Evaluation of alternative policy options.pptx

multicriteria Analysis

1. Evaluation of alternative policy options Analytical Hierarchy Process (AHP) 1

2. Analytical Hierarchy Process (AHP) The analytic hierarchy process (AHP) was developed by Thomas L. Saaty . Saaty , T.L., The Analytic Hierarchy Process , New York: McGraw-Hill, 1980 The AHP is designed to solve complex problems involving multiple criteria . An advantage of the AHP is that it is designed to handle situations in which the subjective judgments of individuals constitute an important part of the decision process.

3. Analytical Hierarchy Process (AHP) Procedure – Judgments and Comparisons Numerical Representation Relationship between 2 elements that share common parent in the hierarchy Comparisons ask 2 questions: – Which is more important with respect to the criterion? – How strongly? Matrix shows results of all such comparisons Typically uses a 1-9 scale Inconsistency may arise

4. Pairwise Comparisons Pairwise comparisons are fundamental building blocks of the AHP. The AHP employs an underlying scale with values from 1 to 9 to rate the relative preferences for two items.

5. Basic AHP Procedure Step 1 Develop a graphical representation of the problem Step 2 Develop the weights for the criteria by (Ranking Criteria) Developing a single pair-wise comparison matrix for the criteria Multiplying the values in each row together and calculating the nth root of said product Normalizing the aforementioned nth root of products to get the appropriate weights and Calculating and checking the Consistency Ration (CR) Step 3 Ranking alternatives Step 4 Develop priority ranking.

6. Example Factor Factor Weighing score Factor More improtant than Equal   C1 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 C2 C2 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 C3 C3 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 C1 Criteria C1 C2 C3 C1 1 4 5 C2 1/4 1 1/2 C3 1/5 2 1 Pair wise comparison matrix which holds the preference values Primary questionnaire design: effective criteria and pair wise comparison

7. Example: Choosing a Remote Area Power Generation System Objective Selecting a Remote Area Power System Criteria Alternatives Diesel Generator only (DG) Diesel and Photovoltaic (DG-PV) Diesel, Photovoltaic and Battery (DG-PV-Battery) Capital Cost O&M NPC COE CO2 Ems Fuel Cons

8. Hierarchy Development The first step in the AHP is to develop a graphical representation of the problem in terms of the overall goal , the criteria , and the decision alternatives . Overall Goal: Criteria: Decision Alternatives: Choose the Best System Capital Cost COE O&M NPC Fuel Consumption CO2 Emission DG Only DG-PV DG-PV-Battery

9. Ranking of Criteria   Capital cost O&M NPC COE CO2 Ems Fuel Cons Priority Capital Cost 1 3 5 7 6 4 0.438 O&M 0.333333 1 3 6 5 2 0.228 NPC 0.2 0.333333 1 5 4 0.5 0.114 COE 0.142857 0.166667 0.2 1 0.5 0.2 0.033 CO2 Ems 0.166667 0.2 0.25 2 1 0.5 0.053 Fuel Cons 0.25 0.5 2 5 2 1 0.135

10. Priority Calculation for Ranking Criteria Matrix A Find Eigen vector (Sum each row) Normalised the Eigen vector (each row /sum of column). e.g. (26/71.44 = 0.364) Calculate Matrix A 2 Find Eigen Vector of Matrix A 2 (Sum of each row) Normalised Eigen Vector for Matrix A 2 Eigenvector   Capital cost O&M NPC COE CO2 Ems Fuel Cons Eigen Vector Normalized Eigen Vector Capital Cost 6 12.03333 29.9 89 58.5 20.9 216.33 0.439 O&M 3.457143 6 14.11667 49.33333 31 10.53333 114.44 0.232 NPC 2.017063 3.15 6 23.9 14.36667 5.466667 54.90 0.111 COE 0.514603 1.028571 2.139286 6 3.890476 1.654762 15.23 0.031 CO2 Ems 0.860714 1.566667 3.333333 10.11667 6 2.591667 24.47 0.050 Fuel Cons 2.114286 3.65 8.25 28.75 18.5 6 67.26 0.137   Capital cost O&M NPC COE CO2 Ems Fuel Cons EigenVector (Sum of row) Normalized Eigen Vector (Sum of row/total of sum of eigenvector) Capital Cost 1 3 5 7 6 4 26.000 0.364 O&M 0.333333 1 3 6 5 2 17.333 0.243 NPC 0.2 0.333333 1 5 4 0.5 11.033 0.154 COE 0.142857 0.166667 0.2 1 0.5 0.2 2.210 0.031 CO2 Ems 0.166667 0.2 0.25 2 1 0.5 4.117 0.058 Fuel Cons 0.25 0.5 2 5 2 1 10.750 0.150 Matrix A Matrix A 2

11. Priority Calculation… Calculate the difference between both Normalised Eigen Vector Sum A 2 Normalised Eigen Vector Row Sum – A Normalised Eigen Vector Row Sum Keep multiplying the matrix until the difference is close to zero or zero. A 2 A Diff. 0.439 0.364 0.075 0.232 0.243 -0.010 0.111 0.154 -0.043 0.031 0.031 0.000 0.050 0.058 -0.008 A 3 A 2 Diff. 0.4378 0.439 -0.001 0.2278 0.232 -0.004 0.1137 0.111 0.002 0.0332 0.031 0.002 0.0527 0.050 0.003

12. Ranking Alternatives Pairwise comparisons among alternatives on O&M   DG Only DG-PV DG-PV-Battery Priority DG Only 1 0.5 0.333333 0.163 DG-PV 2 1 0.5 0.297 DG-PV-Battery 3 2 1 0.540 Pairwise comparisons among alternatives on capital cost   DG Only DG-PV DG-PV-Battery Priority DG Only 1 0.5 0.25 0.143 DG-PV 2 1 0.5 0.286 DG-PV-Battery 4 2 1 0.571 Pairwise comparisons among alternatives on NPC   DG Only DG-PV DG-PV-Battery Priority DG Only 1 2 3 0.540 DG-PV 0.5 1 2 0.297 DG-PV-Battery 0.3333 0.5 1 0.163 'Pairwise comparisons among alternatives on COE   DG Only DG-PV DG-PV-Battery Priority DG Only 1 2 4 0.558 DG-PV 0.5 1 3 0.320 DG-PV-Battery 0.25 0.333333 1 0.122

13. Procedure for Synthesizing Judgments The following three-step procedure provides a good approximation of the synthesized priorities. First: Sum the values in each column of the pairwise comparison matrix. Second: Divide each element in the pairwise matrix by its column total. The resulting matrix is referred to as the normalized pairwise comparison matrix . Third: Compute the average of the elements in each row of the normalized matrix. These averages provide an estimate of the relative priorities of the elements being compared.

14. Example: Synthesizing Step 0: Prepare pairwise comparison matrix Pairwise Comparisons Among Alternatives On Capital Cost   DG Only DG-PV DG-PV-Battery Priority DG Only 1 1/2 1/4 0.143 DG-PV 2 1 1/2 0.286 DG-PV-Battery 4 2 1 0.571

15. Example: Synthesizing Procedure Step 1: Sum the values in each column. Pairwise comparisons among alternatives on capital cost   DG Only DG-PV DG-PV-Battery Priority DG Only 1 0.5 0.25 DG-PV 2 1 0.5 DG-PV-Battery 4 2 1 SUM (column) Pairwise comparisons among alternatives on capital cost   DG Only DG-PV DG-PV-Battery Priority DG Only 1 0.5 0.25 DG-PV 2 1 0.5 DG-PV-Battery 4 2 1 SUM (column)

16. Example: Synthesizing Procedure Continued… Step 2: Divide each element of the matrix by its column total. All columns in the normalized pairwise comparison matrix now have a sum of 1. Pairwise comparisons among alternatives on capital cost   DG Only DG-PV DG-PV-Battery Priority DG Only 1 0.5 0.25 DG-PV 2 1 0.5 DG-PV-Battery 4 2 1 SUM (column) 1+2+4= 7 0.5+1+2= 3.5 0.25+0.5+1= 1.75 Pairwise comparisons among alternatives on capital cost   DG Only DG-PV DG-PV-Battery Priority DG Only 1/7 0.5/3.5 0.25/1.75 DG-PV 2/7 1/3.5 0.5/1.75 DG-PV-Battery 4/7 2/3.5 1/1.75 SUM (column) ( Pairwise comparisons among alternatives on capital cost   DG Only DG-PV DG-PV-Battery Priority DG Only 1/7 0.5/3.5 0.25/1.75 DG-PV 2/7 1/3.5 0.5/1.75 DG-PV-Battery 4/7 2/3.5 1/1.75 SUM (column)

17. Example: Synthesizing Procedure Continued…. Step 3: Average the elements in each row. The values in the normalized pairwise comparison matrix have been converted to decimal form. The result is usually represented as the (relative) priority vector . Pairwise comparisons among alternatives on capital cost   DG Only DG-PV DG-PV-Battery Priority DG Only A = 1/7 B = 0.5/3.5 C = 0.25/1.75 Avg.(A, B, C ) = 0.143 DG-PV D = 2/7 E = 1/3.5 F = 0.5/1.75 Avg.(D, E, F ) = 0.286 DG-PV-Battery G = 4/7 H = 2/3.5 I = 1/1.75 Avg.(G, H, I ) = 0.571 Sum 1 1 1 1

18. Consistency AHP provides a method for measuring the degree of consistency among the pairwise judgments provided by the decision maker. If the degree of consistency is acceptable, the decision process can continue . If the degree of consistency is unacceptable, the decision maker should reconsider and possibly revise the pairwise comparison judgments before proceeding with the analysis. The consistency of pairwise comparison judgments is calculate by computing a Consistency Ratio . The ratio is designed in such a way that values of the ratio exceeding 0.10 are indicative of inconsistent judgments. Although the exact mathematical computation of the consistency ratio is beyond the scope of this text, an approximation of the ratio can be obtained.

19. Procedure: Estimating Consistency Ratio Step 1: Multiply each value in the first column of the pairwise comparison matrix by the relative priority of the first item considered. Same procedures for other items. Sum the values across the rows to obtain a vector of values labeled “ weighted sum .” Step 2: Divide the elements of the vector of weighted sums obtained in Step 1 by the corresponding priority value. Step 3: Compute the average of the values computed in step 2. This average is denoted as l max .

20. Procedure: Estimating Consistency Ratio - 2 St ep 4: Compute the consistency index (CI): Where n is the number of items being compared Step 5: Compute the consistency ratio (CR): Where RI is the random index , which is the consistency index of a randomly generated pairwise comparison matrix. It can be shown that RI depends on the number of elements being compared and takes on the following values.

21. Example: Inconsistency Preferences: If, DG-PV-Battery  DG-Only (4); DG-PV-Battery  DG-PV (2) Then, DG-PV  DG Only (should be 2 , If it is 4 or 6 or 8, it may pose inconsistency) Inconsistency Pairwise comparisons among alternatives on capital cost   DG Only DG-PV DG-PV-Battery DG Only 1 ½ ¼ DG-PV 8 1 ½ DG-PV-Battery 4 2 1

22. Example: Consistency Checking Step 1: Multiply each value in the first column of the pairwise comparison matrix by the relative priority of the first item considered. Same procedures for other items. Sum the values across the rows to obtain a vector of values labeled “ weighted sum .” Pairwise comparisons among alternatives on capital cost   DG Only DG-PV DG-PV-Battery Priority DG Only 1 0.5 0.25 0.143 DG-PV 2 1 0.5 0.286 DG-PV-Battery 4 2 1 0.571   Weighted sum Priority DG Only = 0.4286 DG-PV = 0.8571 DG-PV-Battery = 1.7142   Weighted sum Priority DG Only 0.4286 DG-PV 0.8571 DG-PV-Battery 1.7142

23. Example: Consistency Checking continued…. Step 2: Divide the elements of the vector of weighted sums by the corresponding priority value. Step 3: Compute the average of the values computed in step 2 ( lmax ). Weighted sum Priority value Resultant 0.4286 0.143 3 0.8571 0.286 3 1.7142 0.571 3

24. Example: Consistency Checking – continued…. Step 4: Compute the consistency index (CI). Step 5: Compute the consistency ratio (CR). The degree of consistency exhibited in the pairwise comparison matrix for Capital cost is acceptable.

25. Development of Priority Ranking The overall priority for each decision alternative is obtained by summing the product of the criterion priority (i.e., weight) (with respect to the overall goal ) times the priority (i.e., preference) of the decision alternative with respect to that criterion . Ranking these priority values, will give AHP ranking of the decision alternatives.

26. Priority Ranking Capital Cost O&M NPC COE CO2 Ems Fuel Cons DG 0.1429 0.1634 0.5396 0.5584 0.6548 0.6548 DG-PV 0.2857 0.2970 0.2970 0.3196 0.2499 0.2499 DG-PV-Battery 0.5714 0.5396 0.1634 0.1220 0.0953 0.0953 X   Step 1: Sum the product of the criterion priority (with respect to the overall goal ) times the priority of the decision alternative with respect to that criterion .     Priority ranking of criteria Final Ranking

27. Priority Ranking Step 2: Rank the priority values. Alternative Priority DG 0.302 DG-PV 0.284 DG-PV-Battery 0.414 Total 1.000

28. Choose the Best System Capital Cost COE O&M NPC Fuel Consumption CO2 Emission DG Only DG-PV DG-PV-Battery

29. Random Index Random index (RI) is the consistency index of a randomly generated pairwise comparison matrix. RI depends on the number of elements being compared (i.e., size of pairwise comparison matrix) and takes on the following values: n 1 2 3 4 5 6 7 8 9 10 RI 0.00 0.00 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.49 Back