Linguistic value TFN 1. Average (A) (420.5, 537.7, 654.9) 2. Low 1 (L1) (360.5, 471.1, 581.6) 3. Very low (VL) (214.5, 344, 473.6) 4. Low 2 (L2) (301.9, 406.6, 511.3) 5. Very high (VH) (599.2, 771, 982.8)

Fuzzy Logic and Fuzzy Expert System-Based Material Synthesis Methods

DOI: http://dx.doi.org/10.5772/intechopen.84493

A problem of computational synthesis of Ti-Ni-Pt alloy with predefined characteristics is considered. A big data fragment describing dependence alloy composi-

The following fuzzy IF-THEN rules were obtained by using FCM clustering of

y1 (martensitic start temperature, K)

30 50 20 539 544 20 50 30 833 867 15 50 35 953 1023

10 50 40 1173 1123

No. Linguistic value TFN 1. Very low (VL) (5, 5, 13.75) 2. Low (L) (5, 13.75, 22.5) (2) 3. Average (A) (13.75, 22.5, 31.25) (3) 4. High (H) (22.5, 31.25, 40) (4) 5. Very high (VH) (31.25, 40, 40) (5)

y2 (austenitic start temperature, K)

3.2 Synthesis of Ti-Ni-Pt alloys with given characteristics

tion and the corresponding characteristics is shown in Table 11.

If x1 is VL and x3 is VH THEN y1 is VH and y2 is VH. If x1 is H2 and x3 is L1 THEN y1 is VL and y2 is VL. If x1 is A and x3 is L3 THEN y1 is L2 and y2 is L2. If x1 is L and x3 is H THEN y1 is H and y2 is H. If x1 is H1 and x3 is L2 THEN y1 is L1 and y2 is L.

Composition Transformation temperatures

x3 (Pt, %)

…

Transformation temperatures of Ti-Ni-Pt alloy [23].

the considered big data:

x1 (Ni, %)

Table 11.

Table 12.

83

Codebook for input 1 (x1).

x2 (Ti, %)

Linguistic terms for output 4 (Af).

Table 10.

#### Table 7.

Linguistic terms for output 1 (Mf).


#### Table 8.

Linguistic terms for output 2 (Ms).


#### Table 9.

Linguistic terms for output 3 (As).

The constructed fuzzy model will be used to determine an input vector A<sup>0</sup> ¼ A<sup>0</sup> <sup>1</sup>; …; A<sup>0</sup> n that induces the corresponding output vector <sup>B</sup><sup>0</sup> <sup>¼</sup> <sup>B</sup><sup>0</sup> <sup>1</sup>; …; B<sup>0</sup> m maximally close to the ideal solution <sup>B</sup><sup>∗</sup> <sup>¼</sup> <sup>B</sup><sup>∗</sup> <sup>1</sup> ; B<sup>∗</sup> <sup>2</sup>; B<sup>∗</sup> 3 .

We have found that the fuzzy optimal output vector B<sup>0</sup> induced by the fuzzy input vector A<sup>0</sup> ¼ A<sup>0</sup> <sup>1</sup>; A<sup>0</sup> <sup>2</sup>; A<sup>0</sup> 3 <sup>¼</sup> ð Þ <sup>19</sup>:5; <sup>50</sup>:5; <sup>30</sup> is <sup>B</sup><sup>0</sup> <sup>¼</sup> <sup>B</sup><sup>0</sup> <sup>1</sup>; B<sup>0</sup> <sup>2</sup>; B<sup>0</sup> <sup>3</sup>; B<sup>0</sup> 4 <sup>¼</sup> ð Þ ð Þ 347:78 ;ð Þ 364:86 ;ð Þ 382:17 ;ð Þ 375:22 . It is the closest vector to the considered ideal fuzzy vector <sup>B</sup><sup>∗</sup> <sup>¼</sup> ð Þ ð Þ <sup>302</sup> ;ð Þ <sup>323</sup> ;ð Þ <sup>347</sup> ;ð Þ <sup>313</sup> . The distance is <sup>D</sup> (B,B\*) <sup>=</sup> 94. Thus,

Fuzzy Logic and Fuzzy Expert System-Based Material Synthesis Methods DOI: http://dx.doi.org/10.5772/intechopen.84493


Table 10.

Linguistic terms for output 4 (Af).

the computational synthesis based on the fuzzy model uncovers the following alloy composition: Ni is about 19%, Ti is about 51%, and Pd is about 30% with the characteristics Mf = 347.78, about Ms = 364.86, about Af = 382.17, and As = 375.22.

### 3.2 Synthesis of Ti-Ni-Pt alloys with given characteristics

A problem of computational synthesis of Ti-Ni-Pt alloy with predefined characteristics is considered. A big data fragment describing dependence alloy composition and the corresponding characteristics is shown in Table 11.

The following fuzzy IF-THEN rules were obtained by using FCM clustering of the considered big data:

If x1 is VL and x3 is VH THEN y1 is VH and y2 is VH.

If x1 is H2 and x3 is L1 THEN y1 is VL and y2 is VL.

If x1 is A and x3 is L3 THEN y1 is L2 and y2 is L2.

If x1 is L and x3 is H THEN y1 is H and y2 is H.

If x1 is H1 and x3 is L2 THEN y1 is L1 and y2 is L.


Table 11.

Transformation temperatures of Ti-Ni-Pt alloy [23].


Table 12. Codebook for input 1 (x1).

The constructed fuzzy model will be used to determine an input vector

No. Linguistic value TFN 1. Average (A) (414.3, 527.4, 640.5) 2. Low 1 (L1) (374.6, 466.5, 558.4) 3. Very low (VL) (246.3, 354.8, 463.3) 4. Low 2 (L2) (319.1, 409, 498.9) 5. Very high (VH) (536.5, 730.6, 924.7)

No. Linguistic value TFN 1. Average 1 (A1) (21.28, 30.03, 38.78) 2. Average 2 (A2) (15.9, 24.9, 33.9) 3. Low 1 (L1) (0, 10.58, 28.06) 4. Low 2 (L2) (9.962, 19.04, 28.13) 5. Very high (VH) (28.8, 43.21, 57.62)

No. Linguistic value TFN 1. Average (A) (394.8, 502.1, 609.5) 2. Low 1 (L1) (359.2, 451.3, 543.5) 3. Very low (VL) (199.3, 322.3, 445.2) 4. Low 2 (L2) (294.4, 386.8, 479.2) 5. Very high (VH) (475.4, 674.5, 873.5)

No. Linguistic value TFN 1. Average (A) (417.4, 523.8, 630.2) 2. Low 1 (L1) (369.6, 463.1, 556.6) 3. Very low (VL) (221.4, 338.8, 456.2) 4. Low 2 (L2) (306.8, 400.4, 494) 5. Very high (VH) (532.2, 717.8, 903.5)

We have found that the fuzzy optimal output vector B<sup>0</sup> induced by the fuzzy

ð Þ ð Þ 347:78 ;ð Þ 364:86 ;ð Þ 382:17 ;ð Þ 375:22 . It is the closest vector to the considered ideal fuzzy vector <sup>B</sup><sup>∗</sup> <sup>¼</sup> ð Þ ð Þ <sup>302</sup> ;ð Þ <sup>323</sup> ;ð Þ <sup>347</sup> ;ð Þ <sup>313</sup> . The distance is <sup>D</sup> (B,B\*) <sup>=</sup> 94. Thus,

<sup>1</sup> ; B<sup>∗</sup> <sup>2</sup>; B<sup>∗</sup> 3 .

> <sup>1</sup>; B<sup>0</sup> <sup>2</sup>; B<sup>0</sup> <sup>3</sup>; B<sup>0</sup> 4 <sup>¼</sup>

<sup>1</sup>; …; B<sup>0</sup> m 

that induces the corresponding output vector <sup>B</sup><sup>0</sup> <sup>¼</sup> <sup>B</sup><sup>0</sup>

<sup>¼</sup> ð Þ <sup>19</sup>:5; <sup>50</sup>:5; <sup>30</sup> is <sup>B</sup><sup>0</sup> <sup>¼</sup> <sup>B</sup><sup>0</sup>

A<sup>0</sup> ¼ A<sup>0</sup>

82

Table 9.

Table 6.

Fuzzy Logic

Table 7.

Table 8.

Linguistic terms for input 2 (Pd).

Linguistic terms for output 1 (Mf).

Linguistic terms for output 2 (Ms).

<sup>1</sup>; …; A<sup>0</sup> n

Linguistic terms for output 3 (As).

input vector A<sup>0</sup> ¼ A<sup>0</sup>

maximally close to the ideal solution <sup>B</sup><sup>∗</sup> <sup>¼</sup> <sup>B</sup><sup>∗</sup>

<sup>1</sup>; A<sup>0</sup> <sup>2</sup>; A<sup>0</sup> 3


#### Table 13.

Codebook for input 2 (x3).


desired alloy composition is as follows: Ti is about 50%, Ni is about 37%, Pt is about

No. Linguistic value TFN 1. Very low (VL) (373, 373, 585.5) 2. Low (L) (373, 585.5, 798) 3. Average (A) (585.5, 798, 1010.5) 4. High (H) (798, 1010.5, 1223) 5. Very high (VH) (1010.5, 1223, 1223)

Fuzzy Logic and Fuzzy Expert System-Based Material Synthesis Methods

DOI: http://dx.doi.org/10.5772/intechopen.84493

A series of works exist on material synthesis by using fuzzy models [12, 24, 25]. In this study, to solve material synthesis problem for pressure vessel, two methods are used: possibility measure-based inference method (by ESPLAN shell, Aliev inference) and Mamdani inference method (by MATLAB environment, Fuzzy

Defining the performance index for pressure vessel in material synthesis is a very important problem. The basic problem is to evaluate the performance index by

For determining the performance index, we use data of alloys. There are many

The weighted performance index denoted Out is a compound index built from four characteristics each of which is extracted from the data set. The four characteristics are in1-scaled PREN, in2-scaled yield strength, in3-scaled weldability, and

Using the abovementioned parameters, the performance index model can be

To create this model, we use clustering approach, mainly fuzzy C-means. Data set contains 35 records extracted from big data. For modeling we use two-thirds of the given data and testing one-third. Inputs: x1, scaled PREN; x2, scaled yield; x3, scaled weldability; x4, scaled impact strength. Output: y, performance index. For

1…n are the linguistic input variables, y is the output variable, and Aij

13%, and the obtained characteristics are about Ms = 479,6828 and about

4. Material synthesis by fuzzy expert system

As = 488,1005.

Codebook for output 2 (y2).

Table 17.

Toolbox) [26].

types of alloys.

expressed as.

………

where xj

85

4.1 Statement of the problem

in4-scaled impact strength.

¼ j

and Bi are the fuzzy sets, n = 4, m = 7.

using weighted performance indices.

IF x<sup>1</sup> is A<sup>11</sup> and ….. xn is A1<sup>n</sup> THEN y is B<sup>1</sup> . IF x<sup>1</sup> is A<sup>21</sup> and ….. xn is A2<sup>n</sup> THEN y is B<sup>2</sup> .

IF x<sup>1</sup> is Am<sup>1</sup> and ….. xn is Amn THEN y is Bm.

Fragment of data set is given in Table 18.

simulation FCM-based clustering initial data are:

4.2 Modeling of material data by fuzzy C-means clustering

#### Table 14.

Linguistic terms for input 1 (Ni).


#### Table 15.

Linguistic terms for input 2 (Pt).


Table 16.

Codebook for output 1 (y1).

The codebooks for inputs are shown in Tables 12 and 13.

The linguistic approximation of the inputs is shown in Tables 14 and 15.

The codebooks for the used outputs are shown in Tables 16 and 17.

We have found that the fuzzy optimal output vector B<sup>0</sup> induced by the fuzzy input vector A<sup>0</sup> ¼ A<sup>0</sup> <sup>1</sup>; A<sup>0</sup> <sup>2</sup>; A<sup>0</sup> 3 <sup>¼</sup> ð Þ <sup>40</sup>; <sup>50</sup>; <sup>10</sup> is <sup>B</sup><sup>0</sup> <sup>¼</sup> ð Þ ð Þ <sup>479</sup>:<sup>68</sup> ;ð Þ <sup>488</sup> . It is the closest vector to the considered ideal fuzzy vector <sup>B</sup><sup>∗</sup> <sup>¼</sup> ð Þ ð Þ <sup>363</sup> ;ð Þ <sup>373</sup> . The distance between them is D B<sup>0</sup> ; ; <sup>B</sup><sup>∗</sup> ð Þ¼ 164. The fuzzy model-based results show that the

Fuzzy Logic and Fuzzy Expert System-Based Material Synthesis Methods DOI: http://dx.doi.org/10.5772/intechopen.84493


Table 17.

Codebook for output 2 (y2).

desired alloy composition is as follows: Ti is about 50%, Ni is about 37%, Pt is about 13%, and the obtained characteristics are about Ms = 479,6828 and about As = 488,1005.

## 4. Material synthesis by fuzzy expert system

A series of works exist on material synthesis by using fuzzy models [12, 24, 25]. In this study, to solve material synthesis problem for pressure vessel, two methods are used: possibility measure-based inference method (by ESPLAN shell, Aliev inference) and Mamdani inference method (by MATLAB environment, Fuzzy Toolbox) [26].

## 4.1 Statement of the problem

Defining the performance index for pressure vessel in material synthesis is a very important problem. The basic problem is to evaluate the performance index by using weighted performance indices.

For determining the performance index, we use data of alloys. There are many types of alloys.

The weighted performance index denoted Out is a compound index built from four characteristics each of which is extracted from the data set. The four characteristics are in1-scaled PREN, in2-scaled yield strength, in3-scaled weldability, and in4-scaled impact strength.

Using the abovementioned parameters, the performance index model can be expressed as.

IF x<sup>1</sup> is A<sup>11</sup> and ….. xn is A1<sup>n</sup> THEN y is B<sup>1</sup> . IF x<sup>1</sup> is A<sup>21</sup> and ….. xn is A2<sup>n</sup> THEN y is B<sup>2</sup> . ……… IF x<sup>1</sup> is Am<sup>1</sup> and ….. xn is Amn THEN y is Bm.

where xj ¼ j 1…n are the linguistic input variables, y is the output variable, and Aij and Bi are the fuzzy sets, n = 4, m = 7.

Fragment of data set is given in Table 18.

## 4.2 Modeling of material data by fuzzy C-means clustering

To create this model, we use clustering approach, mainly fuzzy C-means. Data set contains 35 records extracted from big data. For modeling we use two-thirds of the given data and testing one-third. Inputs: x1, scaled PREN; x2, scaled yield; x3, scaled weldability; x4, scaled impact strength. Output: y, performance index. For simulation FCM-based clustering initial data are:

The codebooks for inputs are shown in Tables 12 and 13.

<sup>1</sup>; A<sup>0</sup> <sup>2</sup>; A<sup>0</sup> 3

input vector A<sup>0</sup> ¼ A<sup>0</sup>

Codebook for output 1 (y1).

Table 13.

Fuzzy Logic

Table 14.

Table 15.

Table 16.

84

Codebook for input 2 (x3).

Linguistic terms for input 1 (Ni).

Linguistic terms for input 2 (Pt).

between them is D B<sup>0</sup>

The linguistic approximation of the inputs is shown in Tables 14 and 15. The codebooks for the used outputs are shown in Tables 16 and 17.

No. Linguistic value TFN 1. Very low (VL) (10, 10, 18.75) (1) 2. Low (L) (10, 18.75, 27.5) (2) 3. Average (A) (18.75, 27.5, 36.25) (3) 4. High (H) (27.5, 36.25, 45) (4) 5. Very high (VH) (36.25, 45, 45) (5)

No. Linguistic value TFN 1. Very low (VL) (0, 7.535, 22.65) 2. High 1 (H1) (25.98, 35.17, 44.35) 3. Average (A) (19.27, 26.33, 33.39) 4. Low (L) (8.109, 17.64, 27.17) 5. High 2 (H2) (21.67, 30.06, 38.48)

No. Linguistic value TFN 1. Very high (VH) (27.18, 42.47, 57.75) 2. Low 1 (L1) (5.859, 14.84, 23.82) 3. Low 2 (L2) (14.14, 21.78, 29.42) 4. High (H) (22.63, 32.36, 42.08) 5. Low 3 (L3) (13.71, 19.75, 26.18)

No. Linguistic value TFN 1. Very low (VL) (363, 363, 565.5) 2. Low (L) (363, 565.5, 768) 3. Average (A) (565.5, 768, 970.5) 4. High (H) (768, 970.5, 1173) 5. Very high (VH) (970.5, 1173, 1173)

We have found that the fuzzy optimal output vector B<sup>0</sup> induced by the fuzzy

closest vector to the considered ideal fuzzy vector <sup>B</sup><sup>∗</sup> <sup>¼</sup> ð Þ ð Þ <sup>363</sup> ;ð Þ <sup>373</sup> . The distance

<sup>¼</sup> ð Þ <sup>40</sup>; <sup>50</sup>; <sup>10</sup> is <sup>B</sup><sup>0</sup> <sup>¼</sup> ð Þ ð Þ <sup>479</sup>:<sup>68</sup> ;ð Þ <sup>488</sup> . It is the

; ; <sup>B</sup><sup>∗</sup> ð Þ¼ 164. The fuzzy model-based results show that the


3.IF Scaled PREN = about 26 and Scaled yield = about 5 and Scaled

Fuzzy Logic and Fuzzy Expert System-Based Material Synthesis Methods

4.IF Scaled PREN = about 21 and Scaled yield = about 9 and Scaled

5. IF Scaled PREN = about 25 and Scaled yield = about 3.6 and Scaled

6.IF Scaled PREN = about 47 and Scaled yield = about 4.5 and Scaled

index = about 38.5.

DOI: http://dx.doi.org/10.5772/intechopen.84493

index = about 55.

index = about 53.5.

index = about 83.

Figure 1.

87

Table 19.

Centers of the clusters.

Extracted fuzzy rules (by using fuzzy C-means method).

weldability = about 4.8 and scaled impact strength = about 3 THEN Performance

weldability = about 21.2 and scaled impact strength = about 5 THEN Performance

weldability = about 19 and scaled impact strength = about 6 THEN Performance

weldability = about 18 and scaled impact strength = about 13 THEN Performance

Center 1 18.5215 3.0384 14.7548 10.7647 46.9898 Center 2 27.6395 4.4574 21.3502 12.6412 66.0559 Center 3 26.0329 4.9933 4.8428 3.2598 39.0312 Center 4 20.8528 9.7068 21.3952 5.0337 56.9826 Center 5 25.0287 3.5955 19.0063 6.3912 53.9372 Center 6 46.9418 4.5996 18.4040 13.4963 83.4416 Center 7 24.1544 6.2895 14.0802 9.7481 54.2839

x1 x2 x3 x4 y

#### Table 18.

Fragment of data set (extracted from big data).

Cluster numbers = 7. Max iteration =1000. Exponent = 2. Min. improvement = 0.000001.

Obtained centers of the clusters are given in Table 19. Each row describes a cluster center as five-dimensional vector with coordinates x1 (scaled PREN), x2 (scaled yield), x3 (scaled weldability), x4 (scaled impact strength), and y (performance index). Columns describe the values of the coordinates of the cluster centers.

Representation of the extracted fuzzy rules from big data by using fuzzy c-means method fragment is given below and in Figure 1.


Fuzzy Logic and Fuzzy Expert System-Based Material Synthesis Methods DOI: http://dx.doi.org/10.5772/intechopen.84493



#### Table 19.

Cluster numbers = 7. Max iteration =1000. Exponent = 2.

Table 18.

Scaled PREN

Fuzzy Logic

Scaled yield strength

Scaled weldability

26.60 3.60 18.40 5.00 53.50 29.70 4.40 23.00 8.60 65.60 19.80 3.60 23.00 5.00 51.30 22.30 3.20 23.00 8.60 57.10 26.00 3.60 18.40 6.80 54.70 22.30 5.40 13.80 11.30 52.70 … …… … … 47.00 4.60 18.40 13.50 83.50 29.70 4.40 18.40 15.80 68.30 20.40 12.00 18.40 5.00 55.80 21.00 9.80 23.00 4.50 58.30 23.50 4.60 23.00 13.50 64.60 11.80 2.50 18.40 9.00 41.60 15.50 2.50 18.40 8.80 45.10 22.90 5.80 13.80 7.10 49.50 26.60 6.20 4.60 3.20 40.50 … …… … … 18.60 2.90 18.40 8.80 48.60 32.20 6.20 18.40 6.00 62.70 42.70 4.30 23.00 15.20 85.10 21.00 2.50 18.40 8.80 50.70 21.60 9.50 18.40 4.50 54.00

Scaled impact strength

Performance index

Min. improvement = 0.000001.

Fragment of data set (extracted from big data).

Performance index = about 46.5.

index = about 65.

86

c-means method fragment is given below and in Figure 1.

1. IF Scaled PREN = about 18 and Scaled yield = about 3 and Scaled

2.IF Scaled PREN = about 27 and Scaled yield = about 4.4 and Scaled

weldability = about 14.5 and scaled impact strength = about 10.8,THEN

weldability = about 21 and scaled impact strength = about 12 THEN Performance

Obtained centers of the clusters are given in Table 19. Each row describes a cluster center as five-dimensional vector with coordinates x1 (scaled PREN), x2 (scaled yield), x3 (scaled weldability), x4 (scaled impact strength), and y (performance index). Columns describe the values of the coordinates of the cluster centers. Representation of the extracted fuzzy rules from big data by using fuzzy

Centers of the clusters.

#### Figure 1.

Extracted fuzzy rules (by using fuzzy C-means method).

7. IF Scaled PREN = about 24 and Scaled yield = about 6 and Scaled weldability = about 14 and scaled impact strength = about 10 THEN Performance index = about 54.

Graphical representation of the linguistic terms of inputs and outputs of the rules as trapezoidal fuzzy numbers is given in Figures 2–6.

Figure 2. Linguistic terms of input 1 (scaled PREN).

4.3 Solution of the problem

Linguistic terms of outputs or performance index.

Linguistic terms of input 4 or scaled impact strength.

DOI: http://dx.doi.org/10.5772/intechopen.84493

Fuzzy Logic and Fuzzy Expert System-Based Material Synthesis Methods

performed. TEST 1.

89

Figure 6.

Figure 5.

ANSWER:

(alloy Monel-400).

using the fuzzy model obtained in Section 4.2.

THEN Performance index =?

For solving the problem described in Section 4.1, we will use ESPLAN shell. The problem is to determine material with the given level of performance index

IF Scaled PREN = about 18 and Scaled yield = about 3 and.

The obtained results confirm efficiency of the proposed approach.

Scaled weldability = about 21 and scaled impact strength = about 12.

Solution by using Mamdani inference method. General form of the abovementioned rules are as form (4.13). Mamdani fuzzy inference is most commonly used approximate reasoning methodology for fuzzy modeling. The method

EXPERT system shell ESPLAN's result is "Performance index is about 46.5"

The fuzzy rules were derived from alloy big data by using FCM method, and fuzzy inference within these rules is implemented in expert system shell ESPLAN.

In this context we define basic objects and linguistic terms according to ESPLAN shell. The linguistic terms are described by trapezoidal fuzzy numbers. The rule base given above is put as knowledge base in ESPLAN shell. Then, different tests are

Figure 3. Linguistic terms of input 2 or scaled yield strength.

Figure 4. Linguistic terms of input 3 or scaled weldability.

Fuzzy Logic and Fuzzy Expert System-Based Material Synthesis Methods DOI: http://dx.doi.org/10.5772/intechopen.84493

Figure 5. Linguistic terms of input 4 or scaled impact strength.

7. IF Scaled PREN = about 24 and Scaled yield = about 6 and Scaled

rules as trapezoidal fuzzy numbers is given in Figures 2–6.

index = about 54.

Fuzzy Logic

Figure 2.

Figure 3.

Figure 4.

88

Linguistic terms of input 1 (scaled PREN).

Linguistic terms of input 2 or scaled yield strength.

Linguistic terms of input 3 or scaled weldability.

weldability = about 14 and scaled impact strength = about 10 THEN Performance

Graphical representation of the linguistic terms of inputs and outputs of the

Figure 6. Linguistic terms of outputs or performance index.

## 4.3 Solution of the problem

For solving the problem described in Section 4.1, we will use ESPLAN shell. The problem is to determine material with the given level of performance index using the fuzzy model obtained in Section 4.2.

In this context we define basic objects and linguistic terms according to ESPLAN shell. The linguistic terms are described by trapezoidal fuzzy numbers. The rule base given above is put as knowledge base in ESPLAN shell. Then, different tests are performed.

TEST 1.

IF Scaled PREN = about 18 and Scaled yield = about 3 and. Scaled weldability = about 21 and scaled impact strength = about 12. THEN Performance index =?

## ANSWER:

EXPERT system shell ESPLAN's result is "Performance index is about 46.5" (alloy Monel-400).

The fuzzy rules were derived from alloy big data by using FCM method, and fuzzy inference within these rules is implemented in expert system shell ESPLAN. The obtained results confirm efficiency of the proposed approach.

Solution by using Mamdani inference method. General form of the abovementioned rules are as form (4.13). Mamdani fuzzy inference is most commonly used approximate reasoning methodology for fuzzy modeling. The method works with crisp input which is transformed into a linguistic value using the antecedent membership functions. After the aggregation process of consequents induced by antecedents, obtained final fuzzy set is defuzzified. We can describe fuzzy inference process in algorithmic view as follows:

1. Firing level for each rule is defined as follows:

$$a\_i = \min\_{j=1}^n \left[ \max\_{\mathbf{x}\_j} \left( A\_j'(\mathbf{x}\_j) \land A\_{\vec{\eta}}(\mathbf{x}\_j) \right) \right] \tag{3}$$

In1 = 18.60, In2 = 2.90, In3 = 18.40 and In4 = 8.

Consider other values of the inputs:

Scaled yield strength

DOI: http://dx.doi.org/10.5772/intechopen.84493

Consider also the following input values: In1 = 26, In2 = 3.6, In3 = 18.4, and In4 = 6.8.

Out = 48.60.

Testing data (fragment).

Scaled PREN

Table 20.

in Table 21.

material behavior.

Scaled yield strength

Comparison of computed and given data.

Scaled weldability Scaled impact strength

26.00 3.60 18.40 6.80 54.70 54.8 (alloy

Computed performance index

Given performance index

1925hMo)

Scaled PREN

Table 21.

91

mance index of alloy 317LM.

mance index is computed by using Mamdani fuzzy inference:

Fuzzy Logic and Fuzzy Expert System-Based Material Synthesis Methods

In1 = 18.60, In2 = 2.90, In3 = 18.40 and In4 = 8.80.

For these data, the following defuzzified output describing the alloy perfor-

Scaled weldability

18.60 2.90 18.40 8.80 48.60 21 2.5 18.4 8.8 50.7

Scaled impact strength

Performance index

This value fits the performance index of alloy 317 L (from the given big data).

For these values, the defuzzified output is Out = 50.3. This result fits the perfor-

The computed output (performance index) is 54.7. The performance index computed for the third case and the performance index from big data set are shown

Summarizing the findings in this chapter, we have to conclude that the discovery and design of new materials are driving forces for much of the research that takes place in multiple disciplines, including materials science and engineering, matter physics, materials chemistry, and emerging technologies such as fuzzy logic, soft computing, etc. However, this task is implemented mainly on the basis of timeand resource-consuming experiments. Thus, we consider to shift the approaches to material design investigations from physical experiments to experiments on the basis of fuzzy If-Then rule-based material model. The motivation to use fuzzy model is inspired by the necessity to construct an intuitively well-interpretable material design model based on imperfect and complex data. In this chapter we have considered three material synthesis problems which had shown that instead of carrying out complicated experiments, researchers can use fuzzy model-based computational synthesis approach utilizing digital twins of physical models. Applications of this approach have shown that fuzzy model-based experiments can give better results than physical experiments in terms of desirable characteristics of synthesized materials. The approaches suggested in this chapter are universal and may be applied not only in materials science but also in chemical engineering, drug design, and other fields. Complexity of material design problems mandates to combine fuzzy logic and efficient learning methods as artificial neural networks, evolutionary algorithms, and others to more adequately model and predict possible

Deviation between testing and expert data is 0.18% or 0.0018.

where A<sup>0</sup> <sup>j</sup> xj � � are current input values.

2. Outputs for each rule are calculated:

$$B\_i(\mathbf{y}) = \min(a\_i, B\_i(\mathbf{y})) \tag{4}$$

3. Calculate aggregative output:

$$B'(\boldsymbol{y}) = \max\left(B\_1'(\boldsymbol{y}), B\_2'(\boldsymbol{y}), \dots, B\_m'(\boldsymbol{y})\right) \tag{5}$$

In our example the number of input variables is equal to 4, and for each variable, linguistic value number is equal to 7.

For example, scaled PREN variable is evaluated as (about 18, about 27, about 26, about 21, about 25, about 47, about 24).

Observing the relationship between input and output clusters, we may formulate the following linguistic descriptions—productions rules, for example:


The obtained rules are put into Fuzzy Toolbox to perform tests by using the following data (Table 20):

Below, we provide some test results.

Test results. The following input data are given:



Table 20.

works with crisp input which is transformed into a linguistic value using the antecedent membership functions. After the aggregation process of consequents induced by antecedents, obtained final fuzzy set is defuzzified. We can describe

> max xj

> > 0 <sup>1</sup>ð Þy ; B 0

late the following linguistic descriptions—productions rules, for example:

A0 <sup>j</sup> xj � �∧Aij xj � � � � "

ð Þ¼ y min α<sup>i</sup> ð Þ ; Bið Þy (4)

0 <sup>m</sup>ð Þy

<sup>2</sup>ð Þy ; :…; B

� �

In our example the number of input variables is equal to 4, and for each variable,

For example, scaled PREN variable is evaluated as (about 18, about 27, about 26,

Observing the relationship between input and output clusters, we may formu-

1. IF In1 about 18 and In2 = about 3 and In3 = about 14.5 and In4 = about 10.8

2.IF In1 = about 27 and In2 = about 4.4 and In3 = about 21 and In4 = about 12

4.IF In1 = about 21 and In2 = about 9 and In3 = about 21.2 and In4 = about 5

5. IF In1 = about 25 and In2 = about 3.6 and In3 = about 19 and In4 = about 6

6.IF In1 = about 47 and In2 = about 4.5 and In3 = about 18 and In4 = about 13

7. IF In1 = about 24 and In2 = about 6 and In3 = about 14 and In4 = about 10

The obtained rules are put into Fuzzy Toolbox to perform tests by using the

3.IF In1 = about 26 and In2 = about 5 and In3 = about 4.8 and In4 = about 3 THEN

(3)

(5)

fuzzy inference process in algorithmic view as follows:

1. Firing level for each rule is defined as follows:

� � are current input values.

2. Outputs for each rule are calculated:

B 0

3. Calculate aggregative output:

linguistic value number is equal to 7.

THEN Out = about 46.5.

THEN Out = about 65.

THEN Out = about 55.

THEN Out = about 53.5.

THEN Out = about 83.

THEN Out = about 54.

Below, we provide some test results.

Test results. The following input data are given:

following data (Table 20):

90

Out = about 38.5.

about 21, about 25, about 47, about 24).

where A<sup>0</sup>

Fuzzy Logic

<sup>j</sup> xj

α<sup>i</sup> ¼ min n j¼1

> B 0 i

ð Þ¼ y max B

Testing data (fragment).

In1 = 18.60, In2 = 2.90, In3 = 18.40 and In4 = 8.

For these data, the following defuzzified output describing the alloy performance index is computed by using Mamdani fuzzy inference:

Out = 48.60.

This value fits the performance index of alloy 317 L (from the given big data). Consider other values of the inputs:

In1 = 18.60, In2 = 2.90, In3 = 18.40 and In4 = 8.80.

For these values, the defuzzified output is Out = 50.3. This result fits the performance index of alloy 317LM.

Consider also the following input values:

In1 = 26, In2 = 3.6, In3 = 18.4, and In4 = 6.8.

The computed output (performance index) is 54.7. The performance index computed for the third case and the performance index from big data set are shown in Table 21.

Deviation between testing and expert data is 0.18% or 0.0018.

Summarizing the findings in this chapter, we have to conclude that the discovery and design of new materials are driving forces for much of the research that takes place in multiple disciplines, including materials science and engineering, matter physics, materials chemistry, and emerging technologies such as fuzzy logic, soft computing, etc. However, this task is implemented mainly on the basis of timeand resource-consuming experiments. Thus, we consider to shift the approaches to material design investigations from physical experiments to experiments on the basis of fuzzy If-Then rule-based material model. The motivation to use fuzzy model is inspired by the necessity to construct an intuitively well-interpretable material design model based on imperfect and complex data. In this chapter we have considered three material synthesis problems which had shown that instead of carrying out complicated experiments, researchers can use fuzzy model-based computational synthesis approach utilizing digital twins of physical models. Applications of this approach have shown that fuzzy model-based experiments can give better results than physical experiments in terms of desirable characteristics of synthesized materials. The approaches suggested in this chapter are universal and may be applied not only in materials science but also in chemical engineering, drug design, and other fields. Complexity of material design problems mandates to combine fuzzy logic and efficient learning methods as artificial neural networks, evolutionary algorithms, and others to more adequately model and predict possible material behavior.


Table 21. Comparison of computed and given data.

## 5. Conclusion

In this chapter, we have used data-driven approach to construction of fuzzy model which is more effective than expert-driven approach. Consequently, we have used fuzzy C-means clustering to derive fuzzy If-Then rules from material data that describe material composition and related characteristics. In order to determine the required characteristics, computational experiments on the basis of fuzzy inference and fuzzy expert system were conducted. The expert opinions and some few physical experiments have proven validity of the obtained results. The main advantage of the fuzzy logic-based approach is a high interpretability of fuzzy If-Then rules. However, learning the ability of the fuzzy models is scarce. Thus, combination of fuzzy logic with deep learning methods, mainly, reinforcement learning methods, would help to achieve better results on material synthesis.

References

[1] Agrawal A, Choudhary A.

SR. Exploration of data science

[3] National Institute of Materials Science. Available from: http://smds. nims.go.jp/fatigue/index\_en.html [Accessed: January 12, 2016]

[4] Dieter GE. Mechanical Metallurgy.

[5] Yang ZG, Stevenson JW, Paxton DM, Singh P, Weil KS. Materials Properties Database for Selection of High-Temperature Alloys and Concepts of Alloy Design for SOFC Applications. Richland, Washington: Pacific

Northwest National Laboratory; 2002.

[6] Hill J, Mulholland G, Persson K, Seshadri R, Wolverton C, Meredig B. Materials science with large-scale data and informatics. Unlocking new opportunities. MRS Bulletin. 2016;

[7] Gaultois MW et al. Perspective: Webbased machine learning models for real-

[8] Elishakoff I, Ferracuti B. Fuzzy sets based interpretation of the safety factor. Fuzzy Sets and Systems. 2006;157:

time screening of thermoelectric materials properties. APL Materials. 2016;4(5):053213-1-053213-11. DOI:

New York: McGraw-Hill Book

Company; 1986

78 p

41:399-409

10.1063/1.4952607

2495-2512

93

Perspective: Materials informatics and big data: Realization of the 'fourth paradigm' of science in materials science. APL Materials. 2016;4:1-10

DOI: http://dx.doi.org/10.5772/intechopen.84493

Fuzzy Logic and Fuzzy Expert System-Based Material Synthesis Methods

[9] Lee Y-H, Kopp R. Application of fuzzy control for a hydraulic forging machine. Fuzzy Sets and Systems. 2001;

[10] Rao HS, Mukherjee A. Artificial neural networks for predicting the macromechanical behaviour of ceramicmatrix composites. Computational Materials Science. 1996;5:307-322

[11] Conduit BD, Jones NG, Stone HJ, Conduit GJ. Design of a nickel-base superalloy using a neural network. Materials & Design. 2017;131:358-365

[12] Tajdari M, Mehraban AG, Khoogar AR. Shear strength prediction of Ni–Ti alloys manufactured by powder

metallurgy using fuzzy rule-based model. Materials and Design. 2010;31:1180-1185

[14] Chen DD. Dislocation substructures evolution and an adaptive-network based fuzzy inference system model for constitutive behavior of a Ni-based super alloy during hot deformation. Journal of Alloys and Compounds. 2017;

[15] Babanli MB, Huseynov VM. Znumber-based alloy selection problem. Procedia Computer Science. 2016;102:

[16] Babanli MB, Kolomytsev V, Musienko R, Sezonenko A, Ochin P, Dezellus A, et al. Thermodynamic properties and thermal stability of the multicomponent TiNi based alloy ribbons. Metal Physics and Advanced Technologies. 2001;23:111-124

[17] Zadeh LA. Fuzzy sets. Information

and Control. 1965;8:338-353

[13] Babanli MB. Synthesis of new materials by using fuzzy and big data concepts. Procedia Computer Science.

2017;120:104-111

708:938-946

183-189

99:99-108

[2] Agrawal A, Deshpande PD, Cecen A, Basavarsu GP, Choudhary AN, Kalidindi

techniques to predict fatigue strength of steel from composition and processing parameters. Integrating Materials and Manufacturing Innovation. 2014;3:1-19

In future works, fuzzy materials paradigm may improve processing-structure- property-performance relationship in hierarchy of structural materials levels, from the atomic and electronic to the macrostructural levels. Another important application of fuzzy logic is fuzzy phase diagram construction for different alloy models using uncertain enthalpies and other thermodynamic parameters will be investigated, which opens a door to design new materials.

## Author details

Mustafa B. Babanli Azerbaijan State University of Oil and Industry, Baku, Azerbaijan

\*Address all correspondence to: mustafababanli@yahoo.com

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Fuzzy Logic and Fuzzy Expert System-Based Material Synthesis Methods DOI: http://dx.doi.org/10.5772/intechopen.84493

## References

5. Conclusion

Fuzzy Logic

Author details

Mustafa B. Babanli

92

In this chapter, we have used data-driven approach to construction of fuzzy model which is more effective than expert-driven approach. Consequently, we have used fuzzy C-means clustering to derive fuzzy If-Then rules from material data that describe material composition and related characteristics. In order to determine the required characteristics, computational experiments on the basis of fuzzy inference and fuzzy expert system were conducted. The expert opinions and some few physical experiments have proven validity of the obtained results. The main advantage of the fuzzy logic-based approach is a high interpretability of fuzzy If-Then rules. However, learning the ability of the fuzzy models is scarce. Thus, combination of fuzzy logic with deep learning methods, mainly, reinforcement learning methods,

In future works, fuzzy materials paradigm may improve processing-structure- property-performance relationship in hierarchy of structural materials levels, from the atomic and electronic to the macrostructural levels. Another important application of fuzzy logic is fuzzy phase diagram construction for different alloy models using uncertain enthalpies and other thermodynamic parameters will be investi-

would help to achieve better results on material synthesis.

Azerbaijan State University of Oil and Industry, Baku, Azerbaijan

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: mustafababanli@yahoo.com

provided the original work is properly cited.

gated, which opens a door to design new materials.

[1] Agrawal A, Choudhary A. Perspective: Materials informatics and big data: Realization of the 'fourth paradigm' of science in materials science. APL Materials. 2016;4:1-10

[2] Agrawal A, Deshpande PD, Cecen A, Basavarsu GP, Choudhary AN, Kalidindi SR. Exploration of data science techniques to predict fatigue strength of steel from composition and processing parameters. Integrating Materials and Manufacturing Innovation. 2014;3:1-19

[3] National Institute of Materials Science. Available from: http://smds. nims.go.jp/fatigue/index\_en.html [Accessed: January 12, 2016]

[4] Dieter GE. Mechanical Metallurgy. New York: McGraw-Hill Book Company; 1986

[5] Yang ZG, Stevenson JW, Paxton DM, Singh P, Weil KS. Materials Properties Database for Selection of High-Temperature Alloys and Concepts of Alloy Design for SOFC Applications. Richland, Washington: Pacific Northwest National Laboratory; 2002. 78 p

[6] Hill J, Mulholland G, Persson K, Seshadri R, Wolverton C, Meredig B. Materials science with large-scale data and informatics. Unlocking new opportunities. MRS Bulletin. 2016; 41:399-409

[7] Gaultois MW et al. Perspective: Webbased machine learning models for realtime screening of thermoelectric materials properties. APL Materials. 2016;4(5):053213-1-053213-11. DOI: 10.1063/1.4952607

[8] Elishakoff I, Ferracuti B. Fuzzy sets based interpretation of the safety factor. Fuzzy Sets and Systems. 2006;157: 2495-2512

[9] Lee Y-H, Kopp R. Application of fuzzy control for a hydraulic forging machine. Fuzzy Sets and Systems. 2001; 99:99-108

[10] Rao HS, Mukherjee A. Artificial neural networks for predicting the macromechanical behaviour of ceramicmatrix composites. Computational Materials Science. 1996;5:307-322

[11] Conduit BD, Jones NG, Stone HJ, Conduit GJ. Design of a nickel-base superalloy using a neural network. Materials & Design. 2017;131:358-365

[12] Tajdari M, Mehraban AG, Khoogar AR. Shear strength prediction of Ni–Ti alloys manufactured by powder metallurgy using fuzzy rule-based model. Materials and Design. 2010;31:1180-1185

[13] Babanli MB. Synthesis of new materials by using fuzzy and big data concepts. Procedia Computer Science. 2017;120:104-111

[14] Chen DD. Dislocation substructures evolution and an adaptive-network based fuzzy inference system model for constitutive behavior of a Ni-based super alloy during hot deformation. Journal of Alloys and Compounds. 2017; 708:938-946

[15] Babanli MB, Huseynov VM. Znumber-based alloy selection problem. Procedia Computer Science. 2016;102: 183-189

[16] Babanli MB, Kolomytsev V, Musienko R, Sezonenko A, Ochin P, Dezellus A, et al. Thermodynamic properties and thermal stability of the multicomponent TiNi based alloy ribbons. Metal Physics and Advanced Technologies. 2001;23:111-124

[17] Zadeh LA. Fuzzy sets. Information and Control. 1965;8:338-353

[18] Babanli MB. Theory and practice of material development under imperfect information. In: Advances in Intelligent Systems and Computing. Vol. 896. Springer; 2018. pp. 4-14

[19] Babanli MB, Prima F, Vermaut P, Demchenko LD, Titenko AN, Huseynov SS, et al. Material selection methods: A review. Advances in Intelligent Systems and Computing. 2018;896: 929-936

[20] Babanli MB. Fuzzy modeling of phase diagram under imprecise thermodynamic data. In: Proceedings of the Tenth World Conference "Intelligent Systems for Industrial Automation". b-Quadrat Verlag; 2018. pp. 265-266

[21] Babanli MB, Prima F, Vermaut P, Demchenko LD, Titenko AN, Huseynov SS, et al. Review on the new material design methods. Advances in Intelligent Systems and Computing. Cham, Switzerland: 2018;896:937-944

[22] Frenzel J, Wieczorek A, Opahle I, Maa B, Drautz R, Eggeler G. On the effect of alloy composition on martensite start temperatures and latent heats in Ni–Ti-based shape memory alloys. Acta Materialia. 2015;90:213-231

[23] Vafaeenezhad H, Seyedein SH, Aboutalebi MR, Eivani AR. Application of constitutive description and integrated ANFIS – ICA analysis to predict hot deformation behavior of Sn-5Sb lead-free solder alloy. Journal of Alloys Compounds. 2017;697:287-299

[24] Odejobi OA, Umoru LE. Applications of soft computing techniques in materials engineering: A review. African Journal of Mathematics and Computer Science Research. 2009;2 (7):104-131

[25] Datta S, Chattopadhyay PP. Soft computing techniques in advancement of structural metals. International Materials Reviews. 2013;58:475-504

[26] Aliev RA, Aliyev RR. Soft Computing and Its Applications. New Jersey, London, Singapore, Hong Kong: World Scientific; 2001

Section 6

Control of Electrical Systems

95

Section 6
