IF-MABACMethodforEvaluatingtheIntelligentTransportation ...under intuitionistic fuzzy environment for solving cloud vendorselectionproblem,”inProceedingsofthe2017Inter-national Conference

Research ArticleIF-MABAC Method for Evaluating the Intelligent TransportationSystem with Intuitionistic Fuzzy Information

Yanping Li

Information Engineering School ZhengZhou ShengDa University ZhengZhou HeNan 451191 China

Correspondence should be addressed to Yanping Li 101735shengdaeducn

Received 13 January 2021 Revised 10 March 2021 Accepted 20 March 2021 Published 27 March 2021

Academic Editor Kifayat Ullah

Copyright copy 2021 Yanping Li is is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Intelligent transportation system (ITS) is the development direction of the future traffic system ITS can effectively employ theexisting traffic facilities and ensure the safety of traffic urban traffic and public security management for effective control in order tosatisfy peoplersquos travel demand erefore the results of the system in-depth understanding and objective evaluation are verynecessary And it is frequently regarded as a multiattribute group decision-making (MAGDM) issueus a novelMAGDMmethodis required to tackle it Depending on the conventional multiattributive border approximation area comparison (MABAC) methodand intuitionistic fuzzy sets (IFSs) this article designs a novel intuitive distance-based IF-MABACmethod to assess the performanceof financial management First of all a related literature review is conducted Furthermore some necessary theories related to IFSs arebriefly reviewed In addition since subjective randomness frequently exists in determining criteria weights the weights of criteria aredecided objectively by utilizing the maximizing deviation method Afterwards relying on novel distance measures betweenintuitionistic fuzzy numbers (IFNs) the conventional MABAC method is extended to the IFSs to calculate the final value of eachenterprise erefore all enterprises can be ranked and the one with the best environmental behaviors and awareness can beidentified Eventually an application for evaluating the intelligent transportation system and some comparative analyses have beengiven e results illustrate that the designed algorithm is useful for assessing the performance of financial management

1 Introduction

With our rapid economic development accelerating urban-ization and the rapid rise of motor vehicle ownershipexisting roadsrsquo hardware facilities have failed to meet thedemand of swelling traffic Traffic congestion frequent ac-cidents and serious environmental pollution have becomeincreasingly serious problems It is not a good and effectiveway to solve them by limiting demand increasing supply andexpanding the scale of the roade best strategy to ensure thesustainable development of the urban traffic is adoptingmodern technology to transform the existing transportationsystem and grasp the real-time traffic conditions It can becalled ITSs (intelligent transportation systems) e key ofintelligent transportation systems is to obtain comprehensivereal-time accurate and dynamic traffic information

Like most other phenomena in organizational researchthe intelligent transportation system cannot be observed

directly us for enterprises evaluating the intelligenttransportation system can be regarded as a significantstrategic issue and great challenge To overcome it a novelintuitionistic fuzzy MAGDM method on the basis of theimproved MABAC method is designed to tackle this issueOur workrsquos contributions can be listed as follows

(1) Although Liang He Wang Chen and Li [1] ex-tended the MABAC to the intuitionistic fuzzy en-vironment on the basis of novel generalizedmeasures these measures may generate situationswhich do not consider wavering in IFSs Oppositedepending on the distance measures introduced inthis paper our method can reflect intuitionistic fuzzyinformation more comprehensively Besides thecalculation process of our method is simpler

(2) ere are various criteria in the intelligent trans-portation system evaluation which frequently have

HindawiJournal of MathematicsVolume 2021 Article ID 5536751 10 pageshttpsdoiorg10115520215536751

different weights Since the DMs are restrainedthrough their limited knowledge it not easy to assignthe criteria weights correctly In this paper an ob-jectively weight-determining method is built tocalculate the values of weight

e remainder of this paper proceeds as follows Aliterature review is given in Section 2 e knowledge of IFSsis concisely listed in Section 3 e improved MABACmethod with IFSs is defined for MAGDM in Section 4 Anempirical application for evaluating the intelligent trans-portation system is given and some comparative analyses arealso offered in Section 5 At last the conclusion of this workis given in Section 6

2 Literature Review

Since the process of evaluating the intelligent transportationsystem is filled with uncertainty and ambiguity [2 3] thus inorder to improve the accuracy of MAGDM Zadeh [4] builtthe fuzzy sets (FSs) Atanassov [5] built the intuitionisticfuzzy sets (IFSs) Garg [6] presented the intuitionistic fuzzymultiplicative preference relations and defined severalgeometric operators Gou Xu and Lei [7] built the expo-nential operational law of IFNs Garg [8] defined theintuitionistic fuzzy averaging fused operators with hesitationdegrees He He and Huang [9] integrated the power op-erators with IFSs Liu Liu and Chen [10] built the BMoperator and Dombi operations under IFSs Gupta Aroraand Tiwari [11] built the fuzzy entropy through IFSs andparameter alpha Li and Wu [12] presented the intuitionisticfuzzy cross-entropy distance and grey correlation analysismethod Khan and Lohani [13] defined the similaritymeasure of IFNs through the distance measure of boundedvariation Li Liu Liu Su and Wu [14] built the grey targetdecision-making for IFNs Bao Xie Long and Wei [15]defined the prospect theory and the evidential reasoningmethod under IFSs Chen Cheng and Lan [16] built theTOPSIS method for MCDM through similarity measuresunder IFSs Gan and Luo [17] used a hybrid method with thedecision-making trial and evaluation laboratory (DEMA-TEL) and IFSs Gupta Mehlawat Grover and Chen [18]defined the superiority and inferiority ranking (SIR) methodunder IFSs Hao Xu Zhao and Zhang [19] defined theintuitionistic fuzzy method through the decision fieldKrishankumar Arvinda Amrutha Premaladha and Rav-ichandran [20] integrated AHP with IFSs to design a GDMmethod for effective cloud vendor selection KrishankumarRavichandran and Saeid [21] built the IF-PROMETHEEmethod Luo and Wang [22] built the VIKOR method withdistance measure for IFSs Rouyendegh [23] integrated theELECTRE method under IFSs to tackle some MCDM issuesCali and Balaman [24] extended ELECTRE I with theVIKOR method in the context of intuitionistic fuzzy toreflect the decision makersrsquo preferences Phochanikorn andTan [25] incorporated DEMATEL with ANP to determineuncertainties and interdependencies among criteria andmodified VIKOR to evaluate the sustainable supplier per-formancersquos desired level under the intuitionistic fuzzy

context Liu [26] researched on the teaching quality eval-uation of physical education with the intuitionistic fuzzyTOPSIS method

MABAC method was initially developed throughPamucar and Cirovic [27] to solve MAGDM Comparedwith other MAGDM models MABAC method is used toobtain the alternativesrsquo order by calculating the potentialvalues of gains and losses is method has been extendedto various fuzzy environments For example Sahin andAltun [28] integrated MABAC with the probabilisticneutrosophic hesitant fuzzy environment Wei He LeiWu and Wei [29] defined the probabilistic uncertainlinguistic MABAC Wei et al [30] defined the uncertainprobabilistic linguistic MABAC method Xu Shi Zhangand Liu [31] designed the MABAC with heterogeneouscriteria information Liang He Wang Chen and Li [1] putforward some novel distance measures of IFSs and com-bined them with the MABAC method to tackle MCGDMissues Jia Liu and Wang [32] designed two models whichwere an IF-MABAC and an IFRN-MABAC model re-spectively Liang Zhao Wu and Dai [33] defined theMABAC method related to TFNs

3 Preliminaries

31 IFSs

Definition 1 (see [5]) An IFS on the universe X is defined

I langx μI(x) ]I(x)rang|x isin X1113864 1113865 (1)

where μI(x) isin [0 1] is called the ldquomembership degree of Irdquoand ]I(x) isin [0 1] is called the ldquononmembership degree ofIrdquo and μI(x) ]I(x) meet the mathematical condition0le μI(x) + ]I(x)le 1 forallx isin X

Definition 2 (see [34]) Let I1 (μ1 ]1) and I2 (μ2 ]2) betwo IFNs the operation of them is defined

I1oplusI2 μ1 + μ2 minus μ1μ2 ]1]2( 1113857 (2)

I1 otimes I2 μ1μ2 ]1 + ]2 minus ]1]2( 1113857 (3)

λI1 1 minus 1 minus μ1( 1113857λ ]λ11113872 1113873 λgt 0 (4)

Iλ1 μλ1 1 minus 1 minus ]1( 1113857

λ1113872 1113873 λgt 0 (5)

Definition 3 (see [35]) Let I1 (μ1 ]1) and I2 (μ2 ]2) beIFNs the score and accuracy functions of I1 and I2 can beexpressed

S I1( 1113857 μ1 + μ1 1 minus μ1 minus ]1( 1113857

S I2( 1113857 μ2 + μ2 1 minus μ2 minus ]2( 1113857(6)

H I1( 1113857 μ1 + ]1 H I2( 1113857 μ2 + ]2 (7)

For two IFNs I1 andI2 according to Definition 3

(i) If s(I1)lt s(I2) then I1 lt I2

2 Journal of Mathematics

(ii) If s(I1)gt s(I2) then I1 gt I2

(iii) If s(I1) s(I2) and h(I1)lt h(I2) then I1 lt I2

(iv) If s(I1) s(I2) and h(I1)gt h(I2) then I1 gt I2

(v) If s(I1) s(I2) and h(I1) h(I2) then I1 I2

Definition 4 (see [22]) Let I1 (μ1 ]1) and I2 (μ2 ]2) beIFNs the Hamming distance between two IFNs is defined

IFHD I1 I2( 1113857 16

ℓ1 + ℓ2 + ℓ3( 1113857 (8)

where

ℓ1 μ1 minus μ2

11138681113868111386811138681113868111386811138681113868 + ]1 minus ]2

11138681113868111386811138681113868111386811138681113868 + μ1 + 1 minus ]1( 1113857 minus μ2 + 1 minus ]2( 1113857

11138681113868111386811138681113868111386811138681113868

2

ℓ2 π1 + π2

2

ℓ3 max μ1 minus μ21113868111386811138681113868

1113868111386811138681113868 ]1 minus ]21113868111386811138681113868

1113868111386811138681113868π1 minus π2

11138681113868111386811138681113868111386811138681113868

21113888 1113889

(9)

32 Intuitionistic Fuzzy Aggregation Operators Under thecontext of the IFSs some operators are introduced in-cluding intuitionistic fuzzy weighted averaging (IFWA) andintuitionistic fuzzy weighted geometric (IFWG) operator

Definition 5 (see [34]) Let Ij (μIj ]Ij

)(j 1 2 n) bea set of IFNs the intuitionistic fuzzy weighted averaging(IFWA) operator is defined

IFWAω I1 I2 In( 1113857 oplusn

j1ωjIj1113872 1113873 (10)

where ω (ω1ω2 ωn)T is the weight of Ij(j 1 2

n) and ωj gt 0 1113936nj1 ωj 1

From Definition 5 the following theorem can beobtained

Theorem 1 e fused value by the IFWA operator is also aIFN where

IFWAω I1 I2 In( 1113857 oplusn

j1ωjIj1113872 1113873

1 minus 1113945

n

j11 minus μIj

1113874 1113875ωj

1113945

n

j1]Ij

1113874 1113875ωj

⎛⎝ ⎞⎠

(11)

where ω (ω1ω2 ωn)T is the weight of Ij(j 1 2

n) and ωj gt 0 1113936nj1 ωj 1

Definition 6 (see [34]) Let Ij(j 1 2 n) be a set ofIFNs the IFWG operator is defined

IFWGω I1 I2 In( 1113857 otimesn

j1Ij1113872 1113873

ωj (12)

where ω (ω1ω2 ωn)T is the weight of Ij(j 1 2

n) and ωj gt 0 1113936nj1 ωj 1

From Definition 6 the following theorem can beobtained

Theorem 2 e fused value by the IFWG operator is also anIFN where

IFWGω I1 I2 In( 1113857 otimesn

j1Ij1113872 1113873

ωj

1113945n

j1μIj

1113874 1113875ωj

1 minus 1113945n

j11 minus ]Ij

1113874 1113875ωj

⎛⎝ ⎞⎠

(13)

where ω (ω1ω2 ωn)T is the weight vector of Ij(j

1 2 n) and ωj gt 0 1113936nj1 ωj 1

4 MABAC Method for MAGDM withIntuitionistic Fuzzy Information

Integrating the MABAC method with IFSs the IF-MABACmethod is given by IFNs e calculating procedures of thedesigned method can be listed subsequently Let Z Z11113864

Z2 Zn be a set of attributes and z z1 z2 zn1113864 1113865 bethe weight vector of attributes Zj where rj isin [0 1] j

1 2 n 1113936nj1 rj 1 AssumeH H1 H2 Hl1113864 1113865 is a set

of DMs that have a significant degree of h h1 h2 hl1113864 1113865where hk isin [0 1] k 1 2 l 1113936

lk1 hk 1 Let P P11113864

P2 Pm be a set of alternatives And Q (qij)mtimesn is theoverall decision matrix and qij means the value of alter-native Fi regarding the attribute Rj with IFNs Subsequentlythe corresponding calculating steps will be depicted

Step 1 build the decision makerrsquos decision matrixQ(k) (qk

ij)mtimesn and calculate the overall decisionmatrixQ (qij)mtimesn

Q(k)

qkij1113960 1113961

mtimesn

qk11 q

k12 q

k1n

qk21 q

k22 q

k2n

⋮ ⋮ ⋮ ⋮

qkm1 q

km2 q

kmn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(14)

Q qij1113960 1113961mtimesn

q11 q12 q1n

q21 q22 q2n

⋮ ⋮ ⋮ ⋮qm1 qm2 qmn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (15)

qij 1 minus 1113945l

k11 minus μqk

ij1113874 1113875

hk

1113945l

k1]qk

ij1113874 1113875

hk⎛⎝ ⎞⎠ (16)

where qkij is the assessment value of the alternative

Pi(i 1 2 m) for attribute Zj(j 1 2 n) andDM Hk(k 1 2 l)Step 2 normalize the overall intuitionistic fuzzy matrixQ (qij)mtimesn to QN [qN

ij ]mtimesn

Journal of Mathematics 3

qNij

μij ]ij1113872 1113873 Zj is a benefit criterion

]ij μij1113872 1113873 Zjis a cost criterion

⎧⎪⎨

⎪⎩(17)

Step 3 utilize the maximizing deviation method todetermine the weighting matrix of attributes

e maximizing deviation method will be integratedwith IFSs in this part to determine each attributersquos weightwith completely unknown information is method wasinitially put forward by Wang [36] which took the differ-ences among all alternativesrsquo performance values into

consideration Subsequently the calculating procedures ofthis method are presented

(1) Depending on the normalized overall decision ma-trix QN (qN

ij )mtimesn the deviation of Pi to all the otheralternatives could be calculated

IFDij 1113944m

t1zj middot d q

Nij q

Ntj1113872 1113873 (18)

where

d qNij q

Ntj1113872 1113873

16

μij minus μtj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ]ij minus ]tj

11138681113868111386811138681113868

11138681113868111386811138681113868 + μij + 1 minus ]ij1113872 1113873 minus μtj + 1 minus ]tj1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868

2+πij + πtj

2+ max μij minus μtj

11138681113868111386811138681113868

11138681113868111386811138681113868 ]ij minus ]tj

11138681113868111386811138681113868

11138681113868111386811138681113868πij minus πtj

11138681113868111386811138681113868

11138681113868111386811138681113868

2⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(19)

(2) Calculate the total weighted deviation values of allalternatives

IFDj(z) 1113944m

i1IFDij(z) 1113944

m

i11113944

m

t1zj

16

μij minus μtj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ]ij minus ]tj

11138681113868111386811138681113868

11138681113868111386811138681113868 + μij + 1 minus ]ij1113872 1113873 minus μtj + 1 minus ]tj1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868

2+πij + πtj

2⎛⎝⎛⎝

+ max μij minus μtj

11138681113868111386811138681113868

11138681113868111386811138681113868 ]ij minus ]tj

11138681113868111386811138681113868

11138681113868111386811138681113868πij minus πtj

11138681113868111386811138681113868

11138681113868111386811138681113868

2⎛⎝ ⎞⎠⎞⎠⎞⎠

(20)

(3) Construct a nonlinear programming model withIFNs

(M minus 1)

max D(z) 1113944n

j11113944

m

i11113944

m

t1zj

16

μij minus μtj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ]ij minus ]tj

11138681113868111386811138681113868

11138681113868111386811138681113868 + μij + 1 minus ]ij1113872 1113873 minus μtj + 1 minus ]tj1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868

2+πij + πtj

2⎛⎝⎛⎝

+max μij minus μtj

11138681113868111386811138681113868

11138681113868111386811138681113868 ]ij minus ]tj

11138681113868111386811138681113868

11138681113868111386811138681113868πij minus πtj

11138681113868111386811138681113868

11138681113868111386811138681113868

2⎛⎝ ⎞⎠⎞⎠⎞⎠

st zj ge 0 j 1 2 n 1113944n

j1z2j 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)

To solve this model the Lagrange function can beutilized

L(z ξ) 1113944n

j11113944

m

i11113944

m

t1zj

16

μij minus μtj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ]ij minus ]tj

11138681113868111386811138681113868

11138681113868111386811138681113868 + μij + 1 minus ]ij1113872 1113873 minus μtj + 1 minus ]tj1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868

2+πij + πtj

2⎛⎝⎛⎝

+ max μij minus μtj

11138681113868111386811138681113868

11138681113868111386811138681113868 ]ij minus ]tj

11138681113868111386811138681113868

11138681113868111386811138681113868πij minus πtj

11138681113868111386811138681113868

11138681113868111386811138681113868

2⎛⎝ ⎞⎠⎞⎠⎞⎠ +

ξ2

1113944

n

j1z2j minus 1⎛⎝ ⎞⎠

(22)

4 Journal of Mathematics

where ξ is the Lagrange multiplier en the partialderivatives of L can be calculated

zL

zzj

1113944m

i11113944

m

t1

16

μij minus μtj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ]ij minus ]tj

11138681113868111386811138681113868

11138681113868111386811138681113868 + μij + 1 minus ]ij1113872 1113873 minus μtj + 1 minus ]tj1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868

2+πij + πtj

2⎛⎝⎛⎝

+max μij minus μtj

11138681113868111386811138681113868

11138681113868111386811138681113868 ]ij minus ]tj

11138681113868111386811138681113868

11138681113868111386811138681113868πij minus πtj

11138681113868111386811138681113868

11138681113868111386811138681113868

2⎛⎝ ⎞⎠⎞⎠⎞⎠ + ξzj 0

zL

zξ12

1113944

n

j1z2j minus 1⎛⎝ ⎞⎠ 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)

And then a simple formula for determining theweight can be obtained by solving the aboveequations

zlowastj

1113936mi1 1113936

mt1 16 μij minus μtj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ]ij minus ]tj

11138681113868111386811138681113868

11138681113868111386811138681113868 + | μij + 1 minus ]ij1113872 1113873 minus μtj + 1 minus ]tj1113872 1113873|21113874 1113875πij + πtj2 + max μij minus μtj

11138681113868111386811138681113868

11138681113868111386811138681113868 ]ij minus ]tj

11138681113868111386811138681113868

11138681113868111386811138681113868 πij minus πtj

11138681113868111386811138681113868

1113868111386811138681113868111386821113874 1113875

1113936nj1 1113936

mi1 1113936

mt1 16 μij minus μtj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ]ij minus ]tj

11138681113868111386811138681113868

11138681113868111386811138681113868 + μij + 1 minus ]ij1113872 1113873 minus μtj + 1 minus ]tj1113872 111387311138681113868111386811138681113868

111386811138681113868111386811138682πij + πtj2 + max μij minus μtj

11138681113868111386811138681113868

11138681113868111386811138681113868 ]ij minus ]tj

11138681113868111386811138681113868

11138681113868111386811138681113868 πij minus πtj

11138681113868111386811138681113868

1113868111386811138681113868111386821113874 11138751113874 11138751113874 11138751113874 11138752

1113970

(24)

Finally the normalized weights can be determined

zj 1113936

mi1 1113936

mt1 16 μij minus μtj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ]ij minus ]tj

11138681113868111386811138681113868

11138681113868111386811138681113868 + μij + 1 minus ]ij1113872 1113873 minus μtj + 1 minus ]tj1113872 111387311138681113868111386811138681113868

111386811138681113868111386811138682πij + πtj2 + max μij minus μtj

11138681113868111386811138681113868

11138681113868111386811138681113868 ]ij minus ]tj

11138681113868111386811138681113868

11138681113868111386811138681113868 πij minus πtj

11138681113868111386811138681113868

1113868111386811138681113868111386821113874 11138751113874 11138751113874 1113875

1113936nj1 1113936

mi1 1113936

mt1 16 μij minus μtj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ]ij minus ]tj

11138681113868111386811138681113868

11138681113868111386811138681113868 + μij + 1 minus ]ij1113872 1113873 minus μtj + 1 minus ]tj1113872 111387311138681113868111386811138681113868

111386811138681113868111386811138682πij + πtj2 + max μij minus μtj

11138681113868111386811138681113868

11138681113868111386811138681113868 ]ij minus ]tj

11138681113868111386811138681113868

11138681113868111386811138681113868 πij minus πtj

11138681113868111386811138681113868

1113868111386811138681113868111386821113874 11138751113874 11138751113874 1113875

(25)

Step 4 calculate the weighted matrix O (oij)mtimesn byequation (12)

oij zj middot qNij 1 minus 1 minus μqN

ij1113874 1113875

zj

]zj

qNij

1113874 1113875 (26)

Step 5 compute the border approximation areamatrix G (gi)1timesn e border approximation area(BAA) for every attribute is obtained from the fol-lowing equation

gj 1113945m

i1oij1113872 1113873

1m 1113937

m

i1μoij

1113874 11138751m

1 minus 1113945m

i11 minus ]oij

1113874 11138751m

⎛⎝ ⎞⎠

(27)

Step 6 calculate the distance matrix D (dij)mtimesne alternativesrsquo distances from the BAA are derivedwith the following equation

dij d oij gj1113872 11138731113872 1113873

ϑ if S oij1113872 1113873ge S gj1113872 1113873

minusρ d oij gj1113872 11138731113872 1113873ς if S oij1113872 1113873lt S gj1113872 1113873

⎧⎪⎨

⎪⎩(28)

where the distance measure is defined as equation(8) ϑ and ς are the parameters of DMsrsquo risk attitudesand ρ is the loss aversionrsquos parameter In this articleϑ 088 ς 088 and ρ 225 e values comefrom Tversky and Kahneman [37] who conducted anexperiment to determine the most acceptable valuesfrom numerous researchersNow if dij 0 the alternative Pi will belong to theborder approximation area (G) If dij gt 0 Pi belongsto the upper approximation area (G+) And if dij lt 0Pi belongs to the lower approximation area (Gminus ) G+

is the area involving the positive alternative (P+)whereas Gminus is the area involving the negative al-ternative (Pminus )Step 7 calculate the final value of criterion functionsFi

Journal of Mathematics 5

Fi 1113944n

j1dij i 1 2 m j 1 2 n (29)

Step 8 depending on the calculating results of Fi allthe alternatives could be ranked e larger the valueof Fi is the optimal the alternative will be

5 Numerical Example andComparative Analysis

51 Numerical Example Intelligent transportation system isthe development direction of the future traffic system It isthe advanced information technology data communicationtransmission technology electronic sensor technologycontrol technology and computer technology to effectivelyintegrate with the whole ground traffic management systemand establish a large-range all-round function real-timeaccurate and efficient integrated transportation manage-ment system Not only that the high-tech project is a processfull of unknown by its size complexity of technologyeconomic investment the degree of market demand andother aspects of influence and restriction erefore theproject evaluation plays an important role during the processof investment to project the overall technology evaluationmarket evaluation and economic evaluation risk forecasthas a great impact on the project decision makers for theproject development scheme and is also the key to thesuccess of a project Intelligent transportation systemevaluation could be regarded as the MADM or MAGDMissues [38ndash45] In this section an empirical application ofevaluating the intelligent transportation system is providedwith the IF-MABAC method ere are five potential citiesPi(i 1 2 3 4 5) preparing to evaluate their intelligenttransportation system In order to assess these cities fairlyfive experts H H1 H2 H3 H4 H51113864 1113865 (expertrsquos weight h

(020 020 020 020 020) are invited All experts couldgive their assessment information through four subsequentattributes ① Z1 is the intelligent transportation environ-ment②Z2 is the intelligent transportation cost③Z3 is theintelligent transportation safety and④ Z4 is the intelligenttransportation equipment investment Evidently Z2 is thecost attribute while Z1 Z3 and Z4 are the benefit attributes

Step 1 build each DMrsquos matrix Q(k) (qkij)mtimesn as in

Tables 1ndash5 Derived from the tables and equations(14)ndash(16) the overall decision matrix could be calcu-lated e results are recorded in Table 6Step 2 normalize the matrix Q [qij]mtimesn to QN

[qNij ]mtimesn (see Table 7)

Step 3 decide the attribute weights zj(j 1 2 n)

through the maximizing deviation method (seeTable 8)Step 4 calculate the weighted matrix O (oij)mtimesn byutilizing equation (26) (Table 9)Step 5 determine the BAA matrix G (gj)1timesn

(Table 10)

Table 1 Intuitionistic fuzzy matrix by H1

Z1 Z2 Z3 Z4P1 (063 015) (045 050) (057 031) (026 063)P2 (070 030) (021 069) (072 028) (064 022)P3 (039 051) (038 048) (050 040) (061 030)P4 (053 037) (042 051) (035 056) (055 034)P5 (026 069) (058 035) (055 035) (069 013)

Table 2 Intuitionistic fuzzy matrix by H2Z1 Z2 Z3 Z4

P1 (056 033) (021 053) (049 035) (057 043)P2 (056 033) (028 063) (075 025) (067 025)P3 (052 037) (016 068) (049 051) (058 035)P4 (071 018) (035 057) (045 047) (056 034)P5 (059 039) (026 065) (046 052) (071 011)

Table 3 Intuitionistic fuzzy matrix by H3Z1 Z2 Z3 Z4

P1 (019 065) (030 060) (054 037) (054 035)P2 (080 020) (024 058) (075 015) (077 023)P3 (058 039) (019 066) (044 051) (049 039)P4 (048 047) (023 053) (063 030) (067 020)P5 (054 035) (026 055) (041 057) (069 015)

Table 4 Intuitionistic fuzzy matrix by H4Z1 Z2 Z3 Z4

P1 (056 025) (032 058) (059 035) (058 025)P2 (066 020) (036 064) (055 025) (052 033)P3 (053 031) (043 051) (034 041) (041 035)P4 (043 037) (029 063) (055 030) (049 051)P5 (059 029) (039 055) (027 067) (063 019)

Table 5 Intuitionistic fuzzy matrix by H5Z1 Z2 Z3 Z4

P1 (039 055) (026 068) (047 038) (058 027)P2 (072015) (032 064) (064 025) (070 030)P3 (048 051) (023 058) (054 041) (044 055)P4 (058 033) (036 053) (060 030) (025 061)P5 (044 055) (029 065) (051 039) (039 059)

Table 6 Overall intuitionistic fuzzy matrixZ1 Z2 Z3 Z4

P1 (04874 03382) (03130 05747) (05342 03512) (0518703641)

P2 (07184 02087) (02840 06350) (06906 02309) (0669602628)

P3 (05039 04103) (02863 05766) (04662 04452) (0512303796)

P4 (05575 03284) (03331 05524) (05265 03718) (0521903727)

P5 (04975 04319) (03695 05372) (04479 04860) (0637101889)

6 Journal of Mathematics

Step 6 calculate the distance matrix D (dij)mtimesn (seeTable 11)Step 7 sum up each rowrsquos elements and each alter-nativersquos final value Fi can be determined as in Table 12Step 8 relying on Fi all the alternatives could beranked the larger the value of Fi is the optimal thealternative will be Evidently the rank of all alternativesis P2 gtP1 gtP4 gtP3 gtP5 and P2 is the optimal city

52 ComparativeAnalysis First of all the designed method iscomparedwith IFWAand IFWGoperators [34] For the IFWAoperator the calculating result is S(P1) 05936 S(P2)

07358 S(P3) 05620 S(P4) 05971 and S(P5) 05961us the ranking order isP2 gtP4 gtP5 gtP1 gtP3 For theIFWG operator the calculating result is S(P1) 05922

S(P2) 07336 S(P3) 05573 S(P4) 05963 and S(P5)

05724 So the ranking order is P2 gtP4 gtP1 gtP5 gtP3Furthermore the designed method is compared with the

modified IF-VIKOR method [46] en we can obtain thecalculating result en each alternativesrsquo relative closenessis calculated as DRC1 08683 DRC2 00000 DRC3

10000 DRC4 08878 and DRC5 09366 Hence theorder is P2 gtP1 gtP4 gtP5 gtP3

Besides the designed method is compared with the IF-GRA method [47] en we can obtain the calculatingresult e grey relational grades of every alternative arec1 08065 c2 09800 c3 07847 c4 08274 andc5 08342 erefore the order is P2 gtP5 gtP4 gtP1 gtP3

In the end the designed method is also compared withthe IF-MABAC method [1] en we can obtain the cal-culating result e overall value of every alternative isI1 29135 I2 33834 I3 13719 I4 28685 andI5 10845 erefore the order is P2 gtP1 gtP4 gtP3 gtP5

Eventually the results of these methods are depicted inTable 13

From Table 13 it is evident that the optimal enterprise isP2 while the worst is P3 in most cases In other words thesemethodsrsquo order is slightly different ese methods can ef-fectively solve MAGDM from different angles

Table 7 e normalized intuitionistic fuzzy matrixZ1 Z2 Z3 Z4

P1 (04874 03382) (05747 03130) (05342 03512) (0518703641)

P2 (07184 02087) (06350 02840) (06906 02309) (0669602628)

P3 (05039 04103) (05766 02863) (04662 04452) (0512303796)

P4 (05575 03284) (05524 03331) (05265 03718) (0521903727)

P5 (04975 04319) (05372 03695) (04479 04860) (0637101889)

Table 8 e attribute weights rj

Z1 Z2 Z3 Z4zj 02793 01699 02845 02663

Table 9 Intuitionistic fuzzy weighted normalized performancevalues of alternatives

Z1 Z2 Z3 Z4

P1 (01703 07387) (01352 08209) (01954 07425) (0176907641)

P2 (0298106456) (01574 08074) (02838 06590) (0255407005)

P3 (01778 07797) (01359 08085) (01635 07944) (0174007727)

P4 (02036 07327) (01277 08296) (01916 07547) (0178407689)

P5 (01749 07910) (01227 08444) (01555 08144) (0236606416)

Table 10 BAABAA

Z1 (02002 07422)Z2 (01353 08227)Z3 (01933 07585)Z4 (02015 07341)

Table 11 Distance matrixZ1 Z2 Z3 Z4

P1 minus00948 00144 00289 minus01133P2 01209 minus00801 01186 minus01532P3 minus01165 minus00572 minus01189 minus01273P4 minus00591 minus00498 minus00404 minus01174P5 minus01338 minus00759 minus01529 minus02261

Table 12 e final valueAlternative Final valueP1 minus01647P2 00063P3 minus04200P4 minus02667P5 minus05888

Table 13 Evaluation results of these methods

Methods Ranking ordere

optimalalternative

e worstalternative

IFWA operator[34] P2 gtP4 gtP5 gtP1 gtP3 P2 P3

IFWG operator[34] P2 gtP4 gtP1 gtP5 gtP3 P2 P3

IF-VIKORmethod [46] P2 gtP1 gtP4 gtP5 gtP3 P2 P3

IF-GRA method[47] P2 gtP5 gtP4 gtP1 gtP3 P2 P3

IF-MABACmethod [1] P2 gtP1 gtP4 gtP3 gtP5 P2 P5

e designedmethod P2 gtP1 gtP4 gtP3 gtP5 P2 P5

Journal of Mathematics 7

6 Conclusion

ITS is the trend of future traffic development e problemof traffic jam exists in all the big cities around the worldIntelligent transportation project has made the world attachgreat importance in the development of the intelligenttransportation system which domestic and foreign scholarsin succession of the intelligent transportation managementproject and related research work on performance appraisale performance appraisal of our national public programcurrently has not formed a set of appraising systems ofstandard and systemization and has problems of insufficienttechnology system appraising subjective color and publicparticipation intensity With respect to the intelligenttransportation project carrying on the project expenditureperformance appraisal of the intellectual traffic has the vitalsignificance is paper designs an effective method for thisissue since it designs a novel intuitive distance-based IF-MABAC method for evaluating the intelligent trans-portation system And then a numerical example forevaluating the intelligent transportation system has beengiven to confirm that this novel method is reasonableFurthermore to show the validity and feasibility of thedeveloped method some comparative analyses are alsoconducted However the main drawback of this paper is thatthe number of DMs and attributes is small and interde-pendency of criteria is not taken into consideration whichmay limit the application scope of the developed method tosome extent Furthermore the developed method can beutilized to tackle many other MAGDM issues such as riskevaluation project selection and site selection [48ndash59]

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e author declares that there are no conflicts of interest

References

[1] R X Liang S S He J Q Wang K Chen and L Li ldquoAnextended MABAC method for multi-criteria group decision-making problems based on correlative inputs of intuitionisticfuzzy informationrdquo Computational amp Applied Mathematicsvol 38 p 28 2019

[2] T He G Wei J Lu J Wu C Wei and Y Guo ldquoA novelEDAS based method for multiple attribute group decisionmaking with pythagorean 2-tuple linguistic informationrdquoTechnological and Economic Development of Economy vol 26no 6 pp 1125ndash1138 2020

[3] D-F Li ldquoMultiattribute decision making method based ongeneralized OWA operators with intuitionistic fuzzy setsrdquoExpert Systems with Applications vol 37 no 12 pp 8673ndash8678 2010

[4] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[5] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[6] H Garg ldquoGeneralized intuitionistic fuzzy multiplicative in-teractive geometric operators and their application tomultiplecriteria decision makingrdquo International Journal of MachineLearning and Cybernetics vol 7 no 6 pp 1075ndash1092 2016

[7] X J Gou Z S Xu and Q Lei ldquoNew operational laws andaggregation method of intuitionistic fuzzy informationrdquoJournal of Intelligent amp Fuzzy Systems vol 30 pp 129ndash1412016

[8] H Garg ldquoNovel intuitionistic fuzzy decision making methodbased on an improved operation laws and its applicationrdquoEngineering Applications of Artificial Intelligence vol 60pp 164ndash174 2017

[9] Y He Z He and H Huang ldquoDecision making with thegeneralized intuitionistic fuzzy power interaction averagingoperatorsrdquo Soft Computing vol 21 no 5 pp 1129ndash11442017

[10] P Liu J Liu and S-M Chen ldquoSome intuitionistic fuzzyDombi Bonferroni mean operators and their application tomulti-attribute group decision makingrdquo Journal of the Op-erational Research Society vol 69 no 1 pp 1ndash24 2018

[11] P Gupta H D Arora and P Tiwari ldquoGeneralized entropy forintuitionistic fuzzy setsrdquo Malaysian Journal of MathematicalSciences vol 10 pp 209ndash220 2016

[12] M Li and C Wu ldquoA distance model of intuitionistic fuzzycross entropy to solve preference problem on alternativesrdquoMathematical Problems in Engineering vol 2016 Article ID8324124 2016

[13] M S Khan and Q M D Lohani ldquoA similarity measure forAtanassov intuitionistic fuzzy sets and its application toclusteringrdquo in Proceedings of the 2016 International Workshopon Computational Intelligence (IWCI) Dhaka BangladeshDecember 2016

[14] P Li J Liu S F Liu X Su and JWu ldquoGrey target method forintuitionistic fuzzy decision making based on grey incidenceanalysisrdquo Journal of Grey System vol 28 pp 96ndash109 2016

[15] T Bao X Xie P Long and ZWei ldquoMADMmethod based onprospect theory and evidential reasoning approach withunknown attribute weights under intuitionistic fuzzy envi-ronmentrdquo Expert Systems with Applications vol 88pp 305ndash317 2017

[16] S-M Chen S-H Cheng and T-C Lan ldquoMulticriteria de-cision making based on the TOPSIS method and similaritymeasures between intuitionistic fuzzy valuesrdquo InformationSciences vol 367-368 pp 279ndash295 2016

[17] J W Gan and L Luo ldquoUsing DEMATEL and intuitionisticfuzzy sets to identify critical factors influencing the recyclingrate of end-of-life vehicles in Chinardquo Sustainability vol 92017

[18] P Gupta M K Mehlawat N Grover and W ChenldquoModified intuitionistic fuzzy SIR approach with an appli-cation to supplier selectionrdquo Journal of Intelligent amp FuzzySystems vol 32 no 6 pp 4431ndash4441 2017

[19] Z Hao Z Xu H Zhao and R Zhang ldquoNovel intuitionisticfuzzy decision making models in the framework of decisionfield theoryrdquo Information Fusion vol 33 pp 57ndash70 2017

[20] R Krishankumar S R Arvinda A Amrutha J Premaladhaand K S Ravichandran ldquoA decision making frameworkunder intuitionistic fuzzy environment for solving cloudvendor selection problemrdquo in Proceedings of the 2017 Inter-national Conference on Networks amp Advances in Computa-tional Technologies (NetACT) iruvananthapuram IndiaJuly 2017

[21] K R R Ks and A B Saeid ldquoA new extension to PROM-ETHEE under intuitionistic fuzzy environment for solving

8 Journal of Mathematics

supplier selection problem with linguistic preferencesrdquo Ap-plied Soft Computing vol 60 pp 564ndash576 2017

[22] X Luo and X Z Wang ldquoExtended VIKOR method forintuitionistic fuzzy multiattribute decision-making based on anew distance measurerdquo Mathematical Problems in Engi-neering vol 2017 Article ID 4072486 2017

[23] B D Rouyendegh ldquoe intuitionistic fuzzy ELECTREmodelrdquo International Journal of Management Science andEngineering Management vol 13 no 2 pp 139ndash145 2018

[24] S Cali and S Y Balaman ldquoA novel outranking based multicriteria group decision making methodology integratingELECTRE and VIKOR under intuitionistic fuzzy environ-mentrdquo Expert Systems with Applications vol 119 pp 36ndash502019

[25] P Phochanikorn and C Q Tan ldquoA new extension to a multi-criteria decision-making model for sustainable supplier se-lection under an intuitionistic fuzzy environmentrdquo Sustain-ability vol 11 p 24 2019

[26] S Liu ldquoResearch on the teaching quality evaluation of physicaleducation with intuitionistic fuzzy TOPSIS methodrdquo Journalof Intelligent amp Fuzzy Systems 2021 In press

[27] D Pamucar and G Cirovic ldquoe selection of transport andhandling resources in logistics centers using multi-attributiveborder approximation area comparison (MABAC)rdquo ExpertSystems with Applications vol 42 pp 3016ndash3028 2015

[28] R Sahin and F Altun ldquoDecision making with MABACmethod under probabilistic single-valued neutrosophic hes-itant fuzzy environmentrdquo Journal of Ambient Intelligence andHumanized Computing vol 11 no 5 2020

[29] GWei Y He F Lei J Wu and CWei ldquoMABACmethod formultiple attribute group decision making with probabilisticuncertain linguistic informationrdquo Journal of Intelligent ampFuzzy Systems vol 39 no 3 pp 3315ndash3327 2020

[30] GWWei Y He F Lei J Wu C Wei and Y F Guo ldquoGreensupplier selection in steel industry with intuitionistic fuzzyTaxonomy methodrdquo Journal of Intelligent amp Fuzzy Systemsvol 39 no 5 pp 7247ndash7258 2020

[31] X G Xu H Shi L J Zhang and H C Liu ldquoGreen supplierevaluation and selection with an extended MABAC methodunder the heterogeneous information environmentrdquo Sus-tainability vol 11 p 16 2019

[32] F Jia Y Liu and XWang ldquoAn extendedMABACmethod formulti-criteria group decision making based on intuitionisticfuzzy rough numbersrdquo Expert Systems with Applicationsvol 127 pp 241ndash255 2019

[33] W Liang G Zhao H Wu and B Dai ldquoRisk assessment ofrockburst via an extended MABAC method under fuzzyenvironmentrdquo Tunnelling and Underground Space Technol-ogy vol 83 pp 533ndash544 2019

[34] Z Xu and R R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006

[35] H-W Liu and G-J Wang ldquoMulti-criteria decision-makingmethods based on intuitionistic fuzzy setsrdquo European Journalof Operational Research vol 179 no 1 pp 220ndash233 2007

[36] Y Wang ldquoUsing the method of maximizing deviation tomake decision for multiindicesrdquo Journal of Systems Engi-neering amp Electronics vol 8 pp 21ndash26 1997

[37] A Tversky and D Kahneman ldquoAdvances in prospect theorycumulative representation of uncertaintyrdquo Journal of Risk andUncertainty vol 5 no 4 pp 297ndash323 1992

[38] E K Zavadskas J Antucheviciene and P Chatterjee Mul-tiple-Criteria Decision-Making (MCDM) Techniques for

Business Processes Information Management CRC Press BocaRaton FL USA 2019

[39] T He G Wei J Wu and C Wei ldquoQUALIFLEX method forevaluating human factors in construction project manage-ment with Pythagorean 2-tuple linguistic informationrdquoJournal of Intelligent amp Fuzzy Systems vol 40 no 3pp 4039ndash4050 2021

[40] E K Zavadskas A Cereska J Matijosius A Rimkus andR Bausys ldquoInternal combustion engine analysis of energyecological parameters by neutrosophic MULTIMOORA andSWARA methodsrdquo Energies vol 12 2019

[41] J Li L Wen G Wei J Wu and C Wei ldquoNew similarity anddistance measures of Pythagorean fuzzy sets and its appli-cation to selection of advertising platformsrdquo Journal of In-telligent amp Fuzzy Systems vol 40 no 3 pp 5403ndash5419 2021

[42] E K Zavadskas Z Turskis and J Antucheviciene ldquoSolutionmodels based on symmetric and asymmetric informationrdquoSymmetry-Basel vol 11 2019

[43] M Zhao G Wei C Wei J Wu and Y Wei ldquoExtended CPT-TODIM method for interval-valued intuitionistic fuzzyMAGDM and its application to urban ecological risk as-sessmentrdquo Journal of Intelligent amp Fuzzy Systems vol 40no 3 pp 4091ndash4106 2021

[44] F Lei G Wei J Wu C Wei and Y Guo ldquoQUALIFLEXmethod for MAGDM with probabilistic uncertain linguisticinformation and its application to green supplier selectionrdquoJournal of Intelligent amp Fuzzy Systems vol 39 no 5pp 6819ndash6831 2020

[45] Y Zhang G Wei Y Guo and C Wei ldquoTODIM methodbased on cumulative prospect theory for multiple attributegroup decision-making under 2-tuple linguistic Pythagoreanfuzzy environmentrdquo International Journal of Intelligent Sys-tems 2021 In press

[46] S Zeng S-M Chen and L-W Kuo ldquoMultiattribute decisionmaking based on novel score function of intuitionistic fuzzyvalues and modified VIKOR methodrdquo Information Sciencesvol 488 pp 76ndash92 2019

[47] S-F Zhang and S-Y Liu ldquoA GRA-based intuitionistic fuzzymulti-criteria group decision making method for personnelselectionrdquo Expert Systems with Applications vol 38 no 9pp 11401ndash11405 2011

[48] P Liu and H Xu ldquoGroup decision making method based onhybrid aggregation operator for intuitionistic uncertain lin-guistic variablesrdquo Journal of Intelligent amp Fuzzy Systemsvol 36 no 2 pp 1879ndash1898 2019

[49] M Zhao G Wei J Wu Y Guo and C Wei ldquoTODIMmethod for multiple attribute group decision making basedon cumulative prospect theory with 2-tuple linguistic neu-trosophic setsrdquo International Journal of Intelligent Systemsvol 36 no 3 pp 1199ndash1222 2021

[50] P Liu and X You ldquoBidirectional projection measure oflinguistic neutrosophic numbers and their application tomulti-criteria group decision makingrdquo Computers amp Indus-trial Engineering vol 128 pp 447ndash457 2019

[51] C Wei J Wu Y Guo and G Wei ldquoGreen supplier selectionbased on CODAS method in probabilistic uncertain linguisticenvironmentrdquo Technological and Economic Development ofEconomy 2021 In press

[52] P Liu and X You ldquoImproved TODIM method based onlinguistic neutrosophic numbers for multicriteria group de-cision-makingrdquo International Journal of Computational In-telligence Systems vol 12 no 2 pp 544ndash556 2019

[53] G Wei J Wu Y Guo J Wang and C Wei ldquoAn extendedCOPRAS model for multiple attribute group decision making

Journal of Mathematics 9

based on single-valued neutrosophic 2-tuple linguistic envi-ronmentrdquo Technological and Economic Development ofEconomy 2021 In press

[54] G-F Yu D-F Li J-M Qiu and X-X Zheng ldquoSome op-erators of intuitionistic uncertain 2-tuple linguistic variablesand application tomulti-attribute group decisionmaking withheterogeneous relationship among attributesrdquo Journal ofIntelligent amp Fuzzy Systems vol 34 no 1 pp 599ndash611 2018

[55] M Zhao G Wei C Wei and Y Guo ldquoCPT-TODIMmethodfor bipolar fuzzy multi-attribute group decision making andits application to network security service provider selectionrdquoInternational Journal of Intelligent Systems 2021 In press

[56] G-F Yu D-F Li and W Fei ldquoA novel method for het-erogeneous multi-attribute group decision making withpreference deviationrdquo Computers amp Industrial Engineeringvol 124 pp 58ndash64 2018

[57] S Wang G Wei J Wu C Wei and Y Guo ldquoModel forselection of hospital constructions with probabilistic linguisticGRP methodrdquo Journal of Intelligent amp Fuzzy Systems vol 40no 1 pp 1245ndash1259 2021

[58] M Munir H Kalsoom K Ullah T Mahmood andY-M Chu ldquoT-spherical fuzzy einstein hybrid aggregationoperators and their applications in multi-attribute decisionmaking problemsrdquo Symmetry vol 12 no 3 p 365 2020

[59] T Mahmood K Ullah Q Khan and N Jan ldquoAn approachtoward decision-making and medical diagnosis problemsusing the concept of spherical fuzzy setsrdquo Neural Computingand Applications vol 31 no 11 pp 7041ndash7053 2019

10 Journal of Mathematics