Research ArticleEnhanced Fuzzy Delphi Method in Forecastingand Decision-Making

Majed G Alharbi 1 and Hamiden Abd El- Wahed Khalifa 123

1Department of Mathematics College of Science and Arts Methnab Qassim University Buraydah Saudi Arabia2Department of Operations Research Faculty of Graduate Studies for Statistical Research Cairo University Giza 12613 Egypt3Department of Mathematics College of Science and Arts Al-Badaya Qassim University Buraydah Saudi Arabia

Correspondence should be addressed to Hamiden Abd El- Wahed Khalifa haahmedquedusa

Received 21 May 2021 Revised 12 July 2021 Accepted 4 October 2021 Published 31 October 2021

Academic Editor Bekir Sahin

Copyright copy 2021 Majed G Alharbi and Hamiden Abd El- Wahed Khalifa is is an open access article distributed under theCreative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium providedthe original work is properly cited

eDelphi method is a process where subjective data are transformed into quasi-objective data using statistical analysis and are convergedto stable points e Delphi method was developed by the RAND Corporation at Santa Monica California and is widely used for long-range forecasting inmanagement science It is amethod by which the subjective data of experts aremade to converge using some statisticalanalyses is article proposes a variation of the Delphi method using triangular fuzzy numbers where the communication method withthe experts is the same but the estimation procedure is different e utility of the method is illustrated by a numerical example

1 Introduction

In the Delphi method experts with high qualifications arefirst requested to give their opinions separately without anyintercommunication regarding predicting and realizingcertain events in science and technology en a statisticalanalysis is conducted to assess these subjective data andtheir first second and third quartiles are computed isstatistical information is transmitted to the experts Fur-thermore these new estimates are then analyzed and thefirst second and third quartiles are recalculated e newinformation is sent out to the experts once again and thisprocess of restoration is continued until the process con-verges to a reasonable stable solution Grisham [1] providedproject management students with an example of the Delphiresearch approach applied to a recent doctoral researchthesis reviewed the literature on the Delphi approach andexplained the researchmethod tool Many authors have dealtwith the Delphi method such as Story et al [2] Kauko andPalmroos [3] Skinner et al [4] and Beiderbeck et al andLawnik and Banasik [5 6] Piecyk and McKinnon [7]reviewed applications of the method to the field of freighttransport and logistics assessed the usualness of Delphi-

derived predictions of long-term freight and associatedenvironmental trends ey introduced a case study theexperience of a large two-roundDelphi survey undertaken inthe UK to elicit projections of long-term trends in a numberof road freight and logistics variables Revez et al [8]demonstrated the transformative potential of combiningparticipatory action research approaches with a modifiedDelphi method to understand energy transition issuesparticularly beyond forecasting instruments

In the literature first of all Zadeh [9] proposed thephilosophy of fuzzy sets Decision-making in a fuzzy en-vironment developed by Bellman and Zadeh [10] has beenimproved enhancing themanagement of decision problemsZimmermann [11] introduced fuzzy programming andlinear programming withmultiple objective functionsenseveral researchers have worked on fuzzy set theory etheory and applications of fuzzy sets systems and fuzzymathematical models were studied by many authors [12ndash19]Cheng et al [20] investigated a new fuzzy Delphi method(FDM) to fit membership functions and examined thestability of the process of the technique Roy and Garai [21]used a triangular intuitionistic fuzzy number to introduce anew and improved version of the FDM Moreover Habibi

HindawiAdvances in Fuzzy SystemsVolume 2021 Article ID 2459573 6 pageshttpsdoiorg10115520212459573

et al [22] applied the FDM to the single round for screeningcriteria Hsu et al [23] summarized the key courses in theeducation of construction engineers where the evaluationaspects and index are established using the FDM Ciptonoet al [24] classified and identified scientific research andanalyzed the current literature on the FDM to gain a broadand detailed understanding of education

Furthermore He et al [25] proposed a method to selectthe proper green supplier successfully Huang et al [26] usedthe FDM to identify the key input variables with the deepestinfluence on the prediction of peak particle velocity (PPV)based on the expertrsquos opinions and applied the effective pa-rameters on PPV in hybrid artificial neural network-basedmodels ie genetic algorithm particle swarm optimizationimperialism competitive algorithm artificial bee colony andfirefly algorithm for the PPV prediction Kabir and Sumi [27]proposed a new forecasting mechanism modeled by inte-grating the FDM with an artificial neural network techniqueto manage the demand with incomplete information Manyresearchers have used the FDM in their works [28ndash30]

is article proposed a variation of the Delphi methodusing triangular fuzzy numbers where the communicationmethod with experts is the same but the estimation pro-cedure is different

Having motivation from the above literature in thisstudy a variation of the Delphi method using triangularfuzzy numbers where the communication method with theexperts is the same but the estimation procedure is differentis studied

e rest of the paper is outlined as shown in Figure 1

2 Preliminaries

In order to easily discuss the problem the basic rules andfindings related to fuzzy sets fuzzy numbers triangularfuzzy numbers and arithmetic operations are recalled

Definition 1 (see [31]) Let M X1 times X2 times times Xn be theCartesian product μi be the fuzzy set in Xi and f X⟶ Y

be the mapping en the fuzzy set 1113957A in Y can be definedusing the extension principle as follows

1113957A(y)

Supx1 xn( )isinfminus 1(y)

min μ1 x1( 1113857 μn xn( 11138571113864 1113865 fminus 1

(y)neempty

0 fminus 1

(y) empty

⎧⎪⎪⎨

⎪⎪⎩

(1)

Definition 2 (see [31]) Let 1113957A and 1113957B be two fuzzy sets thealgebraic operations are defined as follows

(1) Addition 1113957A(+)1113957B

μ1113957A(+)1113957B(z) Sup

zx+y

min μ1113957A μ1113957B1113966 1113967 x isin 1113957A y isin 1113957B (2)

(2) Subtraction 1113957A(minus )1113957B

μ1113957A(minus )1113957B(z) Sup

zxminus y

min μ1113957A μ1113957B1113966 1113967 x isin 1113957A y isin 1113957B (3)

(3) Multiplication 1113957A()1113957B

μ1113957A()1113957B(z) Sup

zxy

min μ1113957A μ1113957B1113966 1113967 x isin 1113957A y isin 1113957B (4)

(4) Division 1113957A()1113957B

μ1113957A()1113957B(z) Sup

zxymin μ1113957A

μ1113957B1113966 1113967 x isin 1113957A y isin 1113957B (5)

Definition 3 (see [9] fuzzy number) Fuzzy number 1113957F is afuzzy set with a membership function defined as follows

π1113957F(x) R⟶ [0 1] satisfying the following

(1) 1113957F is fuzzy convex ie π1113957F(δx + (1 minus δ)y)gemin π1113957F(x) πF(y)1113966 1113967 forallx y isin R 0le δ le 1

(2) 1113957F is normal ie existx0 isin R for which π1113957F(x0) 1(3) Supp(1113957F) x isin R π1113957F(x)gt 01113966 1113967 is the support of 1113957F

(4) π1113957F(x) is upper semicontinuous ie for eachα isin (0 1) the αminus cut set 1113957Fα x isin R π1113957F ge α1113966 1113967 isclosed

Definition 4 (see [32]) Fuzzy number 1113957F (a b c) is called atriangular fuzzy number (TFN) if its membership function isgiven by

μ1113957F(x)

0 xlt a

x minus a

b minus a ale xle b

c minus x

c minus bble xle c

0 xge c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

and illustrated by Figure 2 [15]

Definition 5 Let 1113957F (a b c) and 1113957G (e f g) be two tri-angular fuzzy numbers e arithmetic operations on 1113957F and1113957G are as follows

(1) Addition 1113957Foplus 1113957G (a + e b + f c + g)

(2) Subtraction 1113957F⊖ 1113957G (a minus g b minus f c minus e)

(3) Inverse 1113957Fminus 1

(1c) (1b) (1a)

(4) Symmetric (image) minus (1113957F) (minus c minus b minus a)

3 Notations and Remarks

31 Notations In the FDM the following notations may beused

(Pi Qi Ri) sheaf of expertsrsquo opinion(Pi Qi Ri) mean of the sheafδ(Mi Mj) normalized distance between the two tri-angular fuzzy numbers Mi and Mj

2 Advances in Fuzzy Systems

32 Remarks e following important remarks should benoted [15]

(i) A long-range forecasting problem must be consid-ered an uncertain problem but not random edirect use of the probabilisticmethods is not suitablee use of fuzzy numbers and fuzzy methods seemsto be more compatible and well suited

(ii) e experts use their individual competency andsubjectivity which is why we propose the conceptsof fuzzy

(iii) If an expert was asked the following questionConsider the realization of a scientific or technologicalfact in the near future What type of dates for itsrealizationwould you givee expert will certainly bemore comfortable answering this question using threetypes of dates the earliest date the maximum pre-sumption date and the latest date rather than givingan answer regarding the probability of realization

(iv) Using TFNs it is easy for an expert to study therealization dates which are nested within eachother

(v) e sheaf of fuzzy numbers concept is an aggre-gation process that appears to be very convenientfor the objectification of various subjectiveopinions

(vi) Comparisons between expert answers using TFNsare simple and the distance matrices can be easilycomputed is matrix is a dissemblance relationdecomposable into similarity relations Using thisapproach finding a nondisjunction subgroup ofexperts who are favorable to studies in publicopinion is very convenient

(vii) Many forecasts need multistage processes Byconsidering the partial and necessary realizationthe critical pathmethod can be applied to obtain thecomposite extremal dates

4 Fuzzy Delphi Method

In this section the steps of the application of the FDM can beexplained as follows

Step 1 e expert i isin (1 n) is requested to give a possiblerealization date (the earliest date the maximum presump-tion date and the latest date) of the project using a TFN

Pi1 Q

i1 R

i11113872 1113873 (7)

where i indicates the index attached to the expert and 1indicates that this is the first phase of the forecasting process

Step 2 e responses from n experts from a sheaf are asfollows

Pi1 Q

i1 R

i11113872 1113873 i 1 n (8)

e mean of this TFN sheaf is then computed

P1 Q1 R1( 1113857 (9)

Moreover for each expert the divergence is computed asfollows

Di P1 minus Pi1 Q1 minus Q

i1 R1 minus R

i11113966 1113967 (10)

Section 2

Introduces some preliminaries related to the traingular fuzzy numbers andits arithematic operarions

Section 3Notation and remarks

Section 4FDMs steps in forecasting and decision-making problems

Section 5Presented the stability set of the second kind

Section 6a numerical example is given for illustration

Section 7Paper is summarized with future directions

Figure 1 Outline of the paper

0 a b c x

μF (x)~

Figure 2 Triangular fuzzy number representation

Advances in Fuzzy Systems 3

where these divergence numbers can be positive null ornegative in sign is information is then sent to each in-dividual expert

Step 3 Each expert now gives a new TFN

Pi2 Q

i2 R

i21113872 1113873 (11)

Furthermore the process starting with phase 2 isrepeated

Step 4 If necessary a study of partial group opinions isrealized using the distance between the TFN and the de-composition of the dissemblance relations in the maximalsubrelation of similaritye information about these results(nonnominative) can be sent to interested experts

Step 5 When themean TFN becomes sufficiently stable (iethe mean TFN that fixed the solution and decision maker issatisfactory) the process is stopped Of course later ifnecessary or if important discoveries or realizations takeplace the forecasting can be reevaluated by repeating theprocess outlined above

5 Numerical Example

Consider a technological realization problem of a cognitiveinformation processing computer in the near future Let usrequest a group of 12 computer experts to give us a sub-jective estimation using TFNs to realize this new computingtechnology e expertsrsquo estimations are listed in Table 1

e computation from this sheaf gives the mean TFN

P1 Q1 R1( 1113857 (202783 203417 204025) (2027 2034 2040)

(12)

In Figure 3 by comparing (P1 Q1 R1) with the sheaf(P1

1 Q11 R1

1) the divergence yields

P1 minus P11 2027 minus 2024 +3

Q1 minus Q11 2034 minus 2032 +2

R1 minus R11 2040 minus 2049 minus 9

(13)

is information is given to each expert and each re-considers hisher previous forecast and gives a new TFNthus obtaining a new sheaf is process is continued until astable solution according to certain criteria is reached Ofcourse the number of such iteration phases can be limited apriori Many variations of this procedure are possible Forinstance the experts can be advised not to increase thedivergence but this is only a suggestion because an expertmust give their own unbiased opinion

Definition 6 e normalized distance between two TFNs isdefined as follows

δ Mi Mj1113872 1113873 1

2 c2 minus c1( 1113857Δl Mi Mj1113872 1113873 + Δr Mi Mj1113872 11138731113872 1113873

(14)

where Mi and Mj are the TFNs given by experts i and j Δl

and Δr are the left and right distances respectively and c2and c1 are arbitrary values at the right and left respectivelychosen with the condition δ isin [0 1]

Using formula (14) the results of the computation areinserted in Table 2 for the arbitrary values of c1 2021 andc2 2049

As shown in Table 2 we see that theminimum distance isδ(7 11) 003 and the maximum distance isδ(4 11) 087

Let us assume that we are interested in a pair of expertsfor whom the distance δ(Mi Mj)le 015 e matrix inTable 3 gives these pairs e corresponding pairs are shownin Figure 4

A direct examination of Figure 4 gives the maximumsubrelations of similarity For example we have thefollowing

(1 5 8) (1 8 9) (1 10) (2 4) (3 5 8) (3 8 9) (5 6 8) (7 11) (9 12) (15)

Table 1 Sheaf of expertsrsquo opinions for the technological realizationof the cognitive information processing computer

Expert no Earliestdate

Maximum presumptiondate

Latestdate

1 2024 2032 20492 2021 2024 20003 2029 2034 20394 2021 2022 20235 2029 2034 20446 2024 2034 20397 2039 2047 20498 2024 2036 20429 2024 2031 203610 2037 2038 204911 2039 2049 204912 2023 2029 2034

1

0 2024 2029 2034 2039 2049

05

(P1Q1R1)

(P11Q1

1R11)

Figure 3 Comparison of a sheaf (P11 Q1

1 R11) with the mean

(P1 Q1 R1)

4 Advances in Fuzzy Systems

If we consider for instance experts (1 8 9) and (3 8 9)we find that δ(1 3) 016 We can assume therefore thatexperts (1 3 8 9) have given almost the same estimation Asimilar situation appears with (1 5 8) (3 8 9) and(1 3 8 9) that have almost the same estimation ereforewe can aggregate experts (1 3 5 8 9)

6 Discussion of the Results

If we consider δ le 020 we find this aggregation directly Infact using a convenient algorithm the problem can bestudied with various values of δ that is δ le 02 025 If

we wish to consider the similarity problem then a valuesmaller than (1 minus δ) gives the level of similarity which is animportant point in any a posteriori discussion betweenexperts e maximal subrelations of similarity in a re-semblance relation play the same role as classes in a simi-larity relatione second one is always disjointed but this isnot the case for the first one

7 Conclusions and Future Works

A variation of the Delphi method using TFNs has beenproposed in which the method of communication withexperts is the same but the estimation procedure is differentOf note in many cases a forecasted success requires varioussteps of operations therefore using the critical path methodfor forecasting is more efficient in such cases In fact manyother ways can be used for forecasting using fuzzy numberswhich we may investigate in the future Future work mayinclude the further extension of this study to other fuzzy-likestructures i e interval-valued fuzzy set intuitionistic fuzzyset Pythagorean fuzzy set spherical fuzzy set picture fuzzyset neutrosophic set etc with more discussion on real-world problems

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare no conflicts of interest

References

[1] T Grisham ldquoe Delphi technique a method for testingcomplex and multifaceted topicsrdquo International Journal ofManaging Projects in Business vol 2 no 1 pp 112ndash130 2009

[2] V Story L Hurdley G Smith and J Saker ldquoMethodologicaland practical implications of the Delphi technique in mar-keting decision-making a Re-assessmentrdquo 7e MarketingReview vol 1 no 4 pp 487ndash504 2000

[3] K Kauko and P Palmroos ldquoe Delphi method in forecastingfinancial markets- an experimental studyrdquo InternationalJournal of Forecasting vol 30 no 2 pp 313ndash327 2014

Table 2 Distances of δ(Mi Mj)

δ(Mi Mj) 1 2 3 4 5 6 7 8 9 10 11 12

1 0 036 016 043 012 016 04 013 013 013 043 0192 0 035 007 04 044 076 037 023 058 08 0163 0 042 004 0017 041 010 012 023 044 0184 0 047 051 083 046 030 060 087 0245 0 013 034 009 016 018 040 0236 0 033 007 021 026 035 0277 0 039 053 017 003 0598 Symmetry 0 014 021 044 0209 0 035 057 00610 0 021 04111 0 07212 0

Table 3 Pair of experts with arbitrary δ le 015

δij le 015 1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 1 1 12 1 13 1 1 1 14 15 1 1 16 1 17 1 18 1 19 1 110 111 112 1

12

3

4

5

6

78

9

10

11

12

Figure 4 Pairs of experts with δ le 020

Advances in Fuzzy Systems 5

[4] R Skinner R R Nelson W W Chin and L Land ldquoeDelphi method research strategy in studies of informationsystemsrdquo Communications of the Association for InformationSystems vol 37 no 2 pp 31ndash63 2015

[5] D Beiderbeck N Frevel H A VonDer Gracht S L Schmidtand V M Schweitzer ldquoPreparing conducting and analyzingDelphi surveys cross-disciplinary practices new directionsand advancementsrdquo Methods (Orlando) vol 8 Article ID101401 2021

[6] M Lawnik and A Banasik ldquoDelphi method supported byforecasting softwarerdquo Information vol 11 no 2 p 65 2020

[7] M I Piecyk and A C McKinnon ldquoApplication of the Delphimethod to the forecasting of long- term trends in road freightlogistics and related CO2 emissionsrdquo International Journal ofTransport Economics vol 40 no 2 pp 241ndash266 2013

[8] A Revez N Dunphy C Harris G Mullally B Lennon andC Gaffney ldquoBeyond forecasting using a modified Delphimethod to build upon participatory acton research in de-veloping principles for a just and inclusive energy transitionrdquoInternational Journal of Qualitative Methods vol 19 pp 1ndash122020

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

[10] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 no 4 pp Bndash1411970

[11] H-J Zimmermann ldquoFuzzy programming and linear pro-gramming with several objective functionsrdquo Fuzzy Sets andSystems vol 1 no 1 pp 45ndash55 1978

[12] Y Badulescu A-P Hameri and N Cheikhrouhou ldquoEvalu-ating demand forecasting models using multi-criteria deci-sion-making approachrdquo Journal of Advances in ManagementResearch 2021 ahead-of-print

[13] D Dubois and H Prade Fuzzy Sets and Systems 7eory andApplications Academic Press New York NY USA 1980

[14] J Guo C-J Chang Y Huang and K-P Yu ldquoA fuzzy-de-composition grey modeling procedure for management de-cision analysisrdquo Mathematical Problems in Engineeringvol 2021 Article ID 6670196 6 pages 2021

[15] A Kaufmann and M M Gupta Fuzzy Mathematical Modelsin Engineering and Management Science Elsevier SciencePublishing Company INC New York NY USA 1988

[16] Z K Nemaa and G K Aswed ldquoForecasting construction timefor road projects and infrastructure using the fuzzy PERTmethodrdquo IOP Conference Series Materials Science and En-gineering vol 1076 no 1 Article ID 012123 2021

[17] F Tahriri M Mousavi S Hozhabri Haghighi and S ZawiahMd Dawal ldquoe application of fuzzy Delphi and fuzzy in-ference system in supplier ranking and selectionrdquo Journal ofIndustrial Engineering International vol 10 no 3 p 66 2014

[18] H-C Tsai A-S Lee H-N Lee C-N Chen and Y-C LiuldquoAn application of the fuzzy Delphi method and fuzzy AHPon the discussion of training indicators for the regionalcompetition Taiwan National Skills competition in the tradeof joineryrdquo Sustainability vol 12 no 10 p 4290 2020

[19] E Vercher A Rubio and J D Bermudez ldquoFuzzy predictionintervals using credibility distributionsrdquo Engineering Pro-ceedings vol 5 no 1 p 51 2021

[20] P-T Cheng L-C Huang and H-J Lin ldquoe fuzzy Delphimethod via fuzzy statistics and membership function fittingand an application to the human resourcesrdquo Fuzzy Sets andSystems vol 112 no 3 pp 511ndash520 2000

[21] T K Roy and A Garai ldquoIntuitionistic fuzzy Delphi methodmore realistic and iterative forecasting toolrdquo Notes onIntuitionistic Fuzzy Sets vol 18 no 2 pp 37ndash50 2012

[22] A Habibi F F Jahantigh and A Sarafrazi ldquoFuzzy Delphitechnique for forecasting and screening itemsrdquo Asian Journalof Research in Business Economics and Management vol 5no 2 pp 130ndash143 2015

[23] W-L Hsu L Chen L Chen and H-L Liu ldquoe applicationfuzzy Delphi method to summarize key factors in the edu-cation of construction engineersrdquo in Proceedings of the 2ndEurasian Conference on Educational Innovations and Appli-cations- Tijus Meen Chang Singapore January 2019

[24] A Ciptono s Setiyono F Nurhidayati and R VikalianaldquoFuzzy Delphi method in education a mappingrdquo Journal ofPhysics Conference Series vol 1360 no 1360 Article ID012029 2019

[25] 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 no 23 p 6616 2019

[26] J Huang M Koopialipoor and D J Armaghani ldquoA com-bination of fuzzy Delphi method and hybrid ANN-basedsystems to forecast ground vibration resulting from blastingrdquoScientific Reports vol 10 no 1 Article ID 19397 2020

[27] G Kabir and R S Sumi ldquoIntegrating fuzzy Delphi methodwith artificial neural network for demand forecasting of powerengineering companyrdquo Management Science Letters vol 2no 5 pp 1491ndash1504 2012

[28] Y-C Chiang and H-Y Lei ldquoUsing expert decision-making toestablish indicators of urban friendliness for walking envi-ronments a multidisciplinary assessmentrdquo InternationalJournal of Health Geographics vol 15 no 1 pp 15ndash40 2016

[29] A Garai and T Kumar ldquoWeighted intuitionistic fuzzy Delphimethodrdquo Journal of Global Research in Computer Sciencevol 4 no 7 pp 38ndash42 2013

[30] J Heidary Dahooie A R Qorbani and T Daim ldquoProviding aframework for selecting the appropriate method of technol-ogy acquisition considering uncertainty in hierarchical groupdecision-making case Study interactive television technol-ogyrdquo Technological Forecasting and Social Change vol 168Article ID 120760 2021

[31] B Lee and Y Yun ldquoe pentagonal fuzzy numbersrdquo Journalof the Chungcheong Mathematical Society vol 27 no 2 2014

[32] M C J Anand and J Bharatraj ldquoeory of triangular fuzzynumberrdquo in Proceedings of the NCATM Kharghar IndiaMarch 2017

6 Advances in Fuzzy Systems

32 Remarks e following important remarks should benoted [15]

(i) A long-range forecasting problem must be consid-ered an uncertain problem but not random edirect use of the probabilisticmethods is not suitablee use of fuzzy numbers and fuzzy methods seemsto be more compatible and well suited

(ii) e experts use their individual competency andsubjectivity which is why we propose the conceptsof fuzzy

(iii) If an expert was asked the following questionConsider the realization of a scientific or technologicalfact in the near future What type of dates for itsrealizationwould you givee expert will certainly bemore comfortable answering this question using threetypes of dates the earliest date the maximum pre-sumption date and the latest date rather than givingan answer regarding the probability of realization

(iv) Using TFNs it is easy for an expert to study therealization dates which are nested within eachother

(v) e sheaf of fuzzy numbers concept is an aggre-gation process that appears to be very convenientfor the objectification of various subjectiveopinions

(vi) Comparisons between expert answers using TFNsare simple and the distance matrices can be easilycomputed is matrix is a dissemblance relationdecomposable into similarity relations Using thisapproach finding a nondisjunction subgroup ofexperts who are favorable to studies in publicopinion is very convenient

(vii) Many forecasts need multistage processes Byconsidering the partial and necessary realizationthe critical pathmethod can be applied to obtain thecomposite extremal dates

4 Fuzzy Delphi Method

In this section the steps of the application of the FDM can beexplained as follows

Step 1 e expert i isin (1 n) is requested to give a possiblerealization date (the earliest date the maximum presump-tion date and the latest date) of the project using a TFN

Pi1 Q

i1 R

i11113872 1113873 (7)

where i indicates the index attached to the expert and 1indicates that this is the first phase of the forecasting process

Step 2 e responses from n experts from a sheaf are asfollows

Pi1 Q

i1 R

i11113872 1113873 i 1 n (8)

e mean of this TFN sheaf is then computed

P1 Q1 R1( 1113857 (9)

Moreover for each expert the divergence is computed asfollows

Di P1 minus Pi1 Q1 minus Q

i1 R1 minus R

i11113966 1113967 (10)

Section 2

Introduces some preliminaries related to the traingular fuzzy numbers andits arithematic operarions

Section 3Notation and remarks

Section 4FDMs steps in forecasting and decision-making problems

Section 5Presented the stability set of the second kind

Section 6a numerical example is given for illustration

Section 7Paper is summarized with future directions

Figure 1 Outline of the paper

0 a b c x

μF (x)~

Figure 2 Triangular fuzzy number representation

Advances in Fuzzy Systems 3

where these divergence numbers can be positive null ornegative in sign is information is then sent to each in-dividual expert

Step 3 Each expert now gives a new TFN

Pi2 Q

i2 R

i21113872 1113873 (11)

Furthermore the process starting with phase 2 isrepeated

Step 4 If necessary a study of partial group opinions isrealized using the distance between the TFN and the de-composition of the dissemblance relations in the maximalsubrelation of similaritye information about these results(nonnominative) can be sent to interested experts

Step 5 When themean TFN becomes sufficiently stable (iethe mean TFN that fixed the solution and decision maker issatisfactory) the process is stopped Of course later ifnecessary or if important discoveries or realizations takeplace the forecasting can be reevaluated by repeating theprocess outlined above

5 Numerical Example

Consider a technological realization problem of a cognitiveinformation processing computer in the near future Let usrequest a group of 12 computer experts to give us a sub-jective estimation using TFNs to realize this new computingtechnology e expertsrsquo estimations are listed in Table 1

e computation from this sheaf gives the mean TFN

P1 Q1 R1( 1113857 (202783 203417 204025) (2027 2034 2040)

(12)

In Figure 3 by comparing (P1 Q1 R1) with the sheaf(P1

1 Q11 R1

1) the divergence yields

P1 minus P11 2027 minus 2024 +3

Q1 minus Q11 2034 minus 2032 +2

R1 minus R11 2040 minus 2049 minus 9

(13)

is information is given to each expert and each re-considers hisher previous forecast and gives a new TFNthus obtaining a new sheaf is process is continued until astable solution according to certain criteria is reached Ofcourse the number of such iteration phases can be limited apriori Many variations of this procedure are possible Forinstance the experts can be advised not to increase thedivergence but this is only a suggestion because an expertmust give their own unbiased opinion

Definition 6 e normalized distance between two TFNs isdefined as follows

δ Mi Mj1113872 1113873 1

2 c2 minus c1( 1113857Δl Mi Mj1113872 1113873 + Δr Mi Mj1113872 11138731113872 1113873

(14)

where Mi and Mj are the TFNs given by experts i and j Δl

and Δr are the left and right distances respectively and c2and c1 are arbitrary values at the right and left respectivelychosen with the condition δ isin [0 1]

Using formula (14) the results of the computation areinserted in Table 2 for the arbitrary values of c1 2021 andc2 2049

As shown in Table 2 we see that theminimum distance isδ(7 11) 003 and the maximum distance isδ(4 11) 087

Let us assume that we are interested in a pair of expertsfor whom the distance δ(Mi Mj)le 015 e matrix inTable 3 gives these pairs e corresponding pairs are shownin Figure 4

A direct examination of Figure 4 gives the maximumsubrelations of similarity For example we have thefollowing

(1 5 8) (1 8 9) (1 10) (2 4) (3 5 8) (3 8 9) (5 6 8) (7 11) (9 12) (15)

Table 1 Sheaf of expertsrsquo opinions for the technological realizationof the cognitive information processing computer

Expert no Earliestdate

Maximum presumptiondate

Latestdate

1 2024 2032 20492 2021 2024 20003 2029 2034 20394 2021 2022 20235 2029 2034 20446 2024 2034 20397 2039 2047 20498 2024 2036 20429 2024 2031 203610 2037 2038 204911 2039 2049 204912 2023 2029 2034

1

0 2024 2029 2034 2039 2049

05

(P1Q1R1)

(P11Q1

1R11)

Figure 3 Comparison of a sheaf (P11 Q1

1 R11) with the mean

(P1 Q1 R1)

4 Advances in Fuzzy Systems

If we consider for instance experts (1 8 9) and (3 8 9)we find that δ(1 3) 016 We can assume therefore thatexperts (1 3 8 9) have given almost the same estimation Asimilar situation appears with (1 5 8) (3 8 9) and(1 3 8 9) that have almost the same estimation ereforewe can aggregate experts (1 3 5 8 9)

6 Discussion of the Results

If we consider δ le 020 we find this aggregation directly Infact using a convenient algorithm the problem can bestudied with various values of δ that is δ le 02 025 If

we wish to consider the similarity problem then a valuesmaller than (1 minus δ) gives the level of similarity which is animportant point in any a posteriori discussion betweenexperts e maximal subrelations of similarity in a re-semblance relation play the same role as classes in a simi-larity relatione second one is always disjointed but this isnot the case for the first one

7 Conclusions and Future Works

A variation of the Delphi method using TFNs has beenproposed in which the method of communication withexperts is the same but the estimation procedure is differentOf note in many cases a forecasted success requires varioussteps of operations therefore using the critical path methodfor forecasting is more efficient in such cases In fact manyother ways can be used for forecasting using fuzzy numberswhich we may investigate in the future Future work mayinclude the further extension of this study to other fuzzy-likestructures i e interval-valued fuzzy set intuitionistic fuzzyset Pythagorean fuzzy set spherical fuzzy set picture fuzzyset neutrosophic set etc with more discussion on real-world problems

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare no conflicts of interest

References

[1] T Grisham ldquoe Delphi technique a method for testingcomplex and multifaceted topicsrdquo International Journal ofManaging Projects in Business vol 2 no 1 pp 112ndash130 2009

[2] V Story L Hurdley G Smith and J Saker ldquoMethodologicaland practical implications of the Delphi technique in mar-keting decision-making a Re-assessmentrdquo 7e MarketingReview vol 1 no 4 pp 487ndash504 2000

[3] K Kauko and P Palmroos ldquoe Delphi method in forecastingfinancial markets- an experimental studyrdquo InternationalJournal of Forecasting vol 30 no 2 pp 313ndash327 2014

Table 2 Distances of δ(Mi Mj)

δ(Mi Mj) 1 2 3 4 5 6 7 8 9 10 11 12

1 0 036 016 043 012 016 04 013 013 013 043 0192 0 035 007 04 044 076 037 023 058 08 0163 0 042 004 0017 041 010 012 023 044 0184 0 047 051 083 046 030 060 087 0245 0 013 034 009 016 018 040 0236 0 033 007 021 026 035 0277 0 039 053 017 003 0598 Symmetry 0 014 021 044 0209 0 035 057 00610 0 021 04111 0 07212 0

Table 3 Pair of experts with arbitrary δ le 015

δij le 015 1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 1 1 12 1 13 1 1 1 14 15 1 1 16 1 17 1 18 1 19 1 110 111 112 1

12

3

4

5

6

78

9

10

11

12

Figure 4 Pairs of experts with δ le 020

Advances in Fuzzy Systems 5

[4] R Skinner R R Nelson W W Chin and L Land ldquoeDelphi method research strategy in studies of informationsystemsrdquo Communications of the Association for InformationSystems vol 37 no 2 pp 31ndash63 2015

[5] D Beiderbeck N Frevel H A VonDer Gracht S L Schmidtand V M Schweitzer ldquoPreparing conducting and analyzingDelphi surveys cross-disciplinary practices new directionsand advancementsrdquo Methods (Orlando) vol 8 Article ID101401 2021

[6] M Lawnik and A Banasik ldquoDelphi method supported byforecasting softwarerdquo Information vol 11 no 2 p 65 2020

[7] M I Piecyk and A C McKinnon ldquoApplication of the Delphimethod to the forecasting of long- term trends in road freightlogistics and related CO2 emissionsrdquo International Journal ofTransport Economics vol 40 no 2 pp 241ndash266 2013

[8] A Revez N Dunphy C Harris G Mullally B Lennon andC Gaffney ldquoBeyond forecasting using a modified Delphimethod to build upon participatory acton research in de-veloping principles for a just and inclusive energy transitionrdquoInternational Journal of Qualitative Methods vol 19 pp 1ndash122020

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

[10] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 no 4 pp Bndash1411970

[11] H-J Zimmermann ldquoFuzzy programming and linear pro-gramming with several objective functionsrdquo Fuzzy Sets andSystems vol 1 no 1 pp 45ndash55 1978

[12] Y Badulescu A-P Hameri and N Cheikhrouhou ldquoEvalu-ating demand forecasting models using multi-criteria deci-sion-making approachrdquo Journal of Advances in ManagementResearch 2021 ahead-of-print

[13] D Dubois and H Prade Fuzzy Sets and Systems 7eory andApplications Academic Press New York NY USA 1980

[14] J Guo C-J Chang Y Huang and K-P Yu ldquoA fuzzy-de-composition grey modeling procedure for management de-cision analysisrdquo Mathematical Problems in Engineeringvol 2021 Article ID 6670196 6 pages 2021

[15] A Kaufmann and M M Gupta Fuzzy Mathematical Modelsin Engineering and Management Science Elsevier SciencePublishing Company INC New York NY USA 1988

[16] Z K Nemaa and G K Aswed ldquoForecasting construction timefor road projects and infrastructure using the fuzzy PERTmethodrdquo IOP Conference Series Materials Science and En-gineering vol 1076 no 1 Article ID 012123 2021

[17] F Tahriri M Mousavi S Hozhabri Haghighi and S ZawiahMd Dawal ldquoe application of fuzzy Delphi and fuzzy in-ference system in supplier ranking and selectionrdquo Journal ofIndustrial Engineering International vol 10 no 3 p 66 2014

[18] H-C Tsai A-S Lee H-N Lee C-N Chen and Y-C LiuldquoAn application of the fuzzy Delphi method and fuzzy AHPon the discussion of training indicators for the regionalcompetition Taiwan National Skills competition in the tradeof joineryrdquo Sustainability vol 12 no 10 p 4290 2020

[19] E Vercher A Rubio and J D Bermudez ldquoFuzzy predictionintervals using credibility distributionsrdquo Engineering Pro-ceedings vol 5 no 1 p 51 2021

[20] P-T Cheng L-C Huang and H-J Lin ldquoe fuzzy Delphimethod via fuzzy statistics and membership function fittingand an application to the human resourcesrdquo Fuzzy Sets andSystems vol 112 no 3 pp 511ndash520 2000

[21] T K Roy and A Garai ldquoIntuitionistic fuzzy Delphi methodmore realistic and iterative forecasting toolrdquo Notes onIntuitionistic Fuzzy Sets vol 18 no 2 pp 37ndash50 2012

[22] A Habibi F F Jahantigh and A Sarafrazi ldquoFuzzy Delphitechnique for forecasting and screening itemsrdquo Asian Journalof Research in Business Economics and Management vol 5no 2 pp 130ndash143 2015

[23] W-L Hsu L Chen L Chen and H-L Liu ldquoe applicationfuzzy Delphi method to summarize key factors in the edu-cation of construction engineersrdquo in Proceedings of the 2ndEurasian Conference on Educational Innovations and Appli-cations- Tijus Meen Chang Singapore January 2019

[24] A Ciptono s Setiyono F Nurhidayati and R VikalianaldquoFuzzy Delphi method in education a mappingrdquo Journal ofPhysics Conference Series vol 1360 no 1360 Article ID012029 2019

[25] 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 no 23 p 6616 2019

[26] J Huang M Koopialipoor and D J Armaghani ldquoA com-bination of fuzzy Delphi method and hybrid ANN-basedsystems to forecast ground vibration resulting from blastingrdquoScientific Reports vol 10 no 1 Article ID 19397 2020

[27] G Kabir and R S Sumi ldquoIntegrating fuzzy Delphi methodwith artificial neural network for demand forecasting of powerengineering companyrdquo Management Science Letters vol 2no 5 pp 1491ndash1504 2012

[28] Y-C Chiang and H-Y Lei ldquoUsing expert decision-making toestablish indicators of urban friendliness for walking envi-ronments a multidisciplinary assessmentrdquo InternationalJournal of Health Geographics vol 15 no 1 pp 15ndash40 2016

[29] A Garai and T Kumar ldquoWeighted intuitionistic fuzzy Delphimethodrdquo Journal of Global Research in Computer Sciencevol 4 no 7 pp 38ndash42 2013

[30] J Heidary Dahooie A R Qorbani and T Daim ldquoProviding aframework for selecting the appropriate method of technol-ogy acquisition considering uncertainty in hierarchical groupdecision-making case Study interactive television technol-ogyrdquo Technological Forecasting and Social Change vol 168Article ID 120760 2021

[31] B Lee and Y Yun ldquoe pentagonal fuzzy numbersrdquo Journalof the Chungcheong Mathematical Society vol 27 no 2 2014

[32] M C J Anand and J Bharatraj ldquoeory of triangular fuzzynumberrdquo in Proceedings of the NCATM Kharghar IndiaMarch 2017

6 Advances in Fuzzy Systems

where these divergence numbers can be positive null ornegative in sign is information is then sent to each in-dividual expert

Step 3 Each expert now gives a new TFN

Pi2 Q

i2 R

i21113872 1113873 (11)

Furthermore the process starting with phase 2 isrepeated

Step 4 If necessary a study of partial group opinions isrealized using the distance between the TFN and the de-composition of the dissemblance relations in the maximalsubrelation of similaritye information about these results(nonnominative) can be sent to interested experts

Step 5 When themean TFN becomes sufficiently stable (iethe mean TFN that fixed the solution and decision maker issatisfactory) the process is stopped Of course later ifnecessary or if important discoveries or realizations takeplace the forecasting can be reevaluated by repeating theprocess outlined above

5 Numerical Example

Consider a technological realization problem of a cognitiveinformation processing computer in the near future Let usrequest a group of 12 computer experts to give us a sub-jective estimation using TFNs to realize this new computingtechnology e expertsrsquo estimations are listed in Table 1

e computation from this sheaf gives the mean TFN

P1 Q1 R1( 1113857 (202783 203417 204025) (2027 2034 2040)

(12)

In Figure 3 by comparing (P1 Q1 R1) with the sheaf(P1

1 Q11 R1

1) the divergence yields

P1 minus P11 2027 minus 2024 +3

Q1 minus Q11 2034 minus 2032 +2

R1 minus R11 2040 minus 2049 minus 9

(13)

is information is given to each expert and each re-considers hisher previous forecast and gives a new TFNthus obtaining a new sheaf is process is continued until astable solution according to certain criteria is reached Ofcourse the number of such iteration phases can be limited apriori Many variations of this procedure are possible Forinstance the experts can be advised not to increase thedivergence but this is only a suggestion because an expertmust give their own unbiased opinion

Definition 6 e normalized distance between two TFNs isdefined as follows

δ Mi Mj1113872 1113873 1

2 c2 minus c1( 1113857Δl Mi Mj1113872 1113873 + Δr Mi Mj1113872 11138731113872 1113873

(14)

where Mi and Mj are the TFNs given by experts i and j Δl

and Δr are the left and right distances respectively and c2and c1 are arbitrary values at the right and left respectivelychosen with the condition δ isin [0 1]

Using formula (14) the results of the computation areinserted in Table 2 for the arbitrary values of c1 2021 andc2 2049

As shown in Table 2 we see that theminimum distance isδ(7 11) 003 and the maximum distance isδ(4 11) 087

Let us assume that we are interested in a pair of expertsfor whom the distance δ(Mi Mj)le 015 e matrix inTable 3 gives these pairs e corresponding pairs are shownin Figure 4

A direct examination of Figure 4 gives the maximumsubrelations of similarity For example we have thefollowing

(1 5 8) (1 8 9) (1 10) (2 4) (3 5 8) (3 8 9) (5 6 8) (7 11) (9 12) (15)

Table 1 Sheaf of expertsrsquo opinions for the technological realizationof the cognitive information processing computer

Expert no Earliestdate

Maximum presumptiondate

Latestdate

1 2024 2032 20492 2021 2024 20003 2029 2034 20394 2021 2022 20235 2029 2034 20446 2024 2034 20397 2039 2047 20498 2024 2036 20429 2024 2031 203610 2037 2038 204911 2039 2049 204912 2023 2029 2034

1

0 2024 2029 2034 2039 2049

05

(P1Q1R1)

(P11Q1

1R11)

Figure 3 Comparison of a sheaf (P11 Q1

1 R11) with the mean

(P1 Q1 R1)

4 Advances in Fuzzy Systems

If we consider for instance experts (1 8 9) and (3 8 9)we find that δ(1 3) 016 We can assume therefore thatexperts (1 3 8 9) have given almost the same estimation Asimilar situation appears with (1 5 8) (3 8 9) and(1 3 8 9) that have almost the same estimation ereforewe can aggregate experts (1 3 5 8 9)

6 Discussion of the Results

If we consider δ le 020 we find this aggregation directly Infact using a convenient algorithm the problem can bestudied with various values of δ that is δ le 02 025 If

we wish to consider the similarity problem then a valuesmaller than (1 minus δ) gives the level of similarity which is animportant point in any a posteriori discussion betweenexperts e maximal subrelations of similarity in a re-semblance relation play the same role as classes in a simi-larity relatione second one is always disjointed but this isnot the case for the first one

7 Conclusions and Future Works

A variation of the Delphi method using TFNs has beenproposed in which the method of communication withexperts is the same but the estimation procedure is differentOf note in many cases a forecasted success requires varioussteps of operations therefore using the critical path methodfor forecasting is more efficient in such cases In fact manyother ways can be used for forecasting using fuzzy numberswhich we may investigate in the future Future work mayinclude the further extension of this study to other fuzzy-likestructures i e interval-valued fuzzy set intuitionistic fuzzyset Pythagorean fuzzy set spherical fuzzy set picture fuzzyset neutrosophic set etc with more discussion on real-world problems

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare no conflicts of interest

References

[1] T Grisham ldquoe Delphi technique a method for testingcomplex and multifaceted topicsrdquo International Journal ofManaging Projects in Business vol 2 no 1 pp 112ndash130 2009

[2] V Story L Hurdley G Smith and J Saker ldquoMethodologicaland practical implications of the Delphi technique in mar-keting decision-making a Re-assessmentrdquo 7e MarketingReview vol 1 no 4 pp 487ndash504 2000

[3] K Kauko and P Palmroos ldquoe Delphi method in forecastingfinancial markets- an experimental studyrdquo InternationalJournal of Forecasting vol 30 no 2 pp 313ndash327 2014

Table 2 Distances of δ(Mi Mj)

δ(Mi Mj) 1 2 3 4 5 6 7 8 9 10 11 12

1 0 036 016 043 012 016 04 013 013 013 043 0192 0 035 007 04 044 076 037 023 058 08 0163 0 042 004 0017 041 010 012 023 044 0184 0 047 051 083 046 030 060 087 0245 0 013 034 009 016 018 040 0236 0 033 007 021 026 035 0277 0 039 053 017 003 0598 Symmetry 0 014 021 044 0209 0 035 057 00610 0 021 04111 0 07212 0

Table 3 Pair of experts with arbitrary δ le 015

δij le 015 1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 1 1 12 1 13 1 1 1 14 15 1 1 16 1 17 1 18 1 19 1 110 111 112 1

12

3

4

5

6

78

9

10

11

12

Figure 4 Pairs of experts with δ le 020

Advances in Fuzzy Systems 5

[4] R Skinner R R Nelson W W Chin and L Land ldquoeDelphi method research strategy in studies of informationsystemsrdquo Communications of the Association for InformationSystems vol 37 no 2 pp 31ndash63 2015

[5] D Beiderbeck N Frevel H A VonDer Gracht S L Schmidtand V M Schweitzer ldquoPreparing conducting and analyzingDelphi surveys cross-disciplinary practices new directionsand advancementsrdquo Methods (Orlando) vol 8 Article ID101401 2021

[6] M Lawnik and A Banasik ldquoDelphi method supported byforecasting softwarerdquo Information vol 11 no 2 p 65 2020

[7] M I Piecyk and A C McKinnon ldquoApplication of the Delphimethod to the forecasting of long- term trends in road freightlogistics and related CO2 emissionsrdquo International Journal ofTransport Economics vol 40 no 2 pp 241ndash266 2013

[8] A Revez N Dunphy C Harris G Mullally B Lennon andC Gaffney ldquoBeyond forecasting using a modified Delphimethod to build upon participatory acton research in de-veloping principles for a just and inclusive energy transitionrdquoInternational Journal of Qualitative Methods vol 19 pp 1ndash122020

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

[10] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 no 4 pp Bndash1411970

[11] H-J Zimmermann ldquoFuzzy programming and linear pro-gramming with several objective functionsrdquo Fuzzy Sets andSystems vol 1 no 1 pp 45ndash55 1978

[12] Y Badulescu A-P Hameri and N Cheikhrouhou ldquoEvalu-ating demand forecasting models using multi-criteria deci-sion-making approachrdquo Journal of Advances in ManagementResearch 2021 ahead-of-print

[13] D Dubois and H Prade Fuzzy Sets and Systems 7eory andApplications Academic Press New York NY USA 1980

[14] J Guo C-J Chang Y Huang and K-P Yu ldquoA fuzzy-de-composition grey modeling procedure for management de-cision analysisrdquo Mathematical Problems in Engineeringvol 2021 Article ID 6670196 6 pages 2021

[15] A Kaufmann and M M Gupta Fuzzy Mathematical Modelsin Engineering and Management Science Elsevier SciencePublishing Company INC New York NY USA 1988

[16] Z K Nemaa and G K Aswed ldquoForecasting construction timefor road projects and infrastructure using the fuzzy PERTmethodrdquo IOP Conference Series Materials Science and En-gineering vol 1076 no 1 Article ID 012123 2021

[17] F Tahriri M Mousavi S Hozhabri Haghighi and S ZawiahMd Dawal ldquoe application of fuzzy Delphi and fuzzy in-ference system in supplier ranking and selectionrdquo Journal ofIndustrial Engineering International vol 10 no 3 p 66 2014

[18] H-C Tsai A-S Lee H-N Lee C-N Chen and Y-C LiuldquoAn application of the fuzzy Delphi method and fuzzy AHPon the discussion of training indicators for the regionalcompetition Taiwan National Skills competition in the tradeof joineryrdquo Sustainability vol 12 no 10 p 4290 2020

[19] E Vercher A Rubio and J D Bermudez ldquoFuzzy predictionintervals using credibility distributionsrdquo Engineering Pro-ceedings vol 5 no 1 p 51 2021

[20] P-T Cheng L-C Huang and H-J Lin ldquoe fuzzy Delphimethod via fuzzy statistics and membership function fittingand an application to the human resourcesrdquo Fuzzy Sets andSystems vol 112 no 3 pp 511ndash520 2000

[21] T K Roy and A Garai ldquoIntuitionistic fuzzy Delphi methodmore realistic and iterative forecasting toolrdquo Notes onIntuitionistic Fuzzy Sets vol 18 no 2 pp 37ndash50 2012

[22] A Habibi F F Jahantigh and A Sarafrazi ldquoFuzzy Delphitechnique for forecasting and screening itemsrdquo Asian Journalof Research in Business Economics and Management vol 5no 2 pp 130ndash143 2015

[23] W-L Hsu L Chen L Chen and H-L Liu ldquoe applicationfuzzy Delphi method to summarize key factors in the edu-cation of construction engineersrdquo in Proceedings of the 2ndEurasian Conference on Educational Innovations and Appli-cations- Tijus Meen Chang Singapore January 2019

[24] A Ciptono s Setiyono F Nurhidayati and R VikalianaldquoFuzzy Delphi method in education a mappingrdquo Journal ofPhysics Conference Series vol 1360 no 1360 Article ID012029 2019

[25] 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 no 23 p 6616 2019

[26] J Huang M Koopialipoor and D J Armaghani ldquoA com-bination of fuzzy Delphi method and hybrid ANN-basedsystems to forecast ground vibration resulting from blastingrdquoScientific Reports vol 10 no 1 Article ID 19397 2020

[27] G Kabir and R S Sumi ldquoIntegrating fuzzy Delphi methodwith artificial neural network for demand forecasting of powerengineering companyrdquo Management Science Letters vol 2no 5 pp 1491ndash1504 2012

[28] Y-C Chiang and H-Y Lei ldquoUsing expert decision-making toestablish indicators of urban friendliness for walking envi-ronments a multidisciplinary assessmentrdquo InternationalJournal of Health Geographics vol 15 no 1 pp 15ndash40 2016

[29] A Garai and T Kumar ldquoWeighted intuitionistic fuzzy Delphimethodrdquo Journal of Global Research in Computer Sciencevol 4 no 7 pp 38ndash42 2013

[30] J Heidary Dahooie A R Qorbani and T Daim ldquoProviding aframework for selecting the appropriate method of technol-ogy acquisition considering uncertainty in hierarchical groupdecision-making case Study interactive television technol-ogyrdquo Technological Forecasting and Social Change vol 168Article ID 120760 2021

[31] B Lee and Y Yun ldquoe pentagonal fuzzy numbersrdquo Journalof the Chungcheong Mathematical Society vol 27 no 2 2014

[32] M C J Anand and J Bharatraj ldquoeory of triangular fuzzynumberrdquo in Proceedings of the NCATM Kharghar IndiaMarch 2017

6 Advances in Fuzzy Systems

If we consider for instance experts (1 8 9) and (3 8 9)we find that δ(1 3) 016 We can assume therefore thatexperts (1 3 8 9) have given almost the same estimation Asimilar situation appears with (1 5 8) (3 8 9) and(1 3 8 9) that have almost the same estimation ereforewe can aggregate experts (1 3 5 8 9)

6 Discussion of the Results

If we consider δ le 020 we find this aggregation directly Infact using a convenient algorithm the problem can bestudied with various values of δ that is δ le 02 025 If

we wish to consider the similarity problem then a valuesmaller than (1 minus δ) gives the level of similarity which is animportant point in any a posteriori discussion betweenexperts e maximal subrelations of similarity in a re-semblance relation play the same role as classes in a simi-larity relatione second one is always disjointed but this isnot the case for the first one

7 Conclusions and Future Works

A variation of the Delphi method using TFNs has beenproposed in which the method of communication withexperts is the same but the estimation procedure is differentOf note in many cases a forecasted success requires varioussteps of operations therefore using the critical path methodfor forecasting is more efficient in such cases In fact manyother ways can be used for forecasting using fuzzy numberswhich we may investigate in the future Future work mayinclude the further extension of this study to other fuzzy-likestructures i e interval-valued fuzzy set intuitionistic fuzzyset Pythagorean fuzzy set spherical fuzzy set picture fuzzyset neutrosophic set etc with more discussion on real-world problems

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare no conflicts of interest

References

[1] T Grisham ldquoe Delphi technique a method for testingcomplex and multifaceted topicsrdquo International Journal ofManaging Projects in Business vol 2 no 1 pp 112ndash130 2009

[2] V Story L Hurdley G Smith and J Saker ldquoMethodologicaland practical implications of the Delphi technique in mar-keting decision-making a Re-assessmentrdquo 7e MarketingReview vol 1 no 4 pp 487ndash504 2000

[3] K Kauko and P Palmroos ldquoe Delphi method in forecastingfinancial markets- an experimental studyrdquo InternationalJournal of Forecasting vol 30 no 2 pp 313ndash327 2014

Table 2 Distances of δ(Mi Mj)

δ(Mi Mj) 1 2 3 4 5 6 7 8 9 10 11 12

1 0 036 016 043 012 016 04 013 013 013 043 0192 0 035 007 04 044 076 037 023 058 08 0163 0 042 004 0017 041 010 012 023 044 0184 0 047 051 083 046 030 060 087 0245 0 013 034 009 016 018 040 0236 0 033 007 021 026 035 0277 0 039 053 017 003 0598 Symmetry 0 014 021 044 0209 0 035 057 00610 0 021 04111 0 07212 0

Table 3 Pair of experts with arbitrary δ le 015

δij le 015 1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 1 1 12 1 13 1 1 1 14 15 1 1 16 1 17 1 18 1 19 1 110 111 112 1

12

3

4

5

6

78

9

10

11

12

Figure 4 Pairs of experts with δ le 020

Advances in Fuzzy Systems 5

[4] R Skinner R R Nelson W W Chin and L Land ldquoeDelphi method research strategy in studies of informationsystemsrdquo Communications of the Association for InformationSystems vol 37 no 2 pp 31ndash63 2015

[5] D Beiderbeck N Frevel H A VonDer Gracht S L Schmidtand V M Schweitzer ldquoPreparing conducting and analyzingDelphi surveys cross-disciplinary practices new directionsand advancementsrdquo Methods (Orlando) vol 8 Article ID101401 2021

[6] M Lawnik and A Banasik ldquoDelphi method supported byforecasting softwarerdquo Information vol 11 no 2 p 65 2020

[7] M I Piecyk and A C McKinnon ldquoApplication of the Delphimethod to the forecasting of long- term trends in road freightlogistics and related CO2 emissionsrdquo International Journal ofTransport Economics vol 40 no 2 pp 241ndash266 2013

[8] A Revez N Dunphy C Harris G Mullally B Lennon andC Gaffney ldquoBeyond forecasting using a modified Delphimethod to build upon participatory acton research in de-veloping principles for a just and inclusive energy transitionrdquoInternational Journal of Qualitative Methods vol 19 pp 1ndash122020

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

[10] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 no 4 pp Bndash1411970

[11] H-J Zimmermann ldquoFuzzy programming and linear pro-gramming with several objective functionsrdquo Fuzzy Sets andSystems vol 1 no 1 pp 45ndash55 1978

[12] Y Badulescu A-P Hameri and N Cheikhrouhou ldquoEvalu-ating demand forecasting models using multi-criteria deci-sion-making approachrdquo Journal of Advances in ManagementResearch 2021 ahead-of-print

[13] D Dubois and H Prade Fuzzy Sets and Systems 7eory andApplications Academic Press New York NY USA 1980

[14] J Guo C-J Chang Y Huang and K-P Yu ldquoA fuzzy-de-composition grey modeling procedure for management de-cision analysisrdquo Mathematical Problems in Engineeringvol 2021 Article ID 6670196 6 pages 2021

[15] A Kaufmann and M M Gupta Fuzzy Mathematical Modelsin Engineering and Management Science Elsevier SciencePublishing Company INC New York NY USA 1988

[16] Z K Nemaa and G K Aswed ldquoForecasting construction timefor road projects and infrastructure using the fuzzy PERTmethodrdquo IOP Conference Series Materials Science and En-gineering vol 1076 no 1 Article ID 012123 2021

[17] F Tahriri M Mousavi S Hozhabri Haghighi and S ZawiahMd Dawal ldquoe application of fuzzy Delphi and fuzzy in-ference system in supplier ranking and selectionrdquo Journal ofIndustrial Engineering International vol 10 no 3 p 66 2014

[18] H-C Tsai A-S Lee H-N Lee C-N Chen and Y-C LiuldquoAn application of the fuzzy Delphi method and fuzzy AHPon the discussion of training indicators for the regionalcompetition Taiwan National Skills competition in the tradeof joineryrdquo Sustainability vol 12 no 10 p 4290 2020

[19] E Vercher A Rubio and J D Bermudez ldquoFuzzy predictionintervals using credibility distributionsrdquo Engineering Pro-ceedings vol 5 no 1 p 51 2021

[20] P-T Cheng L-C Huang and H-J Lin ldquoe fuzzy Delphimethod via fuzzy statistics and membership function fittingand an application to the human resourcesrdquo Fuzzy Sets andSystems vol 112 no 3 pp 511ndash520 2000

[21] T K Roy and A Garai ldquoIntuitionistic fuzzy Delphi methodmore realistic and iterative forecasting toolrdquo Notes onIntuitionistic Fuzzy Sets vol 18 no 2 pp 37ndash50 2012

[22] A Habibi F F Jahantigh and A Sarafrazi ldquoFuzzy Delphitechnique for forecasting and screening itemsrdquo Asian Journalof Research in Business Economics and Management vol 5no 2 pp 130ndash143 2015

[23] W-L Hsu L Chen L Chen and H-L Liu ldquoe applicationfuzzy Delphi method to summarize key factors in the edu-cation of construction engineersrdquo in Proceedings of the 2ndEurasian Conference on Educational Innovations and Appli-cations- Tijus Meen Chang Singapore January 2019

[24] A Ciptono s Setiyono F Nurhidayati and R VikalianaldquoFuzzy Delphi method in education a mappingrdquo Journal ofPhysics Conference Series vol 1360 no 1360 Article ID012029 2019

[25] 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 no 23 p 6616 2019

[26] J Huang M Koopialipoor and D J Armaghani ldquoA com-bination of fuzzy Delphi method and hybrid ANN-basedsystems to forecast ground vibration resulting from blastingrdquoScientific Reports vol 10 no 1 Article ID 19397 2020

[27] G Kabir and R S Sumi ldquoIntegrating fuzzy Delphi methodwith artificial neural network for demand forecasting of powerengineering companyrdquo Management Science Letters vol 2no 5 pp 1491ndash1504 2012

[28] Y-C Chiang and H-Y Lei ldquoUsing expert decision-making toestablish indicators of urban friendliness for walking envi-ronments a multidisciplinary assessmentrdquo InternationalJournal of Health Geographics vol 15 no 1 pp 15ndash40 2016

[29] A Garai and T Kumar ldquoWeighted intuitionistic fuzzy Delphimethodrdquo Journal of Global Research in Computer Sciencevol 4 no 7 pp 38ndash42 2013

[30] J Heidary Dahooie A R Qorbani and T Daim ldquoProviding aframework for selecting the appropriate method of technol-ogy acquisition considering uncertainty in hierarchical groupdecision-making case Study interactive television technol-ogyrdquo Technological Forecasting and Social Change vol 168Article ID 120760 2021

[31] B Lee and Y Yun ldquoe pentagonal fuzzy numbersrdquo Journalof the Chungcheong Mathematical Society vol 27 no 2 2014

[32] M C J Anand and J Bharatraj ldquoeory of triangular fuzzynumberrdquo in Proceedings of the NCATM Kharghar IndiaMarch 2017

6 Advances in Fuzzy Systems

[4] R Skinner R R Nelson W W Chin and L Land ldquoeDelphi method research strategy in studies of informationsystemsrdquo Communications of the Association for InformationSystems vol 37 no 2 pp 31ndash63 2015

[5] D Beiderbeck N Frevel H A VonDer Gracht S L Schmidtand V M Schweitzer ldquoPreparing conducting and analyzingDelphi surveys cross-disciplinary practices new directionsand advancementsrdquo Methods (Orlando) vol 8 Article ID101401 2021

[6] M Lawnik and A Banasik ldquoDelphi method supported byforecasting softwarerdquo Information vol 11 no 2 p 65 2020

[7] M I Piecyk and A C McKinnon ldquoApplication of the Delphimethod to the forecasting of long- term trends in road freightlogistics and related CO2 emissionsrdquo International Journal ofTransport Economics vol 40 no 2 pp 241ndash266 2013

[8] A Revez N Dunphy C Harris G Mullally B Lennon andC Gaffney ldquoBeyond forecasting using a modified Delphimethod to build upon participatory acton research in de-veloping principles for a just and inclusive energy transitionrdquoInternational Journal of Qualitative Methods vol 19 pp 1ndash122020

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

[10] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 no 4 pp Bndash1411970

[11] H-J Zimmermann ldquoFuzzy programming and linear pro-gramming with several objective functionsrdquo Fuzzy Sets andSystems vol 1 no 1 pp 45ndash55 1978

[12] Y Badulescu A-P Hameri and N Cheikhrouhou ldquoEvalu-ating demand forecasting models using multi-criteria deci-sion-making approachrdquo Journal of Advances in ManagementResearch 2021 ahead-of-print

[13] D Dubois and H Prade Fuzzy Sets and Systems 7eory andApplications Academic Press New York NY USA 1980

[14] J Guo C-J Chang Y Huang and K-P Yu ldquoA fuzzy-de-composition grey modeling procedure for management de-cision analysisrdquo Mathematical Problems in Engineeringvol 2021 Article ID 6670196 6 pages 2021

[15] A Kaufmann and M M Gupta Fuzzy Mathematical Modelsin Engineering and Management Science Elsevier SciencePublishing Company INC New York NY USA 1988

[16] Z K Nemaa and G K Aswed ldquoForecasting construction timefor road projects and infrastructure using the fuzzy PERTmethodrdquo IOP Conference Series Materials Science and En-gineering vol 1076 no 1 Article ID 012123 2021

[17] F Tahriri M Mousavi S Hozhabri Haghighi and S ZawiahMd Dawal ldquoe application of fuzzy Delphi and fuzzy in-ference system in supplier ranking and selectionrdquo Journal ofIndustrial Engineering International vol 10 no 3 p 66 2014

[18] H-C Tsai A-S Lee H-N Lee C-N Chen and Y-C LiuldquoAn application of the fuzzy Delphi method and fuzzy AHPon the discussion of training indicators for the regionalcompetition Taiwan National Skills competition in the tradeof joineryrdquo Sustainability vol 12 no 10 p 4290 2020

[19] E Vercher A Rubio and J D Bermudez ldquoFuzzy predictionintervals using credibility distributionsrdquo Engineering Pro-ceedings vol 5 no 1 p 51 2021

[20] P-T Cheng L-C Huang and H-J Lin ldquoe fuzzy Delphimethod via fuzzy statistics and membership function fittingand an application to the human resourcesrdquo Fuzzy Sets andSystems vol 112 no 3 pp 511ndash520 2000

[21] T K Roy and A Garai ldquoIntuitionistic fuzzy Delphi methodmore realistic and iterative forecasting toolrdquo Notes onIntuitionistic Fuzzy Sets vol 18 no 2 pp 37ndash50 2012

[22] A Habibi F F Jahantigh and A Sarafrazi ldquoFuzzy Delphitechnique for forecasting and screening itemsrdquo Asian Journalof Research in Business Economics and Management vol 5no 2 pp 130ndash143 2015

[23] W-L Hsu L Chen L Chen and H-L Liu ldquoe applicationfuzzy Delphi method to summarize key factors in the edu-cation of construction engineersrdquo in Proceedings of the 2ndEurasian Conference on Educational Innovations and Appli-cations- Tijus Meen Chang Singapore January 2019

[24] A Ciptono s Setiyono F Nurhidayati and R VikalianaldquoFuzzy Delphi method in education a mappingrdquo Journal ofPhysics Conference Series vol 1360 no 1360 Article ID012029 2019

[25] 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 no 23 p 6616 2019

[26] J Huang M Koopialipoor and D J Armaghani ldquoA com-bination of fuzzy Delphi method and hybrid ANN-basedsystems to forecast ground vibration resulting from blastingrdquoScientific Reports vol 10 no 1 Article ID 19397 2020

[27] G Kabir and R S Sumi ldquoIntegrating fuzzy Delphi methodwith artificial neural network for demand forecasting of powerengineering companyrdquo Management Science Letters vol 2no 5 pp 1491ndash1504 2012

[28] Y-C Chiang and H-Y Lei ldquoUsing expert decision-making toestablish indicators of urban friendliness for walking envi-ronments a multidisciplinary assessmentrdquo InternationalJournal of Health Geographics vol 15 no 1 pp 15ndash40 2016

[29] A Garai and T Kumar ldquoWeighted intuitionistic fuzzy Delphimethodrdquo Journal of Global Research in Computer Sciencevol 4 no 7 pp 38ndash42 2013

[30] J Heidary Dahooie A R Qorbani and T Daim ldquoProviding aframework for selecting the appropriate method of technol-ogy acquisition considering uncertainty in hierarchical groupdecision-making case Study interactive television technol-ogyrdquo Technological Forecasting and Social Change vol 168Article ID 120760 2021

[31] B Lee and Y Yun ldquoe pentagonal fuzzy numbersrdquo Journalof the Chungcheong Mathematical Society vol 27 no 2 2014

[32] M C J Anand and J Bharatraj ldquoeory of triangular fuzzynumberrdquo in Proceedings of the NCATM Kharghar IndiaMarch 2017

6 Advances in Fuzzy Systems