Metoda porównywania parami

Transcription

1 Metoda porównywania parami Podejście HRE Konrad Kułakowski Akademia Górniczo-Hutnicza 24 Czerwiec 2014

2 Advances in the Pairwise Comparisons Method Heuristic Rating Estimation Approach Konrad Kułakowski AGH University of Science and Technology 24 June 2014

3 Outline Pairwise comparisons motivation Classical approach Heuristic Rating Estimation (HRE) approach motivation HRE method Possible areas of application Inconsistency and existence of solution in HRE Add-ons Bibliography 3

4 Pairwise comparison motivation FED Museum Atlanta To start the story from the very beginning 4

5 Pairwise comparison motivation Before money - barter (see FED Museum: History of trade is in fact history of pairwise comparisons? 5

6 Pairwise comparison motivation Barter - comparing incomparable it s hard to judge the actual value of things, hence the judgment is always subjective 6

7 1 2 Pairwise comparison motivation Experts judgment implies relative value of goods: 7

8 Pairwise comparison motivation More comparisons a b c d? 8

9 Pairwise comparison motivation The need for methods of synthesis of partial results? 9

10 Pairwise comparison motivation Problems with expert judgments Reciprocity Consistency =? = a, = b? = ab 10

11 Pairwise comparison Method 11

12 Pairwise comparisons (PC) method Result synthesis Pairwise comparisons matrix M = 1 m 12 m 13 m 14 m 15 m 21 1 m 23 m 24 m 25 m 31 m 32 1 m 34 m 35 m 41 m 42 m 43 1 m 45 m 51 m 52 m 53 m 54 1 where m ij R + 12

13 PC method Classical (eigenvalue based) approach Classical approach by Thomas Saaty Assumption matrix must be reciprocal i.e. Value quantification: Method of synthesis: m ij = 1 m ji m ij { a b :a,b {1,,10} } Finding a principal eigenvector x: Mx = λ max x Saaty, T. L., "A scaling method for priorities in hierarchical structures", Journal of Mathematical Psychology (1977),

14 PC method Classical approach Relative value of,,,,, is given as: or even better, as rescaled principal eigenvector: where: x = [ x 1, x 2,, x ] 5 ˆx = [ ˆx 1, ˆx 2,, ˆx ] 5 ˆx i = x i 5 j=1 x j 14

15 PC method Classical approach It is possible to determine the relative value of goods? 15

16 PC Method Problem What happen when some goods are exchanged for money? E.g.: = = 16

17 PC Method Problem What happen when some goods are exchanged for money? E.g.: Then: = $10 = $20 =

18 PC method Standard approach Standard answer Let us adopt that: = 1 2 then calculate the numerical ranking of goods In result, every good will gain some weight (rank position) 18

19 PC method Heuristic Rating Estimation (HRE) approach Let us do not (re) introduce the weights where they are given (here the order is introduced by value in Dollars) but instead Adopt the given values as reference and estimate the others Kułakowski, K., "Heuristic Rating Estimation Approach to The Pairwise Comparisons Method", CoRR 2013, (to be appeared in Fundamenta Informaticae) 19

20 PC method HRE approach Set of concepts: where C = C K C U C K means the set of concepts (alternatives) for which the value µ is initially known C U means the set of concepts (alternatives) for which µ need to be determined 20

21 PC method HRE approach HRE Input Known concepts C K = {, } where µ( ) = 10, µ( ) = 20 Unknown concepts C U = {,, } 21

22 PC method HRE approach result synthesis Pairwise comparisons matrix c 1 c 2 c 3 c 4 c 5 M = c 1 c 2 c 3 c 4 c 5 1 m 12 m 13 m 14 m 15 m 21 1 m 23 m 24 m 25 m 31 m m 35 m 41 m m 45 m 51 m 52 m 53 m 54 1 Known where m ij R + 22

23 PC method HRE approach result synthesis algorithm Input data c 5 m 15 m 52 m 51 m 12 m 25 m 21 c m 53 m 35 m 45 m 54 1 c 2 m 41 m 23 m 13 m 31 m m 24 m m µ(c c c 4 3 ) = 20 3 µ(c 4 ) = 10 23

24 PC method HRE iterative approach result synthesis algorithm First step We may expect that: c 5 µ(c 1 ) = 1 ( 2 m 13µ(c 3 ) + m 14 µ(c 4 )) c 1 c 2 µ(c 2 ) = 1 ( 2 m 23µ(c 3 ) + m 24 µ(c 4 )) c 3 c 4 KNOWN µ(c 5 ) = 1 ( 2 m 53µ(c 3 ) + m 54 µ(c 4 )) 24

25 PC method HRE iterative approach result synthesis algorithm Second step and further Let us take into account all other concepts but the estimated one: µ(c 1 ) = 1 4 and in general: 5 i=2 m 1i µ(c i ) c 5 c 1 c 2 µ(c j ) = 1 4 i {1,,5}\{ j} m ji µ(c i ) c 3 c 4 25

26 PC method HRE iterative approach result synthesis algorithm Update equation in r = 0,1,2, update step where µ r (c j ) = 1 C j r 1 m ji µ r 1 (c i ) r 1 c i C j C j r 1 = {c C : µ r 1 (c) is defined and c c j }, and C j 0 = df C K 26

27 PC method HRE iterative approach result synthesis algorithm Assuming that the relative value of concepts is given as: C K = {c k+1,,c n } µ = [µ(c 1 ),,µ(c k ) ## "## $,µ(c ),,µ(c ) k+1 n ##" ## $ ] C U (HRE procedure) Rescaling (normalization) ˆµ = [ ˆµ(c 1 ),, ˆµ(c n )] C K (a priori known) where ˆµ(c i ) = µ(c ) i n µ(c j ) j=1 27

28 PC method HRE iterative approach What is µ when the number of iterations tends to? 28

29 PC method From iterative procedure to linear equation The algorithm as shown above follows the Jacobi iterative method for the linear equation system in form: where: Aµ = b A is a matrix of r r where r = C C K b is a vector of constant terms µ is a vector: = C U µ T = [ µ(c 1 ),,µ(c k )] 29

30 PC method From iterative procedure to linear equation b - vector of constant terms is given as: b = 1 n 1 m 1,k+1µ(c k+1 ) n 1 m 1,nµ(c n ) 1 n 1 m 2,k+1µ(c k+1 ) n 1 m 2,nµ(c n ) 1 n 1 m k,k+1µ(c k+1 ) n 1 m k,nµ(c n ) note that (assumed for the sake of simplicity): C = {c 1,,c n } and C K = {c k+1,,c n } 30

31 PC method From iterative procedure to linear equation A The matrix used by the HRE approach is: A = 1 1 n 1 m 1,2 1 n 1 m 1,k 1 n 1 m 2,1 1 1 n 1 m 2,3 1 n 1 m 2,k " # " " # " 1 n 1 m k,1 1 n 1 m k,k

32 PC method Direct solving The equation: Aµ = b can also be solved directly. 32

33 PC method Direct solving theorems and facts Theorem 1 Equation Aµ = b has exactly one solution if det(a) 0 Theorem 2 The Jacobi method is convergent if A is strictly diagonally dominant by rows i.e. for i = 1,,k k 1 > a ij j=1, j i 33

34 PC method Direct solving admissibility of solution Observation - result must be positive µ(c i ) > 0 for i = 1,,n i.e. That is because of the form of update equation: µ r (c j ) = 1 C j r 1 A m ji µ r 1 (c i ) r 1 c i C j Observation - is strictly diagonally dominant, then µ obtained using direct method is admissible 34

35 PC method Direct solving solution admissibility Is diagonally dominant? i.e. Observation A A has a high chance to be diagonally dominant when are not to large. a ij Conclusion k 1 > a ij where a ij = 1 n 1 m ij j=1, j i The more concepts in C K and the more similar to each other the better (the more likely is diagonally dominant) A 35

36 PC method Intuitiveness of assumptions The more concepts in Corresponds to the natural desire to have more than the lower number of reference concepts The more similar to each other. Humans (experts) are able to compare (judge) similar things but not different. Observation C K In most cases tested the matrix was diagonally dominant A 36

37 PC method, HRE approach Applications How to? 37

38 PC method, HRE approach How to? Define the problem Specify problem domain, the set of concepts and the partial function µ Define the reference set assign the µ values to the concepts from C K Gather the reference data Apply the HRE method To remember - this is only heuristics above all, common sense C K 38

39 PC method, HRE approach Applications Areas of application Decision Support Systems 39

40 PC method, HRE approach Possible areas of application An optimal drug (treatment) selection Reference set C K Group of drugs with proven efficacy in clinical trials C U = C \ C K Other concepts and Comparative opinions of experts (physicians) based on their clinical experience M 40

41 PC method, HRE approach Possible areas of application Introducing new products into the market Reference set C K existing products with the sales statistics New product candidates C U = C \ C K Comparative opinion survey of the target group 41

42 PC method, HRE approach Possible areas of application Support for assessment of the value of companies Joint stock companies reference set an actual value is determined by stock exchange Companies outside the exchange trading the rest of concepts Comparative company value estimation 42

43 PC method, HRE approach Possible areas of application Real Estate Valuation Reference set C K real estates in the given area with the known transaction price C U = C \ C K Real estates for which the prices are unknown Comparative (paired) judgments of appraiser 43

44 PC method, HRE approach Possible areas of application Painting Valuation Reference set C K paintings from the given period with the known transaction price C U = C \ C K Paintings for which the prices are unknown Comparative (paired) judgments of appraiser 44

45 PC method, HRE approach Possible areas of application Tender assessments Reference set C K group of the winning bids from previous tenders, reference offers investment estimates C U = C \ C K Other concepts and Comparative opinions of experts (offers evaluation committee) based on their knowledge and the lessons learned in the past M Kułakowski, K., J. Szybowski, R. Tadeusiewicz, Tender with success the pairwise comparisons approach, KES 2014, (to be appeared in Procedia in Computer Science, Elsevier) 45

46 intangible criteria tangible criteria PC method, HRE approach Possible areas of application Hierarchical tender assessment model success criterion tangible and intangible criteria Tender Priorities of criteria Criterion HRE..... Criterion Offer Offer Offer Offer Offer Offer 46

47 PC method, HRE approach Possible areas of application anywhere the expert judgment is important for example when accurate model would be very complex and difficult to construct e.g. to estimate the rate of return on investment in green technologies 47

48 PC method, HRE approach Possible areas of application and many more 48

49 Inconsistency and HRE approach Inconsistency, HRE approach and existence of solution 49

50 Inconsistency and HRE approach What is inconsistency? = a, = b ## "## $ = ab + ε The smaller ε the smaller inconsistency 50

51 Inconsistency and HRE approach Inconsistency indices Saaty s inconsistency index: where λ max IC = λ max n n 1 is a principal eigenvalue of some matrix M = (m ij ) m ij + i, j {1,,n} 51

52 Inconsistency and HRE approach Inconsistency indices Koczkodaj s triad (local) inconsistency: κ i, j,k = df min 1 Koczkodaj s inconsistency index: m ij m ik m kj, 1 m ikm kj m ij K(M ) = max { κ } i, j,k i, j,k {1,,n} 52

53 Inconsistency and HRE approach Existence of solution The linear equation system introduced in the HRE approach has exactly one strictly positive solution for 0 < r n 2 if K(M ) < 1 where concepts, whilst concepts. n = C U C K r = C K Aµ = b (n 1)(n r 2) 2(n 1) is the number of all the estimated is the number of the known Kułakowski, K., "Notes on the existence of solution in the pairwise comparisons method using the Heuristic Rating Estimation approach CoRR 2014, 53

54 Inconsistency and HRE approach Proof idea (1), M-matrices M-matrix definition An n n matrix that can be expressed in the form A = si B where B = [b ij ] with b ij 0 for i, j {1,,n} and s ρ(b) is called M-matrix. where: ρ(b) = max{ λ :det(λi B) = 0} is a spectral radius of (the maximum of the moduli of the eigenvalues of B B 54

55 Inconsistency and HRE approach Proof idea (2), M-matrices Plemmons Theorem For every A = [a ij ] - n n matrix over where a ij 0 and i j, i, j = 1,,n each of the following conditions is equivalent to the statement: is a nonsingular M-matrix. is inverse positive. That is exists, and is semi-positive. That is, there exists with There exists a positive diagonal matrix such that has all positive row sums A A A 1 A 1 0 A x 0 Ax > 0 D AD Plemmons, R. J., "M-matrix characterizations. I - nonsingular M-matrices", Linear Algebra and its Applications (1976),

56 Inconsistency and HRE approach Proof idea (3), existence of solution Let us denote: where A = t 1,1 m 1,k m k,1 m 1,k n 1 " " " t k 1,1m k 1,k m k,1 n 1 t k,1m k,1 n 1 1 K(M ) t ij # m k 1,k n K(M ) 56

57 Inconsistency and HRE approach Proof idea (4), Existence of solution Matrix can be written as a product B = where: A t 1,1 m 1,k m 1,k n 1 " # " " t k 1,1 m k 1,k n 1 t k,1 n 1 " t k 1,k 1 m k 1,k m k 1,k n 1 t k,k 1 n 1 A = BC Adopting D = I and applying the third Plemmons criterion leads to the thesis of the Theorem. 1 C = m k, " 0 # # m k,k 1 #

58 Inconsistency and HRE approach Existence of solution Values of K(M ) that guarantees existence of solution in the HRE approach K(M ) r = 1 r = 2 r = 3 r = 4 r = 5 n = 3 n = 4 n = 5 n = 6 n =

59 PC method, HRE approach Exploration areas Add-ons 59

60 PC method, HRE approach Further research, opportunities Non-reciprocal PC matrix heuristics (proposal) Lack of reciprocity: Let us transform: M ˆM where m ij 1 m ji 1 ˆm ij = m ij Observations: ˆM is reciprocal if M is reciprocal then m ji M = ˆM

61 PC method, HRE approach Further research, opportunities Incomplete PC matrix heuristics (proposal) µ(c 1 ) = 1 ( 2 m 13µ(c 3 ) + m 15 µ(c 5 )) Impacts the lack of reciprocity heuristics c 1 c 2? c 5? c 3 c 4 61

62 PC method, HRE approach Further research, opportunities Minimizing estimation error heuristics Average absolute estimation error ê µ = 1 C U Absolute estimation error e µ (c) c C U e µ (c j ) = 1 C j r 1 r 1 c i C j µ(c j ) µ(c i ) m ji 62

63 PC method, HRE approach Wrapping up 63

64 Invitation KES Conference 2014 more at 64

65 PC method, HRE approach Mathematica Package Pairwise Comparisons Mathematica Package powered by Raspberry PI 65

66 Bibliography Papers Kułakowski, K., "A heuristic rating estimation algorithm for the pairwise comparisons method", Central European Journal of Operations Research (2013), Kułakowski, K., "Heuristic Rating Estimation Approach to The Pairwise Comparisons Method", CoRR 2013, (to be appeared in Fundamenta Informaticae) Kułakowski, K., "Notes on the existence of solution in the pairwise comparisons method using the Heuristic Rating Estimation approach CoRR 2014, Kułakowski, K., J. Szybowski, R. Tadeusiewicz, Tender with success - the pairwise comparisons approach, KES 2014, (to be appeared in Procedia in Computer Science, Elsevier) 66

67 One more use of pairwise comparisons (found on the web) 67

68 taken from 68

69 Questions 69