# REDUCE--A Tool Supporting Inconsistencies Reduction in the Decision-Making Process
## 1 Introduction  
Pairwise comparisons (PC) are a critical method used across a wide range of disciplines for evaluating entities by comparing them in pairs to determine which entity is preferred or possesses more of a particular attribute. This technique is foundational in areas such as decision making, psychometrics, machine learning, statistical analysis, and ranking systems. A prominent application of pairwise comparisons is in decision analysis, particularly through [PERSON]'s analytic hierarchy process (AHP). AHP structures complex decision-making problems into a hierarchy, making them easier to handle by breaking them into sub-problems that can be analyzed independently [1]. AHP has found extensive use in resource allocation, project prioritization, and strategic planning [2; 3; 4]. In psychometrics, pairwise comparisons are instrumental in developing scales for measuring attitudes, preferences, and perceptions. [PERSON]'s Law of Comparative judgment is a classical model utilizing pairwise comparisons to quantify subjective experiences [5]. This approach has been used effectively in studies that assessconsumer preferences, social attitudes, and educational assessments [6; 7]. The field of machine learning also benefits significantly from pairwise comparisons, especially in classification and ranking algorithms. Algorithms such as support vector machines (SVM) and RankNet use pairwise comparison principles to train models to distinguish between classes or ranking items based on certain criteria [8]. These techniques have demonstrated substantial effectiveness in bioinformatics, information retrieval, and recommendation systems [9; 10]. In statistical analysis, pairwise comparisons are commonly employed for multiple comparisons among group means in analysis of variance (ANOVA). techniques, such as Tukey's honestly significant difference (HSD) test and Bonferroni correction, and are used to control Type I errors in multiple pairwise tests [11; 12]. These methods are crucial in experimental research for determining significant differences between group means [13; 14]. Sports and gaming also heavily utilize pairwise comparisons for ranking systems. The Elo rating system, initially developed for chess, is a notable application that updates player ratings based on match outcomes [15]. This system has been adapted for various sports and online gaming platforms, ensuring a dynamic and equitable ranking method [16; 17]. Recent advancements in computational techniques and the availability of large datasets have further highlighted the importance of pairwise comparisons. Algorithms that handle large pairwise comparison matrices facilitate more complex applications [18; 19]. For example, in crowdsourcing platforms, pairwise comparisons are used to aggregate human judgments for tasks such as image labeling, sentiment analysis, and quality assessment of content [20; 21]. Pairwise comparisons are also central to preference learning and decision theory. Models predicting individual or group preferences from pairwise comparison data enhance decision support systems in healthcare, finance, and marketing [22; 23]. These models leverage probabilistic and statistical methods to manage inconsistencies and uncertainties in human judgments, thus improving the reliability of derived preferences [24; 25]. Despite their wide application, pairwise comparisons face challenges, particularly inconsistency in judgments due to human error, cognitive biases, or lack of information [26; 27; 28]. The potential for path dependency and its impact on decision outcomes has also been empirically demonstrated in recent studies [29]. If, for pairwise comparisons, a reciprocal matrix \(A=(a_{ij})\) is defined, where \(a_{ij}=\frac{1}{a_{ii}}\), \(a_{ii}=1,\forall i,j\) refers to the value of comparison (e.g., preference of object A over the object B), the pairwise comparisons are consistent if the condition \(a_{ij}\cdot a_{jk}=a_{ik};\forall i,j,k\) is satisfied.