Book Erratum
Understanding The Analytic Hierarchy Process

Konrad Kułakowski

Abstract

Errors found in the book Understanding The Analytic Hierarchy Process [1]

Page 40, line 3 from the top

n_{i j} = {\begin{matrix} 0 & i f i = j o r {a_{i}, a_{j}} \in E \\ 1 & i f i \neq j a n d {a_{i}, a_{j}} \notin E \end{matrix}

should be

n_{i j} = {\begin{matrix} 0 & i f i = j o r {a_{i}, a_{j}} \notin E \\ 1 & i f i \neq j a n d {a_{i}, a_{j}} \in E \end{matrix}

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

and the constant term vector

r

is given as

r = (\begin{matrix} ln 6 \\ ln \frac{21}{8} \\ - ln 3 \\ ln \frac{36}{7} \\ - ln 27 \end{matrix}) .

Solving

G \hat{w} = r

leads to the following logarithmized ranking vector

\hat{w} = (\begin{matrix} \frac{1}{100} (22 ln 3 + 43 ln 6 + 2 ln 27 - 7 ln \frac{36}{7} + 8 ln \frac{21}{8}) \\ \frac{1}{25} (8 ln 3 + 2 ln 6 + 3 ln 27 + 2 ln \frac{36}{7} + 12 ln \frac{21}{8}) \\ \frac{1}{100} (22 ln 3 - 7 ln 6 + 2 ln 27 + 43 ln \frac{36}{7} + 8 ln \frac{21}{8}) \\ \frac{1}{100} (22 ln 3 - 7 ln 6 + 2 ln 27 + 43 ln \frac{36}{7} + 8 ln \frac{21}{8}) \\ \frac{1}{50} (- 4 ln 3 - ln 6 - 14 ln 27 - ln \frac{36}{7} - 6 ln \frac{21}{8}) \end{matrix}) = (\begin{matrix} 1.04 \\ 1.484 \\ - 2.29 \\ 0.963 \\ - 1.19 \end{matrix}) .

Hence, the (unscaled) ranking vector is

w = (\begin{matrix} e^{1.04064} \\ e^{1.48464} \\ e^{- 2.2937} \\ e^{0.96356} \\ e^{- 1.19512} \end{matrix}) = (\begin{matrix} 2.83103 \\ 4.4134 \\ 0.100889 \\ 2.62103 \\ 0.302668 \end{matrix}) .

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} = (\begin{matrix} 0.275 \\ 0.429 \\ 0.0098 \\ 0.255 \\ 0.0294 \end{matrix}) .

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

(\begin{matrix} 1 & \frac{2}{3} & \frac{0.275}{0.0098} & \frac{0.275}{0.255} & 9 \\ \frac{3}{2} & 1 & \frac{0.429}{0.0098} & \frac{7}{4} & \frac{0.429}{0.0294} \\ \frac{0.0098}{0.275} & \frac{0.0098}{0.429} & 1 & \frac{0.0098}{0.255} & \frac{1}{3} \\ \frac{0.255}{0.275} & \frac{4}{7} & \frac{0.255}{0.0098} & 1 & 9 \\ \frac{1}{9} & \frac{0.0294}{0.429} & 3 & \frac{1}{9} & 1 \end{matrix})

results in

w_{g m}

Page 80, line 1 from the top

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

q_{3} = \sum_{i = 1}^{n - 2} \sum_{j = i + 1}^{n - 1} \sum_{k = j + 1}^{n} (2 - \frac{a_{i k}}{a_{i j} a_{j k}} - \frac{a_{i j} a_{j k}}{a_{i k}})

should be

q_{3} = \sum_{i = 1}^{n - 2} \sum_{j = i + 1}^{n - 1} \sum_{k = j + 1}^{n} (2 - \frac{c_{i k}}{c_{i j} c_{j k}} - \frac{c_{i j} c_{j k}}{c_{i k}})

Page 113, line 14 from the bottom

6.3.8.1 Effectiveness of the Koczkodaj index

should be

6.3.8.1 Effectiveness and the Koczkodaj index

Page 123, line 5 from the top

T_{i j k} = (\begin{matrix} 1 & c_{i j} & c_{i k} \\ 1 / c_{i j} & 1 & c_{k j} \\ c_{i k} & 1 / c_{k j} & 1 \end{matrix})

should be

T_{i j k} = (\begin{matrix} 1 & c_{i j} & c_{i k} \\ 1 / c_{i j} & 1 & c_{j k} \\ c_{i k} & 1 / c_{j k} & 1 \end{matrix})

Page 150, line 11 from the bottom

C = (\begin{matrix} 1 & {(\prod_{q = 1}^{r} c_{1, 2, q}^{η_{q}})}^{1 / r} & \dots & {(\prod_{q = 1}^{r} c_{1, n, q}^{η_{q}})}^{1 / r} \\ {(\prod_{q = 1}^{r} c_{2, 1, q}^{η_{q}})}^{1 / r} & 1 & \dots & ⋮ \\ ⋮ & \dots & ⋱ & {(\prod_{q = 1}^{r} c_{n - 1. n, q}^{η_{q}})}^{1 / r} \\ {(\prod_{q = 1}^{r} c_{n, 1, q}^{η_{q}})}^{1 / r} & \dots & \dots & 1 \end{matrix}),

should be

C = (\begin{matrix} 1 & \prod_{q = 1}^{r} c_{1, 2, q}^{η_{q}} & \dots & \prod_{q = 1}^{r} c_{1, n, q}^{η_{q}} \\ \prod_{q = 1}^{r} c_{2, 1, q}^{η_{q}} & 1 & \dots & ⋮ \\ ⋮ & \dots & ⋱ & \prod_{q = 1}^{r} c_{n - 1. n, q}^{η_{q}} \\ \prod_{q = 1}^{r} c_{n, 1, q}^{η_{q}} & \dots & \dots & 1 \end{matrix}),

Page 150, line 7 from the bottom

w (a_{i}) = {(\prod_{k = 1}^{n} {(\prod_{q = 1}^{r} c_{i, k, q}^{η_{q}})}^{1 / r})}^{1 / n}

should be

w (a_{i}) = {(\prod_{k = 1}^{n} \prod_{q = 1}^{r} c_{i, k, q}^{η_{q}})}^{1 / n} .

References

1K. Kułakowski, Understanding the Analytic Hierarchy Process (Chapman and Hall / CRC Press, 2020).

Erratum can also be downloaded in the PDF version here

Book Erratum Understanding The Analytic Hierarchy Process

Page 40, line 3 from the top

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

Page 80, line 1 from the top

Page 81, line 12 from the bottom

Page 113, line 14 from the bottom

Page 123, line 5 from the top

Page 150, line 11 from the bottom

Page 150, line 7 from the bottom

References

Book Erratum
Understanding The Analytic Hierarchy Process