# IF-MABAC Method for Evaluating the Intelligent Transportation System with Intuitionistic Fuzzy Information

Authors: Yanping Li
Journal: Journal of Mathematics
 (2021)

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## Abstract

Intelligent transportation system (ITS) is the development direction of the future traffic system. ITS can effectively employ the existing traffic facilities and ensure the safety of traffic, urban traffic, and public security management for effective control in order to satisfy people’s travel demand. Therefore, the results of the system in-depth understanding and objective evaluation are very necessary. And it is frequently regarded as a multiattribute group decision-making (MAGDM) issue. Thus, a novel MAGDM method is required to tackle it. Depending on the conventional multiattributive border approximation area comparison (MABAC) method and intuitionistic fuzzy sets (IFSs), this article designs a novel intuitive distance-based IF-MABAC method to assess the performance of financial management. First of all, a related literature review is conducted. Furthermore, some necessary theories related to IFSs are briefly reviewed. In addition, since subjective randomness frequently exists in determining criteria weights, the weights of criteria are decided objectively by utilizing the maximizing deviation method. Afterwards, relying on novel distance measures between intuitionistic fuzzy numbers (IFNs), the conventional MABAC method is extended to the IFSs to calculate the final value of each enterprise. Therefore, all enterprises can be ranked, and the one with the best environmental behaviors and awareness can be identified. Eventually, an application for evaluating the intelligent transportation system and some comparative analyses have been given. The results illustrate that the designed algorithm is useful for assessing the performance of financial management.


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## Body

## 1. Introduction

With our rapid economic development, accelerating urbanization, and the rapid rise of motor vehicle ownership, existing roads’ hardware facilities have failed to meet the demand of swelling traffic. Traffic congestion, frequent accidents, and serious environmental pollution have become increasingly serious problems. It is not a good and effective way to solve them by limiting demand, increasing supply, and expanding the scale of the road. The best strategy to ensure the sustainable development of the urban traffic is adopting modern technology to transform the existing transportation system and grasp the real-time traffic conditions. It can be called ITSs (intelligent transportation systems). The key of intelligent transportation systems is to obtain comprehensive, real-time, accurate, and dynamic traffic information.Like most other phenomena in organizational research, the intelligent transportation system cannot be observed directly. Thus, for enterprises, evaluating the intelligent transportation system can be regarded as a significant strategic issue and great challenge. To overcome it, a novel intuitionistic fuzzy MAGDM method on the basis of the improved MABAC method is designed to tackle this issue. Our work’s contributions can be listed as follows:(1)
Although Liang, He, Wang, Chen, and Li [1] extended the MABAC to the intuitionistic fuzzy environment on the basis of novel generalized measures, these measures may generate situations which do not consider wavering in IFSs. Opposite, depending on the distance measures introduced in this paper, our method can reflect intuitionistic fuzzy information more comprehensively. Besides, the calculation process of our method is simpler.(2)
There are various criteria in the intelligent transportation system evaluation which frequently have different weights. Since the DMs are restrained through their limited knowledge, it not easy to assign the criteria weights correctly. In this paper, an objectively weight-determining method is built to calculate the values of weight.The remainder of this paper proceeds as follows. A literature review is given in Section2. The knowledge of IFSs is concisely listed in Section 3. The improved MABAC method with IFSs is defined for MAGDM in Section 4. An empirical application for evaluating the intelligent transportation system is given and some comparative analyses are also offered in Section 5. At last, the conclusion of this work is given in Section 6.


## 2. Literature Review

Since the process of evaluating the intelligent transportation system is filled with uncertainty and ambiguity [2, 3], thus, in order to improve the accuracy of MAGDM, Zadeh [4] built the fuzzy sets (FSs). Atanassov [5] built the intuitionistic fuzzy sets (IFSs). Garg [6] presented the intuitionistic fuzzy multiplicative preference relations and defined several geometric operators. Gou, Xu, and Lei [7] built the exponential operational law of IFNs. Garg [8] defined the intuitionistic fuzzy averaging fused operators with hesitation degrees. He, He, and Huang [9] integrated the power operators with IFSs. Liu, Liu, and Chen [10] built the BM operator and Dombi operations under IFSs. Gupta, Arora, and Tiwari [11] built the fuzzy entropy through IFSs and parameter alpha. Li and Wu [12] presented the intuitionistic fuzzy cross-entropy distance and grey correlation analysis method. Khan and Lohani [13] defined the similarity measure of IFNs through the distance measure of bounded variation. Li, Liu, Liu, Su, and Wu [14] built the grey target decision-making for IFNs. Bao, Xie, Long, and Wei [15] defined the prospect theory and the evidential reasoning method under IFSs. Chen, Cheng, and Lan [16] built the TOPSIS method for MCDM through similarity measures under IFSs. Gan and Luo [17] used a hybrid method with the decision-making trial and evaluation laboratory (DEMATEL) and IFSs. Gupta, Mehlawat, Grover, and Chen [18] defined the superiority and inferiority ranking (SIR) method under IFSs. Hao, Xu, Zhao, and Zhang [19] defined the intuitionistic fuzzy method through the decision field. Krishankumar, Arvinda, Amrutha, Premaladha, and Ravichandran [20] integrated AHP with IFSs to design a GDM method for effective cloud vendor selection. Krishankumar, Ravichandran, and Saeid [21] built the IF-PROMETHEE method. Luo and Wang [22] built the VIKOR method with distance measure for IFSs. Rouyendegh [23] integrated the ELECTRE method under IFSs to tackle some MCDM issues. Cali and Balaman [24] extended ELECTRE I with the VIKOR method in the context of intuitionistic fuzzy to reflect the decision makers’ preferences. Phochanikorn and Tan [25] incorporated DEMATEL with ANP to determine uncertainties and interdependencies among criteria and modified VIKOR to evaluate the sustainable supplier performance’s desired level under the intuitionistic fuzzy context. Liu [26] researched on the teaching quality evaluation of physical education with the intuitionistic fuzzy TOPSIS method.MABAC method was initially developed through Pamucar and Cirovic [27] to solve MAGDM. Compared with other MAGDM models, MABAC method is used to obtain the alternatives’ order by calculating the potential values of gains and losses. This method has been extended to various fuzzy environments. For example, Sahin and Altun [28] integrated MABAC with the probabilistic neutrosophic hesitant fuzzy environment. Wei, He, Lei, Wu, and Wei [29] defined the probabilistic uncertain linguistic MABAC. Wei et al. [30] defined the uncertain probabilistic linguistic MABAC method. Xu, Shi, Zhang, and Liu [31] designed the MABAC with heterogeneous criteria information. Liang, He, Wang, Chen, and Li [1] put forward some novel distance measures of IFSs and combined them with the MABAC method to tackle MCGDM issues. Jia, Liu, and Wang [32] designed two models which were an IF-MABAC and an IFRN-MABAC model, respectively. Liang, Zhao, Wu, and Dai [33] defined the MABAC method related to TFNs.


## 3. Preliminaries

### 3.1. IFSs

Definition 1.
(see [5]). An IFS on the universe X is defined:(1)I=x,μIx,νIx|x∈X,where μIx∈0,1 is called the “membership degree of I” and νIx∈0,1 is called the “nonmembership degree of I,” and μIx, νIx meet the mathematical condition: 0≤μIx+νIx≤1, ∀x∈X.Definition 2.
(see [34]). Let I1=μ1,ν1 and I2=μ2,ν2 be two IFNs; the operation of them is defined:(2)I1⊕I2=μ1+μ2−μ1μ2,ν1ν2,(3)I1⊗I2=μ1μ2,ν1+ν2−ν1ν2,(4)λI1=1−1−μ1λ,ν1λ,λ>0,(5)I1λ=μ1λ,1−1−ν1λ,λ>0.Definition 3.
(see [35]). Let I1=μ1,ν1 and I2=μ2,ν2 be IFNs; the score and accuracy functions of I1 and I2 can be expressed:(6)SI1=μ1+μ11−μ1−ν1,SI2=μ2+μ21−μ2−ν2,(7)HI1=μ1+ν1,HI2=μ2+ν2.
For two IFNsI1 andI2, according to Definition 3,(i)
IfsI1<sI2,thenI1<I2(ii)
IfsI1>sI2,thenI1>I2(iii)
IfsI1=sI2andhI1<hI2,thenI1<I2(iv)
IfsI1=sI2andhI1>hI2,thenI1>I2(v)
IfsI1=sI2andhI1=hI2,thenI1=I2Definition 4.
(see [22]). Let I1=μ1,ν1 and I2=μ2,ν2 be IFNs; the Hamming distance between two IFNs is defined:(8)IFHDI1,I2=16ℓ1+ℓ2+ℓ3,where(9)ℓ1=μ1−μ2+ν1−ν2+μ1+1−ν1−μ2+1−ν22,ℓ2=π1+π22,ℓ3=maxμ1−μ2,ν1−ν2,π1−π22.


### 3.2. Intuitionistic Fuzzy Aggregation Operators

Under the context of the IFSs, some operators are introduced, including intuitionistic fuzzy weighted averaging (IFWA) and intuitionistic fuzzy weighted geometric (IFWG) operator.Definition 5.
(see [34]). Let Ij=μIj,νIjj=1,2,…,n be a set of IFNs; the intuitionistic fuzzy weighted averaging (IFWA) operator is defined:(10)IFWAωI1,I2,…,In=⊕j=1nωjIj,where ω=ω1,ω2,…,ωnT is the weight of Ijj=1,2,…,n and ωj>0,∑j=1nωj=1.
From Definition5, the following theorem can be obtained.Theorem 1.
The fused value by the IFWA operator is also a IFN, where(11)IFWAωI1,I2,…,In=⊕j=1nωjIj=1−∏j=1n1−μIjωj,∏j=1nνIjωj,where ω=ω1,ω2,…,ωnT is the weight of Ijj=1,2,…,n and ωj>0,∑j=1nωj=1.Definition 6.
(see [34]). Let Ijj=1,2,…,n be a set of IFNs; the IFWG operator is defined:(12)IFWGωI1,I2,…,In=⊗j=1nIjωj,where ω=ω1,ω2,…,ωnT is the weight of Ijj=1,2,…,n and ωj>0,∑j=1nωj=1.
From Definition6, the following theorem can be obtained.Theorem 2.
The fused value by the IFWG operator is also an IFN, where(13)IFWGωI1,I2,…,In=⊗j=1nIjωj=∏j=1nμIjωj,1−∏j=1n1−νIjωj,where ω=ω1,ω2,…,ωnT is the weight vector of Ijj=1,2,…,n and ωj>0,∑j=1nωj=1.


## 3.1. IFSs

Definition 1.
(see [5]). An IFS on the universe X is defined:(1)I=x,μIx,νIx|x∈X,where μIx∈0,1 is called the “membership degree of I” and νIx∈0,1 is called the “nonmembership degree of I,” and μIx, νIx meet the mathematical condition: 0≤μIx+νIx≤1, ∀x∈X.Definition 2.
(see [34]). Let I1=μ1,ν1 and I2=μ2,ν2 be two IFNs; the operation of them is defined:(2)I1⊕I2=μ1+μ2−μ1μ2,ν1ν2,(3)I1⊗I2=μ1μ2,ν1+ν2−ν1ν2,(4)λI1=1−1−μ1λ,ν1λ,λ>0,(5)I1λ=μ1λ,1−1−ν1λ,λ>0.Definition 3.
(see [35]). Let I1=μ1,ν1 and I2=μ2,ν2 be IFNs; the score and accuracy functions of I1 and I2 can be expressed:(6)SI1=μ1+μ11−μ1−ν1,SI2=μ2+μ21−μ2−ν2,(7)HI1=μ1+ν1,HI2=μ2+ν2.
For two IFNsI1 andI2, according to Definition 3,(i)
IfsI1<sI2,thenI1<I2(ii)
IfsI1>sI2,thenI1>I2(iii)
IfsI1=sI2andhI1<hI2,thenI1<I2(iv)
IfsI1=sI2andhI1>hI2,thenI1>I2(v)
IfsI1=sI2andhI1=hI2,thenI1=I2Definition 4.
(see [22]). Let I1=μ1,ν1 and I2=μ2,ν2 be IFNs; the Hamming distance between two IFNs is defined:(8)IFHDI1,I2=16ℓ1+ℓ2+ℓ3,where(9)ℓ1=μ1−μ2+ν1−ν2+μ1+1−ν1−μ2+1−ν22,ℓ2=π1+π22,ℓ3=maxμ1−μ2,ν1−ν2,π1−π22.


## 3.2. Intuitionistic Fuzzy Aggregation Operators

Under the context of the IFSs, some operators are introduced, including intuitionistic fuzzy weighted averaging (IFWA) and intuitionistic fuzzy weighted geometric (IFWG) operator.Definition 5.
(see [34]). Let Ij=μIj,νIjj=1,2,…,n be a set of IFNs; the intuitionistic fuzzy weighted averaging (IFWA) operator is defined:(10)IFWAωI1,I2,…,In=⊕j=1nωjIj,where ω=ω1,ω2,…,ωnT is the weight of Ijj=1,2,…,n and ωj>0,∑j=1nωj=1.
From Definition5, the following theorem can be obtained.Theorem 1.
The fused value by the IFWA operator is also a IFN, where(11)IFWAωI1,I2,…,In=⊕j=1nωjIj=1−∏j=1n1−μIjωj,∏j=1nνIjωj,where ω=ω1,ω2,…,ωnT is the weight of Ijj=1,2,…,n and ωj>0,∑j=1nωj=1.Definition 6.
(see [34]). Let Ijj=1,2,…,n be a set of IFNs; the IFWG operator is defined:(12)IFWGωI1,I2,…,In=⊗j=1nIjωj,where ω=ω1,ω2,…,ωnT is the weight of Ijj=1,2,…,n and ωj>0,∑j=1nωj=1.
From Definition6, the following theorem can be obtained.Theorem 2.
The fused value by the IFWG operator is also an IFN, where(13)IFWGωI1,I2,…,In=⊗j=1nIjωj=∏j=1nμIjωj,1−∏j=1n1−νIjωj,where ω=ω1,ω2,…,ωnT is the weight vector of Ijj=1,2,…,n and ωj>0,∑j=1nωj=1.


## 4. MABAC Method for MAGDM with Intuitionistic Fuzzy Information

Integrating the MABAC method with IFSs, the IF-MABAC method is given by IFNs. The calculating procedures of the designed method can be listed subsequently. LetZ=Z1,Z2,…,Zn be a set of attributes and z=z1,z2,…zn be the weight vector of attributes Zj, where rj∈0,1,j=1,2,…,n,∑j=1nrj=1. Assume H=H1,H2,…,Hl is a set of DMs that have a significant degree of h=h1,h2,…,hl, where hk∈0,1, k=1,2,…,l, ∑k=1lhk=1. Let P=P1,P2,…,Pm be a set of alternatives. And Q=qijm×n is the overall decision matrix, and qij means the value of alternative Fi regarding the attribute Rj with IFNs. Subsequently, the corresponding calculating steps will be depicted:Step 1: build the decision maker’s decision matrixQk=qijkm×n and calculate the overall decision matrixQ=qijm×n:(14)Qk=qijkm×n=q11kq12k…q1nkq21kq22k…q2nk⋮⋮⋮⋮qm1kqm2k…qmnk,(15)Q=qijm×n=q11q12…q1nq21q22…q2n⋮⋮⋮⋮qm1qm2…qmn,(16)qij=1−∏k=1l1−μqijkhk,∏k=1lνqijkhk,whereqijk is the assessment value of the alternative Pii=1,2,…,m for attribute Zjj=1,2,…,n and DM Hkk=1,2,…,l.Step 2: normalize the overall intuitionistic fuzzy matrixQ=qijm×n to QN=qijNm×n:(17)qijN=μij,νij,Zjis a benefit criterion,νij,μij,Zjis a cost criterion.Step 3: utilize the maximizing deviation method to determine the weighting matrix of attributes.The maximizing deviation method will be integrated with IFSs in this part to determine each attribute’s weight with completely unknown information. This method was initially put forward by Wang [36] which took the differences among all alternatives’ performance values into consideration. Subsequently, the calculating procedures of this method are presented:(1)
Depending on the normalized overall decision matrixQN=qijNm×n, the deviation of Pi to all the other alternatives could be calculated.(18)IFDij=∑t=1mzj⋅dqijN,qtjN,where(19)dqijN,qtjN=16μij−μtj+νij−νtj+μij+1−νij−μtj+1−νtj2+πij+πtj2+maxμij−μtj,νij−νtj,πij−πtj2.(2)
Calculate the total weighted deviation values of all alternatives:(20)IFDjz=∑i=1mIFDijz=∑i=1m∑t=1mzj16μij−μtj+νij−νtj+μij+1−νij−μtj+1−νtj2+πij+πtj2+maxμij−μtj,νij−νtj,πij−πtj2.(3)
Construct a nonlinear programming model with IFNs.(21)M−1maxDz=∑j=1n∑i=1m∑t=1mzj16μij−μtj+νij−νtj+μij+1−νij−μtj+1−νtj2+πij+πtj2+maxμij−μtj,νij−νtj,πij−πtj2,s.t.zj≥0,j=1,2,…,n,∑j=1nzj2=1.To solve this model, the Lagrange function can be utilized:(22)Lz,ξ=∑j=1n∑i=1m∑t=1mzj16μij−μtj+νij−νtj+μij+1−νij−μtj+1−νtj2+πij+πtj2+maxμij−μtj,νij−νtj,πij−πtj2+ξ2∑j=1nzj2−1,whereξ is the Lagrange multiplier. Then, the partial derivatives of L can be calculated:(23)∂L∂zj=∑i=1m∑t=1m16μij−μtj+νij−νtj+μij+1−νij−μtj+1−νtj2+πij+πtj2+maxμij−μtj,νij−νtj,πij−πtj2+ξzj=0,∂L∂ξ=12∑j=1nzj2−1=0.And then, a simple formula for determining the weight can be obtained by solving the above equations:(24)zj∗=∑i=1m∑t=1m1/6μij−μtj+νij−νtj+|μij+1−νij−μtj+1−νtj|/2πij+πtj/2+maxμij−μtj,νij−νtj,πij−πtj/2∑j=1n∑i=1m∑t=1m1/6μij−μtj+νij−νtj+μij+1−νij−μtj+1−νtj/2πij+πtj/2+maxμij−μtj,νij−νtj,πij−πtj/22.Finally, the normalized weights can be determined:(25)zj=∑i=1m∑t=1m1/6μij−μtj+νij−νtj+μij+1−νij−μtj+1−νtj/2πij+πtj/2+maxμij−μtj,νij−νtj,πij−πtj/2∑j=1n∑i=1m∑t=1m1/6μij−μtj+νij−νtj+μij+1−νij−μtj+1−νtj/2πij+πtj/2+maxμij−μtj,νij−νtj,πij−πtj/2.Step 4: calculate the weighted matrixO=oijm×n by equation (12):(26)oij=zj⋅qijN=1−1−μqijNzj,νqijNzj.Step 5: compute the border approximation area matrixG=gi1×n. The border approximation area (BAA) for every attribute is obtained from the following equation:(27)gj=∏i=1moij1/m=∏i=1mμoij1/m,1−∏i=1m1−νoij1/m.Step 6: calculate the distance matrixD=dijm×n. The alternatives’ distances from the BAA are derived with the following equation:(28)dij=doij,gjϑ,ifSoij≥Sgj,−ρdoij,gjς,ifSoij<Sgj,where the distance measure is defined as equation (8). ϑ and ς are the parameters of DMs’ risk attitudes, and ρ is the loss aversion’s parameter. In this article, ϑ=0.88,ς=0.88, and ρ=2.25. The values come from Tversky and Kahneman [37] who conducted an experiment to determine the most acceptable values from numerous researchers.Now, ifdij=0, the alternative Pi will belong to the border approximation area G. If dij>0, Pi belongs to the upper approximation area G+. And if dij<0, Pi belongs to the lower approximation area G−. G+ is the area involving the positive alternative P+, whereas G− is the area involving the negative alternative P−.Step 7: calculate the final value of criterion functionsFi:(29)Fi=∑j=1ndij,i=1,2,…,m,j=1,2,…,n.Step 8: depending on the calculating results ofFi, all the alternatives could be ranked. The larger the value of Fi is, the optimal the alternative will be.


## 5. Numerical Example and Comparative Analysis

### 5.1. Numerical Example

Intelligent transportation system is the development direction of the future traffic system. It is the advanced information technology, data communication transmission technology, electronic sensor technology, control technology, and computer technology to effectively integrate with the whole ground traffic management system and establish a large-range, all-round function, real-time, accurate, and efficient integrated transportation management system. Not only that, the high-tech project is a process full of unknown by its size, complexity of technology, economic investment, the degree of market demand, and other aspects of influence and restriction. Therefore, the project evaluation plays an important role during the process of investment to project the overall technology evaluation, market evaluation, and economic evaluation; risk forecast has a great impact on the project decision makers for the project development scheme and is also the key to the success of a project. Intelligent transportation system evaluation could be regarded as the MADM or MAGDM issues [38–45]. In this section, an empirical application of evaluating the intelligent transportation system is provided with the IF-MABAC method. There are five potential cities Pii=1,2,3,4,5 preparing to evaluate their intelligent transportation system. In order to assess these cities fairly, five experts H=H1,H2,H3,H4,H5 (expert’s weight h=0.20,0.20,0.20,0.20,0.20 are invited. All experts could give their assessment information through four subsequent attributes: ① Z1 is the intelligent transportation environment; ② Z2 is the intelligent transportation cost; ③ Z3 is the intelligent transportation safety; and ④ Z4 is the intelligent transportation equipment investment. Evidently, Z2 is the cost attribute, while Z1, Z3, and Z4 are the benefit attributes.Step 1: build each DM’s matrixQk=qijkm×n as in Tables 1–5. Derived from the tables and equations (14)–(16), the overall decision matrix could be calculated. The results are recorded in Table 6.Step 2: normalize the matrixQ=qijm×n to QN=qijNm×n (see Table 7).Step 3: decide the attribute weightszjj=1,2,…,n through the maximizing deviation method (see Table 8).Step 4: calculate the weighted matrixO=oijm×n by utilizing equation (26) (Table 9).Step 5: determine the BAA matrixG=gj1×n (Table 10).Step 6: calculate the distance matrixD=dijm×n (see Table 11).Step 7: sum up each row’s elements, and each alternative’s final valueFi can be determined as in Table 12.Step 8: relying onFi, all the alternatives could be ranked; the larger the value of Fi is, the optimal the alternative will be. Evidently, the rank of all alternatives is P2>P1>P4>P3>P5, and P2 is the optimal city.Table 1
Intuitionistic fuzzy matrix byH1.
Z1Z2Z3Z4P1(0.63, 0.15)(0.45, 0.50)(0.57, 0.31)(0.26, 0.63)P2(0.70, 0.30)(0.21, 0.69)(0.72, 0.28)(0.64, 0.22)P3(0.39, 0.51)(0.38, 0.48)(0.50, 0.40)(0.61, 0.30)P4(0.53, 0.37)(0.42, 0.51)(0.35, 0.56)(0.55, 0.34)P5(0.26, 0.69)(0.58, 0.35)(0.55, 0.35)(0.69, 0.13)Table 2
Intuitionistic fuzzy matrix byH2.
Z1Z2Z3Z4P1(0.56, 0.33)(0.21, 0.53)(0.49, 0.35)(0.57, 0.43)P2(0.56, 0.33)(0.28, 0.63)(0.75, 0.25)(0.67, 0.25)P3(0.52, 0.37)(0.16, 0.68)(0.49, 0.51)(0.58, 0.35)P4(0.71, 0.18)(0.35, 0.57)(0.45, 0.47)(0.56, 0.34)P5(0.59, 0.39)(0.26, 0.65)(0.46, 0.52)(0.71, 0.11)Table 3
Intuitionistic fuzzy matrix byH3.
Z1Z2Z3Z4P1(0.19, 0.65)(0.30, 0.60)(0.54, 0.37)(0.54, 0.35)P2(0.80, 0.20)(0.24, 0.58)(0.75, 0.15)(0.77, 0.23)P3(0.58, 0.39)(0.19, 0.66)(0.44, 0.51)(0.49, 0.39)P4(0.48, 0.47)(0.23, 0.53)(0.63, 0.30)(0.67, 0.20)P5(0.54, 0.35)(0.26, 0.55)(0.41, 0.57)(0.69, 0.15)Table 4
Intuitionistic fuzzy matrix byH4.
Z1Z2Z3Z4P1(0.56, 0.25)(0.32, 0.58)(0.59, 0.35)(0.58, 0.25)P2(0.66, 0.20)(0.36, 0.64)(0.55, 0.25)(0.52, 0.33)P3(0.53, 0.31)(0.43, 0.51)(0.34, 0.41)(0.41, 0.35)P4(0.43, 0.37)(0.29, 0.63)(0.55, 0.30)(0.49, 0.51)P5(0.59, 0.29)(0.39, 0.55)(0.27, 0.67)(0.63, 0.19)Table 5
Intuitionistic fuzzy matrix byH5.
Z1Z2Z3Z4P1(0.39, 0.55)(0.26, 0.68)(0.47, 0.38)(0.58, 0.27)P2(0.72,0.15)(0.32, 0.64)(0.64, 0.25)(0.70, 0.30)P3(0.48, 0.51)(0.23, 0.58)(0.54, 0.41)(0.44, 0.55)P4(0.58, 0.33)(0.36, 0.53)(0.60, 0.30)(0.25, 0.61)P5(0.44, 0.55)(0.29, 0.65)(0.51, 0.39)(0.39, 0.59)Table 6
Overall intuitionistic fuzzy matrix.
Z1Z2Z3Z4P1(0.4874, 0.3382)(0.3130, 0.5747)(0.5342, 0.3512)(0.5187, 0.3641)P2(0.7184, 0.2087)(0.2840, 0.6350)(0.6906, 0.2309)(0.6696, 0.2628)P3(0.5039, 0.4103)(0.2863, 0.5766)(0.4662, 0.4452)(0.5123, 0.3796)P4(0.5575, 0.3284)(0.3331, 0.5524)(0.5265, 0.3718)(0.5219, 0.3727)P5(0.4975, 0.4319)(0.3695, 0.5372)(0.4479, 0.4860)(0.6371, 0.1889)Table 7
The normalized intuitionistic fuzzy matrix.
Z1Z2Z3Z4P1(0.4874, 0.3382)(0.5747, 0.3130)(0.5342, 0.3512)(0.5187, 0.3641)P2(0.7184, 0.2087)(0.6350, 0.2840)(0.6906, 0.2309)(0.6696, 0.2628)P3(0.5039, 0.4103)(0.5766, 0.2863)(0.4662, 0.4452)(0.5123, 0.3796)P4(0.5575, 0.3284)(0.5524, 0.3331)(0.5265, 0.3718)(0.5219, 0.3727)P5(0.4975, 0.4319)(0.5372, 0.3695)(0.4479, 0.4860)(0.6371, 0.1889)Table 8
The attribute weightsrj.
Z1Z2Z3Z4zj0.27930.16990.28450.2663Table 9
Intuitionistic fuzzy weighted normalized performance values of alternatives.
Z1Z2Z3Z4P1(0.1703, 0.7387)(0.1352, 0.8209)(0.1954, 0.7425)(0.1769, 0.7641)P2(0.2981,0.6456)(0.1574, 0.8074)(0.2838, 0.6590)(0.2554, 0.7005)P3(0.1778, 0.7797)(0.1359, 0.8085)(0.1635, 0.7944)(0.1740, 0.7727)P4(0.2036, 0.7327)(0.1277, 0.8296)(0.1916, 0.7547)(0.1784, 0.7689)P5(0.1749, 0.7910)(0.1227, 0.8444)(0.1555, 0.8144)(0.2366, 0.6416)Table 10
BAA.
BAAZ1(0.2002, 0.7422)Z2(0.1353, 0.8227)Z3(0.1933, 0.7585)Z4(0.2015, 0.7341)Table 11
Distance matrix.
Z1Z2Z3Z4P1−0.09480.01440.0289−0.1133P20.1209−0.08010.1186−0.1532P3−0.1165−0.0572−0.1189−0.1273P4−0.0591−0.0498−0.0404−0.1174P5−0.1338−0.0759−0.1529−0.2261Table 12
The final value.
AlternativeFinal valueP1−0.1647P20.0063P3−0.4200P4−0.2667P5−0.5888


### 5.2. Comparative Analysis

First of all, the designed method is compared with IFWA and IFWG operators [34]. For the IFWA operator, the calculating result is SP1=0.5936,SP2=0.7358,SP3=0.5620,SP4=0.5971, and SP5=0.5961. Thus, the ranking order isP2>P4>P5>P1>P3. For the IFWG operator, the calculating result is SP1=0.5922,SP2=0.7336,SP3=0.5573,SP4=0.5963, and SP5=0.5724. So, the ranking order is P2>P4>P1>P5>P3.Furthermore, the designed method is compared with the modified IF-VIKOR method [46]. Then, we can obtain the calculating result. Then, each alternatives’ relative closeness is calculated as DRC1=0.8683, DRC2=0.0000, DRC3=1.0000, DRC4=0.8878, and DRC5=0.9366. Hence, the order is P2>P1>P4>P5>P3.Besides, the designed method is compared with the IF-GRA method [47]. Then, we can obtain the calculating result. The grey relational grades of every alternative are γ1=0.8065, γ2=0.9800, γ3=0.7847, γ4=0.8274, and γ5=0.8342. Therefore, the order is P2>P5>P4>P1>P3.In the end, the designed method is also compared with the IF-MABAC method [1]. Then, we can obtain the calculating result. The overall value of every alternative is I1=2.9135, I2=3.3834, I3=1.3719, I4=2.8685, and I5=1.0845. Therefore, the order is P2>P1>P4>P3>P5.Eventually, the results of these methods are depicted in Table13.Table 13
Evaluation results of these methods.
MethodsRanking orderThe optimal alternativeThe worst alternativeIFWA operator [34]P2>P4>P5>P1>P3P2P3IFWG operator [34]P2>P4>P1>P5>P3P2P3IF-VIKOR method [46]P2>P1>P4>P5>P3P2P3IF-GRA method [47]P2>P5>P4>P1>P3P2P3IF-MABAC method [1]P2>P1>P4>P3>P5P2P5The designed methodP2>P1>P4>P3>P5P2P5From Table13, it is evident that the optimal enterprise is P2, while the worst is P3 in most cases. In other words, these methods’ order is slightly different. These methods can effectively solve MAGDM from different angles.


## 5.1. Numerical Example

Intelligent transportation system is the development direction of the future traffic system. It is the advanced information technology, data communication transmission technology, electronic sensor technology, control technology, and computer technology to effectively integrate with the whole ground traffic management system and establish a large-range, all-round function, real-time, accurate, and efficient integrated transportation management system. Not only that, the high-tech project is a process full of unknown by its size, complexity of technology, economic investment, the degree of market demand, and other aspects of influence and restriction. Therefore, the project evaluation plays an important role during the process of investment to project the overall technology evaluation, market evaluation, and economic evaluation; risk forecast has a great impact on the project decision makers for the project development scheme and is also the key to the success of a project. Intelligent transportation system evaluation could be regarded as the MADM or MAGDM issues [38–45]. In this section, an empirical application of evaluating the intelligent transportation system is provided with the IF-MABAC method. There are five potential cities Pii=1,2,3,4,5 preparing to evaluate their intelligent transportation system. In order to assess these cities fairly, five experts H=H1,H2,H3,H4,H5 (expert’s weight h=0.20,0.20,0.20,0.20,0.20 are invited. All experts could give their assessment information through four subsequent attributes: ① Z1 is the intelligent transportation environment; ② Z2 is the intelligent transportation cost; ③ Z3 is the intelligent transportation safety; and ④ Z4 is the intelligent transportation equipment investment. Evidently, Z2 is the cost attribute, while Z1, Z3, and Z4 are the benefit attributes.Step 1: build each DM’s matrixQk=qijkm×n as in Tables 1–5. Derived from the tables and equations (14)–(16), the overall decision matrix could be calculated. The results are recorded in Table 6.Step 2: normalize the matrixQ=qijm×n to QN=qijNm×n (see Table 7).Step 3: decide the attribute weightszjj=1,2,…,n through the maximizing deviation method (see Table 8).Step 4: calculate the weighted matrixO=oijm×n by utilizing equation (26) (Table 9).Step 5: determine the BAA matrixG=gj1×n (Table 10).Step 6: calculate the distance matrixD=dijm×n (see Table 11).Step 7: sum up each row’s elements, and each alternative’s final valueFi can be determined as in Table 12.Step 8: relying onFi, all the alternatives could be ranked; the larger the value of Fi is, the optimal the alternative will be. Evidently, the rank of all alternatives is P2>P1>P4>P3>P5, and P2 is the optimal city.Table 1
Intuitionistic fuzzy matrix byH1.
Z1Z2Z3Z4P1(0.63, 0.15)(0.45, 0.50)(0.57, 0.31)(0.26, 0.63)P2(0.70, 0.30)(0.21, 0.69)(0.72, 0.28)(0.64, 0.22)P3(0.39, 0.51)(0.38, 0.48)(0.50, 0.40)(0.61, 0.30)P4(0.53, 0.37)(0.42, 0.51)(0.35, 0.56)(0.55, 0.34)P5(0.26, 0.69)(0.58, 0.35)(0.55, 0.35)(0.69, 0.13)Table 2
Intuitionistic fuzzy matrix byH2.
Z1Z2Z3Z4P1(0.56, 0.33)(0.21, 0.53)(0.49, 0.35)(0.57, 0.43)P2(0.56, 0.33)(0.28, 0.63)(0.75, 0.25)(0.67, 0.25)P3(0.52, 0.37)(0.16, 0.68)(0.49, 0.51)(0.58, 0.35)P4(0.71, 0.18)(0.35, 0.57)(0.45, 0.47)(0.56, 0.34)P5(0.59, 0.39)(0.26, 0.65)(0.46, 0.52)(0.71, 0.11)Table 3
Intuitionistic fuzzy matrix byH3.
Z1Z2Z3Z4P1(0.19, 0.65)(0.30, 0.60)(0.54, 0.37)(0.54, 0.35)P2(0.80, 0.20)(0.24, 0.58)(0.75, 0.15)(0.77, 0.23)P3(0.58, 0.39)(0.19, 0.66)(0.44, 0.51)(0.49, 0.39)P4(0.48, 0.47)(0.23, 0.53)(0.63, 0.30)(0.67, 0.20)P5(0.54, 0.35)(0.26, 0.55)(0.41, 0.57)(0.69, 0.15)Table 4
Intuitionistic fuzzy matrix byH4.
Z1Z2Z3Z4P1(0.56, 0.25)(0.32, 0.58)(0.59, 0.35)(0.58, 0.25)P2(0.66, 0.20)(0.36, 0.64)(0.55, 0.25)(0.52, 0.33)P3(0.53, 0.31)(0.43, 0.51)(0.34, 0.41)(0.41, 0.35)P4(0.43, 0.37)(0.29, 0.63)(0.55, 0.30)(0.49, 0.51)P5(0.59, 0.29)(0.39, 0.55)(0.27, 0.67)(0.63, 0.19)Table 5
Intuitionistic fuzzy matrix byH5.
Z1Z2Z3Z4P1(0.39, 0.55)(0.26, 0.68)(0.47, 0.38)(0.58, 0.27)P2(0.72,0.15)(0.32, 0.64)(0.64, 0.25)(0.70, 0.30)P3(0.48, 0.51)(0.23, 0.58)(0.54, 0.41)(0.44, 0.55)P4(0.58, 0.33)(0.36, 0.53)(0.60, 0.30)(0.25, 0.61)P5(0.44, 0.55)(0.29, 0.65)(0.51, 0.39)(0.39, 0.59)Table 6
Overall intuitionistic fuzzy matrix.
Z1Z2Z3Z4P1(0.4874, 0.3382)(0.3130, 0.5747)(0.5342, 0.3512)(0.5187, 0.3641)P2(0.7184, 0.2087)(0.2840, 0.6350)(0.6906, 0.2309)(0.6696, 0.2628)P3(0.5039, 0.4103)(0.2863, 0.5766)(0.4662, 0.4452)(0.5123, 0.3796)P4(0.5575, 0.3284)(0.3331, 0.5524)(0.5265, 0.3718)(0.5219, 0.3727)P5(0.4975, 0.4319)(0.3695, 0.5372)(0.4479, 0.4860)(0.6371, 0.1889)Table 7
The normalized intuitionistic fuzzy matrix.
Z1Z2Z3Z4P1(0.4874, 0.3382)(0.5747, 0.3130)(0.5342, 0.3512)(0.5187, 0.3641)P2(0.7184, 0.2087)(0.6350, 0.2840)(0.6906, 0.2309)(0.6696, 0.2628)P3(0.5039, 0.4103)(0.5766, 0.2863)(0.4662, 0.4452)(0.5123, 0.3796)P4(0.5575, 0.3284)(0.5524, 0.3331)(0.5265, 0.3718)(0.5219, 0.3727)P5(0.4975, 0.4319)(0.5372, 0.3695)(0.4479, 0.4860)(0.6371, 0.1889)Table 8
The attribute weightsrj.
Z1Z2Z3Z4zj0.27930.16990.28450.2663Table 9
Intuitionistic fuzzy weighted normalized performance values of alternatives.
Z1Z2Z3Z4P1(0.1703, 0.7387)(0.1352, 0.8209)(0.1954, 0.7425)(0.1769, 0.7641)P2(0.2981,0.6456)(0.1574, 0.8074)(0.2838, 0.6590)(0.2554, 0.7005)P3(0.1778, 0.7797)(0.1359, 0.8085)(0.1635, 0.7944)(0.1740, 0.7727)P4(0.2036, 0.7327)(0.1277, 0.8296)(0.1916, 0.7547)(0.1784, 0.7689)P5(0.1749, 0.7910)(0.1227, 0.8444)(0.1555, 0.8144)(0.2366, 0.6416)Table 10
BAA.
BAAZ1(0.2002, 0.7422)Z2(0.1353, 0.8227)Z3(0.1933, 0.7585)Z4(0.2015, 0.7341)Table 11
Distance matrix.
Z1Z2Z3Z4P1−0.09480.01440.0289−0.1133P20.1209−0.08010.1186−0.1532P3−0.1165−0.0572−0.1189−0.1273P4−0.0591−0.0498−0.0404−0.1174P5−0.1338−0.0759−0.1529−0.2261Table 12
The final value.
AlternativeFinal valueP1−0.1647P20.0063P3−0.4200P4−0.2667P5−0.5888


## 5.2. Comparative Analysis

First of all, the designed method is compared with IFWA and IFWG operators [34]. For the IFWA operator, the calculating result is SP1=0.5936,SP2=0.7358,SP3=0.5620,SP4=0.5971, and SP5=0.5961. Thus, the ranking order isP2>P4>P5>P1>P3. For the IFWG operator, the calculating result is SP1=0.5922,SP2=0.7336,SP3=0.5573,SP4=0.5963, and SP5=0.5724. So, the ranking order is P2>P4>P1>P5>P3.Furthermore, the designed method is compared with the modified IF-VIKOR method [46]. Then, we can obtain the calculating result. Then, each alternatives’ relative closeness is calculated as DRC1=0.8683, DRC2=0.0000, DRC3=1.0000, DRC4=0.8878, and DRC5=0.9366. Hence, the order is P2>P1>P4>P5>P3.Besides, the designed method is compared with the IF-GRA method [47]. Then, we can obtain the calculating result. The grey relational grades of every alternative are γ1=0.8065, γ2=0.9800, γ3=0.7847, γ4=0.8274, and γ5=0.8342. Therefore, the order is P2>P5>P4>P1>P3.In the end, the designed method is also compared with the IF-MABAC method [1]. Then, we can obtain the calculating result. The overall value of every alternative is I1=2.9135, I2=3.3834, I3=1.3719, I4=2.8685, and I5=1.0845. Therefore, the order is P2>P1>P4>P3>P5.Eventually, the results of these methods are depicted in Table13.Table 13
Evaluation results of these methods.
MethodsRanking orderThe optimal alternativeThe worst alternativeIFWA operator [34]P2>P4>P5>P1>P3P2P3IFWG operator [34]P2>P4>P1>P5>P3P2P3IF-VIKOR method [46]P2>P1>P4>P5>P3P2P3IF-GRA method [47]P2>P5>P4>P1>P3P2P3IF-MABAC method [1]P2>P1>P4>P3>P5P2P5The designed methodP2>P1>P4>P3>P5P2P5From Table13, it is evident that the optimal enterprise is P2, while the worst is P3 in most cases. In other words, these methods’ order is slightly different. These methods can effectively solve MAGDM from different angles.


## 6. Conclusion

ITS is the trend of future traffic development. The problem of traffic jam exists in all the big cities around the world. Intelligent transportation project has made the world attach great importance in the development of the intelligent transportation system, which domestic and foreign scholars in succession of the intelligent transportation management project and related research work on performance appraisal. The performance appraisal of our national public program currently has not formed a set of appraising systems of standard and systemization and has problems of insufficient technology system, appraising subjective color, and public participation intensity. With respect to the intelligent transportation project, carrying on the project expenditure performance appraisal of the intellectual traffic has the vital significance. This paper designs an effective method for this issue since it designs a novel intuitive distance-based IF-MABAC method for evaluating the intelligent transportation system. And then, a numerical example for evaluating the intelligent transportation system has been given to confirm that this novel method is reasonable. Furthermore, to show the validity and feasibility of the developed method, some comparative analyses are also conducted. However, the main drawback of this paper is that the number of DMs and attributes is small, and interdependency of criteria is not taken into consideration, which may limit the application scope of the developed method to some extent. Furthermore, the developed method can be utilized to tackle many other MAGDM issues such as risk evaluation, project selection, and site selection [48–59].



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*Source: 5536751-2021-03-27.xml*