理论基础

\quad 模糊层次分析法（$FAHP$）将模糊理论（$FuzzySet$）嵌入到基本层次分析法（$AHP$）中。$AHP$是一种广泛应用于各种多准则决策问题的决策工具,它将不同的备选方案与不同的标准进行成对比较，并为多标准决策问题提供了决策支持工具。在一般的AHP模型中，目标在第一层，标准和子标准分别在第二层和第三层，第四个层次为备选方案。由于基本AHP不包括个人判断的模糊性，它通过受益于模糊逻辑方法得到了改进。在FAHP中，通过语言变量对标准和备选方案进行成对比较，语言变量由三角模糊数等表示。

三角模糊数概念

\quad 定义：三角模糊数$\tilde{A}=(a_l,a_m,a_u)$，$a_l、a_m和a_u$分别为下届、中值和上届，其隶属函数表示如下：
$${\mu }_{\tilde{A}}=\{\begin{array}{l}\frac{x-a_l}{a_m-a_l},a_l\le x\le a_m\\ \frac{a-a_u}{a_m-a_u},a_m\le x\le a_u\\ 0,其他\end{array}$$

\quad 运算方法: $M_1=\left(l_1,m_1,u_1\right)$，$M_2=\left(l_2,m_2,u_2\right)$
$$\{\begin{array}{l}{M}_1⨁M_2=\left(l_1+l_1,m_1+m_2,u_1+u_2\right)\\ M_1⨂M_2\approx \left(l_1{l}_2,m_1{m}_2,u_1{u}_2\right)\\ \frac{1}{M_1}\approx \left(\frac{1}{u},\frac{1}{m},\frac{1}{l}\right)\end{array}$$

表1.语言术语及其对应的三角模糊数

Saaty等级	定义	三角模糊数
1	同等重要(EI)	(1,1,1)
3	稍微重要(WI)	(2,3,4)
5	相当重要(FI)	(4,5,6)
7	非常重要(SI)	(6,7,8)
9	绝对重要(AI)	(9,9,9)
2 4 6 8	两个相邻刻度之间的间歇值	(1,2,3) (3,4,5) (5,6,7) (7,8,9)

\quad 如果C1比C2弱重要，则$C1\to C2$取三角模糊数$(2,3,4)$,$C1\leftarrow C2$取三角模糊数$(\frac{1}{4},\frac{1}{3},\frac{1}{2})$

S1. 建立三角模糊成对比较矩阵

\quad 第k个专家评价结果为：
$${\tilde{A}}^k=\left[\begin{array}{cccc}{\tilde{d}}_{11}^k& {\tilde{d}}_{12}^k& \dots & {\tilde{d}}_{1n}^k\\ {\tilde{d}}_{21}^k& {\tilde{d}}_{22}^k& \dots & {\tilde{d}}_{2n}^k\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{d}}_{n1}^k& {\tilde{d}}_{n2}^k& \dots & {\tilde{d}}_{nn}^k\end{array}\right]\text{\hspace{1em}(1)}$$

S2. 当存在多个决策者时，计算每个决策者的偏好($\widetilde{d_{ij}^k}$)的平均值，$\widetilde{d_{ij}^k}$的计算表达式如下：
$$\widetilde{d_{ij}^k}=\frac{{\sum }_{k=1}^K{\tilde{d}}_{ij}^k}{K}\text{\hspace{1em}(2)}$$

S3. 根据平均偏好，更新成对比较矩阵
$$\tilde{A}=\left[\begin{array}{cccc}\widetilde{d_{11}}& \widetilde{d_{12}}& \dots & \widetilde{d_{1n}}\\ \widetilde{d_{21}}& \widetilde{d_{22}}& \dots & \widetilde{d_{2n}}\\ ⋮& ⋮& \ddots & ⋮\\ \widetilde{d_{n1}}& \widetilde{d_{n2}}& \dots & \widetilde{d_{nn}}\end{array}\right]\text{\hspace{1em}(3)}$$

S4. 根据Buckley，计算每个标准的模糊比较值的几何平均值$\widetilde{r_i}$(三角数)
$$\widetilde{r_i}={\left(\prod _{j=1}^n{\tilde{d}}_{ij}\right)}^{\frac{1}{n}},i=1,2,\dots ,n\text{\hspace{1em}(4)}$$

S5. 计算每个标准的模糊权重：求每个$\widetilde{r_i}$的矢量和；求和向量的$-1$次方,替换模糊三角数，使其按递增顺序排列；计算$\widetilde{r_i}$和它的逆向量的乘积
$${\tilde{w}}_i=\widetilde{r_i}⨂{\left(\widetilde{r_1}⨁\widetilde{r_2}⨁\dots ⨁\widetilde{r_n}\right)}^{-1}=\left(l{w}_i,m{w}_i,u{w}_i\right)\text{\hspace{1em}(5)}$$

S6.反模糊化计算
$$w_i=\frac{l{w}_i+m{w}_i+u{w}_i}{3}\text{\hspace{1em}(6)}$$

S7. 归一化处理
$$n{w}_i=\frac{w_i}{{\sum }_{n=1}^n{w}_i}\text{\hspace{1em}(7)}$$

\quad 然后，通过将每个备选权重与相关标准相乘，计算每个备选的分数。根据这些结果，向决策者建议得分最高的备选方案。

参考

1. Cheng, C.H. (1997). Evaluating Naval Tactical Missile System by Fuzzy AHP Based on the Grade Value of Membership Function. European Journal of Operational Research,96(2),343-350.
2. Cheng, C.H., Yang, L.L., and Hwang, C.L. (1999). Evaluating Attack Helicopter by AHP Based on Linguistic Variable Weight. European Journal of Operational Research,116(2),423-435.
3. Ruoning, X. and Xiaoyan, Z. (1992). Extensions of the Analytic Hierarchy Process in Fuzzy Environment. Fuzzy Sets and System,52(3),251-257.
4. Petkovic, J., Sevarac, Z., Jaksic, M.L., Marinkovic, S. (2012). Application of fuzzy AHP method for choosing a technology within service company. Technics Technologies Education Management,7(1),332-341.
5. Kilic, H.S. (2011). A fuzzy AHP based performance assessment system for the strategic plan of Turkish Municipalities. International Journal of Business and Management Studies, 3(2),77-86.
6. Kilincci, O., & Onal, S. A. (2011). Fuzzy AHP approach for supplier selection in a washing machine company. Expert Systems with Applications, 38(8),9656-9664.
7. Ayhan, M.B.(2013). A Fuzzy AHP Approach for Supplier Selection Problem: A Case Study in A Gearmotor Company. International Journal of Managing Value and Supply Chains (IJMVSC),4(3).

Python源码

import numpy as np# Function: Fuzzy AHP
def fuzzy_ahp_method(dataset):'''dataset: 专家判断矩阵，n*n*3，n表示指标数'''row_sum = []s_row   = []f_w     = []  #模糊权重d_w     = []  #去模糊权重# 一致性检测指标inc_rat  = np.array([0, 0, 0, 0.58, 0.9, 1.12, 1.24, 1.32, 1.41, 1.45, 1.49, 1.51, 1.48, 1.56, 1.57, 1.59])# 模糊判断矩阵转为清晰判断矩阵X       = [(item[0] + 4*item[1] + item[2])/6 for i in range(0, len(dataset)) for item in dataset[i]]  #三角数转为模糊数X       = np.asarray(X)  #列表转数组X       = np.reshape(X, (len(dataset), len(dataset)))  #转为n*n矩阵# S4. 计算每个准则的模糊比较值的几何平均值r_i(公式4)for i in range(0, len(dataset)):a, b, c = 1, 1, 1for j in range(0, len(dataset[i])):d, e, f = dataset[i][j]# 计算公式(4)的括号内部分: r_s = \prod_{j=1}^n d_ija, b, c = a*d, b*e, c*frow_sum.append( (a, b, c) )L, M, U = 0, 0, 0for i in range(0, len(row_sum)):a, b, c = row_sum[i]# 计算公式(4)的括号外部分: s_r = (r_s)^(1/n)a, b, c = a**(1/len(dataset)), b**(1/len(dataset)), c**(1/len(dataset))s_row.append( ( a, b, c ) )# 计算公式(5)中⨁运算部分：R5 = r1⨁r2⨁...⨁rnL = L + aM = M + bU = U + cfor i in range(0, len(s_row)):a, b, c = s_row[i]# 计算公式公式(5)中⨂运算部分：wi = ri⨂R5a, b, c = a*(U**-1), b*(M**-1), c*(L**-1)# 模糊权重f_w.append( ( a, b, c ) )# 计算公式(6):去模糊权重d_w.append( (a + b + c)/3 )# 计算公式(7): 归一化权重n_w      = [item/sum(d_w) for item in d_w]# 计算特征根向量vector   = np.sum(X*n_w, axis = 1)/n_w# 获得平均特征根lamb_max = np.mean(vector)# 计算一致性指标cons_ind = (lamb_max - X.shape[1])/(X.shape[1] - 1)# 一致性判断rc       = cons_ind/inc_rat[X.shape[1]]return f_w, d_w, n_w, rc

测试

dataset = list([[ (  1,   1,   1), (  4,   5,   6), (  3,   4,   5), (  6,   7,   8) ],   #g1[ (1/6, 1/5, 1/4), (  1,   1,   1), (1/3, 1/2, 1/1), (  2,   3,   4) ],   #g2[ (1/5, 1/4, 1/3), (  1,   2,   3), (  1,   1,   1), (  2,   3,   4) ],   #g3[ (1/8, 1/7, 1/6), (1/4, 1/3, 1/2), (1/4, 1/3, 1/2), (  1,   1,   1) ]    #g4])
fuzzy_weights, defuzzified_weights, normalized_weights, rc = fuzzy_ahp_method(dataset)
print('模糊权重')
for i in range(0, len(fuzzy_weights)):print('g'+str(i+1)+': ', np.around(fuzzy_weights[i], 3))
print('清晰权重')
for i in range(0, len(defuzzified_weights)):print('g'+str(i+1)+': ', round(defuzzified_weights[i], 3))
print('归一化权重')
for i in range(0, len(normalized_weights)):print('g'+str(i+1)+': ', round(normalized_weights[i], 3))print('一致性判断')
print('RC: ' + str(round(rc, 2)))
if (rc > 0.10):print('The solution is inconsistent, the pairwise comparisons must be reviewed')
else:print('The solution is consistent')

输出结果为：

模糊权重
g1:  [0.428 0.61  0.859]
g2:  [0.085 0.131 0.218]
g3:  [0.117 0.196 0.309]
g4:  [0.044 0.063 0.099]
清晰权重
g1:  0.632
g2:  0.145
g3:  0.207
g4:  0.068
归一化权重
g1:  0.601
g2:  0.138
g3:  0.197
g4:  0.065
一致性判断
RC: 0.06
The solution is consistent