1. MCE: Eigen Values Calculations from Pair Wise Comparisons. Addition to Exercise 2-8

2. Pair Wise Comparison. The analytic hierarchy process (AHP) is a multi-criteria decision making method that employs a procedure of multiple comparisons to rank order alternative solutions to a multi-objective decision problem.

3. Given Pairwise comparison..

4. Calculate EigenVector by Edrisi.

5. The Calculation of eigenvector from the pairwise comparison values requires a lot of work in terms of matrix algebra. The method presented here is an approximation that suits to a good degree of accuracy in terms of eigenvalues.

6. Step-1 LandfuzzTownfuzzWaterfuzzRoadfuzzSlopefuzzdevelopfuzz Landfuzz 1.000 Townfuzz 3.0001.000 Waterfuzz 3.0001.000 Roadfuzz 3.0007.0003.0001.000 Slopefuzz 5.0003.000 1.000 developfuzz 1.0003.0001.0000.3330.2001.000

7. Step-2 Complete the upper diagonal values of the pair wise matrix. If ‘A’ is 5 time of ‘B’ means ‘B’ is 1/5 times of ‘A’. In the next diagram shaded values shows these values.

8. LandfuzzTownfuzzWaterfuzzRoadfuzzSlopefuzzdevelopfuzz Landfuzz 1.0003.000 5.0001.000 Townfuzz 3.0001.000 7.0003.000 Waterfuzz 3.0001.000 3.000 1.000 Roadfuzz 3.0007.0003.0001.000 0.333 Slopefuzz 5.0003.000 1.000 0.200 developfuzz 1.0003.0001.0000.3330.2001.000

9. Step-4 Calculate the sum of each column. LandfuzzTownfuzzWaterfuzzRoadfuzzSlopefuzzdevelopfuzz Landfuzz 1.0000.333 0.2001.000 Townfuzz 3.0001.000 0.1430.333 Waterfuzz 3.0001.000 0.333 1.000 Roadfuzz 3.0007.0003.0001.000 3.000 Slopefuzz 5.0003.000 1.000 5.000 developfuzz 1.0003.0001.0000.3330.2001.000 16.00015.3339.3333.1433.06711.333

10. Step-5 Divide the values in each cell by its column’s sum. Eg: Col-1: 1/16, 3/16 ….. Col-2: 0.333/15.33, 1/15.333 etc … LandfuzzTownfuzzWaterfuzzRoadfuzzSlopefuzzdevelopfuzz Landfuzz 0.0630.0220.0360.1060.0650.088 Townfuzz 0.1880.0650.1070.0450.1090.029 Waterfuzz 0.1880.0650.1070.1060.1090.088 Roadfuzz 0.1880.4570.3210.3180.3260.265 Slopefuzz 0.3130.1960.3210.3180.3260.441 developfuzz 0.0630.1960.1070.1060.0650.088

11. Step-6 Landfuz zTownfuzzWaterfuzzRoadfuzzSlopefuzzdevelopfuzz Landfuzz 0.0630.0220.0360.1060.0650.0880.379 Townfuzz 0.1880.0650.1070.0450.1090.0290.543 Waterfuzz 0.1880.0650.1070.1060.1090.0880.663 Roadfuzz 0.1880.4570.3210.3180.3260.2651.874 Slopefuzz 0.3130.1960.3210.3180.3260.4411.915 developfuzz 0.0630.1960.1070.1060.0650.0880.625 6.000 Sum the weights in each row. (R1, R2, R3 …) Add the results … Sum = (R1 + R2 + R3 + R4 + R5 + R6).

12. Step-7 0.3790.063 0.5430.091 0.6630.110 1.8740.312 1.9150.319 0.6250.104 6.000 Get the ratio of each of the individual row’s sum to the resulting sum. Eg. 0.379/6 = 0.063 etc.. The Weights we get are the approximated weights as in the values obtained through Idrisi.

13. References: The method for conversion taken from the lecture notes: mat.gsia.cmu.edu/classes/mstc/multiple/node4.html For the exact values calculation refer to the attached paper The Analysis of the Principal Eigenvector of Pairwise Comparison Matrices; András Farkas Excel file with the auto-calculation is also attached. A very good software for auto conversion from pairwise values is; http://www.isc.senshu- u.ac.jp/~thc0456/EAHP/EAHP_manu.html http://www.isc.senshu- u.ac.jp/~thc0456/EAHP/EAHP_manu.html