International Journal of Computational Intelligence Systems

Volume 12, Issue 2, 2019, Pages 557 - 570

Rough Number–Based Three-Way Group Decisions and Application in Influenza Emergency Management

Fan Jia*, Peide Liu
School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan, Shandong, PR China
*Corresponding author. Email:
Corresponding Author
Fan Jia
Received 3 September 2018, Accepted 17 April 2019, Available Online 26 April 2019.
10.2991/ijcis.d.190420.001How to use a DOI?
Three-way decisions; Group decision-making; Rough numbers; Risk preference

Group decision-making can effectively deal with complex decision problems in reality and takes important research status in the field of decision-making. In recent years, three-way decision has been a hot topic in the field of uncertain decision-making, so the models of three-way decision under group environment have become a new direction and problem in decision-making research. Based on the subjectivity and uncertainty of the group judgment, this paper adopts the rough number method to objectively and effectively integrate the group information so as to establish rough number–based three-way decision models. First of all, we use rough numbers to transfer loss functions of individuals into a rough number–based gathering loss function, and verify the rough number–based loss function meets the general characteristics of general loss functions. Then from the perspectives of optimism, pessimism, risk preference of decision-makers, and directly operation of rough numbers, we explore the calculation of thresholds and the acquisition method of three-way rules, and then establish rough number–based three-way group decision models. Finally, influenza emergency management is introduced to verify the effectiveness and advancement of the novel method.

© 2019 The Authors. Published by Atlantis Press SARL.
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Three-way decision-making is a decision-making theory developed on the basis of the decision-theoretic rough set (DTRS). Ever since its introduction by the Canadian scholar Yao in 2009, it has attracted wide attention in academia. To provide a reasonable semantic explanation for the three regions divided by the upper and lower approximate sets in the rough set theory, Yao corresponded the positive, boundary, and negative domains in the rough set model with three decision-making results: acceptance, noncommitment, and rejection [1,2]. The author also expanded the traditional two-way decision-making model with only two outputs (acceptance and rejection) into a three-way model, which more accurately describes the characteristics of human thinking and behavior. Three-way decision-making frequently occurs in real life. For example, a manuscript review by experts based on two-way decision-making leads to one of two results: acceptance or rejection. In the actual review process, however, most manuscripts cannot be directly categorized into the above two results. Instead, a third decision-making result, noncommitment, is selected, in which further revision required. This gives rise to a three-way decision-making problem. Three-way decision-making is proposed based on the idea of the DTRS. The DTRS starts with Bayes' theorem and uses loss functions to describe the risks in decision-making. It selects the decision-making behavior with the minimum risk in response to the expected risks of three-way decision-making results, namely, acceptance, delayed decision-making, and rejection. Two important elements in the DTRS are the conditional probability and the threshold. The final decision-making behavior is determined by comparing their magnitude. Three-way decision-making has already been widely applied in many areas, such as investment decision-making [3,4,5], conflict analysis [6], face recognition [7,8], email filtering [9], medical diagnosis [10,11], and recommendation systems [12,13].

The greatest advantage of three-way decision-making over traditional decision-making methods is that it can reduce the loss caused by incorrect classification. There are many decision-making problems in real life. Once an incorrect decision is made, the resulting loss may be immeasurable, especially for emergencies. In complicated decision-making situations with short decision-making time, incorrect decision-making often results in a series of adverse consequences. The management of an influenza epidemic situation is a typical emergency management event. In recent years, a variety of influenza viruses have caused outbreaks one after another, for example, H1N1. There are both correct and incorrect judgments on influenza epidemic situations. A correct judgment can control the development of the epidemic situation, whereas an incorrect judgment can easily lead to a large-scale influenza outbreak or social disorder. Figure 1 shows four results caused by different decisions aiming at whether the influenza outbreak or not. Suppose there are several influenza cases in a certain city, and the health department needs to determine whether or not this is a harbinger of an influenza outbreak based on the current situation and whether or not preventive measures should be implemented, such as issuing influenza warnings or deploying isolation procedures. The quadrant (ii) and (iv) depict the correct decisions, where quadrant (ii) represents that the city is suffering the influenza and the government decides to take protection actions to reduce the losses, and quadrant (iv) represents that no influenza outbreaks in this city and the government takes no action. While the quadrant (i) and (iii) demonstrate the mistakes in decision-making: the quadrant (iii) shows the type I error (rejection of a true null hypothesis), which means it is currently at the stage of an influenza outbreak, but the government has not taken any measures, which will result in spreading of influenza and more cases; the quadrant (i) shows the type II error (acceptance of a false null hypothesis), meaning that the current situation will not cause an influenza outbreak, but the government takes actions like issuing influenza warnings and deploying isolation procedures, which will cause the general public panic and disturb the normal functioning of society, and even social chaos. In particular, with the current development of the Internet, information spreads much faster, easily leading to the spread of rumors. For instance, a few years ago, a prevailing rumor that H7N9 would outbreak in China caused people to rush out and buy Banlangen (the root of Isatis indigotica) and white vinegar. This resulted in social unrest and consequences even worse than the influenza outbreak itself. Therefore, there is a new research approach in which three-way decision-making is used to cope with practical decision-making problems, especially emergency management problems.

Figure 1

Decision results for influenza emergency management.

As an essential element of three-way decisions, the determination of loss functions is a key procedure and has attracted most attention. It is almost impossible to obtain exact losses by instruments or technologies, so in most realistic problems, losses need to be determined by related experts. For example, the losses of medical diagnosis problems need to be determined by doctors according to their knowledge and experience. While it is undeniable that there exists uncertainty and subjectivity in experts' opinions, therefore, many researches considered uncertainty and constructed three-way decision-making models based on different expression forms of losses. Liu and Liang described the loss using the interval number and constructed DTRS models based on the interval number [14,15], which has provided ideas for later research on three-way decision-making on uncertainty information. Liu introduced randomness into the loss function and constructed a random three-way decision-making model, the relevant characteristics of which were explored based on uniform distributions and normal distributions separately [5]. Subsequently, Liang et al. expanded the three-way decision-making model to fuzzy cases in which the loss function is expressed in the forms of triangular fuzzy numbers [16], intuitionistic fuzzy numbers [17], hesitant fuzzy numbers [18], and other data types. Corresponding three-way decision-making models were constructed, and the threshold calculation method was given to make the optimal decision. Liang et al. used linguistic terms to evaluate the selection and constructed a linguistic three-way decision-making method [19]. Three-way model was introduced into information table with order relation, and then the ordered three-way decisions were constructed as well as the algorithm [20].

As the complexity of decision environment and the limitation of human cognition, opinion of single decision-maker could not describe losses correctly. Therefore, to express losses in a comprehensive way, a few researches have introduced group decision-making approaches to three-way decisions. In traditional group decision-making methods, different aggregating approaches have been discussed according to different forms of evaluating values, some of which have been applied into three-way decision problems. By using the idea of granular computing, Liang aggregated and expressed the loss function as interval numbers and thus established a three-way group decision-making model [21]. After that, Liang expanded group decision-making to a loss function of linguistic value and constructed a three-way group decision-making model based on linguistic evaluation information [19]. Sun constructed a three-way group decision-making model based on a multigranularity fuzzy DTRS [22]. Liu constructed a three-way group decision-making model by taking the decision-makers' attitudes and preferences into consideration and integrating these with the prospect theory [23]. By aggregating group information, an aggregated loss function can be found, and then the thresholds. The conditional probability is always observed by historical experience, and then decision rules can be obtained through comparing conditional probability and thresholds.

Although there exist some studies about three-way group decision-making models, the research for group decision in three-way models is still in the exploratory stage. According to the discussion above, all the approaches for group aggregation in three-way decisions ignores the intrinsic links among evaluation information from decision-makers. Zhai proposed a novel approach, the rough number, which is proposed to quantify experts' cognition based on rough set theory, and can be used for group aggregation by considering the intrinsic relationship among decision-makers [24,25]. Therefore, this paper intends to discuss three-way group decision-making models in the novel perspective of rough number. A rough number is represented as an interval numbers formed by upper and lower limits. It does not require any prior information, but it is determined completely based on the original data. The rough number method can comprehensively take the evaluations from all experts and integrate decisions made by individual experts into a group decision to ensure the objectivity and accuracy of the data and results. The rough number method was originally applied in the development of new products initiated by quality function. Zhai used the rough number method to effectively compile subjective evaluations and judgments of consumers and integrated individual evaluation values into rough numbers, which were then employed for data mining and decision-making [25]. Subsequently, many scholars have integrated the rough number method with the multicriteria decision-making method and have obtained decision-making models, such as rough number AHP (analytic hierarchy process); [26], rough number BWM (best-worst method); [27], and rough number TOPSIS (technique for order preference by similarity to an ideal solution) [28]. Since its introduction, the rough number method has been widely used as an effective way of assembling group information.

Therefore, the aim of this research is to construct a novel three-way group decision-making model based on rough numbers, which can aggregate multiple experts' information in a reasonable way, obtain rough loss functions, and receive more effective three-way rules, and then be used in influenza emergency management problems. The remainder of this paper is organized as follows. Section 2 lists some basic concepts of three-way decisions with DTRSs and rough numbers. Section 3 proposes the construction process of group loss functions based on rough numbers. In section 4, according to group loss functions, we discuss the acquisition methods of three-way rules from four different perspectives. To illustrate the application in influenza emergency management, section 5 presents a practical case in China and discuss the effectiveness of the model. Section 6 concludes this paper and outlines the future work.


This section is composed of two subsections to review some preliminaries about three-way decisions with DTRSs and the method of rough numbers.

2.1. Three-Way Decisions Based on DTRSs

To illustrate the principle of rough sets, Pawlak defined the approximation space as apr=U,R, where U be a finite and nonempty set and R be an equivalence relation. Denote the partition of U induced by R as x. Then Yao proposed probabilistic rough sets (PRSs), where for CU, the probabilistic lower and upper approximations of C can be defined as follows [1]:

where PrC|x is the conditional probability of the classification expressed as
and the thresholds α and β satisfy 0β<α1, where Equations (1) and (2) are degenerated into Pawlak approximations of classical rough sets when α=1 and β=0 [2].

The PRSs divide the universe U into three regions: the positive region, the boundary region, and the negative region, which correspond to acceptance, the noncommitment, and the rejection, the actions of three-way decisions proposed by Yao [2], which can be expressed as


In particular, the models can be degenerated to traditional two-way decisions when α=β, and the positive region and negative region can be written as

where γ=α=β.

In order to give semantic interpretations of the thresholds and three regions in PRSs, Yao proposed DTRSs with the aid of Bayesian theory [29]. A DTRS model is composed of two states and three actions, which can be seen in Table 1. The set of states can be expressed as Ω=C,¬C, which presents that an object is either in C or not in C. The set of actions is denoted as A=aP,aB,aN, in which aP, aB, and aN denote xPOSC, xBNDC, and xNEGC, respectively. The loss function regarding the risks or losses of different actions is given by a 3×2 matrix exhibited in Table 1. λPP, λBP, and λNP indicate the losses of taking actions aP, aB, and aN, respectively, when the object belongs to C, while λPN, λBN, and λNN denote the losses incurred for taking actions aP, aB, and aN, respectively, when the object belongs to ¬C.

aP λPP λPN
aB λBP λBN
aN λNP λNN
Table 1

The loss functions.

For xU, we denote Rai|x as the expected loss when taking the action aii=P,B,N, and it can be calculated as

where PrC|x is interpreted as the conditional probability of an object x belonging to C given that the object is described by its equivalence class [x], and Pr¬C|x is interpreted as the conditional probability of an object x belonging to ¬C given that the object is described by its equivalence class [x].

According to Bayesian theory, the best decision is the one with minimum cost, so decision rules can be indicated as follows:

  • P If RaP|xRaB|x and RaP|xRaN|x, decide xPOSC,

  • B If RaB|xRaP|x and RaB|xRaN|x, decide xBNDC,

  • N If RaN|xRaP|x and RaN|xRaB|x, decide xNEGC.

Since PrC|x+Pr¬C|x=1, we can use 1PrC|x instead of Pr¬C|x in Equations (911). As the losses satisfy the conditions that λPPλBP<λNP and λNNλBN<λPN by a reasonable semantic interpretation, we can obtain the simplified rules (P′)-(N′) as [2]

  • P If PrC|xα and PrC|xγ, decide xPOSC,

  • B If PrC|xα and PrC|xβ, decide xBNDC,

  • N If PrC|xβ and PrC|xγ, decide xNEGC,

where the thresholds α, β, γ are given as

Suppose there exists an assumption

then the thresholds satisfy 0β<γ<α1, so the rules (P′)-(N′) can be further simplified
  • P If PrC|xα, decide xPOSC;

  • B If β<PrC|x<α, decide xBNDC;

  • N If PrC|xβ, decide xNEGC.

On the contrary, if the assumption is

then the thresholds satisfy 0αγβ1, and (B′) is not working in this case because there is no PrC|x satisfying PrC|xα and PrC|xβ simultaneously. So the rules can be rewritten as a two-way decision rules
  • P If PrC|xγ, decide xPOSC,

  • N If PrC|x<γ, decide xNEGC.

The rules (P″)-(N″) and (P‴)-(N‴) indicate that three-way decisions must satisfy the condition Equation (16) or 0β<γ<α1, while two-way decisions satisfy the condition Equation (16) or 0αγβ1.

2.2. Rough Numbers

Inspired by rough set theory, rough number is firstly proposed by Zhai [24] in order to handle subjective preferences of customers in quality function deployment. Similar to the notion of approximates in rough sets, a rough number is constructed by lower and upper limits, which determine a rough boundary interval to characterize imprecise information. Rough numbers merely depend on original data without any prior knowledge, so they can capture the experts' real perception effectively, and aggregate every individual's preference into an objective and consistent group judgement. In this subsection, we review some basic concepts of rough numbers.

Suppose U is the universe containing all the objects, and there are n classes expressed as R=C1,C2,,Cn. If they are ordered as C1<C2<<Cn, then for YU, CiR, the lower approximation Apr_Ci and the upper approximation Apr¯Ci of Ci can be defined as


Then Ci can be transformed to a rough number RNCi, which is expressed as

where RN_Ci and RN¯Ci are the lower limit and upper limit of Ci, and denoted as
where ML and MU are the numbers of objects contained in Apr_Ci and Apr¯Ci, respectively.

To compare the values of rough numbers, Zhai proposed the ranking rules [25]. For any two rough numbers RN1=RN1_,RN¯1 and RN2=RN2_,RN2¯, there are five possible cases, which are shown in Figure 2, where M1 and M2 are the medians of RN1 and RN2, respectively.

Figure 2

Ranking rules for rough numbers.

For an easy-to-operate purpose, we illustrate the ranking rules in an accurate mathematical linguistics rather than a graphical representation. Denote Sa=RNa_+RN¯a as the sum of upper limit and lower limit for a rough number RNa=RNa_,RN¯a, and Da=RN¯aRNa_ as the difference of upper limit and lower limit. For any two rough numbers RN1=RN1_,RN¯1 and RN2=RN2_,RN2¯, we can obtain the ranking rules as follows:

  1. If S1>S2, then RN1>RN2,

  2. If S1<S2, then RN1<RN2,

  3. If S1=S2, then:

    1. if D1<D2, then RN1>RN2,

    2. if D1>D2, then RN1<RN2,

    3. if D1=D2, then RN1=RN2.

Where Sa=RNa_+RN¯a and Da=RN¯aRNa_.

Based on the construction of S and D, we convert the graphical ranking rules to mathematical linguistics, which could facilitate the further research, and it is easy to find out that the two expression forms are equivalent.

For a convenient expression, RN_Ci, RN¯Ci, and RNCi can be denoted as Ci_, Ci¯, and Ci for short, respectively.

Rough numbers are similar to interval numbers in form, so with the aid of arithmetic operations of interval analysis, Zhai proposed the operations of rough numbers [24]. Suppose a=a_,a¯ and b=b_,b¯ are two rough numbers and satisfy a_,a¯,b_,b¯>0, and α>0 is a real number. Then the arithmetic operations of rough numbers can be expressed as follows:



3.1. Description of Three-Way Group Decision-Making

The most important issue of the three-way decision-making model is the determination of the loss function. For losses caused by incorrect classification, researchers usually use subjective experiences and evaluation by relevant experts to determine the loss values. While due to the lack of knowledge and experience, and the limitation of individual subjective knowledge, individuals find it difficult to have systematic and comprehensive understanding of decision-making problems. Therefore, group decision-making has become an important approach in the field. In a group decision-making perspective, h experts can be selected to form a decision-making group, which is represented as E=e1,e2,,eh, among which ek denotes the kth expert. Each expert makes a subjective assessment on various incorrect classifications, and h loss functions can be obtained. The loss function obtained by expert ek through evaluation and judgement is shown in Table 2. The process of three-way group decision-making is to determine the aggregated group loss function based on the individual loss functions given by h individual experts and to describe the opinions of the decision-making group. The aggregated loss function is used to explore the threshold calculation method in three-way decision-making and the acquisition of three-way decision-making rules.

aP λPPk λPNk
aB λBPk λBNk
aN λNPk λNNk
Table 2

The loss function of ek.

3.2. Rough Number–Based Loss Functions

By integrating the evaluation information from h experts, an evaluation summary table for the decision-making group can be obtained, as shown in Table 3.

e1 λPP1 λBP1 λNP1 λPN1 λBN1 λNN1
e2 λPP2 λBP2 λNP2 λPN2 λBN2 λNN2
ek λPPk λBPk λNPk λPNk λBNk λNNk
eh λPPh λBPh λNPh λPNh λBNh λNNh
Table 3

The results of experts' evaluation.

For any λ**, there exist h evaluation results, meaning that the group evaluation result on the loss λ** can be denoted as λ**1,λ**2,,λ**h. With the help of the construction method of rough number, the evaluating values can be aggregated to rough numbers, and then the loss function can be expressed by rough number–based values. By using Equations (1719), each evaluation value λ**k can be converted and expressed as a rough number and represented as λ**k=λ**k_,λ**k¯, where the values of λ**k_ and λ**k¯ are related to the evaluation values of all decision-makers. For example, suppose five experts have evaluated the loss λPP as 2,3,3,4,2, then we can convert each value to a rough number–based expression as


Therefore, the rough number–based evaluation values of λPP is 2,2.8,2.5,3.33,2.5,3.33,2,2.8,2.8,4.

By using Equations (20) and (21), aggregated loss function can be obtained and represented as λ**=λ**_,λ**¯. Table 4 shows the aggregated rough loss function obtained through the rough number method. For the above example, the aggregated rough number can be calculated as


aP λPP=λPP_,λPP¯ λPN=λPN_,λPN¯
aB λBP=λBP_,λBP¯ λBN=λBN_,λBN¯
aN λNP=λNP_,λNP¯ λPP=λPP_,λPP¯
Table 4

Rough loss function.

Therefore, the aggregated rough loss λPP can be denoted as


According to the discussion in section 2, three-way decisions must satisfy the conditions λPPλBP<λNP and λNNλBN<λPN. In order to guarantee the effectiveness of the following research, we have to verify that rough loss functions should also satisfy the conditions.

Theorem 1.

If for any k=1,2,,h there exist λPPkλBPk<λNPk and λNNkλBNk<λPNk, then we can obtain

  1. λPP_λBP_<λNP_, λNN_λBN_<λPN_,

  2. λPP¯λBP¯<λNP¯, λNN¯λBN¯<λPN¯,

  3. λPPλBP<λNP, λNNλBN<λPN.


Firstly, we prove λPP_λBP_<λNP_, λPP¯λBP¯<λNP¯ and λPPλBP<λNP based on λPPkλBPk<λNPk.

It is obvious that there are λPP_=λBP_ and λPP¯=λBP¯ when λPPk=λBPk for any k=1,2,,h. Suppose there exists at least one decision-maker es such that λPPs<λBPs, then there must be λPPs<λBPs according to the construction process of rough numbers, and λPPk=λBPk or λPPk<λBPk for ks. By Equations (20) and (21) there exist λPP_<λBP_ and λPP¯<λBP¯. So λPP_λBP_ and λPP¯λBP¯ are proved when λPPkλBPk, and the equations are satisfied if and only if λPPk=λBPk for any k=1,2,h. Similarly, if λBPk<λNPk for any k=1,2,h, there are λBP_<λNP_ and λBP¯<λNP¯. Therefore, we can obtain that if λPPkλBPk<λNPk, there are λPP_λBP_<λNP_ and λPP¯λBP¯<λNP¯. In the same way, we can prove the corresponding conclusion if λNNkλBNk<λPNk.

For λPPλBP<λNP, we can prove λPPλBP firstly. According to the ranking rules of rough numbers proposed in Section 2, we should explore the relationship between λPP_+λPP¯ and λBP_+λBP¯. The discussion above has proved that if λPPkλBPk there are λPP_λBP_ and λPP¯λBP¯, so there must be λPP_+λPP¯λBP_+λBP¯. As we know, there is λPP_+λPP¯=λBP_+λBP¯ if and only if λPPk=λBPk, so it is easy to obtain that if λPPkλBPk then λPPλBP. Similarly, according to λBPk<λNPk we can obtain λBP<λNP. So λPPλBP<λNP is proved.

According to the discussion, we can prove λPP_λBP_<λNP_, λPP¯λBP¯<λNP¯, and λPPλBP<λNP based on λPPkλBPk<λNPk. And in the same way, we can also prove λNN_λBN_<λPN_, λNN¯λBN¯<λPN¯, and λNNλBN<λPN based on λNNkλBNk<λPNk.

So the theorem has been proved.

Theorem 1 illustrates that the aggregated loss functions based on rough numbers still satisfy the conditions in classical three-way decisions, which guarantee the validity and reasonability of the further research.


By using rough numbers to aggregate the information of decision-makers, the rough loss function, as shown in Table 4, has been obtained, wherein the losses are expressed as rough numbers. In this section, we intend to explore how thresholds, rules, and classifications are obtained in group three-way decisions based on rough loss functions. In consideration of the risk preferences of decision-makers, this section explores decision-making strategies from three perspectives, namely optimism, pessimism, and risk preference coefficient. Then a direct method to acquire the threshold and three-way rule is given based on the algorithm of rough numbers without taking into account the decision-maker's risk preference.

4.1. Three-Way Group Decision Models with Risk Preference

4.1.1. Model 1: An optimistic perspective

The optimistic perspective refers to a situation in which decision-makers believe that the loss caused by incorrect classification is the smallest. Therefore, only the lower limits of rough losses λ**_ are taken as the final decision-making losses. The optimistic loss function can be shown in Table 5.

aP λPP_ λPN_
aB λBP_ λBN_
aN λNP_ λNN_
Table 5

Optimistic loss function.

According to the acquisition procedure of three-way decision rules, the expected losses of different actions ai under optimistic perspective can be denoted as


According to Bayesian theory, the optimistic decision rules can be indicated as follows:

  • Po If RoaP|xRoaB|x and RoaP|xRoaN|x, decide xPOSC,

  • Bo If RoaB|xRoaP|x and RoaB|xRoaN|x, decide xBNDC,

  • No If RoaN|xRoaP|x and RoaN|xRoaB|x, decide xNEGC.

Based on the properties shown in Theorem 1, we can obtain that λPP_λBP_<λNP_ and λNN_λBN_<λPN_. Then because of PrC|x+Pr¬C|x=1, simplified decision rules can be obtained as follows:

  • Po If PrC|xαo and PrC|xγo, decide xPOSC,

  • Bo If PrC|xαo and PrC|xβo, decide xBNDC,

  • No If PrC|xβo and PrC|xγo, decide xNEGC,


Example 1.

Suppose in a group decision-making problem, we have obtained the aggregated rough loss function, and the losses of misclassifications are expressed as λPP=0.2,0.6, λBP=0.7,1, λNP=1.5,2.1, λPN=2.8,3.4, λBN=1.7,2.5, and λNN=0.3,0.8. Based on the optimistic perspective, we choose the lower limits as the final losses, so the losses can be written as λPP=0.2, λBP=0.7, λNP=1.5, λPN=2.8, λBN=1.7, and λNN=0.3. According to Equations (2931), we can calculate the thresholds as α=0.687, β=0.636, and γ=0.658.

4.1.2. Model 2: A pessimistic perspective

The pessimistic perspective means that the decision-maker believes that the loss caused by incorrect classification is the greatest. Hence, for the rough number loss function, only the upper limits of the rough losses λ**¯ are taken as the final losses in decision-making. The pessimistic loss function is shown in Table 6.

aP λPP¯ λPN¯
aB λBP¯ λBN¯
aN λNP¯ λNN¯
Table 6

Pessimistic loss function.

Based on the acquisition procedure of three-way decision rules, the expected losses when choosing different actions ai under pessimistic perspective can be expressed as


According to Bayesian theory, the pessimistic decision rules can be denoted as follows:

  • Pp If RpaP|xRpaB|x and RpaP|xRpaN|x, decide xPOSC,

  • Bp If RpaB|xRpaP|x and RpaB|xRpaN|x, decide xBNDC,

  • Np If RpaN|xRpaP|x and RpaN|xRpaB|x, decide xNEGC.

From Theorem 1, we can obtain that there exist λPP¯λBP¯<λNP¯ and λNN¯λBN¯<λPN¯. Then according to PrC|x+Pr¬C|x=1 we can get the simplified decision rules as

  • Pp If PrC|xαp and PrC|xγp, decide xPOSC,

  • Bp If PrC|xαp and PrC|xβp, decide xBNDC,

  • (Np) If PrC|xβp and PrC|xγp, decide xNEGC,


Example 2.

Continue the Example 1. Based on the pessimistic viewpoint, we determine the upper limits of rough numbers as the final losses, which can be written as λPP=0.6, λBP=1, λNP=2.1, λPN=3.4, λBN=2.5, and λNN=0.8. And according to Equations (3537), the thresholds under pessimistic perspective can be calculated as α=0.692, β=0.607, and γ=0.634.

4.1.3. Model 3: Risk preference coefficient

For many practical problems, the risk preferences of decision-makers are neither pessimistic nor optimistic, and they may be between the two instead. In such a case, a coefficient can be employed to describe the risk preference so that the losses in three-way decisions can be determined.

Definition 1.

For any rough number a=a_,a¯, let θ0,1, and the rough number can be converted into

where θ is the risk preference coefficient of decision-makers and θa is a conversion from a rough number a to a real number.

From Definition 1, we know that θa is a real number. Therefore, it is possible to use this conversion method to convert rough numbers into real numbers. The greater the risk preference coefficient θ is, the more the decision-maker prefers a larger value. On the contrary, the smaller the coefficient θ is, the more the decision-maker prefers a smaller value. The cases θ=0 and θ=1, respectively, represent two special situations corresponding to the previously mentioned optimistic and pessimistic perspectives. Thus, the model based on risk preference coefficient is an extension of the optimistic and pessimistic models.

With the risk preference coefficient, it is possible to determine the loss function expressed using real numbers, which are shown in Table 7.

aP θλPP θλPN
aB θλBP θλBN
aN θλNP θλNN
Table 7

Loss function based on θ.

Where θλ**=1θλ**_+θλ**¯ represents the loss value based on the risk preference coefficient θ.

Based on the loss function derived from risk preference coefficient, the expected losses of different actions ai can be denoted as


According to Bayesian theory, the decision rules under risk preference coefficient model can be expressed as follows:

  • Pθ If RθaP|xRθaB|x and RθaP|xRθaN|x, decide xPOSC,

  • Bθ If RθaB|xRθaP|x and RθaB|xRθaN|x, decide xBNDC,

  • Nθ If RθaN|xRθaP|x and RθaN|xRθaB|x, decide xNEGC.

Theorem 1 illustrates that λPPλBP<λNP and λNNλBN<λPN, and based on the form of risk preference coefficient shown in Equation (38), there are

so according to PrC|x+Pr¬C|x=1, we can obtain the simplified decision rules as follows:
  • Pθ If PrC|xαθ and PrC|xγθ, decide xPOSC,

  • Bθ If PrC|xαθ and PrC|xβθ, decide xBNDC,

  • Nθ If PrC|xβθ and PrC|xγθ, decide xNEGC.



Example 3.

We continue to use the data in Example 1. Denote the risk preference coefficient as θ=0.7, meaning that the decision-makers intend to relatively pessimistic individuals with the degree of 0.7. So based on Equation (38), we can calculate the losses as λPP=0.48, λBP=0.91, λNP=1.92, λPN=3.22, λBN=2.26, and λNN=0.65. And according to Equations (4446), the thresholds of risk preference coefficient model can be calculated as α=0.691, β=0.615, and γ=0.641.

4.2. Three-Way Group Decision Model Based on Rough Numbers

The models discussed above are all based on the viewpoint that converting rough numbers into real numbers. And in this section we intend to solve rough-number based three-way decisions in a direct way. According to the rough loss function shown in Table 4, we can obtain three-way decision rules by utilizing the arithmetic operations and ranking rules of rough numbers. The specific procedure is shown as follows.

Based on rough loss function in Table 4, the expected losses of different actions ai can be denoted as


According to the arithmetic operations of rough numbers, we can obtain the expected rough losses as


Simplify the decision rules based on Bayesian theory and we can get

  • PR If RaP|xRaB|x and RaP|xRaN|x, decide xPOSC,

  • BR If RaB|xRaP|x and RaB|xRaN|x, decide xBNDC,

  • NR If RaN|xRaP|x and RaN|xRaB|x, decide xNEGC.

Where RaP|x, RaB|x, and RaN|x are all rough numbers. According to the ranking rules of rough numbers shown in Section 2.2, we can do some further research about decision rules. Taking RaP|xRaB|x as example, we can obtain the following equations:


If RaP|xRaB|x is expected, SP<SB or SP=SB and DPDB must be satisfied, meaning that


For the two cases: (1) if SP<SB, the Equation (54) can be obtained; (2) if SP=SB and DPDB, the Equation (55) can be obtained.


It is worth noting that, in Equation (55), whether λPN¯λPN_λBN¯λBN_+λBP¯λBP_λPP¯λPP_ is positive or negative cannot be determined at this moment. So it is impossible to simplify Equation (55) and obtain an expression of PrC|x as Equation (54). Therefore, when comparing the relationship between RaP|x and RaB|x, it is convenient if there is SP<SB or SP>SB, and in this case we can get the result by the threshold αS directly. Similarly, if we can compare other expected losses only by S, the thresholds βS and γS can also be obtained, which are expressed as Equations (57) and (58). While if there is SP=SB, deducing that PrC|x=αS, then we need to compare DP and DB. If there is DPDB, then RaP|xRaB|x; while if there is DP<DB, then RaP|x>RaB|x. For the comparison of other expected losses, the process is consistent. Therefore, according to the discussion above, the acquisition process of decision rules under different cases can be illustrated as follows, and the final results are tabulated in Table 8.

Conditions Subconditions Sub-subconditions Results No.
PrC|x>αSPrC|x>γS xPOSC (a)
PrC|x<αSPrC|x>βS xBNDC (b)
PrC|x<βSPrC|x<γS xNEGC (c)
DPDB αS>γS xPOSC (d)
PrC|x=αS αS<γS xNEGC (e)
DP<DB αS>βS xBNDC (f)
αS<βS xNEGC (g)
DBDN βS>αS xPOSC (h)
PrC|x=βS βS<αS xBNDC (i)
DB<DN βS>γS xPOSC (j)
βS<γS xNEGC (k)
DPDN γS>αS xPOSC (l)
PrC|x=γS γS<αS xBNDC (m)
DP<DN γS>βS xBNDC (n)
γS<βS xNEGC (o)
Table 8

Rough three-way decision rules.

In Table 8, three-way decision rules can be classified to three cases: (1) the normal case, where there is PrC|xαS, PrC|xβS, and PrC|xγS, corresponding to No. (a-c); the extreme case I, where PrC|x=αS or PrC|x=βS or PrC|x=γS with αSβSγS, corresponding to No. (d-o); (3) the extreme case II, where PrC|x=αS=βS=γS, corresponding to No. (p-r). The process of decision rules of each case in detail can be explained as follows:

  1. Rules in normal case

    In this case, PrC|xαS, PrC|xβS, and PrC|xγS are satisfied simultaneously, then there is no extreme situation. Theorem 1 has illustrated that λPPλBP<λNP and λNNλBN<λPN, so threshold values αS, βS, and γS can be used directly to determine the three-way decision rules as follows, which are corresponding to the rule (a)-(c), respectively.

    • PR If PrC|x>αS and PrC|x>γS, decide xPOSC,

    • BR If PrC|x<αS and PrC|x>βS, decide xBNDC,

    • NR If PrC|x<γS and PrC|x<βS, decide xNEGC.



  2. Rules in extreme case I

    In this case there exists PrC|x=αS, PrC|x=βS or PrC|x=γS, and αSβSγS. αSβSγS denotes that PrC|x equals to only one of the threshold. Here it is necessary to explore the relations among DP, DBDP, and DN. Taking the comparison of RaP|x and RaB|x, for example.

    Suppose there are PrC|x=αS and αSβSγS, so we can obtain PrC|xβS and PrC|xγS, then

    1. if there is DPDB, we can obtain RaP|xRaB|x, and then

      for αS>γS we have PrC|x>γS, so there is RaP|x<RaN|x, and xPOSC, (Rule (d))

      for αS<γS we have PrC|x<γS, so there is RaP|x>RaN|x, and xNEGC, (Rule (e))

    2. if there is DP<DB, we can obtain RaP|x>RaB|x, and then

      for αS>βS we have PrC|x>βS, so there is RaB|x<RaN|x, and xBNDC, (Rule (f))

      for αS<βS we have PrC|x<βS, so there is RaB|x>RaN|x, and xNEGC. (Rule (g))

      Similarly, the cases when PrC|x=βS or PrC|x=γS can be discussed in the same process, and obtain the rules (h)-(o).

  3. Rules in extreme case II

    In this case, there is PrC|x=αS=βS=γS. According to classical three-way decisions, thresholds only satisfy three situations: α<γ<β, β<γ<α, and α=γ=β, as discussed by Equations (15) and (16). In decision rules (d)-(o), we have not considered the case α=γ=β, so the rule (p)-(r) state the case of PrC|x=αS=βS=γS separately. When PrC|x=αS=βS=γS, there is SP=SB=SN, and to compare the relationship among expected losses, DP, DB, and DN need to be considered. In this situation, the largest D* corresponds to the smallest value of Ra*|x, so we can make the corresponding decision, and yield rule (p)-(r).

    In conclusion, 18 three-way decision rules derived from rough loss functions can be obtained, as shown in Table 8.

Example 4.

We discuss the problem in Example 1 by using the direct model proposed in this subsection. The expected rough losses can be calculated as


According to Equations (5658), we can obtain the values of three thresholds, which are denoted as αS=0.690, βS=0.620, and γS=0.680. If there is PrC|xαS,βS,γS, then the three-way decision rules can be determined by thresholds directly. For example, suppose PrC|x=0.670, then we can decide that xBNDC according to rule (b). While if there is PrC|x=αS, PrC|x=βS, or PrC|x=γS, then it needs further discussion. For example, suppose PrC|x=αS=0.690, then we need to compare the relationship between DRaP|x and DRaB|x. It is easy to find that RaP|x=1.007,1.469 and RaB|x=1.010,1.466, so there is SRaP|x=SRaB|x, and then DRaP|x=0.462 and DRaB|x=0.455, which illustrate that DRaP|x>DRaB|x. As there is αSγS, we can determine that the conditions satisfy rule (d), so when PrC|x=αS we can make the decision as xPOSC.


5.1. Problem Description

As an important practical issue in human society, influenza emergency management is discussed in this section by using rough number–based three-way group decisions. A city in China is suffering from a suspected influenza outbreak. The health department of the city plans to isolate regions where influenza cases have been diagnosed, and takes no preventive measures in regions where there is no influenza outbreak. Based on the judgment on the influenza epidemic, two states Ω=C,¬C are obtained, which represent that an outbreak has occurred and has not yet occurred in this region, respectively. The set of actions taken by the health department and the local government is expressed as A=aP,aB,aN, which denote isolation, continuous monitoring, and nonisolation, respectively. Here, PrC|x represents the probability of an influenza outbreak in region x. There are six administrative regions in this city, denoted as U=x1,x2,x3,x4,x5,x6,x7,x8. Based on past experience as well as data collection and analysis for each administrative region, the conditional probabilities of an influenza outbreak in the six administrative regions are PrC|x1=0.54, PrC|x2=0.21, PrC|x3=0.32, PrC|x4=0.65, PrC|x5=0.58, PrC|x6=0.26, PrC|x7=0.25, and PrC|x8=0.57, respectively.

It is difficult to accurately calculate losses caused by incorrect classification of influenza epidemic situations. Therefore, five experts in influenza emergency management are selected to form a decision-making group E=e1,e2,e3,e4,e5 to estimate losses caused by different classifications. The evaluating results of each expert are depicted in Table 9.

e1 2 13 20 14 7 2
e2 3 10 21 13 6 1
e3 3 8 20 15 5 2
e4 4 7 19 13 7 3
e5 2 6 28 17 8 2
Table 9

Results of experts' evaluation.

Based on the constructing process of rough numbers shown in Equations (1721), the evaluation values of decision group can be aggregated to rough losses, as shown in Table 10.

aP 2.36,3.25 13.50,15.43
aB 7.21,10.63 5.92,7.25
aN 20.15,23.88 1.65,2.35
Table 10

Rough loss function.

5.2. Three-Way Decisions of Influenza Emergency Management

According to the four models proposed in Section 4, that is to say, the optimistic model, the pessimistic model, the risk preference coefficient model, and the rough model, we can obtain different decision results under various environments.

  1. According to model 1 based on the optimistic perspective, the optimistic loss function can be denoted as Table 11. And by using Equations (2931), we can calculate the values of thresholds as αo=0.610, βo=0.248, and γo=0.400.

  2. According to model 2 based on the pessimistic perspective, the pessimistic loss function can be denoted as Table 12. By using Equations (3537), we can obtain the values of thresholds as αp=0.526, βp=0.270, and γp=0.388.

  3. According to model 3 based on the risk preference coefficient perspective, we can obtain the loss function based on θ. In this case we suppose θ=0.7, so the loss function can be denoted as Table 13. Calculating the values of thresholds based on Equations (4446), we can obtain αθ=0.547, βθ=0.264, and γθ=0.391. To illustrate the sensitivity of thresholds to the value of θ, the curves of α, β, and γ with different values of θ are elaborated in Figure 3. It is clear that when θ=0 the values of thresholds correspond to the optimistic model, and when θ=1 they correspond to the pessimistic model.

  4. According to the direct rough model proposed in Section 4.2, we can calculate the values of thresholds by using Equations (5658) based on the data in Table 10, and obtain that αS=0.563, βS=0.264, and γS=0.398.

    The final results derived from four models can be depicted in Table 14 and Figure 4.

aP 2.36 13.50
aB 7.21 5.92
aN 20.15 1.65
Table 11

Optimistic loss function.

aP 3.25 15.43
aB 10.63 7.25
aN 23.88 2.35
Table 12

Pessimistic loss function.

aP 2.98 14.85
aB 9.60 6.85
aN 22.76 2.14
Table 13

Loss function based on θ=0.7.

Figure 3

The thresholds values with different values of θ.

Figure 4

Three-way decisions based on different models.

The above four results have different effects on the determination of the three-way decision-making rules. All results are summarized in Table 14. Table 14 shows that regardless of which decision-making method is selected, the final classifications of x2, x3, and x4 remain unchanged, and they belong to the negative domain, the boundary domain, and the positive domain, respectively. That is to say, in all decision-making modes, x4 is always a region with an influenza outbreak, whereas x2 is not. For x3, no conclusion can be drawn and further data collection is required. For other regions, namely, x1, x5, x6, x7, and x8, when different decision-making models are selected, the three-way decision-making rules obtained are different. Thus, it is necessary to make a choice according to specific decision-making method and specific situation.

Administrative Regions Conditional Probabilities Decision Results
Model 1 Model 2 Model 3 Model 4
Table 14

Results of decision-making.

5.3. Comparison and Discussion

According to previous research, for three-way decision-making models with multiple decision-makers, methods such as taking the maximum value, the minimum value, and the weighted average can be used to aggregate the group information, and the group loss function can be obtained. Then the final three-way decision rules are obtained by using the three-way decision-making process. Here the results obtained by the weighted average method and the three-way group decision-making model proposed in literature [21] were selected and compared with those obtained using the methods in this study, which are shown in Tables 15 and 16.

Methods Thresholds
α β γ
Weighted average method 0.565 0.265 0.397
Method proposed by Liang [21] 0.571 0.218 0.381
Model 1 0.610 0.248 0.400
Model 2 0.526 0.270 0.388
Model 3 0.547 0.264 0.391
Model 4 0.563 0.259 0.394
Table 15

Comparison of thresholds.

Methods Classification
Weighted average method {x4, x5, x8} {x1, x3} {x2, x6, x7}
Method proposed by Liang [21] {x4, x5 } {x1, x3, x6, x7, x8} {x2}
Model 1 {x4 } {x1, x3, x5, x6, x7, x8} {x2}
Model 2 {x1, x4, x5, x8} {x3} {x2, x6, x7}
Model 3 {x4, x5, x8} {x1, x3} {x2, x6, x7}
Model 4 {x4, x5, x8} {x1, x3, x6} {x2, x7}
Table 16

Comparison of classifications.

Table 15 illustrates the values of thresholds obtained by different methods. The values are various according to the selection of methods, but all the thresholds of each method satisfy β<γ<α, so the alternatives can be classified by α and β. For α, the minimum value and the maximum value are obtained by model 2 and model 1, respectively, and the values obtained by other models are restricted in this interval. For β, the minimum value and the maximum value are obtained by method in [21] and model 2, respectively. According to the results of thresholds, we can obtain the classification of each alternative, which is exhibited in Table 16. The three-way decision rules obtained by different methods are diverse because of the difference of thresholds. For the weighted average method, the decision rule is equal to the result derived from model 3, meaning that for the eight alternatives the results are consistent when choosing the weighted average method or model 3. For the method proposed in [21], the decision rule is different from the others, but similar as the result derived from model 1 expect x5. It is easy to find that the negative regions obtained by method in [21] and model 1 are both {x2}, while by other models the negative regions contain more than one alternative. The reason is that in experts' evaluating results, there is an abnormal value of λNP given by e5, while method in [21] and model 1 could not eliminate the effect of the noise in an effective way. The value of β is relevant to λNP, therefore the two models obtain lower values of β and only x2 satisfies that the conditional probability is less than β.

The specific advantages of models in this paper are summarized as follows:

  1. By using the rough number method to aggregate group information, it is possible to take advantage of the rough number method and to fully consider and explore the intrinsic relations among all information in the same evaluation category, which can effectively eliminate the effect of abnormal evaluating values.

  2. The rough number method can reflect the objectivity of the rough set theory. No subjective prior knowledge is required, and only the intrinsic relations among information should be taken into consideration. Thus, the objectivity of the aggregation of group information is guaranteed, and the decision-making rules obtained are also more reliable.

  3. Three-way group decision-making models based on rough numbers can utilize different decision-making modes. For example, the risk preferences of decision-makers in a decision-making situation can be taken into account, or the rough number calculation method can be employed when risk preferences are not considered. Decision-makers could select different three-way group decision-making models in various environments, which can result in decision-making rules that better address problems in real life and can offer greater flexibility.


Group decision-making involving multiple decision-makers is an important way to solve practical decision-making problems. This paper explored the integration of group decision-making with three-way decision-making models and their application. To handle the problem of group information aggregation in the process of group decision-making, the rough number method is adopted. Since the rough number method is objective and systematic, the estimation of the loss functions by all individuals in the group are aggregated to construct the rough number loss function and thus to describe the evaluation information of the group. In the construction stage of the three-way decision-making model, based on whether or not the decision-maker's risk preference should be taken into account, the optimistic method, the pessimistic method, the risk preference coefficient method, and the rough number calculation method were constructed respectively to examine the threshold calculation and the rule acquisition in three-way decision-making models. Finally, the three-way group decision-making model constructed in this paper was applied to an influenza emergency management problem to make the optimal decision with the lowest loss caused by errors. The three-way group decision-making model based on the rough number method is able to effectively aggregate the estimation of the loss function by the decision-making group to reflect the overall preference of the decision-making group and to choose different decision-making modes according to the actual problems to obtain decision-making rules. In future studies, three-way group decision-making models with group uncertain information, such as interval numbers and fuzzy numbers, will be further explored. Meanwhile, other emergency management problems will be selected for application.


This paper is supported by the National Natural Science Foundation of China (Nos. 71771140 and 71471172), the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045), and the Humanities and Social Sciences Research Project of Ministry of Education of China (No. 19YJC630059).


International Journal of Computational Intelligence Systems
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557 - 570
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© 2019 The Authors. Published by Atlantis Press SARL.
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Cite this article

AU  - Fan Jia
AU  - Peide Liu
PY  - 2019
DA  - 2019/04/26
TI  - Rough Number–Based Three-Way Group Decisions and Application in Influenza Emergency Management
JO  - International Journal of Computational Intelligence Systems
SP  - 557
EP  - 570
VL  - 12
IS  - 2
SN  - 1875-6883
UR  -
DO  - 10.2991/ijcis.d.190420.001
ID  - Jia2019
ER  -