Numerous techniques of multi-criteria decision-making (MCDM) have been proposed in a variety of business domains. One of the well-known methods is the Analytical Hierarchical Process (AHP). Various uncertain numbers are commonly used to represent preference values in AHP problems. In the case of multi-granularity linguistic information, several methods have been proposed to address this type of AHP problem. This paper introduces a novel method to solve this problem using shadowed fuzzy numbers (SFNs). These numbers are characterized by approximating different types of fuzzy numbers and preserving their uncertainty properties. The new Shadowed AHP method is proposed to handle preference values which are represented by multi-types of uncertain numbers. The new approach converts multi-granular preference values into unified model of shadowed fuzzy numbers and utilizes their properties. A new ranking approach is introduced to order the results of aggregation preferences. The new approach is applied to solve a supplier selection problem in which multi-granular information are used. The features of the new approach are significant for decision-making applications.

This study is inspired by those of Huang et al. (Soft Comput. 25, 2513--2520, 2021) and Wang et al. (Inf. Sci. 179, 3026--3040, 2009) in which some ranking techniques for interval-valued intuitionistic fuzzy numbers (IVIFNs) were introduced. In this study, we prove that the space of all IVIFNs with the relation in the method for comparing any two IVIFNs based on a score function and three types of entropy functions is a complete chain and obtain that this relation is an admissible order. Moreover, we demonstrate that IVIFNs are complete chains to the relation in the comparison method for IVIFNs on the basis of score, accuracy, membership uncertainty index, and hesitation uncertainty index functions.

Researchers are increasingly focusing on intelligent games as a hot research area.The article proposes an algorithm that combines the multi-attribute management and reinforcement learning methods, and that combined their effect on wargaming, it solves the problem of the agent's low rate of winning against specific rules and its inability to quickly converge during intelligent wargame training.At the same time, this paper studied a multi-attribute decision making and reinforcement learning algorithm in a wargame simulation environment, and obtained data on red and blue conflict.Calculate the weight of each attribute based on the intuitionistic fuzzy number weight calculations. Then determine the threat posed by each opponent's chess pieces.Using the red side reinforcement learning reward function, the AC framework is trained on the reward function, and an algorithm combining multi-attribute decision-making with reinforcement learning is obtained. A simulation experiment confirms that the algorithm of multi-attribute decision-making combined with reinforcement learning presented in this paper is significantly more intelligent than the pure reinforcement learning algorithm.By resolving the shortcomings of the agent's neural network, coupled with sparse rewards in large-map combat games, this robust algorithm effectively reduces the difficulties of convergence. It is also the first time in this field that an algorithm design for intelligent wargaming combines multi-attribute decision making with reinforcement learning.Attempt interdisciplinary cross-innovation in the academic field, like designing intelligent wargames and improving reinforcement learning algorithms.