A Non-overlap-based Conflict Measure for Random Permutation Sets

TL;DR

This paper tackles the problem of measuring conflicts between evidence represented as Random Permutation Sets (RPS) by introducing a non-overlap-based conflict measure grounded in DST and tailored to leverage order information via RPST. Building on DST, RPST, and Rank-biased Overlap (RBO), the authors define a conflict between permutations through a depth-wise non-overlap metric $InC_d$ and then extend it to permutation mass functions with $Conf(RPS_1,RPS_2) = \sum_{F_i^m \neq F_j^n} k_{perm}(F_i^m,F_j^n) Perm_1(F_i^m) Perm_2(F_j^n)$, where $k_{perm}$ aggregates over depths with weights $\alpha_d$. The measure supports arbitrary truncation of depth and includes an adjustable parameter $p$ to control top-weightedness, enabling analysis of infinite-depth conflicts and practical fusion scenarios. The approach addresses limitations of prior RPS conflict measures that rely on empty intersections and fails to reflect order-based inconsistencies, thereby offering a DST-consistent, order-aware tool for multi-source information fusion. Overall, the method provides a flexible, topology-preserving means to quantify conflicts in ordered uncertain information, with clear implications for decision-making under order uncertainty.

Abstract

Random permutation set (RPS) is a new formalism for reasoning with uncertainty involving order information. Measuring the conflict between two pieces of evidence represented by permutation mass functions remains an urgent research topic in order-structured uncertain information fusion. In this paper, a detailed analysis of conflicts in RPS is carried out from two different perspectives: random finite set (RFS) and Dempster-Shafer theory (DST). Starting from the observation of permutations, we first define an inconsistency measure between permutations inspired by the rank-biased overlap(RBO) measure and further propose a non-overlap-based conflict measure method for RPSs. This paper regards RPS theory (RPST) as an extension of DST. The order information newly added in focal sets indicates qualitative propensity, characterized by top-ranked elements occupying a more critical position. Some numerical examples are used to demonstrate the behavior and properties of the proposed conflict measure. The proposed method not only has the natural top-weightedness property and can effectively measure the conflict between RPSs from the DST view but also provides decision-makers with a flexible selection of weights, parameters, and truncated depths.

Figures (7)

• Figure 1: The conflict between RPSs(BBAs) when $X$ takes different values in Example 4.2
• Figure 2: The conflict between RPSs(BBAs) when $X$ changes in Example 4.3
• Figure 3: The conflict between RPSs at different depths in Example 4.4
• Figure 4: The conflict between RPSs when $|X|$ from 1 to $\infty$ in Example 4.3
• Figure 5: Weights with depth from 1 to 100 when $p=0.5$