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

The aim of this study was to develop a novel intuitionistic Type-2 fuzzy inference system (IT-2 FIS) which adopts a parameterized Yager-generating function and particle swarm optimization (PSO). In IT-2 FIS, the intuitionistic Type-2 is set as a fuzzy symmetrical triangular number in which the hesitation degree adopts the Yager-generating function, and the parameters of the proposed IT-2 FIS adopting the PSO are tuned. The intuitionistic and Type-2 fuzzy sets have been proven to be the most effective for handling more uncertainty. Therefore, this study proposes an intuitionistic Type-2 set with a Yager-generating function to enhance the conventional fuzzy inference system. Moreover, PSO can improve the fuzzy inference system by searching for the optimal parameters of IT-2 FIS. In this study, linguistic variables were represented by triangular fuzzy numbers (TFS). Two numerical examples were examined: capacity-planning and medical diagnosis problems. An approaching capacity-loadings example was used to verify that the proposed IT-2 FIS could effectively estimate the results of the capacity loadings. In the medical diagnosis problem, IT-2 FIS could obtain a higher correct rate by revealing experts’ knowledge. In both examples, the proposed IT-2 FIS provided more objective estimated values than traditional fuzzy inference systems (FIS) and Type-2 FIS.


Introduction
Fuzzy inference systems are based on fuzzy IF-THEN rules that connect the fuzzy input and output variables. Therefore, a fuzzy inference system (FIS) can be used as a prediction model that inputs or outputs data with a high uncertainty. In expert systems with approximate reasoning process fields, FIS is one of the popular systems that can approach a single output (scalar) by a vector of several inputs using fuzzy logic [1]. The fuzzy inference process, which effectively maps pattern discerning or decision-making, includes four phases of FIS, which are determining membership functions, constructing reasoning systems (inference rules), and calculating aggregation and defuzzification processes, respectively [1,2].
Olvera-García [3] proposed a new evaluation model using fuzzy inferences combined with the Analytic Hierarchy Process (AHP), to provide a new air quality index. The second phase of the FIS consists of assigning a weight to each environmental parameter, using a priority analysis based on the AHP. The experimental results showed that the proposed air quality index obtained a good performance when compared with those presented in the literature. When weights were assigned This research developed a novel Type-2 fuzzy inference system (IT-2 FIS) for enhancing traditional FIS in uncertain environments. In various situations [11][12][13], the Type-2 fuzzy set has been shown to express uncertainty information effectively. The Type-2 fuzzy set can obtain a higher computational power and better description of uncertainty than traditional FIS. Moreover, traditional degrees of the fuzzy membership function have been determined to be valued between 0 and 1. Atanassov [14] proposed an intuitionistic fuzzy set (IFS) that could express various degrees of membership, non-membership, and hesitation in a fuzzy set for more clearly express uncertain environments. However, degrees of hesitation are not easy to clearly determine. This study adopts the parameterized Yager-generating function to determine degrees of hesitation in a Type-2 fuzzy set. With the help of Atanassov's intuitionistic fuzzy complement with Yager-generating functions [15], the hesitation can be determined to promote efficiency of the proposed fuzzy inference system, and this has been successfully examined in [16,17]. In this study, the parameterized Yager-generating function could tune to optimal target values based on particle swarm optimization (PSO). Therefore, the concept of IFS combined with a Type-2 fuzzy set would obtain a better description of uncertainty, and this study develops an intuitionistic Type-2 FIS with PSO.
The main contributions of this paper are as follows: (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.
The rest of this paper is organized as follows: The proposed IT-2 FIS with the novel intuitionistic Type-2 set and PSO is introduced in Section 2; in Section 3, two numerical examples of capacity-planning and medical diagnosis problems are utilized to demonstrate the performance of different fuzzy inference systems; finally, conclusions are made in Section 4.

IT-2 FIS with a Novel Intuitionistic Type-2 Fuzzy Set
This study developed an IT-2 FIS with a novel intuitionistic Type-2 set. Figure 1 and the following steps illustrate the designed system. First, the intuitionistic Type-2 fuzzy sets are determined and the parameters of the intuitionistic Type-2 fuzzy sets are continuously tuned by the PSO algorithm. Secondly, the fuzzy IF-THEN rules are structured by experts as conventional FIS, and the novel intuitionistic Type-2 fuzzy set with Yager-generating functions is used to substitute traditional triangular fuzzy numbers (TFSs). Thirdly, the intuitionistic fuzzy Type-2 set is type-reduced to obtain the interval consequent set. Fourthly, the interval consequent set is transformed to a crisp value, using the defuzzification technique, and the defuzzification return to the PSO mechanism is calculated with the fitness values for the optimal novel intuitionistic Type-2 fuzzy set, until the maximum number of iterations is reached.
. In a traditional fuzzy operator, the hesitation 163 degree has not been reasonably considered and defined, and it is usually ignored. The parametric 164 intuitionistic fuzzy complement therefore can more effectively define the value of the hesitation 165 degree based on historical or observational data or experiments. Therefore, this study adopts the

Intuitionistic Type-2 Fuzzy Sets with Yager-Generating Functions (Fuzzy Input)
Intuitionistic Type-2 fuzzy sets with Yager-generating functions were employed in all the definitions of fuzzy input in the proposed IT-2 FIS. Firstly, an intuitionistic Type-2 membership function, A IT2 , is determined, and this particular set includes a non-membership function. Hence, the intuitionistic Type-2 membership function can be expressed as µ A (x, u, v), where x ∈ X and u ∈ J x ⊆ [0, 1] are calculated as follows: , with the help of Atanassov's intuitionistic fuzzy complement with Yager-generating functions. In Yager-generating functions for each value of parameter α ∈ (0, ∞), a particular fuzzy complement can be defined, which includes non-membership and hesitation degree. µ A IT2 (x, u, v) can become based on Yager-generating functions, and the hesitation degree is 1/a . In a traditional fuzzy operator, the hesitation degree has not been reasonably considered and defined, and it is usually ignored. The parametric intuitionistic fuzzy complement therefore can more effectively define the value of the hesitation degree based on historical or observational data or experiments. Therefore, this study adopts the parametric intuitionistic fuzzy complement to define the hesitation degree for fuzzy inference system. The Footprint of Uncertainty (FOU) represents that proposed intuitionistic Type-2 fuzzy set that consists of a bounded region, and it is determined as a union of all primary membership. FOU characterizes intuitionistic Type-2 fuzzy sets: The upper membership functions (UMF) and lower membership functions (LMF), which are associated with FOU, as proposed by [18], model the uncertainties in the shape and position of the traditional fuzzy set by using two traditional membership functions. FOU represents the entire Type-2 fuzzy set, and uniform shading denotes the interval sets for the secondary membership functions.
Meanwhile, the membership function, non-membership function, and hesitation degree are determined as intuitionistic fuzzy sets. µ A IT2 (x), v A IT2 (x), and π A IT2 (x) are the upper bounds of the membership, non-membership, and hesitation membership functions, respectively. All membership functions are determined as triangular fuzzy numbers (c − d, c, c + d), where c is the mode, d is the spread of membership functions, and these membership functions can associate with the upper bound of FOU( A IT2 ),NFOU( A IT2 ), and IFOU( A IT2 ). Furthermore, the novel intuitionistic Type-2 membership function includes the membership function and hesitation degree, which are denoted as follows: where Uµ A IT2 (x) is the updated version of the upper membership function. The lower membership function µ A IT2 (x) is associated with the FOU( A IT2 ). All membership functions are determined as triangular fuzzy numbers (c − d, c, c + d), where c is the mode, d is the spread of membership functions, and these membership functions can associate with the lower bound of FOU( A IT2 ), NFOU( A IT2 ), and is the updated version of the lower membership function, which can also be denoted as follows: In this study, the intuitionistic Type-2 fuzzy sets were as shown in Figure 2.
Furthermore, the novel intuitionistic Type-2 membership function includes the membership function and hesitation degree, which are denoted as follows: where x is the updated version of the upper membership function. The lower membership x is the updated version of the lower membership function, which can also be denoted as follows: In this study, the intuitionistic Type-2 fuzzy sets were as shown in Figure 2.

Fuzzy IF-THEN Rules
General fuzzy rules are IF-THEN rules, in which the lth rule has the form "R l : IF x1 is 1 l F and x2

Fuzzy IF-THEN Rules
General fuzzy rules are IF-THEN rules, in which the lth rule has the form "R l : IF x 1 is F l 1 and x 2 is F l 2 and . . . and x q is F l q , THEN y is G l ", where x i represents the inputs, F l i 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 Symmetry 2020, 12, 562 6 of 16 form "R l : IF x 1 is F l 1 and x 2 is F l 2 and . . . and x q is F l q , THEN y is 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.

Type Reduction and Defuzzification
The output of the intuitionistic fuzzy Type-2 inference engine is type-reduced as Type-2 FIS. The type-reduction method is most commonly employed in Type-2 FIS and can determine the center of the set. The type-reduction method can be expressed as follows: . . .
where Y cos is an interval set determined by two end points, y l and y r ; ; Y i is the centroid of the intuitionistic fuzzy Type-2 interval set of 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:

Particle Swarm Optimization in the Proposed FIS
PSO was developed by Kennedy [19] and is a type of stochastic optimization technology. PSO is adopted to identify the three parameters of d, d, and α in the proposed IT-2 FIS. Li et al. [20] mentioned that PSO can model the social behavior of bird flocking and used a population-based search as a genetic algorithm. Based on exploiting a population, the PSO can probe a promising region of the search space. The PSO has been successfully applied in searching optimal parameters [21,22]. The PSO mainly search optimal parameters of intuitionistic Type-2 fuzzy sets, and the performance of PSO is accurate rate of IT-2 FIS based on upper and lower spread of membership functions and α in the proposed IT-2 FIS. Algorithm 1 describes particle swarm optimization in the IT-2 FIS. In this study, the notation for the PSO algorithm is as follows: 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; If f(x i ) < p best then 7.
p best = x i 8.
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.
Calculating new position of the particle x k id =

Numerical Examples
Two numerical examples were examined-capacity-planning and medical diagnosis problems. In the first example, the fuzzy IF-THEN rules are employed to approach capacity loadings. The fuzzy IF-THEN rules of capacity loadings refer to Pai et al. [23], who successfully adopted traditional FIS to approach the actual capacity loadings. Therefore, the example that approached capacity loadings was used to verify that the proposed IT-2 FIS could effectively estimate the results of the capacity loadings. In the second example, the medical diagnosis problem was examined-which is a standard classification problem-by using fuzzy IF-THEN rules. The fuzzy IF-THEN rules could simulate the decision of a radiologist who is an expert at determining the stage of breast cancer. The medical diagnosis problem was also used to verify that the proposed IT-2 FIS could obtain a higher accuracy by revealing expert knowledge. Two different types of problems (estimating values and classification) were examined, and the two examples were real cases from the manufacturing and medical industries, respectively. In IT-2 FIS, the parameters of PSO are shown in Table 2 for the two numerical examples.

Proposed FIS in Capacity-Planning Problems
The capacity-planning is accurately approached as an actual quantity in advance, which can bring many benefits, such as reducing the inventory, improving the quality, allowing on-time deliveries, and reducing the processing time. In the proposed FIS, two stages of approximate reasoning are performed. These data are collected by product-assembly companies in he manufacturing industry. In stage one, the proposed FIS can approach the total capacity requirements (TCR) quantity (production requirement + set up resources) based on the demand (D) and failure rate (FR). Generally, the TCR is estimated by Symmetry 2020, 12, 562 8 of 16 managerial experience in real environment. In stage two, the proposed FIS can approach the capacity loadings (CL) during the production planning period. The fuzzy inputs of D, set-up resources (SR), and capacity constraints (CC) in stage two are considered in capacity-planning systems [23]. A numerical experiment verified that the proposed FIS can effectively approach the capacity loadings. Linguistic variables are presented as Figures 3-5. This study adopted triangular fuzzy sets (TFSs) as linguistic variables, making the study differ from the work of Pai et al. [23], because the proposed FIS adopted the TFSs in this study to promote the performance. The term of "NB" is very less, "NM" is much less, "NS" is less, "O" is normal, "PS" is more, "PM" is much more, and "PB" is very much more in Figures 3-5. The bill of material (BOM) is illustrated in Figure 6. and reducing the processing time. In the proposed FIS, two stages of approximate reasoning are 246 performed. These data are collected by products assembly company in manufacturing industry. In 247 stage one, the proposed FIS can approach the total capacity requirements (TCR) quantity (production 248 requirement + set up resources) based on the demand (D) and failure rate (FR). Generally, the TCR is 249 estimated by managerial experience in real environment. In stage two, the proposed FIS can approach 250 the capacity loadings (CL) during the production planning period. The fuzzy inputs of D, set-up 251 resources (SR), and capacity constraints (CC) in stage two are considered in capacity-planning 252 systems [23]. A numerical experiment verified that the proposed FIS can effectively approach the 253 capacity loadings. Linguistic variables are presented as Fig. 3, 4, and 5. This study adopted triangular 254 fuzzy sets (TFSs) as linguistic variables, which differs from the work of Pai et al. [23], because the 255 proposed FIS adopted the TFSs in this study to promote the performance. The term of "NB" is very 256 less, "NM" is much less, "NS" is less, "O" is normal, "PS" is more, "PM" is much more, and "PB" is 257 very much more in Fig. 3, 4, and 5. The bill of material (BOM) is illustrated in Fig. 6.        The TCR production quantity is inferred in stage one. The input data are the fuzzy D and fuzzy FR of four periods. The rules of the total capacity requirements (TCR) can be referred to in [23], and Figure 7 displays the surface of the fuzzy IF-THEN rules of the total capacity requirement. Table 3 displays the inference results of TCR, which also show the defuzzification of upper membership functions (UMF) and lower membership functions (LMF) of TCR by using Type-2 FIS, and the proposed FIS. The rules of the production requirement with traditional TFSs, fuzzy Type-2 (UMF and    The TCR production quantity is inferred in stage one. The input data are the fuzzy D and fuzzy FR of four periods. The rules of the total capacity requirements (TCR) can be referred to in [23], and Figure 7 displays the surface of the fuzzy IF-THEN rules of the total capacity requirement. Table 3 displays the inference results of TCR, which also show the defuzzification of upper membership functions (UMF) and lower membership functions (LMF) of TCR by using Type-2 FIS, and the proposed FIS. The rules of the production requirement with traditional TFSs, fuzzy Type-2 (UMF and The TCR production quantity is inferred in stage one. The input data are the fuzzy D and fuzzy FR of four periods. The rules of the total capacity requirements (TCR) can be referred to in [23], and Figure 7 displays the surface of the fuzzy IF-THEN rules of the total capacity requirement. Table 3 displays the inference results of TCR, which also show the defuzzification of upper membership functions (UMF) and lower membership functions (LMF) of TCR by using Type-2 FIS, and the proposed FIS. The rules of the production requirement with traditional TFSs, fuzzy Type-2 (UMF and LMF) sets, and intuitionistic Type-2 (UMF, LMF, and α 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. of this stage is the TCR of each period, and shows the approaching values under FIS, Type-2 FIS, and 284 the proposed FIS. The UMF and LMF of the CR for the four periods are presented in Table 4, and the 285 TCR is shown in Table 4 after the exploration of BOM.

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In stage two, the CL is inferred. The rules of the CL [23] are illustrated in Figure 8. The input data 297 of this stage are the fuzzy TCR and fuzzy CC with their UMF and LMF (Type-2 fuzzy sets). The 298 outputs of stage two are the fuzzy CL of four periods. Figure 9 shows the training error of proposed 299 IT-2 FIS via using proposed PSO. The performance has convergence with excellent stability in the 300 capacity-planning problem. Table 5 to 7 indicates the defuzzified CL of the four periods by Eq. (12).

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The positive numbers in Table 5    In stage two, the CL is inferred. The rules of the CL [23] are illustrated in Figure 8. The input data of this stage are the fuzzy TCR and fuzzy CC with their UMF and LMF (Type-2 fuzzy sets). The outputs of stage two are the fuzzy CL of four periods. Figure 9 shows the training error of proposed IT-2 FIS via using proposed PSO. The performance has convergence with excellent stability in the capacity-planning problem. Tables 4-6 indicate the defuzzified CL of the four periods by Equation (12). The positive numbers in Tables 4-6 represent the overloading of the capacity, and the negative numbers are the underloading of the capacity. The proposed IT-2 FIS can provide a more objective estimated value than the traditional FIS because of the fuzzy bounds (UMF and LMF). In the simulated results, period 4 considered the underloading of the capacity for tuning material and production projects.      In this capacity-planning case, the [23] has verified traditional FIS could approach capacity loadings well. Hence, the performance evaluation is difference with results of traditional FIS (capacity loadings is −46, −162, 127, and 7, with period is 1 to 4). Table 7 shows the result of proposed IT-2 FIS is significantly similarity the FIS by T-test and verifies that the proposed IT-2 FIS could successfully estimate the capacity-planning problem, too. A comparison of FIS, Type-2 FIS, and the proposed FIS revealed a number of phenomena. First, the proposed FIS could obtain reasonable capacity loading approach values, similar to [23], in capacity-planning problems. Second, since the proposed FIS could appropriately tune the parameters of the Yager-generating functions using the PSO algorithm, it could obtain a more suitable fuzzy input than FIS or Type-2 FIS, while the case has actual target values. Third, the proposed FIS could be an effective alternative capacity-loading inference system.

Proposed FIS in a Medical Diagnosis Problem
The identification of micro-calcifications can be used in the effective detection of breast cancer, which has been verified in the medical field. The procedure of mammography, a specific type of X-ray radiograph, can digitally capture the contrasting imageries of both micro-calcifications and normal breast tissues. However, different examiners of the same digitized images may arrive at divergent diagnoses due to the varying experience and background of each examiner, as well as his/her own subjectivity. In this numerical example, this study conducted a discussion with a professional physician and investigated the variables of micro-calcifications, calcification density, and calcification abnormal shape to establish the fuzzy decision rules (Table 8). The "H", "M", and "L" respectively represent high (0.5, 0.75, 0.75), median (0.25, 0.5, 0.75), and lower (0.25, 0.25, 0.5) bounds, which are shown in Figure 10. Figure 11 displays the fuzzy decision terms, which are "3" (0, 0.2, 0.4), "4A" (0.2, 0.4, 0.6), "4B" (0.4, 0.6, 0.8), and "4C" (0.6, 0.8, 1), respectively. The fuzzy decision terms mean the breast cancer stages. The "3" stage is when the tumor size is bigger than 50 mm. The "4A", "4B", and "4C" mean different degrees of direct extension to the chest wall. Figure 12 shows the training error of proposed IT-2 FIS via using proposed PSO. The performance also has convergence with excellent stability in a breast cancer diagnosis problem. In the medical diagnosis case, correct rate is measure index. The actual results are shown in Table 9, and the accurate rate of the proposed IT-2 FIS is 100%, which is better than the traditional FIS (50%) and type-2 FIS (80%) in cases A to J. The results show that cases B, C, E, G, and I were incorrect estimations of stages for breast cancer by using traditional FIS; and cases B and G were incorrect estimation of stages for breast cancer by using Type-2 FIS. divergent diagnoses due to the varying experience and background of each examiner, as well as his/her own subjectivity. In this numerical example, this study conducted a discussion with a professional physician and investigated the variables of micro-calcifications, calcification density, and calcification abnormal shape to establish the fuzzy decision rules ( Table 9). The "H", "M", and "L" respectively represent high (0.5, 0.75, 0.75), median (0.25, 0.5, 0.75), and lower (0.25, 0.25, 0.5) bounds, which are shown in Figure 10. Figure 11 displays the fuzzy decision terms, which are "3" (0, 0.2, 0.4), "4A" (0.2, 0.4, 0.6), "4B" (0.4, 0.6, 0.8), and "4C" (0.6, 0.8, 1), respectively. The fuzzy decision terms means the breast cancer stages. The "3" stages is the tumour size is bigger than 50 millimetre. The "4A", "4B", and "4C" means different degrees of direct extension to the chest wall. Figure 12 shows the training error of proposed IT-2 FIS via using proposed PSO. The performance has also convergence with excellent stability in a breast cancer diagnosis problem. In the medical diagnosis case, correct rate is measure index. The actual results shows in Table 10, and the accurate rate of the proposed IT-2 FIS is 100%, which is better than the traditional FIS (50%) and type-2 FIS (80%) in cases A to J. The results can be observed that cases B, C, E, G, I were incorrect estimation of stages for breast cancer by using traditional FIS, and cases B and G were incorrect estimation of stages for breast cancer by using type-2 FIS.       Table 10. Comparison of FIS, Type-2 FIS, and intuitionistic Type-2 FIS in a breast cancer diagnosis problem.

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No. of iterations Error rate Figure 12. The training error of proposed IT-2 FIS in a breast cancer diagnosis problem. Table 9. Comparison of FIS, Type-2 FIS, and intuitionistic Type-2 FIS in a breast cancer diagnosis problem. Finally, the T-test was implemented to test the breast cancer diagnosis results. The test results, presented in Table 10, show significant differences among the FIS and proposed IT-2 FIS models, indicating the breast cancer diagnosis results of the proposed IT-2 FIS model are better than the traditional FIS based on accurate rate. In this example, IT-2 FIS is successfully applied in breast cancer diagnosis problems. The IT-2 FIS is capable of dealing with complex breast cancer diagnosis. Through the IT-2 FIS, managers can obtain more objective estimated values and higher accurate rates than those obtained with traditional FIS and type-2 FIS in medical diagnosis. Cell: standardize value (p-value); * Means p-value less than 0.05 is statistically significant.

Conclusions
In this study, the proposed IT-2 FIS, which adopts intuitionistic Type-2 fuzzy sets with Yager-generating functions as the fuzzy input, was successfully applied in two numerical examples. The proposed IT-2 FIS is capable of dealing with complex capacity loading and medical diagnosis problems where various uncertain variables and incomplete knowledge are involved. It is more suitable for revealing experts' knowledge and constructing fuzzy models in a human tractable form. Through the proposed IT-2 FIS, managers can obtain more flexible approach values compared to traditional FIS and Type-2 FIS in numerical examples. This study was the first to use novel intuitionistic Type-2 fuzzy sets with Yager-generating functions and PSO. However, this study did not examine the ability of the proposed IT-2 FIS to approach other types of actual problems. For instance, the time-series financial data (can refer to [24,25]) may also be examined by using proposed IT-2 FIS in future research. Another unexamined issue is the appropriateness of combining the proposed IT-2 FIS with other approaches, such as different linguistic models, to deal with uncertain data. The state-of-the-art FIS may also adopt the Yager-generating functions and PSO to enhance their fuzzy inference system. The others heuristic algorithm also could be adopted to search optimal parameters of IT-2 FIS, such as improved PSO Algorithm. All issues are worth further investigation.