Sufficient conditions for a Heuristic Rating Estimation Method application

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frontmatter\ead{[email removed]} \ead{[email removed]} \ead{[email removed]} \cortext[cor1]{Corresponding author} \address[wms]{AGH University of Science and Technology, Faculty of Applied Mathematics, Krakow, Poland} \address[weaiib]{AGH University of Science and Technology, Faculty of Electrical Engineering, Automatics, Computer Science and Biomedical Engineering, Krakow, Poland} \address[suo]{Silesian University in Opava, Department of Informatics and Mathematics, School of Business Administration in Karvina, Czech Republic} \begin{abstract} A series of papers has introduced the Heuristic Rating Estimation method, which evaluates a set of alternatives based on pairwise comparisons and the weights of reference alternatives. We formulate the conditions under which the HRE method can be applied correctly. The research considers both arithmetic and geometric algorithms for complete and incomplete pairwise comparison methods. The illustrative examples show that the estimations of inconsistency in the arithmetic variant are optimal. \end{abstract} \begin{keyword} decision-making methods \sep MCDM \sep pairwise comparisons \sep HRE \sep AHP \sep Heuristic rating estimation \end{keyword}

Introduction

Pairwise comparison (PC) methods constitute a well-established class of decision-making tools used to derive priorities or rankings of alternatives based on relative judgments. Among the most prominent representatives of this family are methods such as the Analytic Hierarchy Process (AHP), ELECTRE, PROMETHEE, MACBETH, TOPSIS, PAPRIKA, or the Best--Worst Method BanaECosta1994baip,Brans2005pm,Figueira2005em,Hansen2008anmf,Rezaei2015bwmc,Saaty1977asmfSIMPL,Saaty1980tahp,Yoon1987arad.

According to the classic work Miller1956tmns, humans are able to process only $7\pm2$ "pieces" of information at a time. The fundamental advantage of pairwise comparisons lies in their ability to reduce this cognitive burden, as decision-makers are required to compare only two alternatives at a time rather than evaluate all options simultaneously.

The origins of pairwise comparisons can be traced back to the 13th century, when Ramon Llull applied comparative judgments to select a prelate from a group of candidates Colomer2011rlfa. The method was later formalized in the early 20th century by L. L. Thurstone in his Law of Comparative Judgment Thurstone1927tmop, and gained widespread popularity following the introduction of AHP by T. L. Saaty in the late 1970sSaaty1977asmfSIMPL. Since then, PC methods have been successfully applied across a broad range of disciplines, including engineering, economics, management, and social sciences.

The Heuristic Rating Estimation (HRE) method, proposed in the mid-2010s Kulakowski2014hrea,Kulakowski2015hreg,Kulakowski2016note, represents a modification of classical pairwise comparison techniques. Its distinctive feature is the explicit differentiation between two categories of alternatives: those with known preference values (reference alternatives) and those whose preferences are unknown. The objective of HRE is to estimate the unknown priorities by exploiting pairwise comparisons both among unknown alternatives and between unknown and reference alternatives. This structure makes HRE particularly suitable for decision problems in which partial, yet reliable, preference information is available in advance. Typical situations where this setting occurs include an introduction of a new product on the market while performance of previous products is known, elections where a performance of previous candidates is known, etc.

Two main variants of the HRE method have been developed: the arithmetic and the geometric approaches Kulakowski2014hrea, Kulakowski2016note Kulakowski2015hreg. In the arithmetic version of the HRE, the priority of an alternative is computed as an arithmetic mean of its evaluations relative to other alternatives, whereas the geometric version relies on geometric aggregation. Existing research on HRE has primarily focused on issues related to solution existence Kulakowski2016note, inconsistency assessment Kulakowski2020iifi, and methodological extensions, including multiple-criteria, incomplete or fuzzy pairwise comparison matrices, see also Kedzior2023mchr,Kulakowski2026mbip.

In practical applications, however, decision-makers often face incomplete pairwise comparison data, which raises fundamental questions regarding the solvability and uniqueness of the resulting estimation problem. The aim of this paper is to address this gap by providing sufficient conditions for the applicability of the HRE method in the case of incomplete pairwise comparison matrices. Using tools from linear algebra and spectral theory, we establish conditions under which the arithmetic and geometric HRE formulations admit unique solutions.

While Kulakowski2026mbip shares a common methodological background with studies on mean-based incomplete pairwise comparison methods with reference values, its contribution addresses a distinct and previously underexplored problem. Existing mean-based approaches focus on extending the HRE method by modifying aggregation rules and broadening admissible comparison structures. In contrast, the present study adopts a theoretical perspective and asks under what conditions the standard arithmetic and geometric HRE methods are well-defined. By deriving sufficient conditions for invertibility and uniqueness using spectral radius arguments and matrix theory, this paper fills a theoretical gap by establishing formal guarantees for HRE applicability, independent of any particular extension or optimization-based reformulation.

The remainder of the paper is organized as follows. Section (ref) introduces preliminary concepts related to pairwise comparisons and selected results from linear algebra. Section (ref) discusses the arithmetic and geometric HRE methods in the complete case and derives conditions ensuring solution existence. Section (ref) extends the analysis to incomplete pairwise comparison matrices and presents sufficient conditions for solvability in both arithmetic and geometric HRE variants. The paper concludes with a summary of the main results.

Preliminaries

Pairwise comparisons

The pairwise comparisons (PC) method is the process designed to transform the set of comparisons into a ranking of alternatives. Let $A=\{a_{1},\ldots,a_{n}\}$ denote a set of alternatives while $C=[c_{ij}]$ means set of comparisons in the form of $n\times n$ matrix, where $c_{ij}\in\mathbb{R}_{+}$ for $i,j=1,\ldots,n$. Each $c_{ij}$ means the result of direct comparison between $a_{i}$ and $a_{j}$. When the result of the given comparison $c_{ij}$ is unknown we write that $c_{ij}=c_{ji}=?$.

defnA PC matrix $C$ is said to be reciprocal if $c_{ij}=1/c_{ji}$ for all $i,j=1,\ldots,n$ except when $c_{ij}=?$.

The purpose of PC method is to calculate the ranking.

defnLet the weight function for $A$ be $w:A\rightarrow\mathbb{R}_{+}$, such that if $w(a_{i})>w(a_{j})$ for some $0<i,j\leq n$ then $a_{i}$ is more preferred than $a_{j}.$

The above function prioritizes the various alternatives. The more preferred alternatives have higher weights. The function $w$ is usually represented as a priority vector in the form:

equation[equation omitted — 90 chars of source]

The fact that $c_{ij}$ corresponds to the ratio of the preferential strength of alternatives $a_{i}$ and $a_{j}$ implies that one may expect that $c_{ij}=w(a_{i})/w(a_{j})$. This in turn results in a postulate of transitivity\footnote{As $c_{ik}=w(a_{i})/w(a_{k})$ and $c_{kj}=w(a_{k})/w(a_{j})$.} i.e. $c_{ij}=c_{ik}c_{kj}$ for every triad $i,j,k=1,\ldots,n$. Unfortunately, because the vector $w$ is computed based on all pairwise comparisons, in practice we may expect only that $c_{ij}\approx w(a_{i})/w(a_{j})$, thus $c_{ij}\approx c_{ik}c_{kj}$.

Now we recall the general definition of a consistent PC matrix, which is valid both in a complete and incomplete case.

defnA reciprocal $n\times n$ PC matrix $C$ is said to be consistent if for each $i_{1},\ldots i_{m}\in\{1,\ldots,n\}$ \[ c_{i_{1}i_{2}}\cdot\ldots\cdot c_{i_{m-1}i_{m}}\cdot c_{i_{m}i_{1}}=1, \] provided that all the factors in the above product are known.

It is easy to prove that if $c_{ij}=c_{ik}c_{kj}$ then $c_{ij}=w(a_{i})/w(a_{j})$ Kulakowski2020utahp regardless of the prioritization method providing of course that $c_{ij}$ means the ratio between preferential strength of $a_{i}$ and $a_{j}$.

There are at least a dozen methods for computing a vector $w$ Choo2004acff,Kulakowski2020utahp. The two most popular are the Eigenvalue Method (EVM) and Geometric Mean Method (GMM). The first one was originally proposed by Saaty in his seminal paper Saaty1977asmf is based on the concept of a vector and the eigenvalue of the matrix $C$. So, let $C=[c_{ij}]$ be a PC matrix containing expert judgments for $n$ alternatives, and $w_{\textit{max}}$ be a principal eigenvector of $C$ i.e. \[ Cw_{\textit{max}}=\lambda_{\textit{max}}w_{\textit{max}}, \] where $\lambda_{\textit{max}}$ is a principal eigenvalue (spectral radius) of $C$. Thus, the priority vector $w_{\textit{ev}}$ is a rescaled version of $w_{\textit{max}}$ i.e. \[ w_{\textit{ev}}=\frac{1}{\sum^{n}_{i=1}w_{\textit{max}}(a_{i})}w_{\textit{max}}. \] The second method, although it is based on similar premises Kulakowski2016srot, is easier to calculate. In GMM the priority of the $i$-th alternative is the appropriately rescaled geometric mean of all its direct comparisons. I.e. \[ w_{\textit{gm}}(a_{i})=\alpha\left(\prod^{n}_{j=1}c_{ij}\right)^{1/n}, \]

where \[ \alpha=\left(\sum^{n}_{i=1}\left(\prod^{n}_{j=1}c_{ij}\right)^{1/n}\right)^{-1}. \] Both methods EVM and GMM have their incomplete counterparts. Extending the EVM to incomplete matrices was proposed by Harker Harker1987amoq. Similar extension of GMM for incomplete PC matrices can be found in kulakowski2020otgm or equivalently for Logarithmic Least-Squares method in Bozoki2010ooco,Tone1993llsm.

Inconsistency measures

In the literature we can find a lot of inconsistency indices for complete matrices, however the first one and probably the most popular one is consistency index CI introduced by Saaty Saaty1977asmf. For a given $n\times n$ PC matrix $C$ Consistency Index is defined as \[ CI(C)=\frac{\rho(C)-n}{n-1}, \] where $\rho(C)$ is a spectral radius of $C$, which is also its eigenvalue.

Following the idea of Saaty in Harker1987amoq Harker defined the Consistency Index for incomplete PCMs as \[ \overline{CI}(C)=\frac{\rho(H)-n}{n-1}, \] where $H$ is an auxiliary matrix resulting from $C$:

equation[equation omitted — 164 chars of source]

and $s_{i}$ denotes the number of ? in the $i$-th row.

Some notions and facts from linear algebra

Let us recall some useful definitions and theorems from the linear algebra.

remA square matrix is invertible if and only if $0$ is not its eigenvalue.

defnFor a given $k\times k$ incomplete matrix $A=[a_{ij}]$ we define the corresponding directed graph $G(A)=(V,E)$, with the set of vertices $V=\{1,\ldots,k\}$ such that an edge $i\rightarrow j$ belongs to $E$ if and only if $a_{ij}$ is known.

defnWe call a $k\times k$ matrix $A$ irreducible if its corresponding graph $G(A)$ is strongly connected.

thmIf $\lambda\in\mathbb{C}$ is an eigenvalue of a $k\times k$ matrix $A$ and $\alpha\in\mathbb{C}$ then $\alpha(\lambda-1)$ is an eigenvalue of $\alpha(A-I_{k})$.

defnFor a $k\times k$ matrix $A$ we define its spectral radius as \[ \rho(A)=\max\{|\lambda_{1}|,\ldots,|\lambda_{k}|\}, \] where $\lambda_{1},\ldots,\lambda_{k}$ are eigenvalues of $A$.

thm[Perron-Frobenius] If $A$ is a $k\times k$ non-negative matrix such that for some $p\in\mathbb{N}$ the matrix $(I_{k}+A)^{p}$ is positive then \begin{itemize} • $\rho(A)$ is a positive single eigenvalue of $A$. • there exists an eigenvector corresponding to $\rho(A)$ with positive coordinates. \end{itemize}

For two matrices $A=[a_{ij}]$ and $B=[b_{ij}]$ we will write $A\leq B$ if for each $i,j$ the inequalty $a_{ij}\leq b_{ij}$ holds. Similarly, the notation $A<B$ means that $A\leq B$ and there exist $i,j$ such that $a_{ij}<b_{ij}$.

thmLet $A=[a_{ij}]$ and $B=[b_{ij}]$ be two square nonnegative matrices. If $A\leq B$, then $\rho(A)\leq\rho(B)$. Moreover, if $B$ is irreducible and $A<B$, then $\rho(A)<\rho(B)$.

The next corollary follows immediately from Theorems (ref) and (ref).

corIf $C$ is a $k\times k$ PC matrix and $\alpha>0$ then \[ \rho(\alpha(C-I_{k}))=\alpha(\rho(C)-1). \]

thmIf $B$ is a $k\times k$ matrix and $\rho(B)<1$, then $I_{k}-B$ is invertible, the series \[ \sum^{\infty}_{j=0}B^{j} \] is convergent and \[ (I_{k}-B)^{-1}=\sum^{\infty}_{j=0}B^{j}. \]

For a $k\times k$ matrix we put \[ r_{i}=\sum_{j\neq i}|a_{ij}|. \] and for $i\in\{1,\ldots,k\}$ we define a Gersgorin disc as \[ D_{i}=\{z\in\mathbb{C}:\ |z-a_{ii}|\leq r_{i}\}. \]

thm[Gershgorin] If $A=[a_{ij}]$ is a $k\times k$ matrix and $\lambda\in\mathbb{C}$ is its eigenvalue, then \[ \lambda\in\bigcup^{n}_{i=1}D_{i}. \]

Heuristic Rating Estimation Method - the complete case

We assume that the set of alternatives consists of two disjoint subsets: the alternatives with unknown preferential values $A_{U}=\{a_{1},\ldots,a_{k}\}$ where $1\leq k<n$, and the reference alternatives for which the preferences are known $A_{K}=\{a_{k+1},\ldots,a_{n}\}$. The ranking value $w(a_{l})\in\mathbb{R}_{+}$ for each $l\in\{k+1,\ldots,n\}$ is known from the very beginning. The aim of the HRE process is to derive the unknown priorities of the alternatives from $A_{U}$.

Let \[ C_{k}=\left[

array[array omitted — 151 chars of source]

\right] \] be a $k\times k$ submatrix of an $n\times n$ PC matrix $C$.

The arithmetic HRE

The arithmetic HRE procedure has been introduced in Kulakowski2014hrea. The ranking can be found by solving the equation

equation[equation omitted — 44 chars of source]

where \[ A_{k}=\left[

array[array omitted — 249 chars of source]

\right] \] is a $k\times k$ auxiliary matrix and

\[ b=\left[

array[array omitted — 153 chars of source]

\right] \] is a constant term vector.

We can formulate sufficient conditions for $A_{k}$ to be invertible by means of Saaty consistency index CI of the matrix $C_{k}$.

thmIf $1<k<n$ and $\textit{CI}(C_{k})<\frac{n-k}{k-1}$ or $k=1$ then matrix $A_{k}$ is invertible, so the equation ((ref)) has the unique solution.

proofFor $k=1$ the statement is obvious, so we consider $k>1$. Assume that $\rho(C_{k})=\lambda\geq k$ is the principle eigenvalue for $C_{k}$ (equal to its spectral radius). Since $C_{k}$ is a PC matrix, it follows from Corollary (ref) that \[ \rho(C_{k}-I_{k})=\lambda-1 \] and \[ \rho\left(\frac{1}{n-1}(C_{k}-I_{k})\right)=\frac{\lambda-1}{n-1}. \] Thus, according to Theorem (ref), \[ A_{k}=I_{k}-\frac{1}{n-1}(C_{k}-I_{k}) \] is invertible if \[ \frac{\lambda-1}{n-1}<1, \] which is equivalent to \begin{equation} \lambda<n. \end{equation} As \[ \textit{CI}(C_{k})=\frac{\lambda-k}{k-1}, \] the condition \[ \textit{CI}(C_{k})<\frac{n-k}{k-1} \] is equivalent to ((ref)), which completes the proof.

Theorem (ref) implies the following corollary:

corIf $C_{k}$ is consistent, that is $CI(C_{k})=0$, then the equation ((ref)) has the unique solution.

In particular, if $C$ is consistent then for each $k<n$ matrices $C_{k}$ are consistent, too.

Notice that hardly ever $A_{k}$ is singular, but the following example shows that it is possible.

exampleLet \[ C=\left[\begin{array}{ccccc} 1 & 26+5\sqrt{27} & 1 & * & *\\ 26-5\sqrt{27} & 1 & 1 & * & *\\ 1 & 1 & 1 & * & *\\ * & * & * & 1 & *\\ * & * & * & * & 1 \end{array}\right] \] be a $5\times5$ PC matrix (stars in the last two columns can be replaced with any positive numbers and the ones in the last two rows by their inverses). Then, for $k=3$, in the HRE procedure we obtain the matrix \[ {A}_{3}=\left[\begin{array}{ccc} 1 & -\frac{26+5\sqrt{27}}{4} & -\frac{1}{4}\\ -\frac{26-5\sqrt{27}}{4} & 1 & -\frac{1}{4}\\ -\frac{1}{4} & -\frac{1}{4} & 1 \end{array}\right], \] which is singular. This does not contradict Theorem (ref), because \[ \rho(C_{3})=5, \] which implies that \[ CI(C_{3})=2=\frac{n-k}{k-1}. \] This implies that the threshold of inconsistency in Theorem (ref) is optimal and cannot be improved.

The geometric HRE

According to Kulakowski2015hreg, to calculate the ranking in the case of a geometric approach the equation

equation[equation omitted — 57 chars of source]

must be solved, where

\[ {A}_{k}=\left[

array[array omitted — 121 chars of source]

\right] \] and \[ {b}=\left[

array[array omitted — 208 chars of source]

\right]. \] Since $\widehat{w}(a_{i})$ is the logarithmized value of $w(a_{i})$ i.e. $\widehat{w}(a_{i})=\log_{e}w(a_{i})$ then the final ranking vector is obtained from $\widehat{w}$ by the exponential transformation $w(a_{i})=e^{\widehat{w}(a_{i})}$ for $i=1,\ldots,k$. The solution to a geometric HRE always exists and is optimal Kulakowski2015hreg.

We will show it once more by means of the spectral radius.

thmFor each $k<n$ the matrix ${A}_{k}$ is invertible. Therefore, problem ((ref)) always has the unique solution.

proofLet \[ {\bf 1}_{k}=\left[\begin{array}{ccc} 1 & \cdots & 1\\ \vdots & \ddots & \vdots\\ 1 & \cdots & 1 \end{array}\right] \] be a $k\times k$ matrix of $1$s. Obviously, $\rho({\bf 1_{k}})=k$, so Corollary (ref) implies that \[ \rho\left(\frac{1}{n-1}({\bf 1_{k}}-I_{k})\right)=\frac{k-1}{n-1}<1. \] Therefore, by Theorem (ref) \[ {A}_{k}=(n-1)\left(I_{k}-\frac{1}{n-1}({\bf 1_{k}}-I_{k})\right) \] is invertible.

Heuristic Rating Estimation Method - the incomplete case

In practice it may happen that not all comparisons between alternatives are made, so a PC matrix may be incomplete. Of course, the necessary condition for an incomplete PC matrix to induce a ranking is that it is irreducible (see Koczkodaj2015pcs). From now on we assume that the considered PC matrix $C$ is irreducible. We also assume that \[ c_{pq}=\frac{w(p)}{w(q)}, \] for each $p,q\in\{k+1,\ldots,n\}$.

Let us denote the number of undefined comparisons in the $i$-th row of a PC matrix $C$ by $s_{i}$. The weights we want to determine are denoted as:

equation[equation omitted — 129 chars of source]

The arithmetic incomplete HRE

The original method's idea is to find the unknown weights of alternatives satisfying the system of linear equations:

equation[equation omitted — 210 chars of source]

Let us define the following matrix $k\times k$ matrix $D_{k}$:

equation[equation omitted — 132 chars of source]

The arithmetic procedure ((ref)) can be rewritten in the form of the matrix equation:

equation[equation omitted — 55 chars of source]

where

equation[equation omitted — 428 chars of source]

and

equation[equation omitted — 348 chars of source]

If $C$ is irreducible and consistent we may properly define a complete and consistent matrix $\hat{C}$, whose elements are given as follows: \[ \hat{c}_{ij}=

casesc_{ij}, & if\,\,c_{ij}\neq\ ?\\ c_{i_{1}i_{2}}\cdot\ldots\cdot c_{i_{m-1}i_{m}}, & if\,\,c_{ij}=\ ?,\ i_{1}=i,i_{m}=j\\ & and for each p\in\{1,\ldots,m-1\}\ c_{i_{p}i_{p+1}}\neq\ ?

. \]

thmIf $C$ is an irreducible and consistent PC matrix, then vector ((ref)), whose coordinates are given by \[ w(a_{p})=\hat{c}_{pn}\cdot w(a_{n}) \] for $p\in\{1,\ldots,k\}$ is the unique solution of ((ref)).

proofWe will consider two cases. If $i\leq k$, then \begin{eqnarray*} \frac{1}{n-1}\left(\sum_{\substack{j=1,i\neq j\\ c_{ij}\neq? } }c_{ij}w(a_{j})+\sum_{\substack{j=1,i\neq j\\ c_{ij}=? } }w(a_{i})\right)=\\ =\frac{1}{n-1}\left(\sum_{\substack{j\leq k,i\neq j\\ c_{ij}\neq? } }c_{ij}w(a_{j})+\sum_{\substack{j>k,i\neq j\\ c_{ij}\neq? } }c_{ij}w(a_{j})+\sum_{\substack{j=1,i\neq j\\ c_{ij}=? } }w(a_{i})\right)=\\ =\frac{1}{n-1}\left(\sum_{\substack{j\leq k,i\neq j\\ c_{ij}\neq? } }\hat{c}_{ij}\hat{c}_{jn}w(a_{n})+\sum_{\substack{j>k,i\neq j\\ c_{ij}\neq? } }\hat{c}_{ij}\frac{w(a_{j})}{w(a_{n})}w(a_{n})+\sum_{\substack{j=1,i\neq j\\ c_{ij}=? } }w(a_{i})\right)=\\ =\frac{1}{n-1}\left(\sum_{\substack{j=1,i\neq j\\ c_{ij}\neq? } }\hat{c}_{in}w(a_{n})+\sum_{\substack{j=1,i\neq j\\ c_{ij}=? } }w(a_{i})\right)=\\ =\frac{1}{n-1}\left(\sum_{\substack{j=1,i\neq j\\ c_{ij}\neq? } }w(a_{i})+\sum_{\substack{j=1,i\neq j\\ c_{ij}=? } }w(a_{i})\right)=\frac{1}{n-1}\sum_{\substack{j=1,i\neq j} }w(a_{i})=w(a_{i}). \end{eqnarray*} Similarly, if $i>k$, then \begin{eqnarray*} \frac{1}{n-1}\left(\sum_{\substack{j=1,i\neq j\\ c_{ij}\neq? } }c_{ij}w(a_{j})+\sum_{\substack{j=1,i\neq j\\ c_{ij}=? } }w(a_{i})\right)=\\ =\frac{1}{n-1}\left(\sum_{\substack{j=1,i\neq j\\ c_{ij}\neq? } }\hat{c}_{in}w(a_{n})+\sum_{\substack{j=1,i\neq j\\ c_{ij}=? } }w(a_{i})\right)=\\ =\frac{1}{n-1}\left(\sum_{\substack{j=1,i\neq j\\ c_{ij}\neq? } }\frac{w(a_{i})}{w(a_{n})}w(a_{n})+\sum_{\substack{j=1,i\neq j\\ c_{ij}=? } }w(a_{i})\right)=\\ =\frac{1}{n-1}\sum_{\substack{j=1,i\neq j} }w(a_{i})=w(a_{i}). \end{eqnarray*} The uniqueness of the solution of ((ref)) follows from the consistency of $C$ and the fact that the value of $w(a_{n})$ is fixed.

Similarly to the complete case let us define a $k\times k$ submatrix of $C$: \[ C_{k}=\left[

array[array omitted — 151 chars of source]

\right]. \]

For each $j\in{1,\ldots,k}$ let $s_{j}$ denote the number of missing entries in the $j$-th row of $C_{k}$. Let \[ s_{\mathrm{MAX}}=\max_{j\in\{1,\ldots,k\}}s_{j} \] and \[ s_{\mathrm{MIN}}=\min_{j\in\{1,\ldots,k\}}s_{j}. \]

Now we are ready to formulate a result similar to Theorem (ref).

thmIf $1<k<n$ and $\overline{\mathrm{\mathit{CI}}}(C_{k})<\frac{n-k-s_{\mathrm{MAX}}+s_{\mathrm{MIN}}}{k-1}$ or $k=1$ then matrix $A_{k}$ is invertible, so the equation ((ref)) has the unique solution.

proofFor $k=1$ the statement is obvious. For $k>1$ let us define $k\times k$ matrices $H_{k}$ and $D_{k}$ according to ((ref)) and ((ref)), as well as $R_{k}$ given by \begin{equation} {R}_{k}=\left(\begin{array}{cccc} 1 & \frac{d_{12}}{n-s_{1}-1} & \cdots & \frac{d_{1,k}}{n-s_{1}-1}\\ \frac{d_{2,1}}{n-s_{2}-1} & 1 & \cdots & \frac{d_{2,k}}{n-s_{2}-1}\\ \vdots & \vdots & \vdots & \vdots\\ \frac{d_{k-1,1}}{n-s_{k-1}-1} & \cdots & \ddots & \frac{d_{k-1,k}}{n-s_{k-1}-1}\\ \frac{d_{k,1}}{n-s_{k}-1} & \cdots & \frac{d_{k,k-1}}{n-s_{k}-1} & 1 \end{array}\right). \end{equation} If \[ \overline{\mathrm{\mathit{CI}}}(C_{k})=\frac{\rho(H_{k})-k}{k-1}<\frac{n-k-s_{\mathrm{MAX}}+s_{\mathrm{MIN}}}{k-1}, \] then \[ \rho(H_{k})<n-s_{\mathrm{MAX}}+s_{\mathrm{MIN}}, \] which implies that \[ \rho(D_{k}+s_{\mathrm{MIN}}I_{k})=\rho(D_{k})+s_{\mathrm{MIN}}<n-s_{\mathrm{MAX}}+s_{\mathrm{MIN}}, \] since \[ H_{k}=D_{k}+\left(\begin{array}{cccc} s_{1} & 0 & \cdots & 0\\ 0 & s_{2} & \cdots & 0\\ \vdots & \ddots & \ddots & \vdots\\ 0 & \cdots & 0 & s_{k} \end{array}\right)\geq D_{k}+s_{\mathrm{MIN}}I_{k}. \] As a consequence, \[ \rho(D_{k})<n-s_{\mathrm{MAX}}, \] which is equivalent to \[ \frac{\rho(D_{k})-1}{n-s_{\mathrm{MAX}}-1}<1 \] and to \begin{equation} \frac{1}{2}\left(1+\frac{\rho(D_{k})-1}{n-s_{\mathrm{\mathrm{MAX}}}-1}\right)<1. \end{equation} From ((ref)) we have that \[ R_{k}\leq\left(\begin{array}{cccc} 1 & \frac{d_{12}}{n-s_{\mathrm{MAX}}-1} & \cdots & \frac{d_{1,k}}{n-s_{\mathrm{MAX}}-1}\\ \frac{d_{2,1}}{n-s_{\mathrm{MAX}}-1} & 1 & \cdots & \frac{d_{2,k}}{n-s_{\mathrm{MAX}}-1}\\ \vdots & \vdots & \vdots & \vdots\\ \frac{d_{k-1,1}}{n-s_{\mathrm{MAX}}-1} & \cdots & \ddots & \frac{d_{k-1,k}}{n-s_{\mathrm{MAX}}-1}\\ \frac{d_{k,1}}{n-s_{\mathrm{MAX}}-1} & \cdots & \frac{d_{k,k-1}}{n-s_{\mathrm{MAX}}-1} & 1 \end{array}\right)=I_{k}+\frac{D_{k}-I_{k}}{n-s_{\mathrm{MAX}}-1}, \] so \[ \rho\left(\frac{R_{k}}{2}\right)=\frac{1}{2}\rho(R_{k})\leq\frac{1}{2}\left(1+\frac{\rho(D_{k})-1}{n-s_{\mathrm{MAX}}-1}\right). \] Inequality ((ref)) implies that \[ \rho\left(\frac{R_{k}}{2}\right)<1. \] Since \[ A_{k}=2I_{k}-R_{k}=2\left(I_{k}-\frac{R_{k}}{2}\right), \] and from Theorem (ref) it follows that $A_{k}$ is invertible.

Unlike in the case of Theorem (ref), the right side of the inequality in Theorem (ref) does not have to be positive, so Theorem (ref) not always follows from Theorem (ref). However we can formulate the following corollaries, whose assumptions are sufficient to apply HRE to a consistent incomplete PC matrix.

corIf $s_{\textrm{MAX}}=s_{\textrm{MIN}}$ and $\overline{\textit{CI}}(C_{k})<\frac{n-k}{k-1}$ or $k=1$ then matrix $A_{k}$ is invertible.

corIf $k\leq\frac{n+1}{2}$ and $\overline{\textit{CI}}(C_{k})<\frac{n-2k+2}{k-1}$ or $k=1$ then matrix $A_{k}$ is invertible.

proofDue to irreducibility of $C_{k}$, we have \[ 0\leq s_{\textrm{MIN}}\leq s_{\textrm{MAX}}\leq k-2, \] so \[ \frac{n-2k+2}{k-1}\leq\frac{n-k-s_{MAX}+s_{MIN}}{k-1}, \] and the thesis follows from Theorem (ref).

The next example shows that, similarly to the complete case, the incomplete arithmetic HRE procedure in very special cases may not work. On the other hand, it proves that the inconsistency threshold in Theorem (ref) is optimal.

exampleLet \[ C=\left[\begin{array}{cccccccc} 1 & 1 & ? & ? & ? & 1351-780\sqrt{3} & * & *\\ 1 & 1 & 1 & ? & ? & ? & * & *\\ ? & 1 & 1 & 1 & ? & ? & * & *\\ ? & ? & 1 & 1 & 1 & ? & * & *\\ ? & ? & ? & 1 & 1 & 1 & * & *\\ 1351+780\sqrt{3} & ? & ? & ? & 1 & 1 & * & *\\ * & * & * & * & * & * & 1 & *\\ * & * & * & * & * & * & * & 1 \end{array}\right] \] be an $8\times8$ PC matrix. Then, for $k=6$, in the HRE procedure we obtain the matrix \[ A_{6}=\left[\begin{array}{crrrrc} 1 & -\frac{1}{4} & 0 & 0 & 0 & \frac{780\sqrt{3}-1351}{4}\\ -\frac{1}{4}\;\;\; & 1 & -\frac{1}{4} & 0 & 0 & 0\\ 0 & -\frac{1}{4} & 1 & -\frac{1}{4} & 0 & 0\\ 0 & 0 & -\frac{1}{4} & 1 & -\frac{1}{4} & 0\\ 0 & 0 & 0 & -\frac{1}{4} & 1 & -\frac{1}{4}\;\;\;\\ -\frac{780\sqrt{3}+1351}{4} & 0 & 0 & 0 & -\frac{1}{4} & 1 \end{array}\right], \] which is singular. Let us consider the matrix \[ H_{6}=\left[\begin{array}{crrrrc} 4 & 1 & 0 & 0 & 0 & 1351-780\sqrt{3}\\ 1 & 4 & 1 & 0 & 0 & 0\\ 0 & 1 & 4 & 1 & 0 & 0\\ 0 & 0 & 1 & 4 & 1 & 0\\ 0 & 0 & 0 & 1 & 4 & 1\\ 1351+780\sqrt{3} & 0 & 0 & 0 & 1 & 4 \end{array}\right] \] Its spectral radius equals \[ \rho(H_{4})=8, \] while \[ s_{\textrm{MAX}}=3=s_{\textrm{MIN}}. \] Therefore, \[ \overline{\textit{CI}}(C_{6})=\frac{8-6}{6-1}=\frac{2}{5}=\frac{n-k-s_{\textrm{MAX}}+s_{\textrm{MIN}}}{k-1}, \] which does not violate Theorem (ref), but shows that the estimation of inconsistency cannot be improved.

The geometric incomplete HRE

The geometric procedure for incomplete PC matrices comes down to the solution of the following matrix equation:

equation[equation omitted — 63 chars of source]

where \[ {A}_{k}=\left[

array[array omitted — 162 chars of source]

\right], \] with \[ q_{ij}=

cases-1, & if\,\,c_{ij}\neq\ ?\\ 0, & if\,\,c_{ij}=\ ?

, \] and \[ b=\left[

array[array omitted — 46 chars of source]

\right] \] is a vector depending on the known comparisons $c_{ij}$ and the weights of the reference alternatives.

thmThe matrix ${A}_{k}$ is invertible, thus problem ((ref)) always has the unique solution.

proofLet us note that for all $i\in\{1,\ldots,n\}$ the radius of the $i$-th Gersgorin disc $D_{i}$ for ${A}_{k}$ equals \[ r_{i}=\sum_{\{j\neq i:\ c_{ij}\neq\ ?\}}|-1|=k-1-s_{i}, \] and its center is $n-s_{i}-1$. Since \[ \forall i\in\{1,\ldots,k\}\ |n-s_{i}-1-0|>r_{i}, \] Theorem (ref) implies that $0$ is not an eigenvalue of the matrix ${A}_{k}$, which, by Remark (ref), completes the proof.

Summary

In the paper we have provided sufficient conditions for a Heuristic Rating Estimation method to be used. For the arithmetic HRE we have estimated the inconsistency thresholds which guarantee that the method will work. In the case of a complete PCM it depends only on the total number of alternatives and the number of non-referential ones. In the case of incomplete PCMs also the minimum and maximum numbers of missing entries in rows matters. The estimations of inconsistency which imply the possibility of HRE procedure application are optimal, which has been shown by appropriate examples. As far as the geometric HRE is concerned, both algorithms for complete and incomplete PCMs produce rankings for each input.

Acknowledgments

The research has been supported by the National Science Centre, Poland within the grant VIRGO 2024/55/B/HS4/00860. Jacek Szybowski was also supported by AGH University of Krakow (task no. 11.11.420.004).

References

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