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A~=(al,am,au)\\tilde{A}=(a_l,a_m,a_u)A~=(al​,am​,au​)，al、am和aua_l、a_m和a_ual​、am​和au​分别为下届、中值和上届，其隶属函数表示如下：
μA~={x−alam−al,al≤x≤ama−auam−au,am≤x≤au0,其他\\mu_{\\tilde{A}}=\\begin{cases} \\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{cases} μA~​=⎩⎨⎧​am​−al​x−al​​,al​≤x≤am​am​−au​a−au​​,am​≤x≤au​0,其他​

\\quad运算方法: M1=(l1,m1,u1)M_1=\\left(l_1,m_1,u_1\\right)M1​=(l1​,m1​,u1​)，M2=(l2,m2,u2)M_2=\\left(l_2,m_2,u_2\\right)M2​=(l2​,m2​,u2​)
{M1⨁M2=(l1+l1,m1+m2,u1+u2)M1⨂M2≈(l1l2,m1m2,u1u2)1M1≈(1u,1m,1l)\\begin{cases} M_1 \\bigoplus \\ M_2 = \\left(l_1 + l_1, m_1 + m_2, u_1 + u_2\\right) \\\\ M_1 \\bigotimes \\ 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{cases} ⎩⎨⎧​M1​⨁ M2​=(l1​+l1​,m1​+m2​,u1​+u2​)M1​⨂ M2​≈(l1​l2​,m1​m2​,u1​u2​)M1​1​≈(u1​,m1​,l1​)​

表1.语言术语及其对应的三角模糊数

\\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(14,13,12)(\\frac{1}{4},\\frac{1}{3},\\frac{1}{2})(41​,31​,21​)

S1. 建立三角模糊成对比较矩阵

\\quad第k个专家评价结果为：
A~k=[d~11kd~12k…d~1nkd~21kd~22k…d~2nk⋮⋮⋱⋮d~n1kd~n2k…d~nnk](1)\\tilde{A}^{k}= \\begin{bmatrix} \\tilde{d}_{11}^{k} & \\tilde{d}_{12}^{k} & \\ldots & \\tilde{d}_{1n}^{k} \\\\ \\tilde{d}_{21}^{k} & \\tilde{d}_{22}^{k} & \\ldots & \\tilde{d}_{2n}^{k} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\tilde{d}_{n1}^{k} & \\tilde{d}_{n2}^{k} & \\ldots & \\tilde{d}_{nn}^{k} \\\\ \\end{bmatrix} \\tag{1} A~k=​d~11k​d~21k​⋮d~n1k​​d~12k​d~22k​⋮d~n2k​​……⋱…​d~1nk​d~2nk​⋮d~nnk​​​(1)

S2. 当存在多个决策者时，计算每个决策者的偏好(dijk~\\tilde{d_{ij}^{k}}dijk​~​)的平均值，dijk~\\tilde{d_{ij}^{k}}dijk​~​的计算表达式如下：
dijk~=∑k=1Kd~ijkK(2)\\tilde{d_{ij}^{k}}=\\frac{\\sum_{k=1}^{K}\\tilde{d}_{ij}^{k}}{K}\\tag{2} dijk​~​=K∑k=1K​d~ijk​​(2)

S3. 根据平均偏好，更新成对比较矩阵
A~=[d11~d12~…d1n~d21~d22~…d2n~⋮⋮⋱⋮dn1~dn2~…dnn~](3)\\tilde{A}= \\begin{bmatrix} \\tilde{d_{11}} & \\tilde{d_{12}} & \\ldots & \\tilde{d_{1n}} \\\\ \\tilde{d_{21}} & \\tilde{d_{22}} & \\ldots & \\tilde{d_{2n}} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\tilde{d_{n1}} & \\tilde{d_{n2}} & \\ldots & \\tilde{d_{nn}} \\\\ \\end{bmatrix} \\tag{3} A~=​d11​~​d21​~​⋮dn1​~​​d12​~​d22​~​⋮dn2​~​​……⋱…​d1n​~​d2n​~​⋮dnn​~​​​(3)

S4. 根据Buckley，计算每个标准的模糊比较值的几何平均值ri~\\tilde{r_{i}}ri​~​(三角数)
ri~=(∏j=1nd~ij)1n,i=1,2,…,n(4)\\tilde{r_{i}}={\\left(\\prod_{j=1}^{n}\\tilde{d}_{ij}\\right)}^{\\frac{1}{n}},i=1,2,\\ldots,n \\tag{4} ri​~​=(j=1∏n​d~ij​)n1​,i=1,2,…,n(4)

S5. 计算每个标准的模糊权重：求每个ri~\\tilde{r_{i}}ri​~​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ri~\\tilde{r_{i}}ri​~​和它的逆向量的乘积
w~i=ri~⨂(r1~⨁r2~⨁…⨁rn~)−1=(lwi,mwi,uwi)(5)\\tilde{w}_{i} = \\tilde{r_{i}}\\bigotimes {\\left(\\tilde{r_{1}} \\bigoplus \\tilde{r_{2}} \\bigoplus \\ldots \\bigoplus \\tilde{r_{n}}\\right)}^{-1}=\\left(lw_i,mw_i,uw_i\\right) \\tag{5} w~i​=ri​~​⨂(r1​~​⨁r2​~​⨁…⨁rn​~​)−1=(lwi​,mwi​,uwi​)(5)

S6.反模糊化计算
wi=lwi+mwi+uwi3(6)w_i = \\frac{lw_i + mw_i + uw_i}{3} \\tag{6} wi​=3lwi​+mwi​+uwi​​(6)

S7. 归一化处理
nwi=wi∑n=1nwi(7)nw_i = \\frac{w_i}{\\sum_{n=1}^n w_i} \\tag{7} nwi​=∑n=1n​wi​wi​​(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