# A Novel Regret Theory-Based Decision-Making Method Combined with the Intuitionistic Fuzzy Canberra Distance

Authors: Haiping Ren; Yunxiao Gao; Tonghua Yang
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In practical decision-making, the behavior factors of decision makers often affect the final decision-making results. Regret theory is an important behavioral decision theory. Based on the regret theory, a novel decision-making method is proposed for the multiattribute decision-making problem with incomplete attribute weight information, and the attribute values are expressed by Atanassov intuitionistic fuzzy numbers. At first, a new distance of intuitionistic fuzzy sets is put forward based on the traditional Canberra distance. Then, we utilize it for the definition of the regret value (rejoice) for the attribute value of each alternative with the corresponding values of the positive point (negative point). The objective of this method is to maximize the comprehensive perceived utility of the alternative set by the decision maker. The optimal attribute weight vector is solved, and the optimal comprehensive perceived utility value of each alternative is obtained. Finally, according to the optimal comprehensive perceived utility value, the rank order of all alternatives is concluded.


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## Body

## 1. Introduction

Since Professor Zadeh introduced the concept of fuzzy set in 1965, it has been successfully applied in many fields such as intelligent control, military engineering, economic prediction, and decision-making [1–5]. Zadeh’s fuzzy sets have been proved to be an effective tool to deal with fuzzy and imprecise problems [6, 7]. However, in the process of solving some decision-making problems, due to the limitation of the time, energy, or incomplete knowledge of decision makers, decision makers often hesitate, which makes the evaluation results show three aspects: affirmation, negation, and hesitation. Traditional fuzzy sets cannot describe such problems very well, so Professor Atanassov [8] extended fuzzy sets in 1986 and introduced the concept of intuitionistic fuzzy set (IFS). By introducing the parameter of nonmembership degree, IFS can express the information of affirmation and negation at the same time; furthermore, it can describe the fuzzy concept of “not this or that,” and then it can describe the hesitation and uncertainty of the decision maker’s judgment [9, 10]. Because of this, it can depict the fuzzy essence of the objective world more delicately than Zadeh’s fuzzy set in the processing mode. In recent years, the research on the IFS theory has attracted great attention of scholars and has been applied to the fields of economic decision-making [11, 12], medical diagnosis [13, 14], image processing [15, 16], pattern recognition [17, 18], fault tree analysis [19, 20], and so on. Some other extensions of Zadeh’s fuzzy set such as picture fuzzy set [6] and rough set [21] all have received great attention and have many successful applications in practice.Most of the existing intuitionistic fuzzy MADM methods are based on the expected utility theory, which assumes that decision makers are completely rational. However, decision makers often have subjective preferences, such as psychological and behavioral factors when making decision. So, it is important to consider the subjective preferences of decision makers in the decision process. As an important behavioral decision-making theory, regret theory was firstly proposed by Bell [22] and Loomes and Sugden [23]. In recent years, the research and application of the regret theory have attracted many scholars’ attention [24–27]. Regret theory holds that decision makers are concerned about the possible results if they choose other schemes while considering the results of schemes. If they find that they can get better results by choosing other schemes, they will regret them psychologically. Otherwise, they will be happy. Therefore, when making a decision, the decision maker will estimate the regret or rejoice that the decision may produce in advance and try to avoid choosing the plan of expected regret; that is to say, the decision maker is regret-averse. In this way, the perceived utility value of the decision maker includes two parts: the utility value of the current result and the perceived utility value after comparing with other possible results, “regret-rejoice” value. Chorus [28] and Qu et al. [29] pointed out that the regret theory has some advantages over the cumulative prospect theory in application. For example, in decision-making, reference points need not be given, and few parameters in the calculation formula were involved in decision-making, which makes the calculation simpler [30]. In the decision-making model, regret theory replaces the expected utility theory, and it is in line with the objective reality of human beings [31].Many information measures are proposed for fuzzy sets, such as entropy, similarity, and distance [32]. In the classical regret theory, the deviation involving two numbers can be directly measured in accordance with the absolute value of the difference between two numbers. To measure the difference between two IFSs, we are required to define the distance between two IFSs. Although many intuitionistic fuzzy distance measures have been constructed, some existing distance measures have counter-intuitionistic special cases, so it is very important to develop new improved intuitionistic fuzzy distance measures. Canberra distance, as a classical distance measure, has been widely applied in image processing, pattern recognition, and other fields based on the exact number. Taking into account this distance measure, this paper develops a novel intuitionistic fuzzy distance based on the Canberra distance and further applies it to develop a new intuitionistic fuzzy decision-making method combined with the regret theory.The structure of this paper is as follows:Section 2 first introduces the concept of IFS and then provides some preliminaries of the regret theory. Section 3 puts forward a new intuitionistic fuzzy distance measure based on the traditional Canberra distance. Section 4 develops the intuitionistic fuzzy multiattribute decision-making (MADM) method based on the regret theory combined with the proposed intuitionistic fuzzy Canberra distance. Section 5 provides an example, which explains the new method through the example analysis. Finally, Section 6 is the conclusion of this paper.


## 2. Preliminaries

Some basic concepts and properties of IFSs and regret theory are reviewed in this section.Definition 1 (see [8]).
LetX=x1,x2,…,xn be a universal set. A set U is called an IFS in X if U=<xi,μUxi,νUxi>xi∈X. Here, μUxi and νUxi are the membership degree and nonmembership degree of xi, respectively. They satisfy μUxi, νUxi, μUxi+νUxi∈0,1 for ∀xi∈X. Let πUxi=1−μUxi−νUxi; then, πUxi is called the hesitancy degree of xi. Sometimes, IFS U=<xi,μUxi,νUxi>xi∈X is also expressed as U=<xi,μUxi,νUxi,πUxi>xi∈X.Remark 1.
Grzegorzewski [33] introduced the concept of intuitionistic fuzzy number (IFN) as an extension of the IFS in the continuous case, and intuitionistic fuzzy numbers have many applications in the engineering field [34]. In order to avoid confusion and for convenience, if there is only one element in X, we call U an Atanassov intuitionistic fuzzy number (AIFN). Each AIFN has a physical interpretation, for example, if A=<0.6, 0.2, 0.2>, then μA=0.6, νA=0.2, and πA=0.2, which can be interpreted as “the vote for resolution is 6 in favor, 2 against, and 2 abstention.”
In actual decision-making process, most of the decision makers are not so rational, so the decision maker’s behavior factors need to be considered when making a decision. The prospect theory and regret theory are put forward in this context. In the increasingly complex, modern, political, and economic environment, decision makers need to consider not only the results obtained after choosing a certain scheme but also the possible decision results after assuming that other alternatives are chosen. In the regret theory, the perceived utility function is composed of two parts: the utility function of current decision-making results and the regret-rejoice function compared with other decision-making results. Leta and b, respectively, represent the results that can be obtained by selecting scheme A and scheme B. Then, the perceived utility of decision makers for scheme A is(1)ua,b=υa+Rυa−υb.
Among them,υθ represents the utility value of scheme θ and Rυa−υbis called regret-rejoice value. If Rυa−υb is positive, then it is called a rejoice value, which indicates the extent to which the decision maker is glad to choose the scheme or give up the scheme. If Rυa−υb is negative, then it is called a regret value, which indicates the extent to which the decision maker regrets to choose the scheme or give up the scheme. Obviously, the regret gratification function R⋅ should be monotonically increasing and concave, i.e., it satisfies R′⋅>0, R″⋅<0, and R0=0. Loomes and Sugden [23] pointed out that regret-rejoice function R⋅can be expressed as follows:(2)RΔυ=1−exp−δΔυ.
Here,δ>0 is the regret avoidance coefficient of the decision maker, and the greater δ related to, the larger the regret avoidance degree of the decision maker. Δυ is the difference between the utility value of any two schemes. Figure 1 shows the image of the regret-rejoice function with different values.
LetAii=1,2,…,m be i-th alternatives, and ai is the result of alternative Ai. According to the regret theory, in decision analysis, when the positive ideal point is taken as the reference point, the decision-making evaluation value will not be greater than the positive ideal point, and at this time, the decision maker will regret. When the negative ideal point is taken as the reference point, the decision-making evaluation value will not be less than the negative ideal point, and at this time, the decision maker is happy. Note that xij is the attribute evaluation value of scheme Ai under the evaluation attribute oj given by the decision maker; then, according to Loomes and Sugden [23], the regret value of each attribute evaluation value xij of scheme Ai is related to the corresponding attribute valuexj+ of the positive ideal point, and the joy value of each attribute evaluation value xij of scheme Ai is related to the corresponding attribute value xj− of the negative ideal point. The gratification values can be expressed as(3)Rij1=1−expδxij−xj+,Rij2=1−exp−δxij−xj−,where δ>0 is the regret avoidance coefficient of decision makers.
A large number of psychological studies have shown that regret, as a negative emotion, has a stronger effect on utility than rejoice. Therefore, the decision maker’s comprehensive regret-rejoice value for the evaluation value of schemeAi under the evaluation attribute oj is(4)Rij=Rij1+Rij2=2−expδxij−xj+−exp−δxij−xj−.
According to Loomes and Sugden [23], power function υx=xα,0<α<1, is used as a utility function of the attribute value in this paper. The greater the degree of risk aversion of decision makers is, the smaller α is. α is called the risk aversion coefficient of decision makers. It can be proved that if Δυ0>0, there is R−δΔυ0>RΔυ0. This shows that compared with Δυ0, decision maker’s psychological perception is more sensitive to −Δυ0, that is, decision makers are regret-averse.Figure 1
Regret-rejoice function,RΔυ.


## 3. A New Distance Based on the Canberra Distance

In this section, we will propose a new distance measure between two IFSs based on the Canberra distance. The Canberra distance of two real vectorsx=x1,x2,…,xn and y=y1,y2,…,ynis defined as follows (Perlibakas [35]):(5)dx,y=∑i=1nxi−yixi+yi.As an important information measure, Canberra distance has been successfully applied in image processing, medicine, and other fields [36–38]. Due to the fact that the denominator is zero, the numerical value is meaningless. Then, we propose a revised version of the Canberra distance measure as follows:(6)∑i=1ndx,y=∑i=1nxi−yi2+xi+yi.Note that constant 2 can be changed as any other positive numbers.LetX=x1,x2,…,xn be a universal set. Then, for two given IFSs, A=<xi,μAxi,vAxi,πAxi>xi∈Xand B=<xi,μBxi,vBxi,πBxi>xi∈X, and the new intuitionistic fuzzy information measure based on Canberra distance dA,B is constructed as follows:(7)dA,B=1n∑i=1nμAxi−μBxi2+μAxi−μBxi+vAxi+vBxi2+vAxi+vBxi+πAxi+πBxi2+πAxi+πBxi.Next, we will provedA,B is a valid distance measure. In this section, let R∗ be a set of nonnegative real numbers.Lemma 1.
Leta,b,c∈R∗ and a≤b≤c. Then,(8)ia−c2+a+c≥a−b2+a+b,iia−c2+a+c≥b−c2+b+c,iiia−c2+a+c≤a−b2+a+b+b−c2+b+c.Lemma 2.
Leta,b,c∈R∗ and d1a,b=a−b/2+a+b; then,(9)d1a,b≤d1b,c+d1a,c.Proof.
For the case0≤a≤b≤c, according to Lemma 1, we have(i)
d1a,c≥d1a,b; then, d1a,b≤d1a,c+d1b,c(ii)
d1a,c≥d1b,c; then, d1b,c≤d1a,c+d1a,b(iii)
d1a,c≤d1a,b+d1b,c
That is,d1⋅,⋅ satisfies trigonometric inequality. And we can easily prove that d1⋅,⋅satisfies trigonometric inequality in other cases using a similar reasoning process. Then, the conclusion is proved.Theorem 1.
LetA=<xi,μAxi,vAxi>xi∈X and B=<xi,μBxi,vBxi>xi∈X be two IFSs in X=x1,x2,…,xn. Then, dA,B defined in (7) is a valid distance measure between A and B. That is, dA,B satisfies the following properties:(i)
dA,B≥0(ii)
dA,B=dB,A(iii)
dA,B≤dB,C+dA,C, for any IFSs A,B, and CProof.
(i)
Obviously,dA,B≥0.(ii)
dB,A=1/n∑i=1nμBxi−μAxi/2+μBxi+μAxi+νBxi−νAxi/2+νBxi+νAxi+πBxi−πAxi/2+πBxi+πAxi=1/n∑i=1nμAxi−μBxi/2+μAxi+μBxi+νAxi−νBxi/2+νAxi+νBxi+πAxi−πBxi/2+πAxi+πBxi=dA,B.(iii)
IfA,B,andC are three IFSs, A=<xi,μAxi,vAxi>xi∈X, B=<xi,μBxi,vBxi>xi∈X, and C=<xi,μCxi,νCxi>xi∈X, then by (i) of Lemma 2, we have(10)d1μAxi+μBxi≤d1μBxi+μCxi+d1μAxi+μCxi,d1vAxi+vBxi≤d1vBxi+vCxi+d1vAxi+vCxi,d1πAxi+πBxi≤d1πBxi+πCxi+d1πAxi+πCxi,while(11)dA,B=1n∑i=1nd1μAxi+μBxi+d1vAxi+vBxi+d1πAxi+πBxi.
Consequently,dA,B≤dB,C+dA,C
Then, we complete the proof of Theorem1.
If we consider the important degree ofxii=1,2,…,n and let wii=1,2,…,n be the important degree of xi,i=1,2,…,n, which satisfies wi∈0,1 and ∑i=1nwi=1, then we can get a weighted distance dWA,B between A and B as follows:(12)dWA,B=∑i=1nwiμAxi−μBxi2+μAxi−μBxi+vAxi+vBxi2+vAxi+vBxi+πAxi+πBxi2+πAxi+πBxi.Remark 2.
Ifwi=1/n,i=1,2,…,n, then dWA,B=dA,B. Obviously, dWA,B is also a valid distance, and the proof process is similar to dA,B in Theorem 1.


## 4. A New Regret Theory-Based Decision-Making Method

In this section, we will put forward a new intuitionistic fuzzy MADM method based on the regret theory combined with the above proposed distance. The detail decision process is shown in Figure2.Figure 2
The calculation process of the intuitionistic fuzzy MADM model.For an intuitionistic fuzzy MADM problem, for the convenience of description, the following symbols represent the set or quantity in the decision:X=x1,x2,…,xn: the set of m alternatives m≥2.O=o1,o2,…,on: the set of n attributes n≥2, where oj represents the jth attribute.w=w1,w2,…,wnT: the vector of attributes’ weights. This is because in the decision-making process, different attributes usually have different importance. Here, wj is the weight information of attribute ojj=1,2,…,n, satisfying wj≥0j=1,2,…,n and ∑j=1nwj=1. When the attribute weight information is partially known, the set of mathematical expressions that record the known partial weight information is denoted by H.xij=<μij,vij>: the evaluation value of alternative xi given by the decision maker under attribute oj. The numbers μijandvij show the degree of satisfaction and dissatisfaction of the decision maker with the value of the alternative xi under the index oj, respectively. They satisfy 0≤μij≤1,0≤vij≤1,and0≤μij+vij≤1.Now, we can get an intuitionistic fuzzy decision-making matrixX˜=xijm×n. It is required to determine the order of alternatives and choose the optimal alternative.Now, we propose a new decision-making method based on the regret theory. The decision maker’s comprehensive regret-rejoice value of the evaluation valuexij=<μij,vij> of scheme xi under the evaluation attribute oj is(13)Rij=Rij1+Rij2=2−expδdxij,xj+−exp−δdxij,xj−.In this paper, functionυijxij=Sxijα,0<α<1, is used as a utility function of the attribute value, and Sx is the score function of the AIFN. Then, the decision maker’s perception utility function of the corresponding attribute value xij of scheme xi can be expressed as(14)Fij=υij+Rij=2+Sxijα−expδdxij,xj+−exp−δdxij,xj−.Next, we discuss the method to determine the attribute weight of intuitionistic fuzzy MADM. LetH be the set of known weight information. For each scheme xi, its comprehensive perceived utility function is(15)Fxi=∑j=1nwjFij=∑j=1nwj2+Sxijα−expδdxij,xj+−exp−δdxij,xj−.The weight should be determined so that the greater the comprehensive perceived utility, the better the schemexi is. Therefore, the following optimization model can be established, and its objective function is(16)maxF=Fx1,Fx2,…,Fxm.According to the fact that “the greater the comprehensive perceived utility, the better the scheme” and the fair competition among the schemes aimed at the maximization of the comprehensive perceived utility of the decision maker to the scheme set, the optimization model for solving the attribute weight is established as follows:(17)maxV=∑i=1mFxi=∑i=1m∑j=1nwj2+Sxijα−expδdxij,xj+−exp−δdxij,xj−,s.t.w∈H,∑j=1nwj=1,wj≥0,j=1,2,…,n.The optimal weight vectorw∗=w1∗,w2∗,…,wm∗T can be obtained by solving the above model with MATLAB or LINGO software.Thus, it can be seen that the optimal comprehensive perceived utility value of the decision maker to schemexi is(18)Hi=∑j=1nwj∗2+Sxijα−expδdxij,xj+−exp−δdxij,xj−.Finally, according to the comparison of the optimal comprehensive perceived utility value, the ranking results of all schemes can be obtained. The larger the valueHi is, the better the corresponding alternative xi is.Next, the calculation steps of the MADM method based on the regret theory are given as follows:Step 1: calculate the scoreSij=μij−vij of attribute value xij=<uij,vij>, and get the score matrix S=Sijm×n.Step 2: determine the positive and negative ideal point.The positive ideal point is defined asx+=x1+,x2+,…,xn+, where xj+=<1,0>,j=1,2,…,nThe negative ideal point is defined asx−=x1−,x2−,…,xn−, where xj−=<1,0>,j=1,2,…,nStep 3: calculate the distancesdxij,xj+ and dxij,xj−, where dA,B represents the intuitionistic fuzzy Canberra distance between A=<μA,vA> and B=<μB,vB>. Then,(19)dxij,xj+=1−μij3+μij+νij2+νij+1−μij−νij3−μij−νij,dxij,xj−=μij2+μij+1−νij3+νij+1−μij−νij3−μij−νij.Step 4: according to equation (14), calculate the perceived utility function value Fij of attribute value xij corresponding to each alternative.Step 5: establish optimization model (17), and calculate the optimal weight vector w∗ with the help of MATLAB software.Step 6: the optimal weight obtained from Step 5 is substituted into equation (18), and the comprehensive perceived utility value of each alternative is obtained. The merits and demerits of the scheme are determined according to the comprehensive perceived utility value Hi. The higher the value of Hi, the better the corresponding alternative xi.


## 5. Numerical Example

The effectiveness and practicability of this method are illustrated by an example of assembly parts’ supplier selection in Xu [39]. With the economic globalization and the continuous expansion of enterprise scale, the problem of supplier selection has become an important management decision-making problem that all large enterprises need to seriously consider. Let a manufacturing company prepare to find the best supplier in the world for purchasing the most critical parts in the assembly process.After the primary selection, there are five alternative suppliersxii=1,2,3,4,5. Now, the company will evaluate the suppliers according to the following five evaluation indicators (attributes): product price o1, product quality o2, service performance o3, supplier’s situation o4, and risk factors o5. After experts’ discussion, the evaluation values of each attribute of the candidate suppliers are finally obtained. Suppose the manufacturing company invites N experts to make the judgment. They are expected to answer “Yes” or “No” or “I do not know” to the question whether alternative xi satisfies attribute oj. Let QYi,j and QNi,j denote the sum of “Yes” and “No,” respectively. Then, the degrees to which alternative Ai satisfies and does not satisfy attribute oj can be calculated as(20)μij=QYi,jN,vij=QNi,jN.Then, the evaluation values are expressed by AIFNs, as shown in Table1.Table 1
Attribute evaluation value of each alternative suppliero1.
SuppliersEvaluation attributeo1o2o3o4o5x1<0.449, 0.370><0.565, 0.162><0.705, 0.232><0.730, 0.170><0.646, 0.354>x2<0.719, 0.188><0.630, 0.232><0.448, 0.378><0.557, 0.160><0.597, 0.192>x3<0.546, 0.192><0.727, 0.182><0.641, 0.322><0.399, 0.200><0.658, 0.192>x4<0.520, 0.337><0.630, 0.100><0.539, 0.271><0.679, 0.188><0.708, 0.198>x5<0.727, 0.128><0.520, 0.299><0.619, 0.318><0.618, 0.229><0.609, 0.120>It is assumed that the attribute weight information is partially known, and the attribute weight satisfies(22)H=w=w1,w2,…,w5Tw1≤0.3,0.1≤w2≤0.2,0.2≤w3≤0.5,0.1≤w4≤0.3w3−w1≤0.1,w4≥w1,w5≤0.4,w3−w2≥w5−w4.Next, we use the proposed decision-making method to sort the five suppliers and choose the best desirable supplier.These suppliers are sorted according toHii=1,2,3,4,5 from large to small. The result is x4≻x5≻x1≻x3≻x2, and supplier x4 is the best choice.Table 2
Score matrix of attribute values of each alternative under each attribute.
SuppliersEvaluation attributeo1o2o3o4o5x10.07900.40300.47300.56000.2920x20.53100.39800.07000.39700.4050x30.35400.54500.31900.19900.4660x40.18300.53000.26800.49100.5100x50.59900.22100.30100.39000.4890Step 1: calculate the scoreSij=μij−vij of attribute xij=<μij,vij>. Then, we get the score matrix S=Sij5×5, and the calculation result is shown in Table 2.Step 2: determine the positive and negative ideal point.The positive ideal point is defined as(23)x+=x1+,x2+,…,x5+,wherexj+=<1,0>,j=1,2,…,5.The negative ideal point is defined as(24)x−=x1−,x2−,…,x5−,wherexj−=<0,1>,j=1,2,…,5.Step 3: calculate the Canberra distancesdxij,xj+ and dxij,xj−, and the results are shown in Tables 3 and 4.Step 4: the perceived utility functionFij of attribute xij corresponding to each alternative is calculated. In this paper, α=0.88andδ=0.3 are used for calculation, and the calculation results are shown in Table 5.Step 5: according to equation (17), the following linear programming model is established:(25)maxV=1.8739w1+2.2942w2+1.5653w3+2.2187w4+2.3540w5,s.t.w1≤0.3,0.1≤w2≤0.2,0.2≤w3≤0.5,0.1≤w4≤0.3,w3−w1≤0,w4≥w1,w5≤0.4,w3−w2≥w5−w4,w1+w2+w3+w4+w5=1,w1,w2,…,w5≥0.Solving model (25), we get the optimal attribute weight vector w∗=0.1,0.1,0.2,0.25,0.35T.Step 6: calculate the comprehensive perceived utility value of each alternative, and get(26)H1=0.4190,H2=0.3698,H3=0.3979,H4=0.4728,H5=0.4490.Table 3
Distance set between attribute values of each scheme and corresponding values of PIS.
x1+x2+x3+x4+x5+x10.39890.31710.21410.19830.2475x20.20590.27040.39910.32260.2951x30.33140.20020.25540.43470.2509x40.34730.26850.33640.23550.2137x50.20100.34940.27300.27860.2843Table 4
Distance set between attribute values of each scheme and corresponding values of NIS.
x1+x2+x3+x4+x5+x10.45330.60540.52880.57680.4367x20.56360.54170.44720.60760.5784x30.58340.56720.46500.58330.5705x40.47180.64880.52190.57050.5571x50.61300.50180.47240.54580.6348Table 5
Perceived utility function values of attribute values corresponding to each alternative.
SuppliersEvaluation attributeo1o2o3o4o5x10.07390.39230.55910.67030.2657x20.61860.48260.00600.39750.4545x30.33410.62540.40410.18040.5028x40.11670.58380.31100.60020.5441x50.73070.21010.28500.37020.5868According to the result of Xu [39], the ranking order is x5≻x4≻x1≻x2≻x3, which is different from the result of this paper. This is because the optimization model established by the score function in Xu [39] does not consider the influence of the degree of hesitation on the ranking of IFSs.Most of the existing intuitionistic fuzzy MADM methods are based on the expected utility theory assuming that decision makers are completely rational. However, in the actual MADM process, decision makers often have subjective risk preferences, such as psychological and behavioral factors for alternatives. So, it is important to consider the risk attitude of decision makers in the decision process. In this article, a new regret theory-based decision-making method is proposed for the MADM problem in which attribute values are expressed by AIFNs.The advantage and limitation of the proposed decision-making method can be summarized as follows:(1)
The comprehensive perceived utility value constructed in this paper not only considers the score function but also considers the decision maker’s regret gratification value; therefore, it is more in line with the objective reality(2)
How to determine the most suitable values of parameters in the regret theory is its limitationThe novelty of this paper is the proposition of a new decision-making method based on the regret theory, which can better reflect the psychological and behavioral factors of the decision maker than many existing decision-making methods. This paper also develops a new weighting method based on the intuitionistic fuzzy Canberra distance.


## 6. Conclusion

For the MADM problem in which attribute values are expressed by AIFNs, this paper develops a new decision-making method based on the regret theory combined with an extension of the Canberra distance measure. The main contributions of this article are as follows:(1)
Regret theory can describe humans’ psychological behavior under uncertain conditions more truthfully, and it can explain the phenomena that expected utility theory cannot. Our decision-making method considers the psychological factors of decision makers based on the regret theory, which can be more in line with the reality.(2)
This article first constructs a new distance of IFSs based on the traditional Canberra distance. Then, a new weighting method is put forward by establishing an optimization model, which is a model of the maximum optimal comprehensive perceived utility value under given weighting information. The new method enriches and develops the weight attribute determination method.(3)
A numerical example of supplier selection is utilized to show the effectiveness and feasibility of the proposed method. The proposed MADM method based on the regret theory has advantages of simple calculation process and easy software implementation.In future, we will apply the proposed MADM method to solve other decision-making problems, such as the risk evaluation, system optimization, and material selection. Furthermore, we will develop the new intuitionistic fuzzy distance for the application of clustering analysis and image processing.



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