Part II. S3. intuitionistic fuzzy multi-attribute decision-making method when attribute weight is intuitionistic fuzzy number

3.1 Weighted intuitionistic fuzzy numbers whose attribute weights are intuitionistic fuzzy numbers

   attribute

G

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∈

G

G_{j}\in G

Gj​∈G The weight of is intuitionistic fuzzy number

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\tilde{\omega}_{j} = \left\langle\rho_{j},\tau_{j}\right\rangle(j=1,2,...,n)

ω~j​=*ρj​,τj​*(j=1,2,...,n) Meet the conditions

ρ

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∈

[

0

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\rho_{j}\in[0,1]

ρj​∈[0,1]、

τ

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∈

[

0

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\tau_{j}\in[0,1]

τj​∈[0,1], And

0

≤

ρ

j

+

τ

j

≤

1

0\leq\rho_{j}+\tau_{j}\leq1

0≤ρj​+τj​≤1, be

F

~

i

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\tilde{F}_{ij}

F~ij​ The weighted intuitionistic fuzzy number of is

F

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F

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(3.1)

\begin{aligned} \tilde{F^{'}}_{ij} &= \color{red}{\underline{\left\langle\mu^{'}_{ij},\nu^{'}_{ij}\right\rangle}} \\ &= \tilde{\omega}_{j}\otimes\tilde{F^{'}}_{ij} \\ &= \left\langle\rho_{j},\tau_{j}\right\rangle\left\langle\mu_{ij},\nu_{ij}\right\rangle \\ &= \color{red}{\underline{\left\langle\rho_{j}\mu_{ij},\tau_{j}+\nu_{ij}-\tau_{j}\nu_{ij}\right\rangle}} \tag{3.1} \end{aligned}

F′~ij​​=*μij′​,νij′​*​=ω~j​⊗F′~ij​=*ρj​,τj​**μij​,νij​*=*ρj​μij​,τj​+νij​−τj​νij​*​​(3.1)

   programme

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Y_{i}(1,2,...,m)

Yi​(1,2,...,m) The result of weighted intuitionistic fuzzy comprehensive evaluation is

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⊕

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⊕

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*

\begin{aligned} \tilde{d}_{i} &= \tilde{\omega}_{1} \otimes \tilde{F^{'}}_{i1} \oplus \tilde{\omega}_{2} \otimes \tilde{F^{'}}_{i2} \oplus ... \oplus \tilde{\omega}_{n} \otimes \tilde{F^{'}}_{in}\\ &= \left\langle\rho_{1}\mu_{i1},\tau_{1}+\nu_{i1}-\tau_{1}\nu_{i1}\right\rangle + \left\langle\rho_{2}\mu_{i2},\tau_{2}+\nu_{i2}-\tau_{2}\nu_{i2}\right\rangle + ... + \left\langle\rho_{n}\mu_{in},\tau_{n}+\nu_{in}-\tau_{n}\nu_{in}\right\rangle \\ &= \sum_{j=1}^{n} \left\langle\mu^{'}_{ij},\nu^{'}_{ij}\right\rangle \end{aligned}

d~i​​=ω~1​⊗F′~i1​⊕ω~2​⊗F′~i2​⊕...⊕ω~n​⊗F′~in​=*ρ1​μi1​,τ1​+νi1​−τ1​νi1​*+*ρ2​μi2​,τ2​+νi2​−τ2​νi2​*+...+*ρn​μin​,τn​+νin​−τn​νin​*=j=1∑n​*μij′​,νij′​*​

   that ,

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(3.2)

\color{red} { \begin{aligned} \tilde{d}_{i} &= \left\langle\mu^{'}_{i},\nu^{'}_{i}\right\rangle \\ &= \sum_{j=1}^{n} \left\langle\mu^{'}_{ij},\nu^{'}_{ij}\right\rangle \\ &= \color{red}{\underline{\left\langle1-(1-\mu^{'}_{i1})(1-\mu^{'}_{i2})...(1-\mu^{'}_{in}),\nu^{'}_{i1}\nu^{'}_{i2}...\nu^{'}_{in}\right\rangle}} \tag{3.2} \end{aligned} }

d~i​​=*μi′​,νi′​*=j=1∑n​*μij′​,νij′​*=*1−(1−μi1′​)(1−μi2′​)...(1−μin′​),νi1′​νi2′​...νin′​*​​(3.2)
   programme

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Y_{i}\left(i=1,2,...,m\right)

Yi​(i=1,2,...,m) Weighted comprehensive evaluation results of

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\tilde{d}_{i}

d~i​ The score value and the exact value of are ：

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s\left(\tilde{d}_{i}\right) = \mu^{'}_{i}-\nu^{'}_{i}\left(i=1,2,...,m\right),h\left(\tilde{d}_{i}\right) = \mu^{'}_{i}+\nu^{'}_{i}\left(i=1,2,...,m\right)

s(d~i​)=μi′​−νi′​(i=1,2,...,m),h(d~i​)=μi′​+νi′​(i=1,2,...,m)
   Using intuitionistic fuzzy number sorting rules , The scheme can be determined

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Y_{i}\left(i=1,2,...,m\right)

Yi​(i=1,2,...,m) Order of advantages and disadvantages .

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3.2 The analysis steps of multi-attribute decision-making method whose attribute weight is intuitionistic fuzzy number

   The steps of multi-attribute decision-making method whose attribute weight is intuitionistic fuzzy number are as follows ：
   step S1  Determining the scheme set of multi-attribute decision making problem

Y

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{

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Y=\left\{Y_{1},Y_{2},...,Y_{m}\right\}

Y={ Y1​,Y2​,...,Ym​} And property sets

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{

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n

}

G=\left\{G_{1},G_{2},...,G_{n}\right\}

G={ G1​,G2​,...,Gn​};
   step S2  Obtain the scheme in multi-attribute decision-making problem

Y

i

∈

Y

Y_{i} \in Y

Yi​∈Y About attributes

G

j

∈

G

G_{j} \in G

Gj​∈G Intuitionistic fuzzy characteristic information , Construct intuitionistic fuzzy decision matrix

F

F

F;
   step S3  Determine the intuitionistic fuzzy weight of each attribute of the multi-attribute decision-making problem , Get the intuitionistic fuzzy weight vector of the attribute

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T

\tilde{\omega} = {\left(\tilde{\omega}_{1},\tilde{\omega}_{2},...,\tilde{\omega}_{n}\right)}^{T}={\left(\left\langle\rho_{1},\tau_{1}\right\rangle,\left\langle\rho_{2},\tau_{2}\right\rangle,...,\left\langle\rho_{n},\tau_{n}\right\rangle\right)}^{T}

ω~=(ω~1​,ω~2​,...,ω~n​)T=(*ρ1​,τ1​*,*ρ2​,τ2​*,...,*ρn​,τn​*)T;
   step S4  Utilization

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\tilde{F^{'}}_{ij} = \left\langle\mu^{'}_{ij},\nu^{'}_{ij}\right\rangle=\tilde{\omega}_{j}\otimes\tilde{F^{'}}_{ij}= \left\langle\rho_{j},\tau_{j}\right\rangle\left\langle\mu_{ij},\nu_{ij}\right\rangle=\left\langle\rho_{j}\mu_{ij},\tau_{j}+\nu_{ij}-\tau_{j}\nu_{ij}\right\rangle

F′~ij​=*μij′​,νij′​*=ω~j​⊗F′~ij​=*ρj​,τj​**μij​,νij​*=*ρj​μij​,τj​+νij​−τj​νij​* Calculate the weighted intuitionistic fuzzy decision matrix of multi-attribute decision-making problem

F

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(

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i

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m

×

n

F^{'}={\left(\tilde{F^{'}}_{ij}\right)}_{m×n}

F′=(F′~ij​)m×n​;
   step S5  Utilization

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\tilde{d}_{i} = \left\langle\mu^{'}_{i},\nu^{'}_{i}\right\rangle = \sum_{j=1}^{n} \left\langle\mu^{'}_{ij},\nu^{'}_{ij}\right\rangle = \left\langle1-(1-\mu^{'}_{i1})(1-\mu^{'}_{i2})...(1-\mu^{'}_{in}),\nu^{'}_{i1}\nu^{'}_{i2}...\nu^{'}_{in}\right\rangle

d~i​=*μi′​,νi′​*=∑j=1n​*μij′​,νij′​*=*1−(1−μi1′​)(1−μi2′​)...(1−μin′​),νi1′​νi2′​...νin′​* Calculation scheme

Y

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Y_{i}

Yi​ Weighted intuitionistic fuzzy comprehensive attribute value

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\tilde{d}_{i}\left(i=1,2,...,m\right)

d~i​(i=1,2,...,m);
   step S6  Calculation scheme

Y

i

Y_{i}

Yi​ Weighted intuitionistic fuzzy comprehensive attribute value

d

~

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\tilde{d}_{i}

d~i​ Score value of

s

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s\left(\tilde{d}_{i}\right)

s(d~i​) And the exact value

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h\left(\tilde{d}_{i}\right)

h(d~i​), determine

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\tilde{d}_{i}\left(i=1,2,...,m\right)

d~i​(i=1,2,...,m) Do not increase sort order , The ranking results are used to evaluate the scheme

Y

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{

Y

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m

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Y=\left\{Y_{1},Y_{2},...,Y_{m}\right\}

Y={ Y1​,Y2​,...,Ym​} Sort the pros and cons .

  

in

example

3.1

\color{red}{ Example 3.1}

in example 3.1

   Consider the evaluation of the effectiveness of the enterprise quality management system . The main purpose of evaluating the operation effectiveness of the quality management system is to find that the operation of the quality management system is imperfect or does not adapt to environmental changes , Improve the management ability and business performance of the organization . Usually from the quality policy objectives （

G

1

G_{1}

G1​）、 Product quality stability （

G

2

G_{2}

G2​）、 Quality improvement and innovation （

G

3

G_{3}

G3​）、 Resource management （

G

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G_{4}

G4​）、 Financial performance （

G

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G_{5}

G5​） And so on . Suppose that by Market Research and expert consultation Obtain five enterprises

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Y_{i}\left(i=1,2,3,4,5\right)

Yi​(i=1,2,3,4,5) About attributes

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G_{j}\left(i=1,2,3,4,5\right)

Gj​(i=1,2,3,4,5) The intuitionistic fuzzy evaluation results are shown in the following table .

	$G_1$	$G_2$	$G_3$	$G_4$	$G_5$
$Y_1$	$*0.3,0.4*$	$*0.2,0.2*$	$*0.2,0.4*$	$*0.3,0.5*$	$*0.4,0.5*$
$Y_2$	$*0.4,0.2*$	$*0.4,0.3*$	$*0.3,0.4*$	$*0.6,0.2*$	$*0.8,0.1*$
$Y_3$	$*0.3,0.5*$	$*0.5,0.2*$	$*0.6,0.3*$	$*0.5,0.2*$	$*0.9,0.0*$
$Y_4$	$*0.6,0.3*$	$*0.7,0.2*$	$*0.4,0.4*$	$*0.4,0.1*$	$*0.7,0.2*$
$Y_5$	$*0.6,0.1*$	$*0.3,0.1*$	$*0.1,0.4*$	$*0.7,0.1*$	$*0.5,0.2*$

   according to Expert consultation 、 The questionnaire survey And so on

G

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G_{j}\left(i=1,2,3,4,5\right)

Gj​(i=1,2,3,4,5) The intuitionistic fuzzy weight vector of is

ω

~

=

(

*

0.35

,

0.45

*

,

*

0.65

,

0.20

*

,

*

0.45

,

0.25

*

,

*

0.40

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0.30

*

,

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0.25

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0.45

*

)

T

\tilde{\omega}={\left(\left\langle0.35,0.45\right\rangle,\left\langle0.65,0.20\right\rangle,\left\langle0.45,0.25\right\rangle,\left\langle0.40,0.30\right\rangle,\left\langle0.25,0.45\right\rangle\right)}^{T}

ω~=(*0.35,0.45*,*0.65,0.20*,*0.45,0.25*,*0.40,0.30*,*0.25,0.45*)T. Then enterprise

Y

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Y_{i}\left(i=1,2,3,4,5\right)

Yi​(i=1,2,3,4,5) What is the ranking result of the effectiveness of the quality management system ？

   The code is as follows ：

import numpy as np

#  Calculate the weighted intuitionistic fuzzy decision matrix F'
def calculate_weight_IF_matrix(IFD_matrix,IFW_attr):
    # IFD_matrix Of shape by (m×2×n), respectively m A plan 、n Intuitionistic fuzzy number of attributes 
    # IFW_attr Of shape by (2×n) Express n Intuitionistic fuzzy number of attributes 
    #  Calculation  [u_ij_'] = [p_j] * [u_ij]
    #  Calculation  [v_ij_'] = [t_j] + [v_ij] - [t_j] * [v_ij]
    weighted_IFD_matrix = np.zeros(IFD_matrix.shape)
    for i in range(IFD_matrix.shape[0]):
        weighted_IFD_matrix[i][0] = IFW_attr[0]*IFD_matrix[i][0]
        weighted_IFD_matrix[i][1] = IFW_attr[1] + IFD_matrix[i][1] - IFW_attr[1]*IFD_matrix[i][1]
    return weighted_IFD_matrix

#  Calculate comprehensive evaluation results 
def calculate_assessment_result(weighted_IFD_matrix):
    # weighted_IFD_matrix Is a weighted intuitionistic fuzzy decision matrix , Size is (m×2×n), among m Number of solutions 、n Represents the number of attributes 
    #  Initialize the result matrix , Size is (2×m)
    result_matrix = np.zeros((2,weighted_IFD_matrix.shape[0]))
    
    for i in range(weighted_IFD_matrix.shape[0]):
        mu_i = weighted_IFD_matrix[i][0]  #  The first i It's a plan mu value 
        nu_i = weighted_IFD_matrix[i][1]  #  The first i It's a plan nu value 
        mu_r = 1
        nu_r = 1
        #  Traverse mu_i, Calculation mu_tmp = (1-mu_1)*(1-mu_2)*...*(1-mu_n)
        for value in mu_i:
            mu_r *= 1-value
        #  Traverse nu_i, Calculation nu_result = nu_1*nu_2*...*nu_n
        for value in nu_i:
            nu_r *= value
        # mu_result = 1 - mu_tmp
        result_matrix[0][i] = 1 - mu_r
        result_matrix[1][i] = nu_r
    #  Return the result matrix 
    return result_matrix

#  Calculate the score value and sort the schemes 
def calculate_scores_and_ranks(result_matrix):
    #  Calculate the score ：
    scores = result_matrix[0] - result_matrix[1]
    #  Get sort index 
    index = np.argsort(-scores)
    return scores,index

# S1. Intuitionistic fuzzy decision matrix F
IFD_matrix = np.array([[[0.3,0.2,0.2,0.3,0.4],[0.4,0.2,0.4,0.5,0.5]],
                       [[0.4,0.4,0.3,0.6,0.8],[0.2,0.3,0.4,0.2,0.1]],
                       [[0.3,0.5,0.6,0.5,0.9],[0.5,0.2,0.3,0.2,0.0]],
                       [[0.6,0.7,0.4,0.4,0.7],[0.3,0.2,0.4,0.1,0.2]],
                       [[0.6,0.3,0.1,0.7,0.5],[0.1,0.1,0.4,0.1,0.2]]])
print(' Intuitionistic fuzzy decision matrix F:\n',IFD_matrix)
# S2. Intuitionistic fuzzy weight vector of each attribute 
IFW_attr = np.array([[0.35,0.65,0.45,0.40,0.25],[0.45,0.20,0.25,0.30,0.45]])
print('\n Intuitionistic fuzzy weight vector of each attribute :\n',IFW_attr)
# S3. Calculate the weighted intuitionistic fuzzy decision matrix F'
weighted_IFD_matrix = calculate_weight_IF_matrix(IFD_matrix,IFW_attr)
print('\n Weighted intuitionistic fuzzy decision matrix F:\n',weighted_IFD_matrix)
# S4. Calculate the intuitionistic fuzzy comprehensive evaluation results R
Result = calculate_assessment_result(weighted_IFD_matrix)
print('\ Intuitionistic fuzzy comprehensive evaluation results R:\n',Result)
# S5. Calculate the score value and sort the schemes 
scores,index = calculate_scores_and_ranks(Result)
print('\n Score of each scheme :\n',scores)
print(' Sorting result :\n',index+1)

   The calculation results are as follows ：

Intuitionistic fuzzy decision matrix F:
[[[0.3 0.2 0.2 0.3 0.4]
[0.4 0.2 0.4 0.5 0.5]]
[[0.4 0.4 0.3 0.6 0.8]
[0.2 0.3 0.4 0.2 0.1]]
[[0.3 0.5 0.6 0.5 0.9]
[0.5 0.2 0.3 0.2 0. ]]
[[0.6 0.7 0.4 0.4 0.7]
[0.3 0.2 0.4 0.1 0.2]]
[[0.6 0.3 0.1 0.7 0.5]
[0.1 0.1 0.4 0.1 0.2]]]
Intuitionistic fuzzy weight vector of each attribute :
[[0.35 0.65 0.45 0.4 0.25]
[0.45 0.2 0.25 0.3 0.45]]
Weighted intuitionistic fuzzy decision matrix F:
[[[0.105 0.13 0.09 0.12 0.1 ]
[0.67 0.36 0.55 0.65 0.725]]
[[0.14 0.26 0.135 0.24 0.2 ]
[0.56 0.44 0.55 0.44 0.505]]
[[0.105 0.325 0.27 0.2 0.225]
[0.725 0.36 0.475 0.44 0.45 ]]
[[0.21 0.455 0.18 0.16 0.175]
[0.615 0.36 0.55 0.37 0.56 ]]
[[0.21 0.195 0.045 0.28 0.125]
[0.505 0.28 0.55 0.37 0.56 ]]]
\ Intuitionistic fuzzy comprehensive evaluation results R:
[[0.43881137 0.66530451 0.72657302 0.75533566 0.61738068]
[0.06251603 0.03011254 0.02454705 0.02523074 0.01611394]]
Score of each scheme :
[0.37629535 0.63519197 0.70202597 0.73010491 0.60126674]
Sorting result :
[4 3 2 5 1