New Stochastic Ranking Method Brings Probability to Group Decisions

When a panel of experts must choose between competing options, the final ranking often hides a crucial question: how certain is it? A new study published in Applied Intelligence addresses that gap with a multi-stage stochastic preference analysis framework for group decision-making, built on interval multiplicative preference relations and the Multi-Attributive Border Approximation Area Comparison (MABAC) method. The work, authored by Xueqian Zhao, Ligang Zhou, Hao Li and Xianchao Dai, was published online on 26 October 2026 and appears in Volume 56 of the journal as article number 518.

The core problem the researchers tackle is a familiar one in decision science. Experts frequently cannot express their judgments as precise numbers. Instead, they use interval multiplicative preference relations, where each pairwise comparison between two alternatives is given as a range, such as a preference strength between 6 and 7 on a 1-to-9 scale. According to the authors, previous research on such interval judgments typically focused on the interval endpoints, performing calculations only with the lower and upper bounds. That practice, they argue, ignores the countless values lying inside the interval, causing information loss and reducing the reliability of the resulting rankings.

The new method replaces endpoint-based calculation with stochastic simulation. In the first stage, sample multiplicative preference relations are generated by randomly drawing values from within each expert’s interval judgments, producing complete pairwise comparison matrices that satisfy the multiplicative consistency condition, in which each entry and its transpose multiply to one. From each sampled matrix, a priority vector is computed using the logarithmic least squares method, a technique originally proposed by Crawford and Williams that derives priority weights by minimizing the squared logarithmic deviations between the observed comparisons and the ratios of the implied weights. These priority vectors are then assembled into a sample priority matrix of experts, capturing how each expert’s relative weighting of the alternatives looks across many random draws.

The second stage introduces randomness in a second dimension: the experts’ own importance weights. Rather than fixing expert weights in advance, the framework assigns them randomly at each sampling iteration. Combining the randomly assigned expert weights with the sample priority matrix yields what the authors call a stochastic comprehensive ranking decision matrix. From this matrix, the MABAC machinery is applied in stochastic form. A stochastic boundary comparison area vector is computed as the geometric mean of the weighted standardized values for each expert dimension, and the difference between each alternative’s value and this boundary produces a superiority distance matrix. Summing those distances gives the stochastic total distance superiority degree for each alternative, where a larger value indicates a better rank. This process is repeated for 10,000 iterations, generating a large population of possible rankings that reflect both the uncertainty inside the intervals and the uncertainty about expert importance.

The third stage applies ideas from Stochastic Multicriteria Acceptability Analysis, specifically the SMAA-2 method, which was first proposed by Lahdelma and colleagues in 1998 to handle decision problems where criterion weights and values are uncertain. In the new framework, MABAC effectively acts as the measure function while SMAA-2 supplies the acceptability analysis. For each alternative, the method identifies the favorable preference weight vector space, the set of expert weight vectors under which that alternative outranks all others. Integrating over these spaces yields several quantities: a stochastic acceptability index for each possible rank, an SM stochastic acceptability index for ranking first, a stochastic central preference weight vector representing the center of gravity of the favorable weight space, and a stochastic preference confidence factor that measures the probability that an alternative is the preferred one when the central weight vector is chosen.

These outputs are designed to answer two distinct questions that a single point ranking cannot. The stochastic acceptability indexes determine the ranking results, indicating how probable each alternative is to occupy each rank position, while the stochastic preference confidence factors check whether the sorting result is credible. A higher confidence factor indicates a more reliable final rank. The authors emphasize that this dual reporting gives decision makers a more intuitive and credible form of decision support than a bare ordering, because it shows not only which alternative wins but how likely that outcome is across the full space of plausible expert judgments and weights.

To demonstrate the method, the paper presents a case study in which an electric vehicle manufacturer must select one long-term partner from five potential new energy battery suppliers: CATL, BYD, LG Energy, SK On and Panasonic. Three procurement experts provided interval multiplicative preference relations based on their experience, reflecting uncertainties in technical specifications, delivery times and costs. After running the three-stage analysis, the ranking emerged as LG Energy first, followed by CATL, Panasonic, SK On and BYD. The SM stochastic acceptability index for the top-ranked supplier was 0.9969, meaning that in nearly all of the 10,000 simulated scenarios that supplier ranked first. The reported stochastic preference confidence factors for the ranked positions were high, with values including 0.9964 for the first-ranked alternative, 0.9966 for the second, 0.9192 for the third, 0.9235 for the fourth and 0.9946 for the fifth, which the authors interpret as indicating that the result is quite credible.

The authors also report comparison analyses intended to illustrate the effectiveness and credibility of the new approach. Against a method by Zhou and colleagues that used a group continuous compatibility model with a continuous ordered weighted geometric averaging operator, the best alternative was the same, and the rankings differed only in the interchange of the alternatives placed third and fourth. Against a method by Zhang and colleagues that combined the TODIM approach with SMAA-2 for interval multiplicative preference relations, the ranking was identical. The framework was additionally tested by replacing MABAC within it with three other multi-attribute decision-making methods: ELECTRE III, VIKOR and TOPSIS. According to the paper, all four methods produced the same final ranking, and the proposed method showed a higher stochastic acceptability index in selecting the best supplier. The authors note that the comparison methods have known limitations: ELECTRE III requires extensive calculations and threshold choices that introduce subjectivity, VIKOR does not fully account for downside information about options, PROMETHEE III can confuse decision makers when choosing preference functions and thresholds, and TOPSIS assumes a unique and fixed ideal solution that may not hold in changing environments.

The motivation for the work, as the authors lay it out, rests on three claims about earlier research. First, prior studies on interval multiplicative preference relations in group decision-making used interval endpoint values and overlooked the values within the interval, leading to information loss. Second, past methods provided only a ranking without determining the possibility and acceptability of that ranking, leaving no visual indication of how certain the sorting was. Third, existing group decision-making methods relied on subjectively determined expert weights or failed to measure and assign expert weights systematically, which did not adequately account for expert judgment and uncertainty. The random allocation of expert weights in the new framework, in which weights are assigned randomly each time a sample matrix is drawn, is presented as a way to eliminate subjective human influence on that step of the process.

The paper positions itself within a broader family of stochastic preference analysis research, noting that SMAA has been combined with decision models including the Choquet integral, the ELECTRE method, the analytic hierarchy process and data envelopment analysis, and that the MABAC method offers advantages in handling uncertainty and fuzziness, reducing subjectivity, and producing easily interpretable results. The authors state that future research will explore combining MABAC with other approaches, extending the framework to interval additive preference relations and interval intuitionistic fuzzy preference relations, and applying it to three-way decision-making and large-scale group decision-making problems, along with further study of consistency and consensus under uncertain preferences. For now, the contribution is a methodological one: a demonstration, through a numerical example and comparisons, that a three-stage stochastic analysis can convert interval expert judgments into rankings accompanied by explicit probability measures of their own reliability.

Subject of Research: A stochastic group decision-making method combining MABAC and SMAA-2 to rank alternatives from interval multiplicative preference relations with probabilistic acceptability measures.

Article Title: A multi-stage stochastic preference analysis on group decision-making with interval multiplicative preference relations based on the MABAC method

Article References: Zhao, X., Zhou, L., Li, H., & Dai, X. (2026). A multi-stage stochastic preference analysis on group decision-making with interval multiplicative preference relations based on the MABAC method. Applied Intelligence, 56(16), Article 518. https://doi.org/10.1007/s10489-026-07544-z

Image Credits: AI Generated

DOI: 10.1007/s10489-026-07544-z

Keywords: group decision-making, interval multiplicative preference relations, MABAC method, SMAA-2, Monte Carlo simulation, stochastic acceptability index, logarithmic least squares, expert weights, multi-criteria decision analysis, ranking reliability, Applied Intelligence, preference relations