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Inconsistency in the analytic hierarchy process under data uncertainties

Przemysław Rutka, Ryszard Smarzewski

arXiv:2610.29418v1cs.CE

TL;DR

The paper asks how likely a pairwise-comparison matrix close to a given inconsistent matrix is to fall below an inconsistency threshold. It combines probabilistic product measures with Koczkodaj-type indicators to derive formulas for selected matrix orders and strict estimates for others.

  • Problem

    The paper addresses estimating the probability that a randomly generated matrix close to a given inconsistent matrix stays below a specified inconsistency threshold.

    Source: §1 · p. 2, §2 · p. 4

  • Method

    The study combines Lebesgue-Stieltjes product measures with Koczkodaj indicators and, for n > 3, introduces a Steiner-triple-system-based indicator.

    Source: §1 · p. 2, §6 · p. 12

  • Results

    Explicit inconsistency-probability formulas are derived for n ≡ 1 or 3 (mod 6), with strict estimates for the remaining values of n.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The probability formulas can estimate the frequency of random matrices within an assumed inconsistency level and suggest a way to set an acceptance threshold for expert assessments.

    Source: §7 · p. 15

    For the threshold-adjustment algorithm, larger matrix order requires more experts; the paper notes that alternatives can instead be divided into smaller groups.

    Source: §7 · p. 16

Abstract

from arXiv · show

In this paper, we study the inconsistency of matrices in multicriteria group decision making under the assumption of uncertainty in their entries. The study binds together an $n$-dimensional probabilistic product measure $P$ and the Koczkodaj indicator $K_{n}$ of order $n$ used in pairwise comparisons. In the case of $n$ greater than 3, it starts with a new definition of $K_{n}$ based on the Steiner triple system of order $n$. It is of interest that the new indicator requires only $O \left( n ^{2} \right)$ evaluations, while the old one $O \left( n ^{3} \right)$. Our study results in analytic formulae for the probabilities of events $K_{n} \leq \varepsilon$ corresponding to uniform, exponential, and normal distribution functions, whenever the remainder of $n$ divided by 6 equals 1 or 3. For the remaining values of $n$, we present analytic formulae for their strict bounds. The formulae lead to much more efficient algorithms when compared with Monte Carlo simulations. This unexpected feature is illustrated by a few numerical experiments.

1. Introduction and preliminaries

The paper frames inconsistency as a probabilistic question: how likely is a nearby pairwise-comparison matrix to remain below a chosen threshold? It develops probability formulas and estimates, and introduces a Steiner-triple-system-based indicator for larger matrices.

1. Introduction and preliminaries

  • Pairwise comparisons collect experts’ subjective relative preferences into a square matrix whose order equals the number of alternatives.The method aims to derive a priority vector ranking those alternatives by importance.Source: §1 · p. 1
  • Because human assessments commonly depart from perfect consistency, matrices must be checked, and insufficiently consistent ones repaired or excluded.The paper identifies consistency assessment as a practical concern in decision making.Source: §1 · pp. 1–2
  • The central problem is estimating the chance of generating a matrix close to a given inconsistent matrix while keeping its probabilistic inconsistency below a fixed threshold.The paper notes that this probability can inform choices of alternatives and thresholds for accurate approximations.Source: §1 · p. 2
  • For n > 3, the paper introduces a Koczkodaj-type indicator based on a Steiner triple system of order n.The introduction reports that the analytic formulas for 3 × 3 perturbations under uniform, exponential, and normal distributions are compared with Monte Carlo simulations requiring 10 million samples for comparable results.Source: §1 · p. 2
  • Explicit inconsistency-probability formulas are derived for n ≡ 1 or 3 (mod 6), with strict estimates for the remaining values of n.The introduction also describes a threshold-adjustment algorithm based on the number of experts.Source: §1 · p. 2

2. Indicator measure of inconsistency

This section sets up pairwise-comparison matrices and an inconsistency indicator, then formulates the probability problem under product measures. It also identifies a coverage hypothesis that does not hold for the original indicator construction when n > 3.

2. Indicator measure of inconsistency

  • A pairwise-comparison matrix collects experts’ relative preferences, with its order matching the number of alternatives.The priorities sought from these comparisons rank alternatives by importance.Source: §2 · p. 3
  • The Koczkodaj indicator has K3 = K, lies between 0 and 1, and is zero if and only if the matrix is consistent.The section introduces the order-3 indicator as a basis for measuring inconsistency.Source: §2 · p. 3
  • Hypothesis 1 requires each above-diagonal matrix entry to occur in exactly one selected triad submatrix.The section states that this hypothesis fails for the original construction when n > 3.Source: §2 · p. 3
  • The section formulates the task as estimating the chance that a random n × n pairwise-comparison matrix is close to a given inconsistent matrix and satisfies an inconsistency threshold.The probability is considered for Lebesgue-Stieltjes product measures.Source: §2 · p. 4
  • The probability problem is to estimate the chance of randomly generating a matrix close to a given inconsistent matrix while satisfying the stated inequality.The indicator is considered as a function of positive entries m_ij for i < j.Source: §2 · p. 4

3. Uniform perturbation of 3 × 3 matrices

The section derives uniform-measure inconsistency probabilities for perturbed 3 × 3 pairwise-comparison matrices, including scaled supports. Explicit formulae cover the stated parameter range and support efficient probability computation.

3. Uniform perturbation of 3 × 3 matrices

  • The probability measure is built from independent uniform variables, with scaled support Q_a = aQ for perturbations beyond the unit cube.The three variables define the 3 × 3 pairwise-comparison matrix.Source: §3 · pp. 4, 6
  • The probability approaches 0 and 1 at ε = 0 and ε = 1, respectively, with endpoint convergence orders stated in a corollary.The supplied passage also gives an asymptotic bound as ε approaches 1.Source: §3 · p. 5
  • For 0 < ε = 1−τ < 1 < a, Theorem 3 gives explicit uniform probabilities with a branch at aτ ≤ 1.The section also states a complementary-threshold formula for P_a(K ≤ 1−ε).Source: §3 · pp. 6–7
  • Theorem 2 and 3 formulae enable efficient computation of P(K ≤ ε) and P_a(K ≤ ε), which Algorithm 1 uses to compute P_a.The algorithm takes ε and scale a as inputs and returns the probability.Source: §3 · p. 7

Numerical experiments

The numerical experiments evaluate Algorithm 1 over a range of scales and inconsistency thresholds. Their probabilities agree closely with Monte Carlo estimates based on 10^7 random matrices.

Numerical experiments

  • Algorithm 1 was evaluated for a = 1, 2, …, 10 and ε = 0.05, 0.1, 0.2, …, 0.9, 0.95.The reported probabilities were rounded to four decimal places.Source: Numerical experiments · p. 7
  • Monte Carlo estimates based on N = 10^7 random matrices differed from the formula-derived probabilities by no more than 0.0004.Each matrix used independently sampled x, y, and z values from (0, a].Source: Numerical experiments · p. 7
  • The experiments compare each formula-derived probability with a classical Monte Carlo approximation of P_a(K ≤ ε).The Monte Carlo estimate was the fraction of sampled matrices whose inconsistency did not exceed ε.Source: Numerical experiments · p. 7

4. Exponential perturbation of 3 × 3 matrices

For positive data drawn from an exponential distribution, the section derives an analytic inconsistency probability. It expresses the result using special functions and describes efficient numerical evaluation.

4. Exponential perturbation of 3 × 3 matrices

  • The exponential-case probability can be expressed using the exponential integral E1 or the confluent hypergeometric function U.The passage identifies U(1, 1, t) = e^tE1(t).Source: §4 · p. 7
  • Theorem 4 gives an analytic expression for P(K ≤ ε) under the exponential distribution, for 0 < ε = 1−τ < 1 and λ > 0.The density is λe^(−λt) for t > 0.Source: §4 · pp. 7–8
  • The probability can be efficiently approximated with Gauss–Laguerre quadrature, whose weights and nodes can be precomputed.The passage also gives a MATLAB evaluation route using expint(t) ≈ E1(t).Source: §4 · p. 9

Numerical experiments

The exponential experiments compute probabilities across parameter values and compare them with Monte Carlo estimates. The largest reported difference is 0.0003.

Numerical experiments

  • The exponential probabilities were computed for parameter values and selected results are shown in Figure 2.The figure presents P(K ≤ ε) for different λ values.Source: Numerical experiments · p. 9, Figure 2 · p. 10
  • Each Monte Carlo estimate used 10 million exponentially distributed random PC matrices, with inconsistency checked for each matrix.The maximum difference from the corresponding computed probabilities was 0.0003.Source: Numerical experiments · p. 9
  • The maximum difference between corresponding computed and Monte Carlo probabilities was 0.0003.The probabilities were rounded to four decimal places.Source: Numerical experiments · p. 9

5. Normal perturbation of 3 × 3 matrices

For normal perturbations of 3 × 3 matrices, the paper expresses the conditional inconsistency probability using an arctan-exponential integral and describes its numerical approximation.

5. Normal perturbation of 3 × 3 matrices

  • The normal model assumes mean μ = 0 without loss of generality and uses standard deviation σ > 0.The density is described in terms of mean μ and standard deviation σ.Source: §5 · p. 10
  • For 0 < ε < 1 and σ > 0, Theorem 5 gives a formula for the conditional probability.The formula is expressed in terms of the arctan-exponential integral.Source: §5 · pp. 10–11
  • The conditional-probability expression is derived using integration by parts and repeated application of an identity.The derivation concludes by invoking the arctan-exponential integral.Source: §5 · p. 11
  • The paper recommends Gauss-Hermite quadrature to approximate the probability numerically.Its weights and nodes can be precomputed using established algorithms.Source: §5 · p. 11

Numerical experiments

The normal-perturbation experiments compare computed probabilities with Monte Carlo estimates; the maximum reported discrepancy is 0.0003.

Numerical experiments

6. Perturbation of n × n matrices for n > 3

For matrices of order n > 3, the paper defines inconsistency probabilities using selected triads and product measures. Steiner triple systems give pairwise coverage for certain orders; other orders use minimum triangle covers.

6. Perturbation of n × n matrices for n > 3

  • The section considers P_n(K_n ≤ ε) under a Lebesgue-Stieltjes product measure on R^n for n ≥ 3.For n = 3, the notation is P = P_3.Source: §6 · p. 12
  • For n > 3, triad submatrices are not generally independent, so the refined indicator assumes independence only for disjoint index sets.Under Hypothesis 1, the theorem uses independence of the selected submatrices to obtain the probability formula.Source: §6 · p. 13
  • When n ≡ 1 or 3 (mod 6), a Steiner triple system selects n(n−1)/6 triads so every pair of indices occurs together exactly once.Such systems exist precisely for those congruence classes.Source: §6 · p. 13
  • For other n, Q(n) is a minimum triangle cover containing every above-diagonal matrix entry in at least one selected triad.Corollary 6.1 gives a bound on the cover size.Source: §6 · p. 14
  • The paper gives cover sizes |Q(4)| = 3, |Q(5)| = 4, |Q(6)| = 6, and |Q(8)| = 11.Table 2 shows the minimal triangle cover for n = 8.Source: §6 · p. 15, Table 2 · p. 15

7. Conclusions

The probability formulae estimate how often random pairwise-comparison matrices fall within an inconsistency tolerance and suggest a way to set that tolerance for expert assessments. Algorithm 2 adjusts the threshold according to the number of experts, while its applicability depends on the matrix order.

7. Conclusions

  • The formulae estimate the frequency of random matrices within a specified inconsistency level and suggest a threshold for accepting experts’ pairwise-comparison matrices.The proposed threshold is the largest value for which r < 1.Source: §7 · pp. 15–16
  • Only pairwise-comparison matrices with Kn ≤ ε proceed to later stages of decision making.Source: §7 · p. 15
  • Algorithm 2 adjusts the inconsistency threshold by divide-and-conquer using the expert count, an initial threshold, and a required precision.It compares mPn(Kn ≤ ε) with 1 while narrowing the threshold interval.Source: §7 · p. 16
  • Threshold adjustment requires more experts as matrix order n increases; in practice n is usually at most 7, and larger alternative sets may be split into smaller groups.Source: §7 · p. 16
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