层次分析法和灰色聚类分析法在绩效评估中的应用.docx

考核指标比值考核指标*12*3*111.1332*0.8912.673*0.330.381相对重要程度定义解释1同等重要目标i和j同样重要3略微重要目标i比j略微重要5相当重要目标i比j重要7明显重要目标i比j明显重要9绝对重要目标i和j绝对重要2、4、6、8介于两相邻重要程度间标记为1,2,3。层次分析法和灰色聚类分析法在绩效评估中的应用施狄峰摘要绩效考核的评估是帮助企业维持和提高生产力、实现企业经营目标的手段之一，它一个复杂的大系统，一般企业的绩效评估是建立在关键考核指标得分乘以权系数的线性关系的基础上，但如果有两个下属分公司考核得分分别是97分和94分，究竟它们都属于优，还是一个是优、一个是良，原先的方法显然无法判断。笔者运用运筹学决策分析法的层次分析法和灰色系统理论的灰聚类法两种方法对绩效加以评估，能将被考核企业的经营情况很清楚地区分开来，分类排序出来。关键词绩效评估层次分析法灰色聚类分析法设以某公司下属11个分公司绩效考核情况数据为例，记为i=A,B,CLK;并选取经营效绩考核中三个指*一、用层次分析法：1、权重设置：根据三个指标的权重系数A1，A2，A3两两比较，得到数值aij,其定义和解释见表1，得出表2所示系数。表1表2得到矩阵A=(aij)33矩阵A为经营效绩的判断矩阵。A=相应的特征向量为：B3=(0.450.400.15)T得出3个考核指标权重分别为0.45、0.40、0.151ABCDEFGHIJKA1.000.881.131.020.971.291.061.071.021.241.15B1.141.001.291.161.101.471.211.221.161.411.30C0.880.781.000.900.861.140.940.950.901.101.01D0.980.861.111.000.951.271.041.051.001.221.12E1.030.911.171.051.001.331.091.111.051.281.18F0.770.680.880.790.751.000.820.830.790.960.89G0.940.831.070.960.911.221.001.010.961.171.08H0.930.821.060.950.901.200.991.000.951.161.07I0.980.861.111.000.951.271.041.051.001.221.12J0.800.710.910.820.781.040.850.860.821.000.92K0.870.770.990.890.851.130.930.940.891.091.00下属分公司代号*12*3*A1.020.971.05B1.161.101.07C0.900.791.01D1.000.981.13E1.051.061.26F0.790.641.10G0.961.001.06H0.951.111.07I1.001.140.92J0.820.710.98K0.891.001.11ABCDEFGHIJKA1.000.881.230.990.921.520.970.870.851.370.97B1.131.001.391.121.041.721.100.990.961.551.10C0.810.721.000.810.751.230.790.710.691.110.79D1.010.891.241.000.921.530.980.880.861.380.98E1.090.961.341.081.001.661.060.950.931.491.06F0.660.580.810.650.601.000.640.580.560.900.64G1.030.911.271.020.941.561.000.900.881.411.00H1.141.011.411.131.051.731.111.000.971.561.11I1.181.041.441.161.081.781.141.031.001.611.142、类似地根据表3可用特征向量法求下属11个分公司相对于上述3个指标中每一个的权系数。成对比较的结果如表4、表5、表6所示：表3指标1：*表4指标2：*2ABCDEFGHIJKA1.000.981.040.930.830.950.990.981.141.070.95B1.021.001.060.950.850.971.011.001.161.090.96C0.960.941.000.890.800.920.950.941.101.030.91D1.081.061.121.000.901.031.071.061.231.151.02E1.201.181.251.121.001.151.191.181.371.291.14F1.051.031.090.970.871.001.041.031.201.120.99G1.010.991.050.940.840.961.000.991.151.080.95H1.021.001.060.950.850.971.011.001.161.090.96I0.880.860.910.810.730.840.870.861.000.940.83J0.930.920.970.870.780.890.920.921.071.000.88K1.061.041.100.980.881.011.051.041.211.131.00J0.730.650.900.720.671.110.710.640.621.000.71K1.030.911.271.020.941.561.000.900.881.411.00表5指标3：*表63、由此可求出3个指标的相应特征向量，按列组成矩阵B3。B3=若记Bk为第k层次上所有因素相对于上一层上有关因素的权向量按列组成的矩阵，则第k层次的组合权系数向量Wk满足：Wk=BkBk-1B2B1由W3=B3B2=(0.09380.10500.08150.09440.10130.07210.09260.09650.09790.07450.0903)T可以得出以下11个分公司经营绩效排名：3排名分公司考核结果向量1B0.10502E0.10133I0.09794H0.09655D0.09446A0.09387G0.09268K0.09039C0.081510J0.074511F0.0721上例经营效绩考核中三个指标为聚类指标，记为1,2,3；以好、中、差为三个考核结果等级，记为k表7以上矩阵特征向量的计算是根据方根法近似计算。二、用灰色聚类分析法：*=1,2，3；以下按11个分公司3个考核指标(表3数据)进行聚类。第一步、给出聚类白化值dij(如果灰类的白化值相差很大，应先作无量纲处理)1.161.00dij=0.790.961.000.891.131.060.921.111.020.901.050.950.820.971.100.790.981.060.641.001.111.140.711.001.051.071.011.261.101.070.98第二步、确定灰类的白化函数设fjk为j项考核指标第k个灰类的白化函数。1*考核指标好的灰数为111.1,)，表示1*考核指标在1.1以上就算好；2*考核指标好的灰数为211.1,)，表示2*考核指标在1.1以上就算好；43*考核指标好的灰数为311.2,)，表示1*1*考核指标在0.9左右为中，120.9x；2*考核指标在0.9左右为中，220.9x；3*考核指标在1.0左右为中，321.0x；1*考核指标在0.8以下为差，130,0.8；2*考核指标在0.8以下为差，230,0.8；3*考核指标在0.9以下为差，330,0.9；考核指标在1.2以上就算好；n第3步、求标定聚类权rjk，利用公式rjk=ljk*lj=1*jk,k=1,2,3L,m可求得1*考核指标属于好的权r11为lr11=l113*j=1*j1=l11l11+l21+l31=1.11.1+1.1+1.2=0.324同理，2考核指标属于好的权r21为0.324；*3*考核指标属于好的权r31为0.353；1*考核指标属于中的权r12为lr12=l123*j=1*j2=l12l12+l22+l32=0.90.9+0.9+1=0.321同理，2考核指标属于中的权r22为0.321；*3*考核指标属于中的权r32为0.357；1*考核指标属于差的权r13为lr13=l133*j=1*j3=l13l13+l23+l33=0.80.8+0.8+0.9=0.32同理，2考核指标属于差的权r23为0.32；*53*考核指标属于差的权r33为0.36；f第四步、求聚类系数sik，利用公式sik=n*j=1*jk(dij)rjk可得第1个分公司属于好的聚类系数s11s11=fj1(d1j)rj1=f11(d11)r11+f21(d12)r21+f31(d13)r313*j=1*=f11(1.02)0.324+f21(0.97)0.324+f31(1.05)0.353s12=fj2(d1j)rj2=f12(d11)r12+f22(d12)r22+f32(d13)r32图1查图1(1)，(4)，(7)，有f11(1.02)=0.927，f21(0.97)=0.882，f31(1.05)=0.875s11=0.9270.324+0.8820.324+0.8750.353=0.8950第1个分公司属于中的聚类系数s123*j=1*=f12(1.02)0.321+f22(0.97)0.321+f32(1.05)0.3576=31313)(jjjrdfs331333231223131113)()()(rdfrdfrdf+查图1(2)，(5)，(8)，有f12(1.02)=0.867，f22(0.97)=0.922，f32(1.05)=0.95s12=0.8670.321+0.9220.321+0.950.357=0.9134第1个分公司属于差的聚类系数s133*j=1*=f13(1.02)0.32+f23(0.97)0.32+f32(1.05)0.36查图1(3)，(6)，(9)，有f13(1.02)=0.8，f23(0.97)=0.845，f33(1.05)=0.875s13=0.80.32+0.8450.32+0.8750.36=0.8414第五步、构造聚类行向量对于第1个分公司的聚类行向量记为s1s1=s11s12s13=0.89500.91340.8414第六步：聚类由于第1个分公司聚类向量中，以s12为最大，所以第1个分公司属于第2类，即考核结果为中，同理可求：s2=s21s22s23=0.96290.81000.7569s3=s31s32s33=0.79490.95630.9378s4=s41s42s43=0.91570.88840.8202s5=s51s52s53=0.97480.79540.7438s6=s61s62s63=0.74490.83140.9399s7=s71s72s73=0.88910.92040.8475s8=s81s82s83=0.91880.88120.8151s9=s91s92s93=0.88930.84910.8368s10=s101s102s103=0.73880.89560.97017排序下属分公司代号*12*3*总分得分乘以权重45%后得分得分乘以权重40%后得分得分乘以权重15%后得分1B1.1652.201.1044.001.0716.05112.252E1.0547.251.0642.401.2618.90108.553I1.0045.001.1445.600.9213.80104.404H0.9542.751.1144.401.0716.05103.205D1.0045.000.9839.201.1316.95101.156A1.0245.900.9738.801.0515.75100.457G0.9643.201.0040.001.0615.9099.108K0.8940.051.0040.001.1116.6596.709C0.9040.500.7931.601.0115.1587.2510J0.8236.900.7128.400.9814.7080.0011F0.7935.550.6425.601.1016.5077.65s11=s111s112s113=0.88320.92