The Random Permutation Set (RPS) is a new type of set proposed recently,
which can be regarded as the generalization of evidence theory. To measure the
uncertainty of RPS, the entropy of RPS and its corresponding maximum entropy
have been proposed. Exploring the maximum entropy provides a possible way of
understanding the physical meaning of RPS. In this paper, a new concept, the
envelope of entropy function, is defined. In addition, the limit of the
envelope of RPS entropy is derived and proved. Compared with the existing
method, the computational complexity of the proposed method to calculate the
envelope of RPS entropy decreases greatly. The result shows that when $N \to
\infty$, the limit form of the envelope of the entropy of RPS converges to $e
\times (N!)^2$, which is highly connected to the constant $e$ and factorial.
Finally, numerical examples validate the efficiency and conciseness of the
proposed envelope, which provides a new insight into the maximum entropy
function.

RPS 的熵的包络是通过计算 RPS 熵的上限获得的，与常数 e 和阶乘高度相关，通过降低计算复杂度提供了一种理解 RPS 物理意义的新方法。

最大随机排列集熵的极限

Limit of the Maximum Random Permutation Set Entropy

Evidence theory is widely used in decision-making and reasoning systems. In
previous research, Transferable Belief Model (TBM) is a commonly used
evidential decision making model, but TBM is a non-preference model. In order
to better fit the decision making goals, the Evidence Pattern Reasoning Model
(EPRM) is proposed. By defining pattern operators and decision making
operators, corresponding preferences can be set for different tasks. Random
Permutation Set (RPS) expands order information for evidence theory. It is hard
for RPS to characterize the complex relationship between samples such as
cycling, paralleling relationships. Therefore, Random Graph Set (RGS) were
proposed to model complex relationships and represent more event types. In
order to illustrate the significance of RGS and EPRM, an experiment of aircraft
velocity ranking was designed and 10,000 cases were simulated. The
implementation of EPRM called Conflict Resolution Decision optimized 18.17\% of
the cases compared to Mean Velocity Decision, effectively improving the
aircraft velocity ranking. EPRM provides a unified solution for evidence-based
decision making.

提出了一种证据模式推理模型（EPRM），并引入了随机图集（RGS）和随机排列集（RPS）以建模复杂关系和表示更多事件类型，通过冲突解决决策实施的 EPRM 相对于平均速度决策优化了 18.17％的案例，有效改进了飞机速度排序。

随机图集与证据模式推理模型

Random Graph Set and Evidence Pattern Reasoning Model

Random permutation set (RPS), as a recently proposed theory, enables powerful
information representation by traversing all possible permutations. However,
the repetition of items is not allowed in RPS while it is quite common in real
life. To address this issue, we propose repeatable random permutation set ($\rm
R^2PS$) which takes the repetition of items into consideration. The right and
left junctional sum combination rules are proposed and their properties
including consistency, pseudo-Matthew effect and associativity are researched.
Based on these properties, a decision support system application is simulated
to show the effectiveness of $\rm R^2PS$.

提出一种考虑项的重复性的 repeatable random permutation set ($m R^2PS$) 模型，并构建一个基于其属性的决策支持系统模拟来展示其有效性。

可重复的随机排列集合

Repeatable Random Permutation Set

Recently, a new type of set, named as random permutation set (RPS), is
proposed by considering all the permutations of elements in a certain set. For
measuring the uncertainty of RPS, the entropy of RPS is presented. However, the
maximum entropy principle of RPS entropy has not been discussed. To address
this issue, in this paper, the maximum entropy of RPS is presented. The
analytical solution for maximum entropy of RPS and its corresponding PMF
condition are respectively proofed and discussed. Numerical examples are used
to illustrate the maximum entropy RPS. The results show that the maximum
entropy RPS is compatible with the maximum Deng entropy and the maximum Shannon
entropy. When the order of the element in the permutation event is ignored, the
maximum entropy of RPS will degenerate into the maximum Deng entropy. When each
permutation event is limited to containing just one element, the maximum
entropy of RPS will degenerate into the maximum Shannon entropy.

本研究提出了随机置换集（RPS）的最大熵原理，并通过数值实验验证，结果表明最大熵的 RPS 与最大邓（Deng）熵和最大香农熵相容。当排列事件中元素的顺序被忽略时，RPS 的最大熵将退化为最大邓熵；当每个排列事件被限制为仅包含一个元素时，RPS 的最大熵会退化为最大香农熵。