Underlying ideas
Three-way decision consists of three primary components, namely, a philosophy of thinking in threes, a methodology of working in threes, and a mechanism of processing in threes. A TAO model of three-way decision is made up of trisecting (T), acting (A), and outcome (O). In such a model, we divide a whole into three parts, three units, or three perspectives, devise strategies, and apply them to the three parts in order to obtain an optimized outcome. The research article “Sec-theoretic models of three-way decision”, proposed by Dr. Yiyu Yao, investigated set-theoretic models in the framework of three-way decision.
The concept is one of the fundamental tools which are used by people to understand the real world and problem-solving. It could be interpreted from two aspects: intension and extension. The intension of a concept is considered as the total of inherent properties and attributes, while the extension of it is viewed as a set of real instances or objects belonging to the concept. Set theory is able to process definite and explicit concepts. Generally, a concept possibly has a gradually changing border. Lack of information and limitation of cognition make it challenging to define a concept precisely. Non-standard sets provide us with a suitable and effective tool to process uncertainty. In the proposed paper, Dr. Yao reviewed the understanding, representation, and approximation of concepts in set theory, explored set-theoretic models of three-way decision, and discussed three primary contributions.
First:
There are many types of the uncertainty of concepts and there are a great number of factors that produce uncertainty. According to the sources, uncertainty can be basically categorized into two classes, namely, objective uncertainty and subjective uncertainty. From an objective perspective, if a concept comes with a clear border, then it is considered certain; if the border of it is continuously changing, then it is uncertain. Subjectively, even though the extension of a concept can be characterized definitely, the intension of it is possibly difficult to be defined. Table 1 in the proposed paper concludes different types of uncertainty represented by a standard set and six non-standard sets.
Although the representations of some concepts have unclear or gradually changing boundaries, we still need to distinguish them. Three-way decision here, instead of conventional two-way decision, introduces three decision rules: acceptance, rejection, and non-commitment, and helps us understand concepts under uncertainty.
Second:
This paper introduced a framework of evaluation-based three-way, proposed a classification of trisections, and investigated the structure of evaluating space. The interpretation of trisection is to approach a universal set into three subsets. The union of the three subsets must be the whole. In the TAO model, trisecting is especially the triplet consisting of the three subsets. Among the three sets, one or two can be empty. Based on the relationship between the three subsets and the properties of them, trisections are classified into seven classes, namely, general trisection, nonempty trisection, distinctive trisection, disjoint trisection, tri-covering, distinctive & disjoint trisection, and tri-partition. Figure 2 in the paper depicts a Hasse diagram to present the seven classes and the connections between them. The majority of three-way decision models use tri-covering or tri-partition. The remaining five types of trisections provide new directions for researchers.
There are two kinds of evaluation-based models to construct the trisection, one is using a pair of evaluations and the other one uses a single evaluation. The proposed paper explained the two models based on bipolarity. Regarding bipolarity (positivity and negativity), evaluation functions evaluate the positive information and negative information provided by each object in the domain. The model of a pair of evaluations is viewed as two independent evaluation processes using two opposite landmark values, for example, true or false, good or bad, hospitable or hostile. While the model of single evaluation can be explained as an aggregating process by considering simultaneously positivity and negativity. Under numerical scales, Figure 3 in the paper describes the two corresponding structures of the two trisecting models.
Third:
In the proposed framework, this paper systematically investigates three-way decision in several types of sets including rough sets, interval sets, fuzzy sets, shadowed sets, rough fuzzy sets, interval fuzzy sets, and soft sets. Many non-standard sets use multi-valued and infinite-valued membership functions. It leads to difficulties in understanding and processing. The main superiority of three-way decision is the simplicity of thinking in threes. Three-way decision theory applies evaluation-based models to obtain three-way approximations of non-standard sets by treating membership functions as evaluations. Two steps are required to construct three-way approximations of non-standard sets: (i) define the evaluation function(s) based on information provided by non-standard sets; (ii) apply evaluation-based three-way decision models to trisect the objects in the universe.
Through constructing set-theoretic models with three-way decision, the paper demonstrates the power of the TAO model. Three-way approximations enable us to deploy a third decision, non-commitment, besides acceptance and rejection. It offers an effective method and tool to make a decision under uncertainty. Many researchers have made valuable contributions to three-way decision. Investigating new models of three-way decision reveals the direction of future research.
The original post of this article is written in Chinese and posted in WeChat Official Account “三支决策与三支计算”.
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Reference
Yao, Y.Y.: Set-theoretic models of three-way decision. Granul. Comput. (2020).
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