

M | 3 | |||||

L | 4A | |||||

L | H | 3 | ||||

M | 3 | |||||

L | 3 | |||||

M | H | H | 4A | |||

M | 3 | |||||

L | 3 | |||||

M | H | 4A | ||||

M | 3 | |||||

L | 3 | |||||

L | H | 3 | ||||

M | 3 | |||||

L | 3 | |||||

L | H | H | 4A | |||

M | 3 | |||||

L | 3 | |||||

M | H | 3 | ||||

M | 3 | |||||

L | 3 | |||||

L | H | 3 | ||||

M | 3 | |||||

L | 3 |

**Table 9.**Comparison of FIS, Type-2 FIS, and intuitionistic Type-2 FIS in a breast cancer diagnosis problem.

Fuzzy Input | Actual Results | Estimated Results | |||||
---|---|---|---|---|---|---|---|

Micro-Calcification Clusters | Calcification Density | Calcification Abnormal Shape | FIS | Type-2 FIS | IT-2 FIS (α = 0.04) | ||

Case A | 0.8 | 0.9 | 0.8 | 4C | 4C | 4C | 4C |

Case B | 0.6 | 0.6 | 0.3 | 4A | 4B | 4B | 4A |

Case C | 0.3 | 0.6 | 0.3 | 3 | 4A | 3 | 3 |

Case D | 0.4 | 0.7 | 0.6 | 4B | 4B | 4B | 4B |

Case E | 0.3 | 0.5 | 0.3 | 3 | 4A | 3 | 3 |

Case F | 0.2 | 0.3 | 0.2 | 3 | 3 | 3 | 3 |

Case G | 0.4 | 0.5 | 0.4 | 4B | 4A | 4A | 4B |

Case H | 0.2 | 0.5 | 0.2 | 3 | 3 | 3 | 3 |

Case I | 0.7 | 0.6 | 0.3 | 4A | 4B | 4A | 4A |

Case J | 0.8 | 0.6 | 0.7 | 4B | 4B | 4B | 4B |

Accurate rate (%) | 50% | 80% | 100% |

Type-2 FIS | IT-2 FIS (α = 0.04) | |
---|---|---|

FIS | 0.3 (0.08) | 0.5 (0.015) * |

Type-2 FIS | 0.8 (0.167) |

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**MDPI and ACS Style**

Yu, C.-M.; Lin, K.-P.; Liu, G.-S.; Chang, C.-H.
A Parameterized Intuitionistic Type-2 Fuzzy Inference System with Particle Swarm Optimization. *Symmetry* **2020**, *12*, 562.
https://doi.org/10.3390/sym12040562

**AMA Style**

Yu C-M, Lin K-P, Liu G-S, Chang C-H.
A Parameterized Intuitionistic Type-2 Fuzzy Inference System with Particle Swarm Optimization. *Symmetry*. 2020; 12(4):562.
https://doi.org/10.3390/sym12040562

**Chicago/Turabian Style**

Yu, Chun-Min, Kuo-Ping Lin, Gia-Shie Liu, and Chia-Hao Chang.
2020. "A Parameterized Intuitionistic Type-2 Fuzzy Inference System with Particle Swarm Optimization" *Symmetry* 12, no. 4: 562.
https://doi.org/10.3390/sym12040562