Notes on the existence of solutions in the pairwise comparisons method using the Heuristic Rating Estimation approach

Pairwise comparisons are a well-known method for modelling of the subjective preferences of a decision maker. A popular implementation of the method is based on solving an eigenvalue problem for M - the matrix of pairwise comparisons. This does not take into account the actual values of preference. The Heuristic Rating Estimation (HRE) approach is a modification of this method in which allows modelling of the reference values. To determine the relative order of preferences is to solve a certain linear equation system defined by the matrix A and the constant term vector b (both derived from M). The article explores the properties of these equation systems. In particular, it is proven that for some small data inconsistency the A matrix is an M-matrix, hence the equation proposed by the HRE approach has a unique strictly positive solution.


Introduction
The first written evidence of the use of pairwise comparisons (PC) dates back to the thirteenth century [3]. After a period of growth in the first half of the twentieth century, the pairwise comparisons method solidified in the form of the Analytic Hierarchy Process (AHP) proposed by Saaty [17]. Starting as a voting method, PC has become a way of deciding on the relative importance (relative utility) of concepts (alternatives), used in decision theory [17], economics [14], psychometrics and psychophysics [18] and others. The utility of the method has been confirmed many times by various researchers [6]. The theory of paired comparison is growing all the time. Examples of such exploration are the Rough Set approach [5], fuzzy PC relation handling [13], incomplete PC relation [1,4,9], data inconsistency reduction [10] and non-numerical rankings [8]. A recent contribution to the pairwise comparisons method includes the Heuristic Rating Estimation (HRE) approach [11,12] that allows the user to explicitly define a reference set of concepts, for which the utilities (the ranking values) are known a priori. The base heuristics used in HRE proposes to determine the relative values of a single non-reference concept as a weighted average of all the other concepts. Such a proposition leads to formulation a linear equation system defined by the matrix A and the strictly positive vector of constant terms b . As will be shown later, in the most interesting cases the matrix A is an M-matrix as defined in [15]. The sufficient condition for A to be an M-matrix is formulated using the notion of inconsistency referring to the quantitative relationship between entries of the pairwise comparisons matrix W . In particular it is shown that the fully consistent PC matrix W implies that A is an M-matrix.

Pairwise comparisons method
The input to the PC method is M = (m i j ) ∧ m i j ∈ + ∧ i , j ∈ {1, . . . , n } a PC matrix that expresses a quantitative relation R over the finite set of concepts C df = {c i ∈ ∧ i ∈ {1, . . . , n }} where is a non empty universe of concepts and R(c i , c j ) = m i j , R(c j , c i ) = m j i . The values m i j and m j i represent subjective expert judgment as to the relative importance, utility or quality indicators of the concepts c i and c j . Thus, according to the best knowledge of experts it should hold that c i = m i j c j .

Definition 1.
A matrix M is said to be reciprocal if ∀i , j ∈ {1, . . . , n } : m i j = 1 m j i and M is said to be consistent if ∀i , j , k ∈ {1, . . . , n } : m i j · m j k · m k i = 1.
Since the data in the PC matrix represents the subjective opinions of experts, they might be inconsistent. Hence, there may exist a triad m i j , m j k , m k i of entries in M for which m i k ·m k j = m i j . This leads to a situation in which the relative importance of c i with respect to c j is either m i k · m k j or m i j . This observation gives rise to both a priority deriving method that transforms even an inconsistent matrix of pairwise comparisons into a consistent priority vector, and an inconsistency index describing to what extent the matrix M is inconsistent. There are a number of priority deriving methods and inconsistency indexes [2,7]. For the purpose of the article Koczkodaj's inconsistency index is adopted.

Definition 2. Koczkodaj's inconsistency index
of n × n and (n > 2) reciprocal matrix M is equal to The result of the pairwise comparisons method is a ranking -a function that assigns values to concepts. Formally, it can be defined as follows: Definition 3. The ranking function for C (the ranking of C ) is a function µ : C → + that assigns to every concept from C ⊂ a positive value from + .
Thus, µ(c ) represents the ranking value for c ∈ C . The µ function is usually defined as a vector of weights µ T . According to the most popular eigenvalue based approach, proposed by Saaty [17], the final ranking µ ev is determined as the principal eigenvector of the PC matrix M , rescaled so that the sum of all its entries where µ ev -the ranking function, µ max -the principal eigenvector of M . A more complete overview including other methods can be found in [2,7].

Heuristic Rating Estimation approach
In the classical pairwise comparisons approach the ranking function µ for all the concepts c ∈ C is initially unknown. Hence every µ(c ) need to be determined by the priority deriving procedure. In real life this is not always true. Sometimes decision makers have extra knowledge about the group of elements C K ⊆ C that allows them to determine µ(c ) for C K in advance. For example, let c 1 , c 2 and c 3 be a goods that the company X intends to place on the market, whilst c 4 and c 5 have been available for some time in stores. In order to choose the most profitable and promising product out of c 1 , . . . , c 3 the company X want to calculate the function µ for c 1 , c 2 and c 3 . Due to some similarities between c 1 , . . . , c 3 and the pair c 4 , c 5 the company X want to include them in the ranking treating as a reference. Of course it makes no sense to ask experts about how profitable c 4 and c 5 are. The values µ(c 4 ) and µ(c 5 ) can be easily determined based on sales reports. The situation as outlined in this simple example leads to the Heuristic Rating Estimation (HRE) model proposed in [11,12]. The main heuristics of the HRE model assume that the set of concepts C is composed of the unknown concepts C U = {c 1 , . . . , c k } and known (reference) concepts C K = {c k +1 , . . . , c n }. Of course, only the values µ j for c ∈ C U need to be estimated, whilst the values µ(c i ) for c i ∈ C K are considered to be known. The adopted heuristics assumes that for every unknown c j ∈ C U the value µ(c j ) should be estimated as the arithmetic mean of all the other values µ(c i ) multiplied by factor m j i : Thus, the value µ(c i ) for each unknown concept c j ∈ C U is calculated according to the following formulas: Since the values µ(c k +1 ), . . . , µ(c n ) are known and constant (c k +1 , . . . , c n are the reference concepts), they can be grouped together. Let us denote: Thus (3) could be written as the linear equation system Aµ = b where: T . It is worth noting that b > 0, since every b i for i = 1, . . . , k is a sum of strictly positive components. According to (Def. 3) the ranking results must be strictly positive, hence only strictly positive vectors µ are considered to be feasible.

M-matrices
The answer to the question concerning the existence of solution of the linear equation system Aµ = b requires knowledge of certain properties of the M-matrix [16]. For this purpose, let us denote (n ) -the set of n × n matrices over , . . , j }. Moreover, for every matrix A ∈ (n ) and vector b ∈ n the notation A ≥ 0 and b ≥ 0 will mean that each entry of A and b is non-negative and neither A nor b equals 0. The spectral radius of A is defined as ρ(A) = max{|λ| : det(λI − A) = 0}. In practice, solving many problems in the biological and social sciences can be reduced to problems, involving M-matrices [15]. For this reason, M-matrices have been of interest to researchers for a long time and many of their properties have already been proven. Following [15] some of them are recalled below in the form of the Theorem 1.

Theorem 1. For every A ∈
(n ) each of the following conditions is equivalent to the statement: A is a nonsingular M-matrix.

1.
A is inverse positive. That is, A −1 exists and A −1 ≥ 0 2. A is semipositive. That is, there exists x > 0 with Ax > 0 3. There exists a positive diagonal matrix D such that AD has all positive row sums.
In the context of equation Aµ = b it is worth noting that if A is nonsingular then A −1 is also nonsingular, and thus the the vector µ could be determined as A −1 b . Moreover for b > 0 (every entry of vector b is a sum of strictly positive values) and A -M-matrix, due to the theorem above A −1 ≥ 0, the vector µ also must be strictly positive i.e. µ = A −1 b > 0.

Inconsistency based condition for the existence of a solution
The entries of M = [m i j ] represent comparative opinions of experts, they are thus inherently strictly positive, that is M > 0. For the same reason the matrix A (5), formed on the basis of M , has positive entries only on the diagonal, i.e. A ∈ (n ). Therefore proving that A satisfies any of the conditions of the Theorem 1, implies that A is an Mmatrix. Thus, due to the remarks below the Theorem 1, and the fact that in the HRE approach b > 0, the equation Aµ = b has only one strictly positive solution µ.
The sufficient condition for A to be an M-matrix is formulated with the help of the inconsistency index (M ) (Def. 2). Using an inconsistency index simplifies the evaluation of Aµ = b and enables linking the reliability of expert assessments (the paired ranking for which the inconsistency index is too high are considered as unreliable [17]) with the solution existence problem.

Theorem 2. The linear equation system Aµ = b introduced in the HRE approach has exactly one strictly positive solution if
where n = |C U ∪C K | -is the number of all the estimated concepts, r = |C K | -is the number of the known concepts.
Proof. Following (Def. 2), Koczkodaj's inconsistency index (M ), in short , means that the maximal inconsistence for some maximal triad m p q , m qr and m p r is . Thus, in the case of an arbitrarily chosen triad m i k , m k j , m i j it must hold that: . This means that either: we obtain the result that either It is easy to see that 0 ≤ < 1, thus 0 < α ≤ 1. Thus, both these assertions lead to the common conclusion: for every i , j , k ∈ {1, . . . , n }. This mutual relationship between entries of M can be written as the parametric equation m i j = t · m i k m k j where α ≤ t ≤ 1 α . Using this equation the matrix A (5) can be written as: Hence, finally the matrix A can be written as the matrix product A = BC in the form: Since both t i j and m i j are strictly positive, it holds that B ∈ (n ). Therefore, due to the third condition of the Theorem 1 where D df = I , B is a nonsingular M-matrix if sums of all its rows are positive. In other words B is an M-matrix if each of the following inequalities (10) are true.
Due to the constraints introduced by the inconsistency (M ) the minimal and the maximal value of every t i j is α and 1 α correspondingly. Thus the inequalities (10) are true if the following two inequalities are satisfied: where r = n − k is the number of elements in C K . In other words B is an M-matrix if the following two conditions are met: By solving f (α) = 0 and choosing the larger root 1 we obtain the result that: whilst the right, linear inequality g (α) > 0 leads to In order to decide which of these criteria are more restrictive and which should therefore be chosen, the following two cases need to be considered: (1) r = n − 2 and (2) 0 < r ≤ n − 3 (it must obviously hold that r ≤ n − 2).
When r = n − 2 it is easy to see that f (α) = αg (α). Thus both functions f (α) and g (α) take the 0 value for the same values of argument α. Hence, both criteria (13) and (14) are equal.
If 0 < r ≤ n − 3 it is easy to see 2 that the first condition (13) is more restrictive than (14), i.e. wherever (13) holds (14) is also true. In other words, to provide a guarantee that B is an M-matrix it is enough to consider the more restrictive condition (13).
The fact that B is an M-matrix implies that there exists an inverse matrix B −1 ≥ 0. Hence, due to the form of the matrix C it is easy to see that the inverse matrix C −1 exists, thus A −1 exists and A −1 = C −1 B −1 ≥ 0. Thus, due to the first condition of the Theorem 1, A is an M-matrix, which means that the equation Aµ = b has a unique strictly positive solution. This conclusion completes the proof of the theorem.
Of course, the theorem proven above does not address the case r = n − 1. This is because r = n − 1 implies A ∈ (1), hence solving Aµ = b is trivial. When M is fully consistent, i.e. (M ) = 0 and α = 1, it is easy to see that both conditions (11) are satisfied. Thus, in such a case A is an M-matrix, and what follows Aµ = b always has strictly positive solution. Several upper bounds for (M ) related to parameters n and r arising from the above theorem are gathered in the Table 1.  Let us note that for any combination of r and n where 0 < r ≤ n − 2, where r, n ∈ + the right side of (13) is greater than 0. In other words for a sufficiently low inconsistency the equation Aµ = b always has a feasible solution. To prove this it is enough to show that for n = 3, 4, . . . holds 1 + 1 + 4(n − 1)(n − r − 2) /2(n − 1) < 1.

Summary
The reliability of the results achieved in the PC method are inseparably linked to the degree of inconsistency of the input data [17]. The lower the inconsistency the better and more reliable the results might be expected to be. Therefore, most practical applications of the PC method seek to construct the PC matrix with the smallest possible inconsistency. The theorem proven in this article is in line with the tendency to seek PC solutions with low inconsistency. It shows that for an appropriately small inconsistency (M ) the linear equation proposed in the HRE approach always has a feasible solution. 1 The smaller root 1− 1+4(n −1)(n −r −2) 2(n −1) ≤ 0 for any n = 3, 4 . . . and 0 < r ≤ n − 2, so it does not need to be taken into account. 2 To demonstrate this please consider the sequence of inequalities Despite the use of (M ), only those entries of M that make up the matrix A are important for the proof of the theorem 2. Therefore, in practice the inconsistency can be checked only for the minor of M whose rows and columns correspond to the elements from the set of unknown concepts C U . This observation also may suggest that the provided estimation could be improved so that all the entries of M contribute to the final result.