IJAHP Article: Mizuno, Kinoshita/ An algebraic representation via differential equations for 

pairwise comparisons of AHP 

 

 International Journal of the 
Analytic Hierarchy Process 

136 Vol. 9 Issue 1 2017 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v9i1.278 

AN ALGEBRAIC REPRESENTATION VIA DIFFERENTIAL 

EQUATIONS FOR PAIRWISE COMPARISONS OF AHP 

 
Takafumi Mizuno 

Faculty of Urban Science 

Meijo University 

tmizuno@urban.meijo-u.ac.jp 

 

Eizo Kinoshita 

Faculty of Urban Science 

Meijo University 

kinoshit@urban.meijo-u.ac.jp 

 

 

ABSTRACT 

 

We propose a simple algebraic representation for pairwise comparisons of AHP. 

The representation is an associative relation between the importances of elements 

and consists of basic arithmetic operations. First, we define a ratio, which is 

estimated by decision makers by comparing the importances of elements, as a 

partial differentiation of importances (Section 2). Then, we construct systems of 

differential equations. Algebraic representations of the importances are derived as 

formal solutions of the equations. We analyze pairwise comparisons and the 

construction of the importances from them with the representations (Section 3). 

The validity of using eigenvectors and C.I. in AHP is illustrated by deriving a 

particular solution of the equations. https://doi.org/10.13033/ijahp.v9i1.278 
 

 

Keywords: Pairwise comparison method; AHP; partial differentiation 

 

 

1. Introduction 

Pairwise comparisons are primitive procedures in AHP (Saaty, 1977, 1980). Decision 

makers construct relative importances of elements from ratios of pairs of elements. Let 𝑎1, 
⋯, 𝑎𝑛 be the elements, and 𝑥𝑖 be an importance of an element 𝑎𝑖. Decision makers want 
to obtain 𝑥𝑖, but they can only estimate ratios 𝑥𝑖/𝑥𝑗 by pairwise comparisons for all pairs 

(𝑎𝑖, 𝑎𝑗). There are many methods to derive importances from the set of ratios (Cogger & 

Yu, 1985). In the actual usage of AHP, relative importances are often obtained by 

applying the principal eigenvector method (Saaty, 1980). In this method, a ratio 𝑟𝑖𝑗 which 

is an estimation of 𝑥𝑖/𝑥𝑗  is arranged in the 𝑖-th row 𝑗-th column cell in the pairwise 

comparison matrix 𝑅, which is 𝑛 × 𝑛 square matrix. The importances which decision 
makers want are obtained as elements of the principal eigenvector of 𝑅 ; a detected 
relative importance 𝑥�̂� is an element of vector �̂� = [𝑥1̂, ⋯ , 𝑥�̂�]

𝑡 which holds R�̂� = λ𝑚𝑎𝑥�̂�. 
Harker and Vargas (1987) discussed why we can regard the vector as the approximation 

of importances. Their illustrations, however, are correct but quite difficult because of 

https://doi.org/10.13033/ijahp.v9i1.278
https://doi.org/10.13033/ijahp.v9i1.278


IJAHP Article: Mizuno, Kinoshita/ An algebraic representation via differential equations for 

pairwise comparisons of AHP 

 

 International Journal of the 
Analytic Hierarchy Process 

137 Vol. 9 Issue 1 2017 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v9i1.278 

their analyses of eigenvectors. In a decision making process, we have to make decision 

makers intuitively understand the usefulness of the methods. We also want to construct 

useful semantics which treat mental measurements and physical models with the same 

scheme. In this paper, we propose a representation which simply illustrates the validity of 

calculations for relative importances from pairwise comparison.  

 

 

2. Hypotheses 

We presume that the importance 𝑥𝑖  of an element 𝑎𝑖  can be represented in a multi-
variable function whose arguments are 𝑥𝑗, 𝑗 ≠ 𝑖; 

 

x𝑖 ≡ 𝑥𝑖(𝑥1, ⋯ , 𝑥𝑖−1, 𝑥𝑖+1, ⋯ , 𝑥𝑛),    𝑖 = 1, ⋯ , 𝑛. (1) 

  

In the pairwise comparisons of the AHP, for all pairs (𝑥𝑖, 𝑥𝑗), decision makers give an 

estimated ratio 𝑟𝑖𝑗 which means that 𝑥𝑖 is 𝑟𝑖𝑗 times as large as 𝑥𝑗. We make the further 

assumption that the ratio is an estimation of the partial differentiation of these functions; 

 

𝑟𝑖𝑗 ≡ (𝑛 − 1)
𝜕𝑥𝑖
𝜕𝑥𝑗

. 
(2) 

 

 

It means that if decision makers enlarge the estimate of the 𝑎𝑗, then the 𝑎𝑖 will be larger, 

and the 𝑎𝑖 growth rate of the estimate will be 𝑟𝑖𝑗 times larger than that of the 𝑎𝑗. There is 

a term (𝑛 − 1) in Equation (2), because decision makers estimate the ratio of 𝑥𝑖  as a 
single-variable function whose argument is 𝑥𝑗 in spite of the former assumption that the 

function is an (𝑛 − 1)-variable function. 
 

 

3. An analysis of the pairwise comparison method 

With the hypotheses in the previous section, we can write the pairwise comparison matrix 

𝑅 as follows: 

𝑅 = [

𝑟11 ⋯ 𝑟1𝑛
⋮ ⋱ ⋮

𝑟𝑛1 ⋯ 𝑟𝑛𝑛
] ≡ (𝑛 − 1)

[
 
 
 
 
𝜕𝑥1
𝜕𝑥1

⋯
𝜕𝑥1
𝜕𝑥𝑛

⋮ ⋱ ⋮
𝜕𝑥𝑛
𝜕𝑥1

⋯
𝜕𝑥𝑛
𝜕𝑥𝑛]

 
 
 
 

= (𝑛 − 1)𝜕𝒙𝜕𝒙𝑡 (3) 

 

where ∂𝒙 = [𝜕𝑥1, ⋯ , 𝜕𝑥𝑛]
𝑡, and ∂𝒙 = [1/𝜕𝑥1, ⋯ ,1/𝜕𝑥𝑛]

𝑡. 

 

Let 𝑑𝒙 = [𝑑𝑥1, ⋯ , 𝑑𝑥𝑛]
𝑡, and let us consider a product 𝑅𝑑𝒙. Combining the formula of 

total differentiation, we obtain a relation 

 

𝑑𝒙 = (∂𝒙 ∂𝒙𝑡 − 𝐼)𝑑𝒙 =
1

(𝑛 − 1)
(𝑅 − 𝐼)𝑑𝒙, 

(4) 

 

 

𝑑𝑥𝑖 =
1

(𝑛 − 1)
[𝑟𝑖1𝑑𝑥1 + ⋯ + 𝑟𝑖,𝑖−1𝑑𝑥𝑖−1 + 𝑟𝑖,𝑖+1𝑑𝑥𝑖+1 + ⋯ + 𝑟𝑖𝑛𝑑𝑥𝑛]. 

(5) 

 

https://doi.org/10.13033/ijahp.v9i1.278


IJAHP Article: Mizuno, Kinoshita/ An algebraic representation via differential equations for 

pairwise comparisons of AHP 

 

 International Journal of the 
Analytic Hierarchy Process 

138 Vol. 9 Issue 1 2017 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v9i1.278 

 

where 𝐼  is the identity matrix. Notice that the total differentiation of 𝑥𝑖  is 𝑑𝑥𝑖 =
∂𝑥𝑖/𝜕𝑥1𝑑𝑥1 + ⋯ + ∂𝑥𝑖/𝜕𝑥𝑖−1𝑑𝑥𝑖−1 + ∂𝑥𝑖/𝜕𝑥𝑖+1𝑑𝑥𝑖+1 + ⋯ + ∂𝑥𝑖/𝜕𝑥𝑛𝑑𝑥𝑛 . We can 
represent the importances 𝒙 as a system of total differential equations. 
 

We obtain an algebraic representation of 𝑥𝑖 by integrating Equation (5). 
 

𝑥𝑖 = ∫ 𝑑𝑥𝑖 =
1

(𝑛 − 1)
[∫ 𝑟𝑖1 𝑑𝑥1 + ⋯ + ∫ 𝑟𝑖𝑛 𝑑𝑥𝑛]

=
1

(𝑛 − 1)
[𝑟𝑖1𝑥1 + ⋯ + 𝑟𝑖,𝑖−1𝑥𝑖−1 + 𝑟𝑖,𝑖+1𝑥𝑖+1 + ⋯ + 𝑟𝑖𝑛𝑥𝑛] − 𝑑𝑖, 

(6) 

 

 

 

𝒙 =
1

𝑛 − 1
(𝑅 − 𝐼)𝒙 − 𝒅 

(7) 

 

 

where 𝒅 = [𝑑1, ⋯ , 𝑑𝑛]
𝑡  is a constant of integration. To determine the constant, we 

reformulate Equation (7). 

 

𝑅𝒙 = 𝑛𝒙 + (𝑛 − 1)𝒅. (8) 
 

This is an algebraic representation for importances. It has a degree of freedom caused by 

the constant of integration 𝒅.  
 

To find particular solutions by determining the constant of integration, let �̂�  be an 
eigenvector of 𝑅, and λ  its corresponding eigenvalue. Thus the representation can be 
transformed to: 

 

𝑅�̂� = λ�̂� = 𝑛�̂� + (𝑛 − 1)𝒅,  (9) 
 

𝒅 =
λ − 𝑛

(𝑛 − 1)
�̂�. 

(10) 

 

 

We obtain a representation of importances as the system of equations: 

 

𝒙 =
1

𝑛 − 1
(𝑅 − 𝐼)𝒙 −

λ − 𝑛

𝑛 − 1
�̂�, 

(11) 

 

 
1

𝑛 − 1
(𝑅 − 𝑛𝐼)𝒙 =

λ − 𝑛

𝑛 − 1
�̂�. 

(12) 

 

 

If the null-space of the matrix (𝑅 − 𝑛𝐼) has the same dimensions, then the solution will 
be 

𝒙 = 𝒚 + �̂�. (13) 
 

A vector 𝒚 is the solution of the equation (𝑅 − 𝑛𝐼)𝒚=0. We can confirm that �̂� is also the 
solution of the Equation (12). 

 

 

https://doi.org/10.13033/ijahp.v9i1.278


IJAHP Article: Mizuno, Kinoshita/ An algebraic representation via differential equations for 

pairwise comparisons of AHP 

 

 International Journal of the 
Analytic Hierarchy Process 

139 Vol. 9 Issue 1 2017 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v9i1.278 

 

 

4. Conclusions 

We propose an algebraic representation for the pairwise comparisons of the AHP. A key 

idea is that we regard ratios of importances as partial differentiations of them. Relations 

between importances are derived directly from these differentiations. In  

Section 3, we also naturally introduced why eigenvectors are needed and what C.I. the 

term (λ − 𝑛)/(𝑛 − 1) , is. Eigenvectors are particular solutions of the system of 
differential equations, and C.I. is a coefficient of the nonhomogeneous term of the 

equations. 

 

In this paper, we demonstrate that estimated ratios can be regarded as differentials of 

importances without any fault. This means that we can include physical models in the 

pairwise comparisons of the AHP. We expect that mental measurements, which are 

obtained using ordinary pairwise comparisons, and physical models are treated using the 

same scheme. And we can apply the semantics to machine learnings, or can retrieve 

importances of any element automatically to put in the AHP. In real databases of physical 

models, there are many numeric calculations for extracting differentiations.  

  

https://doi.org/10.13033/ijahp.v9i1.278


IJAHP Article: Mizuno, Kinoshita/ An algebraic representation via differential equations for 

pairwise comparisons of AHP 

 

 International Journal of the 
Analytic Hierarchy Process 

140 Vol. 9 Issue 1 2017 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v9i1.278 

 

REFERENCES 

 

Saaty, T. (1980). The Analytic Hierarchy Process, New York: McGraw-Hill. Doi: 
http://dx.doi.org/10.1080/00137918308956077 

 
Saaty, T. (1977). A scaling method for priorities in hierarchical structure. Journal of 

Mathematical Psychology, 15, 234-281. Doi: http://dx.doi.org/10.1016/0022-

2496(77)90033-5 

 

Cogger, K. O. & Yu., P. L. (1985). Eigenweight vectors and least-distance approximation 

for revealed preference in pairwise weight ratios. Journal of Optimization Theory and 

Applications, 46(4), 483-491. Doi: 10.1007/BF00939153 

 

Harker, P. and Vargas, L. (1987). The theory of ratio scale estimation: Saaty's analytic 

hierarchy process. Management Science, 33(11), 1383–1403. Doi: 
http://dx.doi.org/10.1287/mnsc.33.11.1383 

https://doi.org/10.13033/ijahp.v9i1.278
http://dx.doi.org/10.1080/00137918308956077
http://dx.doi.org/10.1016/0022-2496(77)90033-5
http://dx.doi.org/10.1016/0022-2496(77)90033-5
http://dx.doi.org/10.1287/mnsc.33.11.1383