Leader Culling Using AHP - PROMETHEE Methodology

Transcription

1 Chapter 4 Leader Culling Using AHP - PROMETHEE Methodology 4.1 Introduction In earlier two chapters we worked on Multi Objective Decision Making (MODM) models where multiple objectives are simultaneously considered for optimization purposes. We now focus on different Multi Attribute Decision Making (MADM) approaches which deal with the issues in which the decision maker needs to choose an attribute among available ones which are evaluated with different characteristics. Advances in the decision sciences have led to the development of a number of Multiple Criteria Decision Making (MCDM) methods. These MCDM methods are designed to help people make better choices when faced with complex decisions 96

2 involving several dimensions. They are especially helpful when there is a need to combine hard data with subjective preferences, to make trade-offs between desired outcomes, and to involve multiple decision makers. The candidate selection process involves thoughtful empirical decisions which determine the best fit from a pool of contesting candidates. candidate is of vital importance for a responsible citizen. Selecting a suitable In this chapter, we demonstrate the implementation of an integrated approach that employs Analytical Hierarchy Process (AHP) and Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE) together for selecting the most suitable candidate for the election purpose. AHP is used to determine weights of the criteria and PROMETHEE is used to obtain the final ranking of the candidates. A wide range of criteria is used to assess the candidates. An improved PROMETHEE II methodology is utilized which efficiently establishes its applicability and potentiality to solve such types of decision-making problems with multiple conflicting criteria and alternatives. The employed methodology utilizes systematic approach to screen and ultimately select the most suitable candidate. The factors considered here explore personal as well as professional aspects of the aspiring candidates. The developed methodology facilitates the selection procedure in a rational and transparent way that can be examined and understood by the concerned voters. The study is substantiated using Simple Additive Weighting (SAW) technique. PROMETHEE is a multi criteria decision making method developed by Brans(1982). 97

3 It is a ranking method well adapted to problems where a finite number of alternatives are to be ranked considering several, sometimes conflicting, criteria. Since then it has successively been applied in many fields especially in the investment analysis and performance evaluation. Albadvi(2007), Babic & Plazibat(1998), Bouri & Martel(2000), Mareschal(1988), Mareschal & Brans(1991) and Vrangel, Stanojevic, Stevanovic & Luein(1996), all applied PROMETHEE as a decision making tool to solve the different problems in the field of finance. There are many competing candidates and several critical criteria for choosing the most suitable one. By using AHP introduced by Saaty (1980), suitable weights are allocated to all the associated criteria which are crucial for a felicitous selection. Two surveys, a general survey and the AHP survey had been conducted to achieve these objectives. The first general survey was performed to collect general views of the citizens to identify the perceived critical selection criteria, while the AHP survey was conducted to prioritize and assign the important weightings to the perceived criteria in the general survey. PROMETHEE II is used to obtain the best choice from a finite set of contesting candidates. SAW methodology as described by Belton & Stewart(2002) has been used to affirm the ranking. The study includes Indian citizens above 18 years of age. The ranking shows the selection of the most preferred and capable politician. 98

4 4.2 Problem Description and Development of Hierarchical Structure A group of voters has to select a candidate among a set of contesting candidates. Each voter has a personal ranking of the candidates according to his/her preferences. The problem of the selection or the ranking of alternatives submitted to a multi criteria evaluation is a controversial task. Usually there is no optimal solution as no alternative is the best one on each criterion. Problem is to rank the candidates according to their credibility on a set of weighted judgment criteria. The development of a ranking procedure requires a consensus on a set of criteria for evaluation, upon which the selection of candidates will be based. Potentially large number criteria are to be judged on a valid scale that is acceptable to all the participants. The most critical phase in designing the selection process model is to structure the decision problem. The main goal is to select the best candidate, according to a set of criteria for evaluation. As conflicting views may arise among different communities in determining the most important criteria of evaluation in a given decision making setting, a general survey was conducted to develop the main criteria, sub criteria and categories for each sub criteria for achieving the goal. Figure 4.1. shows the developed hierarchical structure. These criteria include personal and professional attributes at the main level based on survey results. These are further divided into sub criteria and sub-sub criteria as shown in the given hierarchy. 99

5 Figure 4.1: Hierarchy for Leader Selection 4.3 Solution Procedure Allotting Weights Using AHP The scenarios for this survey analysis were crafted after a study of the responses from our previous surveys and of the predictions made to workout the hierarchy using AHP. To evaluate the hierarchy (Figure 4.1), various surveys were conducted to rate each attribute to others in a series of pair wise comparisons using a scale from 1 to 9 (Table 1.1). The study scrutinized Indian citizens above eighteen years of age from all communities. For this purpose, sixty one responsible people rep- 100

6 resenting all categories like teachers, students, house wives, doctors, professionals and politicians etc. were asked to rate various attributes in the prepared questionnaire forms based on 9 point scale given by Saaty(1980). The respondents were requested to mark each response carefully after discussing the procedure thoroughly to obtain convincing judgments. An approach of AHP (explained in section 1.2.1) is applied to find out the weights of each criterion at different levels of hierarchy. Survey results were analyzed for each level of the hierarchy and reciprocal matrices were generated. An aggregated reciprocal matrix was developed for each set of matrices having C.R < 0.1, by taking geometric mean of corresponding values for further calculations. The local priority weights were derived from a series of pairwise comparisons involving all the nodes. The nodes at each level were compared pairwise with respect to their contribution to the nodes above them to find their respective global weights. We rank each of the criterion in the final set by evaluating it with respect to upper level attributes separately. The evaluation process finally generates the global weights for each requisite criterion of interest. Table 4.1 shows calculation of weights for criteria at level 1 where a ij entries in the matrix are geometric mean of all values from similar matrices having C.R. < 0.1. Table 4.1: Relative Importance of Criteria at Main Level Ranking Personal Professional Weights Personal Professional λ max =2, C.I. =0, C.R. =0 Tables 4.2 and 4.3 respectively show criteria weights at level 2 of hierarchy. 101

7 Table 4.2: Relative Importance of Sub Criteria at Personal Level Personal Education Religion Gender Age Reputation Public Speaking Weights Education Religion Gender Age Reputation Public Speaking λ max = , C.I. = , C.R. = Table 4.3: Relative Importance of Sub Criteria at Professional Level Professional Manifesto Party Social ActivitiesPrevious ExperienceBackgroundWeights Manifesto Party Social Activities Previous Experience Background λ max = , C.I. = , C.R. = Table 4.4 reveals calculations of criteria weights at level 3 of hierarchy. Table 4.4: Relative Importance of Sub Criteria Relative to Criteria (Education) Education Illiterate School Level Graduate Post Graduate Doctorate Weights Illiterate School Level Graduate Post Graduate Doctorate λ max = , C.I. = , C.R. =

8 Similar matrices are worked out for all the criteria at level 3. Also global priority weights for each criterion are calculated by multiplying its local priority weight by its upper level criterion s global priority weight. Figure 4.2 shows local and global priorities of the criteria and its sub criteria. Figure 4.2: Local and Global Priority Weights for Each Criterion Here global priority weights for each one are preceded by an asterisk ( ) symbol to differentiate them from local priority weights. Above exercise provides a structure to implement the survey analysis. However AHP can be further extended to another level to allocate priority weights to various decision alternatives. Yet tedious comparisons at an advanced level due to a large number of criteria can be supplemented by using PROMETHEE II methodology to trim down the calculations to a moderate level. 103

9 4.3.2 Ranking of Alternatives Using PROMETHEE II Methodology After having weight allocations to the set of perceived criteria via AHP, the problem of selection of suitable candidate is submitted to PROMETHEE II methodology for a pertinent ranking, explained explicitly in section Alternatives A i (i =1,...,m), criteria C j (j =1,...,n) and weights W j (j = 1,...,n) are sited in matrix form similar to that of Table 1.3. An evaluation x ij corresponding to every f j (A i ) is taken as a crisp score for each alternative corresponding to the respective criterion. It is conspicuous to mention that instead of using any standard scale, we are using local priority weights calculated via AHP for each evaluation as crisp scores. Each column of the decision matrix is normalized using equation 1.3 as all the criteria considered here are beneficial criteria. In order to take the deviations and the scales of the criteria into account, a preference function is associated to each criterion. For this purpose, a preference function P j (A i,a í ) is defined, representing the degree of the preference of alternative A i over A í for criterion C j. Preference functions for all the pairs of alternatives are evaluated using equations 1.5 and 1.6. Global intensity preference function between the couples of alternatives defined by π(a i,a í ) is found out to associate weights with each preference function as given by equation 1.7 in section Now the precedence flows represented by φ + (A i )andφ (A i ) are determined using 104

10 equations 1.8 and 1.9 to conceptualize the net outranking flow given in equation Strengthening PROMETHEE with ideas of AHP: In the present text, some ideas of AHP have been applied in the PROMEETHE methodology viz. level 2 criteria global weights given by hierarchy in Figure 4.2 are being taken as normalized criteria weights to ascertain their adaptability to the condition n j=1 W j = 1. Also instead of using any arbitrary scale to convert the ratings of various alternatives with respect to different criteria to crisp scores, we are using local priority weights of level 3 criteria calculated via AHP as requisite x ij measures Ranking of Alternatives Using SAW As described in section 1.2.3, the total score for each alternative is computed by multiplying the comparable rating (or utility) for each attribute by the importance weight assigned to the attribute and then summing these products over all the attributes as given by equation Strengthening SAW with ideas of AHP: In this paper, we have used the global weights of level 3 criteria calculated via AHP as the requisite criteria weights. Also an alternative is rated 1 or 0 for each criterion subject to respective yes or no ratings. 105

11 4.4 Illustration We will now illustrate the proposed methodologies via an example in which three candidates say A 1, A 2 and A 3 are contesting an election. The important criteria at personal as well as professional levels were probed. We have identified eleven criteria to be used in choosing the fitting person as already discussed in section 4.2. Reputation is the highest weighted criteria followed by education and public speaking. Candidate s political party is also prominently weighed proceeded by social background and participation in social activities as indicated in Figure 4.2 of hierarchy. Summary of this inquisition is shown in Table 4.5. Table 4.5: Subjective Data for Ranking Procedure Criteria (Level 2) Alternatives A 1 A 2 A 3 Education Doctorate (D) Graduate (GR) School Level (SC) Religion Sikh (S) Hindu (H) Hindu (H) Gender Male (M) Male (M) Female (F) Age 68 (60A) 55 (40-60) 39 (40U) Reputation Very Good (VG) Good (G) Bad (B) Public Speaking Average (A) Strong (S) Weak (W) Manifesto Middle class (MC) All Class Types (AL) Weaker Section (WS) Party National (N) National (N) Regional (R) Social Activities International Level (IL) National Level (NL) State Level (SL) Previous Experience Strong (S) Strong (S) Average (A) Social Background Average (A) Strong (S) Weak (W) 106

12 4.4.1 Ranking Using PROMETHEE II Now to rank the alternatives using PROMETHEE II, crisp evaluation x ij corresponding to every f j (A i ) is accessed using local priority weights calculated via AHP. Also criteria weights (w j ) are their global priority weights calculated via AHP incorporating survey analysis as shown in Table 4.6. Table 4.6: Objective Data for Ranking Procedure Edu. Rel. Gender Age Reput. Pub. Manif. Party Part.In Pre.Exp. Soc. (A i ) Speak. Soc.Act. In Pol. Back A 1 D S M 60A VG A MC N IL S A A 2 GR H M G S AL N NL S S A 3 SC H F 40U B W WS R SL A W Now to calculate normalized value r ij against each x ij, using equation 1.3, as an example r 11 =( )/( ) = 1. Similarly other r ij s are calculated. Table 4.7 depicts the normalized decision matrix hence obtained. Table 4.7: Normalized Decision Matrix Edu. Rel. Gender Age Reput. Pub. Manif. Party Part.In Pre.Exp. Soc. (A i ) Speak. Soc.Act. In Pol. Back A A A

13 The preference function P 1 (A 1,A 2 )=r 11 r 21, asr 11 r 21. Likewise (P Ai,P Aí ) s are worked out for all possible pairs of alternatives and are given in Table 4.8. Table 4.8: Preference Functions for All the Pairs of Alternatives Edu. Rel. Gender Age Reput. Pub. Manif. Party Part.In Pre.Exp. Soc. (P Ai,P Aí ) Speak. Soc.Act. In Pol. Back (P A1,P A2 ) (P A1,P A3 ) (P A2,P A3 ) (P A2,P A1 ) (P A3,P A1 ) (P A3,P A2 ) Table 4.9 exhibits the aggregated preference function values for all the paired alternatives, as calculated using Equation 1.7. As an example π(a 1,A 2 )=W 1.P 1 (A 1,A 2 )+W 2.P 2 (A 1,A 2 ) W 11.P 11 (A 1,A 2 ). Likewise all the π(a i,a í ) s are determined. Table 4.9: Aggregated Preference Function Preference P A1 P A2 P A3 P A P A P A The leaving and the entering flows for different alternatives are now computed 108

14 using Equations 1.8 and 1.9 respectively. Also net outranking flow is calculated using equation 1.10 as illustrated in Table φ + (A 1 )= 1[π(A 2 1,A 2 )+π(a 1,A 3 )] = 1 [ ] = φ (A 1 )= 1[π(A 2 2,A 1 )+π(a 3,A 1 )] = 1 [ ] = φ(a 1 )=[φ + (A 1 ) φ (A 1 )] = [ ] = φ(a 2 )andφ(a 3 ) are computed in the similar manner. Table 4.10 shows net outranking flows for different alternatives. Table 4.10: Net Outranking Flow Values for Different Alternatives Alternative Leaving Flow(φ + ) Entering Flow(φ ) Net Flow(φ) Rank A A A Ranking Using SAW : To each of the attributes in SAW, the decision maker assigns importance weights which become the coefficients of the variables. The decision maker can then obtain a total score for each alternative simply by multiplying the scale rating for each attribute value by the importance weight assigned to the attribute and then summing these products over all attributes. Now to rank the alternatives using SAW, we use the global weights of different criteria at level 3, calculated via AHP as the requisite criteria weights. Also an alternative is rated 1 or 0 subject to respective yes or no ratings for each criterion. Table 4.11 shows the implementation of SAW methodology for ranking procedure. 109

15 Table 4.11: Ranking Using Saw Criteria (Level 1) Criteria (Level 2) Criteria (Level 3) Weight Alternatives A 1 A 2 A 3 Illiterate School Level Education Graduate Post Graduate Doctorate Hindu Religion Muslim Sikh Cristian Gender Male Personal Female Under Age Above Very Good Reputation Good Average Bad Strong Public Speaking Average Weak

16 Criteria (Level 1) Criteria (Level 2) Criteria (Level 3) Weight Alternatives A 1 A 2 A 3 Weaker Section Manifesto Middle Class Upper Class All Class Types National Party Regional Independent Local Level Professional Participation in State Level Social Activities National Level International Level Previous Experience Strong In Politics Average Weak Strong Social Background Average Weak Alternative Weight Alternative Ranking From the above illustration, it is prominently established that alternative ranking corresponds almost perfectly by both the methodologies. 111