Book erratum

Book Erratum
Understanding The Analytic Hierarchy Process

Konrad Kułakowski
Abstract
Errors found after publishing Understanding The Analytic Hierarchy Process [1]

Page 40, line 3 from the top

is
n i j = { 0 if i = j or { a i , a j } ∈ E 1 if i ≠ j and { a i , a j } ∉ E should be
n i j = { 0 if i = j or { a i , a j } ∉ E 1 if i ≠ j and { a i , a j } ∈ E

Page 75, the numerical example is faulty. The correct version is:

and the constant term vector r is given as r = ( ln 6 ln 21 8 − ln 3 ln 36 7 − ln 27 ) .
Solving G w ^ = r leads to the following logarithmized ranking vector
w ^ = ( 1 100 ( 22 ln 3 + 43 ln 6 + 2 ln 27 − 7 ln 36 7 + 8 ln 21 8 ) 1 25 ( 8 ln 3 + 2 ln 6 + 3 ln 27 + 2 ln 36 7 + 12 ln 21 8 ) 1 100 ( 22 ln 3 − 7 ln 6 + 2 ln 27 + 43 ln 36 7 + 8 ln 21 8 ) 1 100 ( 22 ln 3 − 7 ln 6 + 2 ln 27 + 43 ln 36 7 + 8 ln 21 8 ) 1 50 ( − 4 ln 3 − ln 6 − 14 ln 27 − ln 36 7 − 6 ln 21 8 ) ) = ( 1.04 1.484 − 2.29 0.963 − 1.19 ) .
Hence, the (unscaled) ranking vector is w = ( e 1.04064 e 1.48464 e − 2.2937 e 0.96356 e − 1.19512 ) = ( 2.83103 4.4134 0.100889 2.62103 0.302668 ) .
The last step to receive the ranking in the usual form is scaling so that the entries of the ranking vector sum up to 1 . The final form of the ranking vector is as follows: w g m = ( 0.275 0.429 0.0098 0.255 0.0294 ) . According to the computed ranking, the most preferred alternative is a 1 with the ranking value w ( a 2 ) = 0.429 . The second place is taken by a 4 with w ( a 1 ) = 0.275 , then a 4 , a 5 and a 3 .
Of course, one may verify that GMM applied to the following matrix ( 1 2 3 0.275 0.0098 0.275 0.255 9 3 2 1 0.429 0.0098 7 4 0.429 0.0294 0.0098 0.275 0.0098 0.429 1 0.0098 0.255 1 3 0.255 0.275 4 7 0.255 0.0098 1 9 1 9 0.0294 0.429 3 1 9 1 ) results in w g m .

Page 80, line 1 from the top

is
where p i is the number of existing comparisons in the i-th row of C ,
should be
where p i is the number of existing comparisons in the i-th row of C except the diagonal

Page 81, line 12 from the bottom

is
q 3 = ∑ i = 1 n − 2 ∑ j = i + 1 n − 1 ∑ k = j + 1 n ( 2 − a i k a i j a j k − a i j a j k a i k ) should be
q 3 = ∑ i = 1 n − 2 ∑ j = i + 1 n − 1 ∑ k = j + 1 n ( 2 − c i k c i j c j k − c i j c j k c i k )

Page 113, line 14 from the bottom

is
6.3.8.1 Effectiveness of the Koczkodaj index
should be
6.3.8.1 Effectiveness and the Koczkodaj index

Page 96, line 2 from the top

is
Let w : A → 𝕣 + be a priority function assigning priorities to alternatives and C = [ c i j ] be an n × n PC matrix.
should be
Let w : A → 𝕣 + be a priority function assigning priorities to alternatives where C = [ c i j ] be an n × n PC matrix and the relative priority of the two alternatives corresponds to their direct comparison.

Page 123, line 5 from the top

is
T i j k = ( 1 c i j c i k 1 / c i j 1 c k j c i k 1 / c k j 1 ) should be T i j k = ( 1 c i j c i k 1 / c i j 1 c j k c i k 1 / c j k 1 )

Page 150, line 11 from the bottom

is
C = ( 1 ( ∏ q = 1 r c 1 , 2 , q η q ) 1 / r ⋯ ( ∏ q = 1 r c 1 , n , q η q ) 1 / r ( ∏ q = 1 r c 2 , 1 , q η q ) 1 / r 1 ⋯ ⋮ ⋮ ⋯ ⋱ ( ∏ q = 1 r c n − 1. n , q η q ) 1 / r ( ∏ q = 1 r c n , 1 , q η q ) 1 / r ⋯ ⋯ 1 ) ,
should be
C = ( 1 ∏ q = 1 r c 1 , 2 , q η q ⋯ ∏ q = 1 r c 1 , n , q η q ∏ q = 1 r c 2 , 1 , q η q 1 ⋯ ⋮ ⋮ ⋯ ⋱ ∏ q = 1 r c n − 1. n , q η q ∏ q = 1 r c n , 1 , q η q ⋯ ⋯ 1 ) ,

Page 150, line 7 from the bottom

is
w ( a i ) = ( ∏ k = 1 n ( ∏ q = 1 r c i , k , q η q ) 1 / r ) 1 / n
should be
w ( a i ) = ( ∏ k = 1 n ∏ q = 1 r c i , k , q η q ) 1 / n .

References

1Kułakowski, K., Understanding the Analytic Hierarchy Process (6000 Broken Sound Parkway, Boca Raton, FL, 33487, USA: Chapman and Hall / CRC Press, 2020).

Erratum can also be downloaded as the PDF file here