Effectiveness measure in change-based three-way decision

The trisecting–acting–outcome model of a three-way decision is a practical approach to solving complex problems, consisting of three components: trisecting, acting, and outcome. Trisecting means dividing a complex problem into three parts based on an evaluation function, and acting means designing the corresponding action to apply to these three parts. The outcome denotes evaluation effectiveness based on the first two steps. Measuring this outcome is a challenging and moving question after trisecting-and-acting (T &A). This paper proposes a method to measure outcome based on object changes before and after T &A based on top-down and bottom-up perspectives, depending on the decision needs. First, evaluation functions are constructed that extend the traditional evaluation functions to measure changes. Then, based on the introduced evaluation functions, qualitative and quantitative of three-way changes are introduced and analyzed. Furthermore, we present the processes that determine the optimal strategy with the quantitative analysis. The analysis of several examples shows the effectiveness and practicality of our proposed method.


Introduction
In 2009, Yao (2009) first introduced the notion of three-way decision, whose main objective is to provide a sound semantics interpretation of the three types of decision/classification rules induced from the positive, boundary, and negative regions of rough set approximations; correspondingly, three types of rules are referred to as the acceptance, noncommitment, and rejection rules. Formally, Yao (2012) in 2012 proposed the concept of three-way decision, whose basic idea is the trisecting a universal set for producing accurate classifications. Many authors have been applied three-way decision to different fields such as three-way clustering (Yu et al. 2020(Yu et al. , 2010, three-way recommendation (Qian et al. 2019a;Xu et al. 2017), three-way concept analysis (Zhi et al. 2019;Yao 2020b;Qian et al. 2019b), three-way concept learning (Fan et al. 2019;Ganter and Wille 1999;Yuan et al. 2022;Xu et al. 2022), three-way approximation B Chunmao Jiang hsdrose@126.com 1 (Yao 2008(Yao , 2004, three-way attribute reduction (Ma et al. 2014), and many more (Zhou et al. 2010;Jiang et al. 2019a, b;Jia and Shang 2014;Hu et al. 2018;Zhang and Yao 2020;Jia et al. 2019;Xu and Li 2014).
Nowadays, three-way decision has developed from a narrow sense into a wide sense. Unlike the narrow sense of three-way decision based on the probabilistic rough set, the core idea of the wide sense of three-way decision is working (thinking, processing) in threes. 'Thinking in threes' is a common human problem-solving practice in many disciplines and has a sound cognitive foundation. Dividing a whole into three relatively independent and related parts would reduce cognitive and information overload by turning a complex whole into the simplicity of three parts. The triadic thinking model includes three parts, three elements, three components, three categories, three perspectives, three views, three dimensions, three levels (Yao 2020a), three layers, three generations, three phases, three stages, three steps, and so on. This variety of interpretations of triads makes the three-way decision a practical and applicable theoretical workable computational paradigm.
A typical representation of the wide sense of three-way decision, the trisecting-acting-outcome (TAO) model consists of three stages: the first is to divide a whole into three relatively independent and interdependent parts; the second is to design strategies aiming at these three parts. The third is to analyze and evaluate the effects of the first two steps. Extensive research has focused on trisecting and acting, and relatively less research has been done on the analysis and measurement of effectiveness. Jiang and Yao proposed a basic method of evaluation by calculating the quantitative of the trisection before and after applying a strategy a, that is, Q(π a π ) = Q(π ) − Q(π ) (Jiang and Yao 2018). Jiang and Guo proposed a probabilistic movement model of three-way decision and give a strategy selection mechanism based on information entropy . In their work, conditional entropy and cross entropy are used to select strategies by measuring these probabilities according to the user's preferences and structural features of the three partitions. Based on the utility theory, Jiang and Guo introduced the proportional utility function (PUF), representing the ratio between an object's initial and final quantity, to measure the outcomes from two different perspectives for movementbased three-way decision (Jiang et al. 2021a(Jiang et al. , 2022. Based on their program, the decision-maker may adopt a series of strategies in action under these investment budgets. Furthermore, in the Jiang et al. (2021b), the authors presented an effectiveness measure framework consisting of three parts: a specific TAO model-change-based TAO model, interval sets, and utility functions with unique characteristics. Interval sets were used to represent these changes when a strategy or an action is applied. Then, the utility measurement method was used to figure out three change intervals. Finally, it aggregates the toll utility through the joint of the three utilities mentioned above. Aiming to consider the conflicting information widely existing in the information system, Jiang and Zhao proposed a novel three-way decision model for action strategy set, which can analyze and classify strategies by introducing credibility and coverage. The model can remove noisy strategies and choose strategies more suitable for the need of decision makers by introducing the probabilistic preference in a movement-based three-way decision (Jiang and Zhao 2020).
Different from previous works, this work measures outcome based on object changes before and after T&A based on top-down and bottom-up perspectives, depending on the decision needs. we extend the traditional evaluation functions to construct an evaluation space. For a given pair of state, we introduce three-way changes and analyze it qualitative and quantitative. Furthermore, we present the processes that determine the optimal strategy with the quantitative analysis.
The rest of the paper is organized as follows. Section 2 reviews related work. In Sect. 3, we introduce the concept of three-way decision and propose a model of three-way changes. To illustrate the optimization of the qualitative and quantitative analysis, a case is studied in Sect. 4. In Sect. 5, we present the conclusion and future work.

Related work
In this section, we review the related works of trisectingacting-outcome model of three-way decision. Figure 1 shows the whole processes of TAO of three-way decision (Yao 2021). The TAO model consists of three components, i.e., trisecting, acting, and outcome. The TAO model also shows the relationships of the three parts after the trisection and action. The three parts are independent, but they have some connections. First, the integration of the three parts is the whole. Second, the three parts are not the same, and they are disjoint or weak joint. Meanwhile, it also shows that the three parts are divided by trisecting procedure. From another perspective, trisecting and acting are the essential operators of the TAO model. The outcome step provides the most valuable evaluation information for trisecting and acting procedures, and the feedback information will impact the trisecting and the acting in the future.

Overview of TAO model
To be more specific, trisecting is one of the steps of threeway decision and is the most effective procedure of three-way decision. The main reason is that various trisecting methods causes different results of three-way decision. Trisecting is the primary procedure to accomplish a trisection, which is a core notion in three-way decision. Acting is another essential procedure of the TAO framework, and its function is to act upon the three parts based on strategies for achieving our expectations. In other words, the acting step impacts the effectiveness of the three-way decision. Different strategies will cause different acting results, so the procedure also may influence the effectiveness of three-way decisions.
Further, both trisecting and acting impact the effectiveness of three-way decisions. The usage of the trisection-driven or Fig. 1 The TAO of three-way decision (Yao 2021) action-driven methods of the TAO model depends on application. Trisection-driven method pays attention to trisecting, and action and strategies are constructed on the results of trisecting. The trisection-driven method focuses more on finding out the most effective trisecting ways. Action-driven mode concentrates on choosing strategies in a strategy set to influence the effectiveness of three-way decision. The selected strategies decide how to trisect a whole into three regions. Sometimes we combine the two methods to apply the three-way decision to pursue more beneficial results or to fit some specific requirements.
Outcome evaluation differs from the previous two procedures since it usually evaluates trisecting and acting. The outcome evaluation measures the effectiveness of the trisecting and acting or three-way decision. In other words, it considers the efforts of the trisection and the chosen strategies, and it gives a feedback to the trisection and selected strategies.
Therefore, there are three directions to apply the three-way decision with the TAO model. The one direction is from top to bottom in the TAO model, and it is trisection-driven mode. In trisection-driven mode, the trisecting is more important than the acting, so the mode focuses on trisection strategy and trisection process. The second direction is from bottom to top in the TAO mode, and it is acting-driven mode, so the mode focuses on acting strategy and acting process. Some applications develop from the middle to upper and bottom of the TAO model, so the name is middle-out mode. The middle-out mode considers advantages and disadvantages of the trisection-driven and action-driven modes, so it is from the middle of the TAO model to apply three-way decision.

Evaluation-based trisecting
Trisecting is a core concept in the TAO framework, and the goal of trisecting is to construct a trisection. How to divide a whole into three parts becomes an important issue. The evaluated-based methods to build trisection are the most popular approach. Yao (2021) constructed two evaluation based models, one uses two evaluations and the other uses one evaluation. Two evaluation mode uses of a positive and a negative part; positive evaluation assesses objects' degree of belongingness to the positive area; whereas a negative evaluation assesses objects' degree of belongingness to the negative area. One evaluation is a convenience method, and it mixes the positive and negative evaluations into one evaluation. On the same scale, one evaluation may assess objects' degree of belongingness to the positive or negative area.
The evaluation methods map objects in a non-empty set into evaluation values. According to the evaluation values, trisecting, acting, and the outcome will be executed. The main ideas of one evaluation method are described as follows. Sup- posing e is a function that maps objects in a non-empty set U into evaluation values, e : U → (L, ≺) (Yao 2021), where ≺ is a total order on L. According to the evaluation values, by setting a set of a pair of threshold (L, ≺), i.e., one can construct a trisection. A simple example is used to demonstrate the ideas of trisecting in the one evaluation based model. Table 1 describes the application of evaluation to construct a trisection. A pair of thresholds on grades are 60 and 80. The set of marks from 1 to 100 will be trisected into three parts, one consists of marks that are equal or lower than 60 percent, another consists of marks that are equal or higher than 80 percent, and the rest consists of marks between 60 and 80.
In this work, an extended evaluation is constructed set. The extended evaluation maps object in different states in a set into evaluation values.
Definition 1 Suppose that e : U × S → (L, ≺) is an evaluation function on a finite non-empty universal set of objects U and a finite non-empty states S, where (L, ≺) is totally ordered of set. For x ∈ U and s ∈ S, e(x, s) is a value of evaluation of x in the states. Given a pair low and high thresholds (α, β), α, β ∈ L and α ≺ β, according to the evaluation and thresholds, we construct a trisection as following: (1) According to Definition 1, the evaluation values that objects' state in a set are mapped into ordered numbers, the trisecting procedure is built on the evaluation value. Based on the extensional evaluation space and methods, a model of analysis of changes based on the three-way decision will become the core of our work, and its name is the three-way changes. The three-way changes trisect whole things based on the extended evaluation methods to achieve the three-way decision. It trisects a whole according to the change amounts of objects by the evaluation methods. We will provide an elaborate introduction in the next section.

Actions and strategies
Acting is one of the three components of the TAO model of three-way decision. Acting profoundly impacts the results of the three-way decision. Therefore, evaluating an action or strategy is significant. Choosing an appropriate strategy will produce more effective results. In measuring action and strategy, one way is to utilize movement-based effectiveness measures (Jiang and Yao 2018). A superior action will produce more excellent performance than the other actions or strategies with respect to object's movement.
One should choose a strategy that leads to more preferred movements and less non-preferred movements. The acting step involves several tasks. First, it is necessary to form a set of strategies. In the top-down model, one forms the set of strategies based on a trisection. In the bottom-up model, one forms a set of strategies, and they use strategies to guide the process of trisection. Second, it is necessary to apply the set of strategies to process the three parts. Third, in order to choose the best strategy, there is a need for designing measures to evaluate different strategies. For example, the movementbased model calculates the number of moved objects under a strategy, and the number shows the performance of the strategy. In this work, we evaluate the performance of a strategy through computing changes of objects in different states under the strategy.

Evaluation of outcome
The outcome's task is to evaluate the effectiveness of the three-way decision. The essential principle is to offer the difference before and after trisecting and acting steps. According to the difference, we may evaluate the quality of the three-way decision, so we have a general formula below to represent the change.
where Q(π a π ) refers to the difference of the effect before and after applying a strategy a. Q(π ) and Q(π ) represents the initial quantified result and subsequent quantitative results. The measurement may evaluate benefit, cost, or utility.
Well, in the existing researches they focus on tripartition and action, and few documents discuss the problem of outcome evaluation (Jiang and Yao 2018). In this first paper, Yao and Jiang proposed a general metric framework to measure the outcome based on trisecting and acting as follows: where Q(π ) denotes results (e.g., cost, benefit, and risk), Q(P i ) denotes a result of one region P i , w i denotes the weight of Q(P i ), i = 1, 2, 3.

Two running process of TAO model
If attaining three regions by trisecting, the evaluation of objects is essential in the trisecting step. The evaluation of objects refers to satisfying criteria that mean some specific conditions, and the evaluation depicts degrees of satisfiable of objects regarded as the criteria. Dividing a universe OB into three regions has many evaluation-based methods. Most of them are based on cost risk and probability effectiveness of strategies. For example, they minimized cost risk, information entropy, Gini coefficient, and maximized variance. The rest of them point to specific situations or applications, such as fuzzy membership functions, similarity measures, and granularity-based measures. In this research, the primary goal is to solve the general situation, so the research object is evaluation-based methods. In these methods, a pair of thresholds are required to construct and interpret. We use a GPA management of students' example to explain the essential ideal and process of the TAO model, as shown in Fig. 2. To enhance the students' GPA, instructors typically divide a set of students OB into three levels by their score, the high-GPA, middle-GPA, and low-GPA regions. According to two normal thresholds, 70 and 85. The region of low-GPA is comprised of students whose scores are below 70; students whose scores are over 85 comprise the region of high-GPA. And it divided the rest students into the middle-GPA region. The three regions are given by: Through this method, instructor may choose and execute proper teaching strategies for improving students' scores accord to the trisection.
Both top-bottom and bottom-top are two approaches to process and understanding three-way decision. They are applied in various fields, such as software development, humanistic, and clinical theories. Both of them have distinct features.
The top-bottom view is from top to bottom to design and process three-way decision in the TAO model, as shown in Fig. 2. The three-way decision in this view is dividing the whole into trisection. Decision-makers design strategies and achieve expected outcomes according to features of each region, and the outcome is connected to trisecting and acting steps. Different trisecting causes different outcomes under dissimilar strategies. The evaluation of outcome has to consider both trisecting and acting and the results of their interaction. In the example of GPA management, how The bottom-top view is from bottom to top to design and understand the three-way decision in the TAO model, as shown in Fig. 3. The decision-maker has an expected outcome for the procedure of trisecting and acting. These expected outcomes will translate into objects' needed changes in regions. For instance, teachers expect the proportion of students in three regions, low, middle, and high, to be 20:30:50. Hence, these demands are translated into what changes need to take place. According to the changes, the teacher needs to design teaching strategies and sequence adjust a trisection to implement the strategy better. They are guiding the decision-maker to reach the teacher's expectations.
Both top-bottom and bottom-top have distinct respective, so they can serve various requirements when users utilize three-way decision. Top-bottom fits to improve original performance, and bottom-top fits to achieve specific trisection. In addition, from the two views, we can easily understand and apply the three-way decision.

A formulation of a model of three-way changes
This section introduces the concept of three-way changes and proposes a model of three-way changes.

An example of motivation
Before giving the formal formulation, let us consider a simple example to have some general ideas. Suppose that a universe U consists of three objects, that is, Figure 4 shows the values of the three objects in two states s a and s b . By looking at the figure, we can easily observe three types of changes. In Fig. 4, the three objects changes are described as follows.
• Object O 1 : its values in state s a and s b are 8 and 13, respectively. There is an increase of 5 from state s a to s b , that is, 13 − 8 = 5. • Object O 2 : its values in state s a and s b are 13 and 11, respectively. There is a decrease of −2 from state s a to s b , that is, 11 − 13 = −2. If we want to classify objects in U , we can immediately have three regions, namely an increasing region INC = {O 1 } consists of objects whose values have been increased, a decrement region DEC = {O 2 } consists of objects whose values have been decreased, and an unchanged region UNC = {O 3 } consisting of objects whose values have not changed. Essentially, three-way classification of changes models such three-way changes.

The construction of evaluation functions for three-way changes
The first step is to construct an evaluation function for developing a model of three-way changes, which is done by introducing a set of states to the standard evaluation-based model.
Definition 2 An evaluation space is defined by a triplet E = (U , S, e), where U is a finite non-empty set of objects, S is a set of states, and e is an evaluation from U × S to the set of real number , namely, e : U × S −→ . For a pair of an object x ∈ U and a state s ∈ S, e(x|s) denotes the evaluation value of x in state s.
As a special case of , we may use a three-valued set {− 1, 0, + 1} to qualitatively indicate some properties of objects. We may also use the unit interval [0, 1]. The evaluation value e(x|s) represents the object in states. By looking at the two states, we can observe changes.
In the definition above, we produce a new evaluation method. It does not merely evaluate an object in one state, and it uses the object's values in two different states by calculating their difference. The new evaluation extends the traditional evaluation methods. More specially, we employ the difference between two values as changes. It does not relate this difference to the initial value e(x|s). Sometimes it needs to consider changes relative to the value e(x|s).

Definition 4
In an evaluation space E = (U , S, e), suppose e(x|s) = 0 for all x ∈ U and s ∈ S. Given a pair of states (s, s ) ∈ S × S, the relative change of values of an object s ∈ U is defined by: rc(x|(s, s )) = e(x|s ) − e(x|s) abs(e(x|s)) , where abs (.) denotes the absolute value of a real number.
In some situations, we may want to evaluate the overall changes of a set of objects. The simplest way is to sum up the changes of objects in the set.
The summation collects the differences of all objects in the set X in two states. If the summation result is over zero, the strategy achieves a positive effect; if the result is zero, there is no change; otherwise, the approach will offer a negative effect, so the summation can provide a qualitative characterization for a strategy.
Definitions 3, 4, and 5 give three different change. The change e(x|(s, s )) is not related to the initial value e(x|s), the relative change is related to e(x|s), and the total change x∈X c(x|(s, s )) and the total relative change x∈X rc (x|(s, s )) are the measures of change of a set of objects. Based on these measures, we can discuss qualitative and quantitative threeway changes.

Qualitative three-way changes
According to the three-way classification model, one can analyze changes with respect to increase, decrease, or unchanged of values in two states. If an object's change, c(x|(s, s )) is positive, the object belongs to the INC region. The object belong to the DEC if the value c(x|(s, s )) is negative. Otherwise, the objects belong to BND region.

Definition 6
Suppose U is a set of objects, and x is an object in U . The evaluation value change of x between two states, from s to s , is c(x|(s, s )). According to the value of c(x|(s, s )), the set of objects U is divided in three regions as follow: The region INC collects objects whose values of change are more than 0, DEC collects objects whose values of change are less than 0, and UNC collects objects whose values of change are equal to 0. The trisection is based on changed values of objects in two states, and the three regions illustrate objects' qualitative change. In other words, if an object is in INC the change values of the object are increased, so it means that the changing status of the object is positive. In contrast, if an object is in DEC, the change values of the object are decreased, so it means that the changing status of the object is negative. If an object is in UNC, it means the object does not change between two states because the change value of the object is 0. Hence, the trisection provides a qualitative evaluation for each object in a set. It offers a method to analyze data.
The trisecting method based on the three-way decision classification of changes differs from the previous trisecting way of three-way decisions. The traditional method maps an object in a state into a value. The new way trisects according to the differences of values from one state to another. For instance, a teacher desires to discover the distribution of students' marks, so the traditional way trisect all students by their mark range. A new way trisects them by the differences in students' scores in two examinations.

Quantitative three-way changes
A quantitative analysis of three-way classification in two states trisects objects of a set into three parts. According to the percentage of changes relative to the initial value, a pair of thresholds is chosen for constructing a trisection. It avoids the drawbacks of using absolute change value to trisect and uses zero as the trisection standard. Although qualitative analysis can obverse objects' differences, the quantitative analysis provides more information according to distinct changes. Unlike the qualitative analysis based on the absolute change of objects to divide a set into three parts, the quantitative analysis is according to the change percentage to trisect a set into three parts. This way can produce a more meaningful trisection. In the real world, trisection standards are hard to reach zero, so the quantitative analysis adopts a pair of thresholds as a trisection standard. The quantitative analysis chooses a pair of thresholds according to the actual requirements and objects' distribution features.
Definition 7 Let U be a set of objects. The relative change of x between two states is rc(x|(s, s )). Given a pair of thresholds α and β with Min(rc(x|(s, s ))) < β < α < Max(rc(x|(s, s ))). A trisection of U is given by: Unlike the qualitative analysis, which uses absolute values of objects' change to trisect a set, the quantitative analysis focuses on the relative change to trisect. Therefore, the quantitative analysis generates more meaningful trisection than the qualitative analysis, improving the effectiveness of three-way decisions. The quantitative analysis's principal application is to discover an optimal trisection strategy in all the current strategies for improving the performance of three-way decisions. The application processes of the quantitative analysis include four steps.
1. The classification model calculates the change ratio of each object according to its formulas. 2. The model trisects all objects of a set into three parts according to a pair of thresholds chosen by actual requirements and distribution of objects' change. 3. The quantitative analysis measures the performance of the trisection in a strategy according to distribution of objects in three regions. 4. Comparing performances of all strategies reveals the optimal strategy.
Throughout the entire process, the classification model of changes of three-way decisions can find the optimal strategy based on comparing distributions of objects' change under different strategies with the quantitative analysis.

Case study
To illustrate the optimization of the qualitative and quantitative analysis, we consider still the simple case-students' GPA management. Table 2 offers typical students' grades record in two tests in English language. It records students' GPA and corresponding grades in two semesters in the English language.
All students who registered in the course is given by U = {2003830001, 2003847002, 2004816002, 2004716001, 200 4717002, 2004715004, 2003818007, 2004812009, 200371 3001, 2003773020}. Each row represents a student record that notes his or her scores and GPAs of two tests in the class. The section will show the elaborated application from two levels. The first level depicts the basic usage that refers to trisect and generates a trisection.

The classification with qualitative and quantitative three-way changes
The subsection illustrates the idea of the three-way classification model through qualitative and quantitative analysis. The goal is to introduce the common usages of the three-way classification with qualitative and quantitative analysis. The first demonstration uses the qualitative analysis of the three-way classification, which trisects the student's GPA set. According to the qualitative analysis of the three-way classification in two states, four students whose grades are increased. Their student IDs are 2003830001, 2004816002, 2004716001, 2003818007, 2004812009, and 2003713001 in Table 3. The student's scores with the ID of 2003773020 are the same in the two semesters. The grades of the rest of the students are decreased. Therefore, the three regions are as  follows. , 2004816002, 2004716001, 2003818007, 2004812009, 2003713001}, Obversely, six of ten students moved into a positive region, three of ten students moved into a decreased region, and one of ten students moved into the boundary region. Sixty percent changed to positive region, thirty percent changed to negative region, and ten percent changed to boundary region. Therefore, the qualitative analysis of the three-way classification in two states is an excellent way to analyze the performance of three-way decision. Additionally, The summation of all students' changes is over zero. The result also demonstrates the current strategy is positive.
The second demonstration is that using the quantitative analysis of the three-way classification model trisects the students' GPA set. When the classification model adopts the quantitative analysis to trisect the set, it calculates change relative to their original values. Table 4 represents the information of the set as follows.
According to Table 4, the relative of changes are given by 3.4, − 1. 0, 24.2, 34.5, − 2.8, − 4.7, 10.3, 7.5, 1.1, 0. The range of relative change is between 34.5 and − 4.7. There are six students over 0, one is equal to 0, and three students are less than 1. The current strategy performs well because the changed values of sixty percent of students in the set are positive. Only three students' changes are less than 0, and they are − 1.0, − 2.8, and − 4.7. The increased scale and number of students are bigger than the decreased scale and number of students. Some change in students' scale is negligible, close to zero-for example, two students whose relative change are 1.1 and − 1.0. If trisecting still chooses zero as the trisection standard, the trisection produced would not be appropriate. Using quantitative analysis is better than qualitative analysis. On the other hand, the current strategy causes most students changes towards the better. If the trisection uses the zero in the qualitative method, it only shows that an overview of the strategy is sound. It ignores the analysis of how many objects have significant changes. Therefore, the quantitative analysis method provides a flexible pair of thresholds that support a trisection according to students' change distribution and the actual requirements. The relative changes, {3.4, − 1.0, 24.2, 34.5, − 2.8, − 4.7, 10.3, 7.5, 1.1, 0}, or {− 4.7, − 2.8, − 1.0, 0, 3.4, 7.5, 10.3, 24.2, 34.5}, describe the distribution of the relative grades changes of students. Suppose that (− 1, 2) is chosen as the pair of thresholds. Five students' changes are over 2, three students' changes are less than or equal to −1, and the rest is between −1 and 2. The trisection is as follows: , 2004816002, 2004716001, 2003818007, 2004812009}, Suppose that the pair of thresholds is chosen to be (0, 10). The students whose scale changes are over 10 will appear in the positive region. The three regions are given as follows: Therefore, the quantitative analysis method provides a more flexible and worthy way to evaluate and analyze data by using the three-way classification of changes.

Selection of the optimal strategy with quantitative three-way changes
As mentioned earlier, the three-way classification of changes improves trisection. They can discover the optimal strategy by evaluating multiple strategies. The quantitative analysis is more worthy for the real applications. This subsection will present the processes that determine the optimal strategy with the quantitative analysis. Table 5 presents relative changes with respect to three strategies. The relative change, "RC1", is produced under strategy 1, "RC2" is generated under strategy 2, and "RC3" is the relative change of objects under strategy 3.
According to the distribution of objects in different strategies, the distribution is shown in Fig. 5. The objects' change scale is small between − 1 and 1, close to zero. Therefore, choosing a pair of thresholds is (− 1, 1) in this case β is equal to − 1, and α is the same as 1.  Their distribution is D3 INCRC (U ) = 2 10 , D3 UNCRC (U ) = 2 10 , and D3 DECRC (U ) = 6 10 . In the three strategies, their D UNCRC is equal to 2, D1 INCRC is more than D2 INCRC , and D2 INCRC is more than D3 INCRC . Therefore, the strategy 1 is the best among the three strategies.

Conclusion and future work
We have noticed that while there has been a large number of research conducted on the trisecting and acting, much less work has been done at the outcome evaluate. In this paper, a general model of three-way classification of changes is proposed. First, evaluation functions are constructed that extend the traditional evaluation functions to measure changes. Second, based on the introduced evaluation functions, qualitative and quantitative of three-way changes are introduced and analyzed. Finally, an example of changes of students' grades is used as example to illustrate the main ideas. The threeway change model provides an entirely new view to analyze objects' change to reveal meanings behind those changes.
How to measure the benefits of thinking in threes, or rather, thinking in others, e.g., thinking in twos is a critical limitation of the current state of the art in three-way decision research. The evaluation of outcomes includes benefit-cost analysis, effect, efficiency, and other metrics. The future work includes more measurement methods to evaluate the effectiveness of the three-way decision, data-based decision-making effect evaluation, etc.