Effectiveness measure for TAO model of three-way decisions with interval set

Abstract

The basic idea of the three-way decisions (3WD) is ‘thinking in threes.’ The TAO (trisecting-acting-outcome) model of 3WD includes three components, trisect a whole into three reasonable regions, devise a corresponding strategy on the trisection, and measure the effectiveness of the outcome. By reviewing existing studies, we found that only a few papers touch upon the third component, i.e., measure the effect. This paper’s principal aim is to present an effectiveness measure framework consisting of three parts: a specific TAO model - Change-based TAO model, interval sets, and utility functions with unique characteristics. Specifically, the change-based TAO model provides a method to measure effectiveness based on the difference before and after applying a strategy or an action. First, we use interval sets to represent these changes when a strategy or an action is applied. These changes correspond to three different intervals. Second, we use the utility measurement method to figure out three change intervals. Namely, different utility measures correspond to the different intervals, concave utility metric, direct utility metric, and convex utility metric, respectively. Third, it aggregates the toll utility through the joint of the three utilities mentioned above. The weights among these three are adjusted by a dual expected utility function that conveys the decision-makers’ preferences. We give an example and experiment highlighting the validity and practicability of the utility measure method in the change-based TAO model of three-way decisions.

Keywords

Three-way decisions change-based TAO model interval set utility measurement dual expected utility

1 Introduction

People unconsciously utilize the information granularity when they think in general, such as 7 ± 2, 4 ± 1, 3 ± 0 [3 –6]. In particular, ‘Thinking in threes’ is widely and frequently employed in many disciplines and fields, including computer science, medicine, management, psychology, and cognitive science. As a model for thinking, problem-solving, and information processing, Yao [1] first proposed three-way decisions (3WD), the core idea of which is to ‘Thinking in 3s’, such as three aspects, three parts, three classifications, three views, three levels [2]. The generalized concept of 3WD is to divide a whole into three parts and take an effective strategy to further deal with each piece for obtaining an expected outcome. Related studies have shown that three-way decisions is more consistent with human cognition [7].

Numerous applications and theoretical studies of three-way decisions have emerged one after another by replacing the word ‘decisions’ with other terms, such as processing, analysis, solution, calculation. From the perspective of theoretical research, include three-way decisions space [8], three-way clustering [9 –11], three-way game theory [12, 13], three-way logic [14], three-way fuzzy set [15, 16], three-way conflict analysis [17, 18], attribute reduction based on three-way decisions [19 –21], three-way concept analysis [22 –28], three-way concept learning [29, 30], sequential three-way decisions [31, 32], cost-sensitive three-way decisions [33, 34] and movement-based three-way decisions model [35 –37]. In application, Li [38] introduced the three-way decisions into facial recognition. Hu [39] proposed a new three-way decisions model for online diagnosis and medical selection. Zhou [34] successfully applied the results of the cost-sensitive three-way decisions to the spam filtering system, which reduced the occurrence of spam misclassification. Zhang and Min [40, 41] applied the three-way decisions to the recommendation system based on regression and random forest methods, which improved the accuracy of the recommendation system. Liu [42] applied the decision-based rough sets to government decision analysis and proposed three-way government decisions. Jiang [43] introduced three-way decisions into cloud computing and successfully applied the results to cloud energy optimization and cloud task scheduling [44, 45]. Zhang et al. established the tri-level granular structure of neighborhood system by three-way decisions, and it further researches double-quantitative distance measurement and classification learning [46]. The existing research has expanded from the initial trisecting-acting model to the trisecting-acting-outcome (TAO) model of three-way decisions [47, 48]. “Outcome" has gradually drawn attention from researchers. Existing studies on effective measures for the TAO model concentrate on movement-based three-way decisions [35 –37]. This paper further explores the “Outcome" in the TAO model and proposed a change-based TAO model of three-way decisions.

Measuring the change in outcome before and after an action is an easy-to-understand method in the TAO model. These circumstances may include the region change of objects located in, the quantity of the object itself, or the proportion of objects in the three regions. These differences may be changes in utility, efficiency, and effect before and after an action. The utility reflects the subjective consciousness of decision-makers [49, 50], and it often is an essential means to evaluate the effectiveness of one model. The major contributions of this work are summarized as follows. (1) Developed a new TAO model - change-based TAO model, and (2) Using interval sets to represent these changes. (3) The utility approach is used to measure the effectiveness. In short, this work designed a new effectiveness measure method for the change-based TAO model.

The remainder of the paper is organized as follows: Section 2 reviews the TAO model of three-way decisions. In Section 3, we present the measurement framework based on the change-based TAO model from the perspective of the interval set. Section 4 gives the details of the utility measurement. The simulation work and the results obtained are presented and discussed in Section 5. Finally, Section 6 concludes the paper and gives some future works directions.

2 Related works of trisecting-acting-outcome model

The basic concept of the three-way decisions has been continuously improved, from the narrow sense based on the rough set to the broad sense three-way decisions model, such as the TAO model. The theory has formed a vast research network. In this section, we briefly review the related works of the TAO model.

2.1 Trisecting

Figure 1 shows a basic view of the TAO model. The relevant parts of TAO can be briefly described subsequently. Level 1 represents the whole set of objects, attributes, operations, or others, denoted by U, which represent the ‘Whole’. The U is trisected into three disjoint parts, P₁, P₂, P₃, respectively, denoted in level 2. We give a formal notation as follow: $\begin{matrix} P_{1} \cup P_{2} \cup P_{3} = U, \\ P_{1} \cap P_{2} = \emptyset, P_{1} \cap P_{3} = \emptyset, P_{2} \cap P_{3} = \emptyset . \end{matrix}$ (1)

Fig. 1

The TAO model of three-way decision (adapted from [47]).

How to divide a whole set into three parts according to two thresholds is an important research topic. In rough set models, it is necessary to determine which elements belong to the positive region (acceptance rule), negative region (rejection rule), and boundary region (non-commitment rule). Some methods proposed to accomplish these tasks, such as risk [64], information entropy [65], Gini coefficient [66], chi-square statistic [67], etc. From the perspective of a generalized three-way decisions, constructing the trisection needs to consider specific subject regions. A recent study proposed many constructing methods of three-way decisions, such as with rough sets, interval sets, fuzzy sets, shadowed sets, rough fuzzy sets, and so on [51].

The concept of an interval set was presented by Yao [52, 53]. The trisecting using interval set refers to divide objects into three parts based on the degree of partially known. Let 2^U be the power set of U, the two subsets of $\underline{X}$ and $\bar{X}$ , satisfying $\underline{X} \subseteq \bar{X}$ . We can use the interval set $[\underline{X}, \bar{X}] = {Y \in 2^{U} | \underline{X} \subseteq Y \subseteq \bar{X}}$ to denote a trisection of one set, in which $\underline{X}$ and $\bar{X}$ are called the lower and upper bounds of the interval set, respectively. Therefore, the trisection is, $\begin{matrix} A = (\bar{X})^{c}, B = \bar{X} - \underline{X}, C = \underline{X} . \end{matrix}$ (2)

2.2 Acting

In the phase of ‘Acting’ as shown in Fig. 1, the decision-maker needs to devise valuable strategy and action on the trisection for pursuing an expected outcome. We call the process of taking strategy and action on the three regions is “three-way acting". There are two main research aspects of the three-way acting: one is to regional transform, the other is the application. We consider a particular type of three-way decisions for the former in which strategies facilitate movements of objects from unfavorable regions to favorable regions. Further, we study the outcome of strategies and actions based on regional transformation. The latter is to apply the ‘Acting’ in a specific situation. Much research has been done related to the ‘acting,’ such as cloud computing, medical decision-making, spam filtering, recommendation system, government decision-making, etc. The effectiveness of the Trisecting-Acting-Outcome(TAO) relies not only on trisection but also on the strategy and action.

Let still consider the example of an interval set. For the three regions, $A = (\bar{X})^{c}, B = \bar{X} - \underline{X}, C = \underline{X}$ , we will devise different actions. As a result, the region the object in will change with the object’s knowledge. An object may transfer among the three regions. The decision-maker has his preference for results. If the object’s change between the three intervals is ideal, then this strategy is expected by the decision-maker in advance. Otherwise, the opposite is exact.

2.3 Outcome

The TAO model’s outcome depends on a proper match of trisecting methods and strategies for actions, which is also one of the fundamental problems of the three-way decisions. Let U be a finite set of objects, and Π denote the set of all tri-partitions of U. Suppose a mapping function $Q : Π \to R$ represents the measure function of the tri-partitions. Therefore, given a tri-partition π = {P₁, P₂, P₃}, a framework of basic metrics is given by, $Q (π) = w_{1} Q (P_{1}) + w_{2} Q (P_{2}) + w_{3} Q (P_{3}),$ (3) where Q(P_i) is the quality of the region P_i, and i = 1, 2, 3, w_i represents the weight by different regions P_i, which denote the importance of the three regions, respectively. Therefore, a three-way division that optimizes the quality of the tri-partitions can be found as an optimization problem: $arg \underset{P_{1}, P_{2}, P_{3} \subseteq U}{opt} (w_{1} Q (P_{1}) + w_{2} Q (P_{2}) + w_{3} Q (P_{3})),$ (4) where w_i ∈ [0, 1] and w₁ + w₂ + w₃ = 1. The existing outcome evaluation content mainly includes: measuring the risk of loss of trisecting based on Bayesian risk probability [54], measuring the uncertainty of trisecting based on information entropy [55] and Gini coefficient [56], measuring the trisecting based on chi-square statistics relevance [57], etc.

The outcome evaluation of the TAO model is also related to the design and selection of optimal strategies. Suppose A = {a₁, a₂, ⋯ , a_n} represents a set of finite n actions, when an action a ∈ A is applied to a certain region, the objects in the region will cause changes under the influence of the action a. In turn, it leads to the overall change of the tri-partitions. Therefore, given a tri-partition π represents the region before the strategy is applied, and π′ represents the tri-partitions after the strategy is applied, a basic measurement framework is, $Q (π^{'} | π) = Q (π^{'}) - Q (π),$ (5) where Q(π′|π) represents the change quality before and after the application of the strategy. Q(π) and Q(π′) represent the quality of tri-partitions π and the quality of tri-partitions π′ after the strategy. Therefore, an optimized outcome of strategy can be found as the following optimization problem: $arg \underset{π^{'}, π \in Π}{opt} (Q (π^{'}) - Q (π)) .$ (6)

However, the existing research on the outcome evaluation of the three-way acting was relatively rare. Based on the above measurement framework, Gao and Yao [35] proposed the A-3WD model based on actions, Jiang and Yao [36] proposed the M-3WD model based on object movement, Jiang and Guo [37] proposed the PM-3WD model based on probability movement. A-3WD, or actionable 3WD, is from the perspective of action strategy development or selection, corresponding to the phase of Acting in the TAO model. An actionable rule can be used as a guideline for action and strategy design. Given an actionable rule, many actions can be developed because many options may exist. M-3WD, or movement-based 3WD, evaluates the outcome from measuring the object’s movement after an action, further to determine the strategy’s goodness or badness. As a particular case of M-3WD, PM-3WD describes that objects’ movement occurs with a certain probability under an action.

3 Change-based TAO model of three-way decision

This section presents the change-based TAO model of three-way decisions, and introduces the interval set to represent the result caused by change when a strategy or an action is applied. The symbols and definitions in this paper are shown in Table 1.

Table 1
Symbols and definitions

Symbol Definition

OB Objetc set

x Object x in OB

π Tri-partition of OB

π _c Change tri-partition of OB

P _i Region i on π

R ⁺ Desirable-change region on π

R ^- Undesirable-change region on π

R ⁰ Indifferent-change region on π

A Action set

a _i Action i in A

q Change quantity

q(x) Change quantity of x

⇝_a Change process when applying action a

M _l Lower bounds of change

M _u Upper bounds of change

[M_l, M_l] Change interval

e Change evaluation function

e_{[M_l,M_u]}(q(x)) Change evaluation value of q(x) on [M_l, M_u]

P_i⇝_aR^• The process of P_i change to R^•(• = + , 0, -) when applying action a

P_i⇝_aπ_c The process of P_i change to π_c when applying action a

π⇝_aπ_c The process of π change to π_c when applying action a

E(P_i⇝_aR^•) Effectiveness of P_i⇝_aR^•

E(P_i⇝_aπ_c) Effectiveness of P_i⇝_aπ_c

E(π⇝_aπ_c) Effectiveness of π⇝_aπ_c

q(P_i⇝_aR^•) Change quantity of P_i⇝_aπ_c

q(π⇝_aπ_c) Change quantity of π⇝_aπ_c

w _i Weight of E(P_i⇝_aπ_c)

v _j Weight of E(P_i⇝_aR^•)(j = + , 0, -)

u Utility function

u(q) Utility value of q

CP _i Interval states of change

f _t Probability of change

g Non-decreasing function

g(x) Function value of x

η Weigh of change interval

Symbol	Definition
OB	Objetc set
x	Object x in OB
π	Tri-partition of OB
π _c	Change tri-partition of OB
P _i	Region i on π
R ⁺	Desirable-change region on π
R ^-	Undesirable-change region on π
R ⁰	Indifferent-change region on π
A	Action set
a _i	Action i in A
q	Change quantity
q(x)	Change quantity of x
⇝_a	Change process when applying action a
M _l	Lower bounds of change
M _u	Upper bounds of change
[M_l, M_l]	Change interval
e	Change evaluation function
e_{[M_l,M_u]}(q(x))	Change evaluation value of q(x) on [M_l, M_u]
P_i⇝_aR^•	The process of P_i change to R^•(• = + , 0, -) when applying action a
P_i⇝_aπ_c	The process of P_i change to π_c when applying action a
π⇝_aπ_c	The process of π change to π_c when applying action a
E(P_i⇝_aR^•)	Effectiveness of P_i⇝_aR^•
E(P_i⇝_aπ_c)	Effectiveness of P_i⇝_aπ_c
E(π⇝_aπ_c)	Effectiveness of π⇝_aπ_c
q(P_i⇝_aR^•)	Change quantity of P_i⇝_aπ_c
q(π⇝_aπ_c)	Change quantity of π⇝_aπ_c
w _i	Weight of E(P_i⇝_aπ_c)
v _j	Weight of E(P_i⇝_aR^•)(j = + , 0, -)
u	Utility function
u(q)	Utility value of q
CP _i	Interval states of change
f _t	Probability of change
g	Non-decreasing function
g(x)	Function value of x
η	Weigh of change interval

3.1 Change-based TAO model

For constructing the TAO model from the perspective of the change, we consider a special type of three-way decisions in which strategies facilitate the change of objects from undesirable regions to desirable regions. As shown in Fig. 2, the universe U is divided into three parts P₁, P₂, and P₃ that do not intersect or weak intersect each other. At the same time, the corresponding strategy is devised and applied for achieving an ideal change. Depending on the outcome of change, these objects are divided into three parts, R⁺, R^-, and R⁰, which denote that desirable change, undesirable change, and indifferent change, respectively. In the first stage, the solid line represents objects in universe U are divided into three regions P₁, P₂, and P₃; In the second stage, the objects represented by solid line in the region P_i(i = 1, 2, 3) generate the desired change to region R⁺, objects represented by the dotted line in the region P_i(i = 1, 2, 3) produce the indifferent change to part R⁰, objects represented by the solid line with a slash in the region P_i(i = 1, 2, 3) bring the undesirable change to region R^-. As a specific example of the TAO model [47], the formation of the new three partitions is based on the objects’ changes under the effect of one strategy. To describe specific ideas, we give relevant formalization:

Fig. 2

Change-based TAO model of three-way decision.

Definition 1. Let OB be a finite non-empty set, a tri-partition π = {P₁, P₂, P₃} and an action set A = {a₁, a₂, . . . , a_m}. Let $q : π \times A \to R$ be a measurement function from a given tri-partition and a strategy set to an outcome $R$ . The outcome $R$ is a real number set. Therefore, q(x) denotes the change quantity of x ∈ OB under an action a ∈ A.

Definition 2. Given the tri-partition π = {P₁, P₂, P₃} and the change quantity q(x) of the object x ∈ OB, the change-based three-way decisions model map a tri-partition π into R⁺, R⁰, and R^- region by mapping function f. Let these three regions are given by tri-partition π_c = {R⁺, R⁰, R^-}. In addition, the three regions satisfy the following three properties: $\begin{matrix} π_{c} = R^{+} \cup R^{0} \cup R^{-}; \\ R^{+} \cap R^{0} = \emptyset, R^{+} \cap R^{-} = \emptyset, R^{-} \cap R^{0} = \emptyset; \\ R^{+} ≻ R^{0} ≻ R^{-} . \end{matrix}$ (7)

Definition 3. Assume that the Π is a tri-partition set of OB. For any one action a ∈ A, which converts one tri-partition π = {P₁, P₂, P₃} to another tri-partitions π_c = {R⁺, R⁰, R^-}. The process of change caused by the action a ∈ A is denoted as L_a : π⇝_aπ_c. For an object x ∈ OB, the change is denoted as L_a(x) : P_i(x)⇝_aR^•(x).

Compared with the general TAO model, the change-based TAO model’s three partitions are with different preferences, as shown in Fig. 2. Based on this, we introduce the interval number to represent the character of changes and, in turn, analyze and measure the effect of changes for the three partitions using different utility measures.

3.2 Using interval sets to represent changes

In the previous section, we gave a qualitative description of the change-based TAO model. The tri-partition will produce different effects under a strategy and action. That is, the effectiveness is different even if objects have the same change. From the perspective of tri-partition R⁺, R⁰, and R^-, it is different because of the changes produced from different regions P_i. To further study the relationship between the tri-partitions π = {P₁, P₂, P₃} and the tri-partitions π_c = {R⁺, R⁰, R^-}, we give a quantitative description of the model in this section.

For an incomplete information system, some concepts often cannot be accurately defined and expressed, and there are only partly known concepts. The interval set [52] gives the scope of the extension of a partially known set defined by a lower and upper bound. Its original purpose is to solve partly known concepts because the interval set has good universality, and its relevant research works had attracted many attentions [53 , 60]. The interval number is introduced to construct the quantitative change model. Tri-partition π = {P₁, P₂, P₃} is changed by taking a series of strategy and action. A concept of upper and lower bounds is used to define the changing quality. Let M_l and M_u represent the lower and upper bounds of change, respectively.

An action a prompts an undesirable change if the change quantity q(x) of object at the lower bound M_l. An action a prompts a desirable change if the change quantity q(x) of object at the upper bound M_u. And an action a prompts an indifferent change if the change quantity q(x) of object between the upper bound M_u and the lower bound M_l. The interval set [M_l, M_u] consisting of the upper and lower bounds is defined by interval sets. Between them, interval of change [M_l, M_u] is a subset of 2^U defined as follows: $= {M \in 2^{U} | M_{l} \subseteq M \subseteq M_{u}},$ (8) where M_l ⊆ M_u.

Definition 4. Given the change quantity q(x) of object x ∈ OB. Let C = {c^-, c⁰, c⁺} be a set of truth values, and satisfy the total order c^- ⩽ c⁰ ⩽ c⁺, the interval of change [M_l, M_u] be equivalently defined by the e(•) as follows: $e_{[M_{l}, M_{u}]} (q (x)) = {\begin{matrix} c^{-}, & q (x) \in {(M_{u})}^{C}, \\ c^{0}, & q (x) \in M_{l} - M_{u}, \\ c^{+}, & q (x) \in M_{l} . \end{matrix}$ (9) where e(•) represents a change evaluation value of a change quantity q(x) for an object x ∈ OB.

Definition 5. Given change quantity q(x) of the universes set OB, through a pair of change thresholds (c⁺, c^-) and a change evaluation function e(•), objects can be quantitatively divided into three regions as follows: $\begin{matrix} \underset{(c^{+}, c^{-})}{R^{+}} ([M_{l}, M_{u}]) & = {x \in OB | e_{[M_{l}, M_{u}]} (q (x)) ⩾ c^{+}}, \\ = M_{l} . \\ {R^{0}}_{(c^{+}, c^{-})} ([M_{l}, M_{u}]) & = {x \in OB | c^{-} < e_{[M_{l}, M_{u}]} (q (x)) < c^{+}}, \\ = M_{l} - M_{u} . \\ \underset{(c^{+}, c^{-})}{R^{-}} ([M_{l}, M_{u}]) & = {x \in OB | e_{[M_{l}, M_{u}]} (q (x)) ⩽ c^{-}}, \\ = (M_{u})^{c} . \end{matrix}$ (10)

Given an interval of change [M_l, M_u] and a pair of change thresholds (c⁺, c^-), we quantitatively divide the object x ∈ OB from trisection π to tri-partition $R_{(c^{+}, c^{-})}^{+} ([M_{l}, M_{u}])$ , $R_{(c^{+}, c^{-})}^{0} ([M_{l}, M_{u}])$ , $R_{(c^{+}, c^{-})}^{-} ([M_{l}, M_{u}])$ .

The Example of GPA Management. According to GPA grades, the teacher generally classifies students into three parts: high-GPA, medium-GPA, and low-GPA. For a different level GPA region, they take the corresponding teaching strategies. After the trisecting and acting, the level of GPA for students changed under some teaching strategies. These changes may be desirable, undesirable, and indifferent. The level of change of GPA intuitively reflects the effectiveness of teaching strategies. The effectiveness is different based on different changes in different regions. Even if they happen the same change in different regions, the effect is still not the same. It is much more challenging for students in the high-GPA region to improve their scores than students in the low-GPA region. Students with high GPA have less room for improvement in their grades, and the cost of raising their scores is usually high, while students with very low GPA have more room for improvement in grades, and their rise in cost is generally less. For the students with a high GPA, the performance improvement is relatively few, and they usually are divided into the region where the GPA changes are desirable. However, students with low GPA need a significant increase. We will only divide them into the region where GPA changes are desirable.

The interval sets are used to construct the tri-partition π_c by representing changes in students’ GPA. For example, if the interval of change of students in the high-GPA region is [2,-3], then the students with a higher GPA score of 2 points or higher in the high-GPA region are reclassified to the region where the GPA improved. The students whose GPA drops by over 3 points are re-divided into the region where GPA had deteriorated. The rest is divided into the region where GPA remained virtually unchanged. Similarly, students in the medium-GPA region and the low-GPA region can be re-divided into the region where the GPA becomes improved, the GPA becomes worse than before, and the GPA is unchanged according to an interval of change and a change evaluation function.

4 Utility measurement based on change preference

This section presents a separation and integration method for measuring the effectiveness of the change-based TAO model, which introduces the concept of utility to calculate the effectiveness of change when a strategy or an action is applied. In the separation stage, the three regions are P_i⇝_aR⁺, P_i⇝_aR⁰, P_i⇝_aR^-, where i = 1, 2, 3, respectively. In the integration stage, three outcome are P₁⇝_aR^•, P₂⇝_aR^•, P₃⇝_aR^•, where • ∈ {+ , 0, -}, respectively.

4.1 Change-based utility measurement framework

Separation and integration are two essential facets suggested by Yao in three levels thinking [2]. By separation, we are focusing on a particular part of the whole at different levels, using different methods and means to explain and deal with each level, to reduce the overall research complexity. By integration, all levels’ results are comprehensively collected to provide a full understanding of the whole. The same method is applied to study the effectiveness measure in the change-based TAO model, which we call measure effectiveness. We use different methods to divide the overall objects according to their change quantity q(x), and then calculate the effectiveness brought the respective parts. The integration method integrates each part’s results to study the effectiveness of trisecting and acting with lower complexity.

In the change-based TAO model of three-way decisions, different changes in the region have different effects. This paper fully considers the outcome of trisecting and acting in the effectiveness measurement. As shown in Fig. 3, the model’s evaluation process focuses on what is indicated by the dashed rectangle, based on the object changes produced. It aggregates these results into “Effectiveness” to produce the outcome, further measuring the model’s effectiveness.

Fig. 3

The evaluation process of change-based TAO model.

Let E(•) denotes the effectiveness result, and E(π⇝_aπ_c) denotes the effectiveness result derived from a tri-partition π converted to a new tri-partition π_c under a strategy or an action a. The calculation is given as follows:

At the point level, the effectiveness E(π(x)⇝_aπ_c(x)) of the change of the object x ∈ OB is performed for the change of the state P_i⇝_aR^• in different intervals.

At the line level, the effectiveness results of changes of each object x ∈ OB in the joint point level are combined to evaluate the effectiveness of each interval, that is, $E (P_{i} ⇝_{a} R^{•}) = \sum_{x \in P_{i} \land x \in R^{•}} E (π (x) ⇝_{a} π_{c} (x)),$ (11) where E(P_i⇝_aR^•) indicates the effectiveness of the change from region P_i to region R^•.

In the area level, for the tri-partation π = {P₁, P₂, P₃}, combined with the calculation result E(P_i⇝_aR^•) in the line level, the effectiveness measure of the change in each region P_i is given by: $\begin{matrix} E (P_{i} ⇝_{a} π_{c}) = v_{1} E (P_{i} ⇝_{a} R^{+}) \\ + v_{2} E (P_{i} ⇝_{a} R^{0}) + v_{3} E (P_{i} ⇝_{a} R^{-}), \end{matrix}$ (12) where E(P_i⇝_aπ_c) denotes the effectiveness of the change come from region P_i to the tri-partition π_c, v_i denotes its corresponding weight.

Therefore, the effectiveness E(π⇝_aπ_c) is given by a weighted summation function of the three regions P₁, P₂, P₃: $\begin{matrix} E (π ⇝_{a} π_{c}) & = w_{1} E (P_{1} ⇝_{a} π_{c}) + w_{2} E (P_{2} ⇝_{a} π_{c}) \\ + w_{3} E (P_{3} ⇝_{a} π_{c}), \end{matrix}$ (13) where E(π⇝_aπ_c) represents the outcome of the change from tri-partition π to π_c, v_i represents its corresponding weight.

In this calculation process joint with point-line-area three-level, the evaluation of effectiveness is mainly measured by weighing the changes caused by the action a. Compared with the M-3WD model [36], which uses the quality E(π) - E(π′) represents the effectiveness evaluation result, the significant difference of the measurement framework in this paper is that it directly measures the change in quantity E(π⇝_aπ_c).

4.2 Utility measurement

In the change-based TAO model, the change quantity q(π(x)⇝_aπ_c(x)) may be interpreted as the utility, benefit, movement, efficiency, etc. The object was changed from one tri-partition to another tri-partition accompanied by changes in utility. The utility function u we define as the form of a quaternion u(x, π, π_c, q), represents different utility value between the object x ∈ OB in the tri-partition π = {P₁, P₂, P₃} and π_c = {R⁺, R⁰, R^-}, denoted by u(q(π(x)⇝_aπ_c(x))). The value reflects the change in direction and quantity. Different changes correspond to different intervals and utility. Without losing generality, we define utility values as ranging from 0 to 1. The maximum change corresponding to the maximum utility, i.e., 1. The minimum change corresponding to the minimum utility, i.e., 0. In this model, objects are divided into three regions R⁺, R⁰, R^- by a pair of acceptance-rejection thresholds (c⁺, c^-), represent the decision maker’s intuitive perception of the amount of change, corresponding to the desired change, indifferent change, undesirable change, respectively. Therefore, we measure the effects separately for different regions and then compare and rank the TAO model’s effectiveness. Without losing generality, we take the specified interval of change in the region R⁺ as an example. We assume that q₁ and q₂ are two changes in the specific region R⁺, and their utility values are u(q₁) and u(q₂).

Case (1). Given a change quantity q and q₁ < q < q₂. Object x ∈ R⁺ change quantity q₁ with the probability of p, and q₂ with the probability of (1 - p). When the decision-maker judges that q₁ obtained with a probability of 1 is equivalent to getting q₂ with a possibility of p and q with a probability of (1 - p), the outcome of the q is given by: $u (q) = pu (q_{1}) + (1 - p) u (q_{2}) .$ (14)

Case (2). Given a change quantity q and q₁ < q₂ < q. Object x ∈ R⁺ change quantity q₁ with the probability of p, and q with the probability of (1 - p). When the decision-maker judges that q₂ obtained with a probability of 1 is equivalent to getting q₁ with a possibility of p and q₂ with a probability of (1 - p), the outcome of the q₂ is given by: $u (q_{2}) = pu (q_{1}) + (1 - p) u (q),$ (15) where $u (q) = \frac{u (q_{2}) - pu (q_{1})}{1 - p} .$

Case (3). Given a change quantity q and q < q₁ < q₂. Object x ∈ R⁺ change quantity q with the probability of p, and q₂ with the probability of (1 - p). When the decision-maker judges that q₁ obtained with a probability of 1 is equivalent to getting q with a possibility of p and q₂ with a probability of (1 - p), the outcome of the q₁ is given by: $u (q_{1}) = pu (q) + (1 - p) u (q_{2}),$ (16) where $u (q) = \frac{u (q_{1}) - (1 - p) u (q_{2})}{p} .$

In summary, baseline utility values are used to determine the utility functions at the different interval changes. At the same time, we use the rectangular coordinate system to describe the corresponding metric utility curves and divide the metric utility curves into three basic types. As shown in Fig. 4, if u(q) is concave, we call it a concave utility measure; if u(q) is straight, we call it a direct utility measure; if u(q) is convex, we call it a convex utility measure.

Fig. 4

Curve of the measurement function.

According to different types of utility curves, we calculate the utility value u(q(π(x)⇝_aπ_c(x))) produced by the change quantity q(π(x)⇝_aπ_c(x)) of each object in the region. From the perspective of the line of point-line-area, a basic utility measurement framework $U : OB \times π \times π_{c} \to R$ of a tri-partition P_i is given by: $U (P_{i} | π ⇝_{a} R^{•} | π_{c}) = \sum_{x \in (P_{i} \cap R^{•})} u (q (π (x) ⇝_{a} π_{c} (x)))$ (17)

4.3 Dual expectation utility measurement method based on change preferences

In the change model, objects are divided to the three regions R⁰, R⁺, R^- according to their change intervals. Thus, each region has three different interval states, denoted as CP_i = {P_i⇝_aR⁺, P_i⇝_aR⁰, P_i⇝_aR^-}, and there is a strict preference relationship between them, $(P_{i} ⇝_{a} R^{+}) ≻ (P_{i} ⇝_{a} R^{0}) ≻ (P_{i} ⇝_{a} R^{-}) .$ (18)

The traditional expected utility theory approach does not take into account the subjective preferences of decision-makers in determining the weights when jointly calculating utility outcomes of different intervals [61]. Therefore, we introduce the dual expected utility function theory [62] to adjust the weights of these intervals of changes so that they are arranged in the order of change preferences. The dual expected utility function based on change preference is constructed. We arrange the preferences in order from low to high and change the order of preferences by t = 1, 2, 3: P_i⇝_aR^-(t = 1), P_i⇝_aR⁰(t = 2), P_i⇝_aR⁺(t = 3), and f_t represent the probability of corresponding changes, that is, $f_{t} = \frac{| P_{i} \cap R^{•} |}{| P_{i} |}$ . The function is given by, $CPEU = \sum_{t = 1}^{3} U (P_{i} | π ⇝_{a} R^{•} | π_{c}) η (f_{1}, \dots, f_{t}) .$ (19) Where $η (f_{1}, \dots, f_{t}) = g (\sum_{k = 1}^{t} f_{t}) - g (\sum_{k = 1}^{t - 1} f_{t})$ , g(x) is a non-decreasing function and satisties g(0) =0, g(1) =1. When g(x) = x, η(f₁, ⋯ , f_t) = f_t, CPEU degenerates to the traditional expected utility function. When g(x) is a concave function, for the changes smaller than the reference value V, g(x) is used to assign a higher weight, which indicates that the change of utility u(CP_i) lower than V in CPEU plays a more significant role. When g(x) is a convex function, for the changes higher than the reference value V, g(x) is used to assign a higher weight, which indicates that the change of utility u(CP_j) higher than V in CPEU plays a more significant role.

According to the unevenness of g(x), η(f₁, ⋯ , f_t) allows CPEU to focus on changes to be front or back. We discuss a case where g(x) is a convex function. At this time, decision-makers are more focused on changes with high preferences, as shown in Fig. 5. We assume that $g (\sum_{k = 1}^{t} f_{t})$ has the following functional form: $\begin{matrix} g (\sum_{k = 1}^{t} f_{t}) = \\ {\begin{matrix} B \sum_{k = 1}^{t} f_{t}, & 0 ⩽ \sum_{k = 1}^{t} f_{t} ⩽ Pr (u (C P_{i}) ⩽ V), \\ A \sum_{k = 1}^{t} f_{t} + C, & Pr (u (C P_{i}) ⩽ V) ⩽ \sum_{k = 1}^{t} f_{t} ⩽ 1 . \end{matrix} \end{matrix}$ (20) Where V is a constant given in advance, $\begin{matrix} 1 ⩽ A ⩽ \frac{1}{Pr (u (C P_{i}) ⩽ V)}, \\ B = \frac{1 - A Pr (u (C P_{i}) ⩽ V)}{1 - Pr (u (C P_{i}) ⩽ V)}, \\ C = (B - A) Pr (u (C P_{i}) ⩽ V) . \end{matrix}$

Fig. 5

g(x)-function.

The probability weight η(f₁, ⋯ , f_t) is re-expressed as follows according to the above definition, $η (f_{1}, \dots, f_{t}) = {\begin{matrix} B f_{i}, & \sum_{k = 1}^{t} f_{t} ⩽ Pr (u (C P_{i}) ⩽ V) \\ A f_{i}, & else \end{matrix}$ (21)

In the area level, the outcome from the tri-partition π = {P₁, P₂, P₃} is given by, $\begin{matrix} CPEU (P_{i} | π ⇝_{a} π_{c}) \\ = \sum_{t = 1}^{3} U (P_{i} | π ⇝_{a} R^{•} | π_{c}) η (f_{1}, \dots, f_{t}), \\ = U (P_{i} | π ⇝_{a} R^{-} | π_{c}) g (f_{1}) + U (P_{i} | π ⇝_{a} R^{0} | π_{c}) (g (f_{1} + f_{2}) \\ - g (f_{1})) + U (P_{i} | π ⇝_{a} R^{+} | π_{c}) (g (f_{1} + f_{2} + f_{3}) - g (f_{1} + f_{2})) . \end{matrix}$ (22) In summary, the effectiveness of the chang-based 3WD based on the utility of tri-partition π is, $EU (π ⇝_{a} π_{c}) = \sum_{i} v_{i} \cdot CPEU (P_{i} | π ⇝_{a} π_{c}),$ (23) where v_i represents the weight occupied by the region P_i.

The effectiveness of change relies on an evaluation criterion given by the utility function, which completes the model’s evaluation by fixing the desired utility at a baseline utility and then weighting the weighted average of the utility values for different states of change overestimated or underestimated.

4.4 An illustrative example

The example of student GPA management continues to be used in this section, which introduces the utility measurement method based on change preferences. Suppose the student set is OB = {o₁, o₂, ⋯ , o₈, o₉}, three regions are constructed: P₁ = {o₁, o₃} , P₂ = {o₂, o₄, o₆, o₉} , P₃ = {o₅, o₇, o₈}, according to their GPAs. Through the promotion of some teaching strategies, students’ GPAs are changed, therefore, students are re-divided into three new regions according to specific criteria: R⁺ = {o₁, o₂, o₅, o₉} , R⁰ = {o₃, o₄, o₇} , R^- = {o₆, o₈}. According to the students’ GPAs and their changes, two tri-partition with different standards can be divided, that is, 9 different changes. Assume that the utility value u(CP_i) given by the results of different changes is shown in Table 2:

Table 2
Utility value corresponding to different interval change

Acting Change state

Region R ⁺ R ⁰ R ^-

P ₁ 0.7 0.4 0

P ₂ 1.2 0.5 0.2

P ₃ 0.8 0.4 0.1

Acting	Change state
P ₁	0.7	0.4	0
P ₂	1.2	0.5	0.2
P ₃	0.8	0.4	0.1

Correspondingly, we calculate the probability of occurrence of different changes according to the base of the students. The results are given in Table 3.

Table 3

Probability corresponding to different interval change

Acting	Change state
Region	R ⁺	R ⁰	R ^-
P ₁	$\frac{1}{2}$	$\frac{1}{2}$	0
P ₂	$\frac{1}{2}$	$\frac{1}{4}$	$\frac{1}{4}$
P ₃	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$

The g(x) is a convex function, as discussed in Section 4.3. The corresponding values are assumed to be: $A = 2, B = \frac{1}{2}, C = \frac{5}{6}, V = \frac{1}{3}$ , respectively. The probability weights η_i(f₁, ⋯ , f_t) of different changes in the region P_i are calculated separately by, $\begin{matrix} η_{1} (f_{1}) = 0; η_{1} (f_{1}, f_{2}) = 2 \times \frac{1}{2} = 1; \\ η_{1} (f_{1}, \dots, f_{3}) = 2 \times \frac{1}{2} = 1; η_{2} (f_{1}) = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}; \\ η_{2} (f_{1}, f_{2}) = 2 \times \frac{1}{4} = \frac{1}{2}; η_{2} (f_{1}, \dots, f_{3}) = 2 \times \frac{1}{2} = 1; \\ η_{3} (f_{1}) = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}; η_{3} (f_{1}, f_{2}) = 2 \times \frac{1}{3} = \frac{2}{3} \\ η_{3} (f_{1}, \dots, f_{3}) = 2 \times \frac{1}{3} = \frac{2}{3} . \end{matrix}$

Further, we get the expected utility values CPEU of the region P_i, which are 1.1, 1.475, 0.8167, according to Equation 22. Finally, the effect of change EU(π⇝_aπ_c) is: $EU (π ⇝_{a} π_{c}) = 1 . 1 + 1 . 475 + 0 . 8167 = 3 . 391 .$ According to 23 when the weight of region P_i is uniformly set to v_i = 1 in this example.

5 Experiment and analysis

We use python programming to validate the effectiveness of change-based 3WD in this section based on data set [63], which is derived from the UCI database and focuses on the comparison of the utility of the same change in different regions. The experiment’s purpose is to compare the utility results for each interval change and obtain results for changes in different directions.

The data set contains 3000 records, and each includes 21 attributes. We pre-process the data: data cleaning, which is to initialize the absenteeism time as null to 0, integrate the 36 employee ids, and calculate the absenteeism time (hours) for each season. We calculate employees’ absenteeism time in spring-autumn and summer-winter separately and integrate 72 late arrival change data and record it as 72 objects. As shown in Fig. 6, the dots show the object’s absenteeism time in the spring and autumn period. The triangle represents the absenteeism time of the object in the summer and winter periods.

Fig. 6

The absenteeism time of employees.

We record the absenteeism of employees during the spring and autumn period as the initial quality of the object before the strategy and use this construct the initial tri-partition π = {P₁, P₂, P₃} and the absenteeism of employees during the summer and winter period as the final quality. The difference in the absenteeism time between spring-summer and autumn-winter is used as the changes. That is, the difference between the initial quality and the final quality is used as the change amount. The specific quality is: change _ quantity = initial _ quantity - final _ quantity.

In the experiment, the change quantity before and after the action strategy is calculated: changed q(t_i) = q₀(t_i)- q₁(t_i). According to each employee’s the absenteeism before the strategy, we divided the employees with absenteeism time less than or equal to 2 into the region P₁, which represents a good employee region. The employees with absenteeism time between 2 and 7 were divided into the region P₂, representing the neutral employee region. Employees are divided into region P₃ according to absenteeism time seven or more, representing poor employee region.

Assume that the thresholds of P₁, P₂, and P₃ regions are: (-4, 2), (-3, 1), (-6, 3). To facilitate comparison using the L-A simulation method, the utility range for each interval of change is [0, 1]. The utility function is y = a(x + c) ^b. For each interval of change, three utility points are given, as shown in Table 4. Finally, according to the utility value points, utility functions fitted to the nine change intervals are as follows and the corresponding curves are illustrated in Fig. 7: $\begin{matrix} y_{P_{1} \cap R^{+}} = 0.5 (x - 1)^{1}; y_{P_{1} \cap R^{0}} = 0.5 (x + 3)^{0.6309}; \\ y_{P_{1} \cap R^{-}} = 0.5 (x + 6)^{1}; y_{P_{2} \cap R^{+}} = (x - 1)^{0.4307}; \\ y_{P_{2} \cap R^{0}} = (x + 2)^{1.9434}; y_{P_{2} \cap R^{-}} = 0.5 (x + 7)^{0.5}; \\ y_{P_{3} \cap R^{+}} = 0.1667 (x + 3)^{1}; y_{P_{3} \cap R^{0}} = 0.25 (x + 2)^{1}; \\ y_{P_{3} \cap R^{-}} = 0.0011 (x + 16)^{2.6419} . \end{matrix}$

Table 4

Utility points of interval-change

Acting	Change state
Region	R ⁺	R ⁰	R ^-
P ₁	(1,0)(2,0.5)(3,1)	(-3,0)(-2,0.5)(0,1)	(-6,0)(-5,0.5)(-4,1)
P ₂	(1,0)(1.2,0.5)(-2,1)	(-2,0)(-1.2,0.5)(-1,1)	(-7,0)(-6,0.5)(-3,1)
P ₃	(3,0)(6,0.5)(9,1)	(3,0)(6,0.5)(9,1)	(-16,0) (-6,0.5)(-3,1)

Fig. 7

The utility curve of interval-change.

The different curves represent the subjective preferences of the decision-makers in terms of the effects of their actions strategies, and the convexity of each curve reflects the degree of preference. We can calculate the utility value for each employee who changes under this action strategy. The utility values obtained for the nine interval changes are given by: $\begin{matrix} u_{P_{1} \cap R^{+}} = 2; u_{P_{1} \cap R^{0}} = 14.87; \\ u_{P_{1} \cap R^{-}} = 0.5; u_{P_{2} \cap R^{+}} = 1; \\ u_{P_{2} \cap R^{0}} = 0; u_{P_{2} \cap R^{-}} = 5.07; \\ u_{P_{2} \cap R^{-}} = 5.07; u_{P_{3} \cap R^{+}} = 4.83; \\ u_{P_{3} \cap R^{0}} = 0.25; u_{P_{3} \cap R^{-}} = 1.57 . \end{matrix}$

According to the result of the utility value, the priority relationship of changed under a strategic action a can be compared: $(P_{3} ⇝_{a} R^{+}) ≻ (P_{1} ⇝_{a} R^{+}) ≻ (P_{2} ⇝_{a} R^{+});$ $(P_{1} ⇝_{a} R^{0}) ≻ (P_{3} ⇝_{a} R^{0}) ≻ (P_{2} ⇝_{a} R^{0});$ $(P_{2} ⇝_{a} R^{-}) ≻ (P_{3} ⇝_{a} R^{-}) ≻ (P_{1} ⇝_{a} R^{-}) .$

The experiments validate the TAO model’s effectiveness measurement using interval sets, which characterize the change of objects in the light of the strategy and action. The fitted utility curves indicate that they are consistent with the subjective preferences of decision-makers.

6 Conclusion

The effectiveness measure of the TAO model is a new subject. In this paper, we propose the first time to measure the TAO model’s effect by examing objects’ changes under a strategy action, which is called the change-based TAO model of three-way decisions. We first use an interval set to represent objects’ changes, thereby re-divide objects into a new tri-partition. Utility functions are then used to characterize the effectiveness of these changes. By aggregating these utility values, the outcome from the strategy and action is obtained. This paper studies the TAO model’s effectiveness evaluation methods by introducing the idea of interval sets into the change-based TAO model. It is more convenient for the model to analyze and calculate the different intervals of change. The utility measurement method is also more consistent with the decision-makers’ cognitive preferences. The research further enriches the research on outcome in the TAO model of three-way decisions.

In future work, we will consider the change models based on specific sets, such as those based on fuzzy sets, shaded sets, interval sets, etc. Then, the effectiveness measures will be studied based on these change models. Another perspective is that making strategy selection based on effectiveness measures is also essential work in three-way decisions.

Footnotes

Acknowledgments

This work was supported in part by Natural Science Foundation of Heilongjiang Provincial (No. LH2020F031) and Postgraduate Innovation Project of Harbin Normal University (HSDSSCX2020-30).

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Effectiveness measure for TAO model of three-way decisions with interval set

Abstract

Keywords

1 Introduction

2 Related works of trisecting-acting-outcome model

2.1 Trisecting

2.3 Outcome

4.1 Change-based utility measurement framework

Table 2 Utility value corresponding to different interval change Acting Change state Region R + R 0 R - P 1 0.7 0.4 0 P 2 1.2 0.5 0.2 P 3 0.8 0.4 0.1

Footnotes

Acknowledgments

References

Table 2
Utility value corresponding to different interval change

Acting Change state

Region R ⁺ R ⁰ R ^-

P ₁ 0.7 0.4 0

P ₂ 1.2 0.5 0.2

P ₃ 0.8 0.4 0.1