Decision Making in Complex Environments. Lecture 2 Ratings and Introduction to Analytic Network Process

Transcription

1 Decision Making in Complex Environments Lectre 2 Ratings and Introdction to Analytic Network Process

2 Lectres Smmary Lectre 5 Lectre 1 AHP=Hierar chies Lectre 3 ANP=Networks Strctring Complex Models with BOCR Week 7 Final Projects Lectre 2 Ratings=Larg e nmber of alternatives Convert AHP to ANP model Lectre 4 Benefits Opportnities Costs Risks Lectre 6 Resorce Allocation Grop Decisions

3 Main points Ratings vs Pairwise Comparisons Analytic Network Process ANP Use SperDecisions software to learn how to Create a ratings model Convert an AHP model to ANP model

4 Otcomes of this lectre Understand what a ratings model is Be able to design a ratings model Understand what an ANP model is Be able to desing an ANP model Learn how to se SperDecisions to handle Ratings and ANP models

5 Ratings

6 What are ratings? In AHP/ANP we have two ways of creating priorities: By comparing the alternative in pairs (pairwise comparisons) By RATING the alternatives one at a time with respect to an ideal or standard This kind of measring is analogos to measring something with a physical device, like measring length with a yardstick When do we se ratings? When alternatives are thogh to be independent of one another The presence or absence of an alternative mst have no effect on how one rates any of the others When we can have an IDEAL alternative in mind to compare with When the nmber of alternatives make the pairwise comparisons too time consming e.g. if we want to evalate 50 employees, then 1,225 (50(49)/2) pairwise comparisons, wold be reqired for each criterion if we sed a pairwise comparison model and not a ratings model.

7 Ratings - Methodology Step 1 Create and AHP or ANP model WITHOUT any alternatives No alternatives clster will appear in the model Step 2 Do all the pairwise comparisons needed to evalate the criteria Step 3 Select all the bottom level criteria and se them as criteria in the ratings model Step 4 For each criterion define a cstom scale Step 5 Step 6 Rate each alternative against the criteria sing the respective scale Calclate each alternative s score by mltiplying the scale vale to the criterion weight and sm throgh the line Step 7 Idealize all reslts to find what is the rank of each alternatives

8 Ratings and Rank reversal When sing rating we don t have rank reversals That means that adding alternatives to the model will not create changes on the relative ranks of the existing alternatives In pairwise comparisons this is not the case Rank reversal

9 Ratings scales One mst be able to say how high or low an alternative rate on each criterion To do that yo mst have something in mind called an IDEAL so that yo get the feeling abot how close or far the alternative is from the ideal and allocate it to one of varios intensity slots of ranking we call them SCALES Scales A scale is a grop of intensities sed to evalate an alternative against a specific criterion We calclate each scale item s intensity by pairwise comparing the scale items or by directly entering vales We have one scale for each decision criterion The same scale can be sed for more than one criteria in the same model The same scale can be re-sed in different models

10 Ratings Example: Best City to Live in

11 Step 2 Pairwise compare criteria We pairwise compare the criteria to find their weights Cltral 1 Cltral Family Hosing Jobs Transportation Family 1 Hosing 1 Jobs 1 Transportation 1

12 Step 3 Create Scale for each criterion For each criterion we create a scale, to do so we Find the scale items that are appropriate for that kind of criterion Pairwise compare them to calclate the priorities or Directly enter the vales

13 Step 4 Rate Alternatives Each alternative is rated having in mind how this alternative compares to the ideal alternative We calclate the final reslts by mltiplying the vale of the cell to the weight of the criterion and sm across the row

14 Analytic Network Process

15 AHP Linear Hierarchy

16 Analytic Network Process (ANP) The ANP is a mathematical theory that makes it possible for one to deal systematically with dependence and feedback, and incldes the AHP as a special case. The reason for its sccess is the way it elicits jdgments and ses measrement to derive ratio scales. Real life problems involve dependence and feedback. Sch phenomena can not be dealt with in the framework of a hierarchy bt we can by sing a network with priorities. With feedback the alternatives can depend on the criteria as in a hierarchy bt may also depend on each other. The criteria themselves can depend on the alternatives and on each other as well. Feedback improves the priorities derived from jdgments and makes prediction more accrate.

17 Analytic Network Process Feedback Network with components having Inner and Oter Dependence among Their Elements C 4 C 1 Arc from component C 4 to C 2 indicates the oter dependence of the elements in C 2 on the elements in C 4 with respect to a common property. Feedback C 2 C 3 Loop in a component indicates inner dependence of the elements in that component with respect to a common property.

18 AHP vs ANP AHP: What is more preferred or more important? Both are more or less sbjective and personal. ANP: What has greater inflence? This reqires factal observation and knowledge to yield valid answers and ths is more objective. Decisions with the ANP shold be more stable becase one can consider their effect on and srvival in the face of other inflences.

19 The qestions we answer in ANP Given a criterion, which of two elements has greater inflence (is more dominant) with respect to that criterion? Given an alternative, which of two criteria or properties is more dominant in that alternative? Given a criterion and given an element X in any clster, which of two elements in the same clster or in a different clster has greater inflence on X with respect to that criterion? The entire decision mst se the idea of something inflencing another. Otherwise it mst se the idea of inflenced by throghot the analysis as follows: Given a criterion and given an element X in any clster, which of two elements in the same or in a different clster is inflenced more by X with respect to that criterion.

20 Main Operations of the ANP Relative measrement: Reciprocal relation Jdgments: Homogeneity Hierarchy or Network: Strctre of problem; the control hierarchy Priorities, Dominance and Consistency: Eigenvector Weighting the components Composition, Additive to also handle dependence throgh the spermatrix Spermatrix: Interdependence; raising the spermatrix to powers

21 Weighting The Components In the ANP one often needs to prioritize the inflence of the components themselves on each other component to which the elements belong. This inflence is assessed throgh paired comparisons with respect to a control criterion. The priority of each component is sed to weight the priorities of all the elements in that component. The reason for doing this is to enable s to perform feedback mltiplication of priorities by other priorities in a cycle, an infinite nmber of times. The process wold not converge nless the reslting matrix of priorities is colmn stochastic (each of its colmns adds to one). To see that one mst compare clsters in real life, we note that if a person is introdced as the president it makes mch difference, for example, whether he or she is the President of the United States or the president of a local labor grop.

22 Inner vs. Oter Dependence In a network, the elements in a component may inflence other elements in the same component (inner dependence) may inflence elements in other components (oter dependence) with respect to each of several properties We want to determine the overall inflence of all the elements. To do so we: organize the properties or criteria define connections among criteria and alternatives perform comparisons synthesize to obtain the priorities of these properties derive the inflence of elements in the feedback system weight the reslting inflences obtain the overall inflence of each element.

23 c 1. c 4 c 2 Networks and the Spermatrix C 1 C 2 C N C 3 Feedback. e 11 e 12 e 1n1 e 21 e 22 e 2n2 e N1 e N2 e NnN C 1 e 11 e 12 W 11 W 12 W 1N e 1n1 W = C 2 e 21 e 22 W 21 W 22 W 2N e 2n2 C N e N1 e N2 W N1 W N2 W NN e NN

24 where (j 1 ) (j 2 ) (jn j ) W i1 W i1 W i1 W ij = (j 1 ) (j 2 ) (jn j ) W i2 W i2 W i2 (j W 1 ) (j ini W 2 ) (jn ini W j ) ini

25 Three Spermatrices in ANP 1) The original spermatrix of colmn eigenvectors obtained from pairwise comparison matrices of elements 2) Weighted spermatrix in which each block of colmn eigenvectors belonging to a component is weighted by the priority of inflence of that component. This renders the weighted spermatrix colmn stochastic. 3) The limit spermatrix obtained by raising the weighted spermatrix to large powers.

26 Spermatrix of a Hierarchy C 1 C 2 C N-2 C N-1 C N e 11 e 1n1 e 21 e 2n2 e (N-2)1 e (N-2) nn-2 e N1 e NnN W = C 1 C 2 e 11 e 1n1 e 21 e 2n2 e (N-1)1 e (N-1) nn W W C N e N1 e NnN 0 0 W n-1, n W n, n-1 I

27 W k = W n,n-1 W n-1,n-2 W 32 W 21 W n,n-1 W n-1,n-2 W W n,n-1 W n-1,n-2 W n,n-1 I for k>n-1

28 Smmarizing ANP In ANP we have criteria groped in clsters and alternatives groped in a clster sally named Alternatives We can have inner and oter dependencies among the criteria, the alternatives and the criteria and the alternatives We can have feedback (self loop) in any clster We can pairwise compare the clsters like we do the criteria and the alternatives The final reslts are given by the limit spermatrix We do sensitivity in the same way that we do the AHP sensitivity

29 ANP Example - US Economy Trnarond

30 ANP Example Hambrger Model