Expert Systems with Applications 42 (2015) 2198–2204

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Expert Systems with Applications

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Comparison of some aggregation techniques using group analytichierarchy process

http://dx.doi.org/10.1016/j.eswa.2014.09.0600957-4174/� 2014 Published by Elsevier Ltd.

⇑ Corresponding author. Tel.: +386 1 32 03 604; fax: +386 1 25 72 297.E-mail addresses: petra.groselj@bf.uni-lj.si (P. Grošelj), lidija.zadnik@bf.uni-lj.si

(L. Zadnik Stirn), nadiray@istanbul.edu.t (N. Ayrilmis), manja.kuzman@bf.uni-lj.si(M.K. Kuzman).

Petra Grošelj a, Lidija Zadnik Stirn a, Nadir Ayrilmis b, Manja Kitek Kuzman a,⇑a University of Ljubljana, Biotechnical Faculty, Jamnikarjeva 101, 1000 Ljubljana, Sloveniab Istanbul University, Forestry Faculty, Bahcekoy 34473, Sariyer, Istanbul, Turkey

a r t i c l e i n f o

Article history:Available online 13 October 2014

Keywords:Decision analysisDecision support systemsAnalytic hierarchy processGroup decision makingConstructionWood

a b s t r a c t

Group decision making is an important part of multiple criteria decision making and the analytic hierar-chy process (AHP). The aim of this paper was to compare group AHP methods. Seven simple group AHPaggregation techniques that could be attractive for applications selected from the vast array of group AHPmodels proposed in the literature were selected for evaluation. We developed three new measures ofevaluation: group Euclidean distance, group minimum violations, and distance between weights forthe purpose of evaluation. The results of seven group AHP methods of the theoretical example were eval-uated by three new evaluation measures, satisfactory index and fitting performance index. Furthermore,a case study of a decision making problem from the construction engineering field was performed andnine group AHP aggregation techniques, seven of them formerly presented and two new two stage groupapproaches were applied. Finally, the case study was evaluated using all five measures for each of thenine group decision making methods. The results showed that not all group AHP methods are equallyconvenient and that the selection of the method depended on the specific application.

� 2014 Published by Elsevier Ltd.

1. Introduction

Group decision making is becoming an increasingly importantpart of multiple criteria decision making (Ahmad, Saman,Mohamad, Mohamad, & Awang, 2014; De Brucker, Macharis, &Verbeke, 2013; Ishizaka & Labib, 2011a; Kuzman, Grošelj,Ayrilmis, & Zbasnik-Senegacnik, 2013; Ren, Fedele, Mason,Manzardo, & Scipioni, 2013; Skorupski, 2014; Wang, Peng, Zhang,& Chen, 2014; Yu & Lai, 2011). Multiple stakeholders can contributea variety of experiences, expertise and perspectives, and a group canbetter deal with the complexity of the problem than a singledecision maker (DM). The analytic hierarchy process (AHP) (Saaty,1980) is deemed to be one of the most appropriate methods forgroup multiple criteria decision making (Peniwati, 2007). In groupAHP, four basic approaches for deriving the group priority vectorfrom comparison matrices of DMs are suggested (Dyer & Forman,1992; Ishizaka & Labib, 2011b; Lai, Wong, & Cheung, 2002). Thegroup can try to reach a consensus on a meeting, first in developingthe hierarchy and then in generating pairwise comparisons. If theycannot reach a consensus regarding a particular judgment, they can

vote or try to achieve a compromise. Social choice theory withvoting systems (Taylor & Pacelli, 2008) can be combined with AHP(Srdjevic, 2007). The aggregation of individual priorities (AIP) andthe aggregation of individual judgments (AIJ) are two main mathe-matical aggregating methods (Forman & Peniwati, 1998). The mostwidely used aggregation technique is the weighted geometric meanmethod for AIJ (WGM–AIJ), which has been applied in numerousapplications (Ananda & Herath, 2008; Cortés-Aldana, García-Melón, Fernández-de-Lucio, Aragonés-Beltrán, & Poveda-Bautista,2009; de Luca, 2014; Lee, Chang, & Lin, 2009; Srdjevic, Lakicevic, &Srdjevic, 2013; Sun & Li, 2009).

The decision maker is satisfied if the final group priorities are asclose as possible to his judgments, priorities or his ranking ofcriteria. Unlike the single DM case, in the group case there arenot many studies comparing the results of different AHP groupapproaches, which results in a lack of measures for comparinggroup methods (Hosseinian, Navidi, & Hajfathaliha, 2012; Huang,Liao, & Lin, 2009).

The main objective of this paper is to develop new measures forevaluating AHP group methods. We proposed three new measures:group Euclidean distance (GED), group minimum violations (GMV),and distance between weights (WD). The second aim of the presentstudy was to select the most appropriate group AHP method foremployment in the applications. Although WGM–AIJ is the mostoften applied method it is not necessary the most suitable method.

P. Grošelj et al. / Expert Systems with Applications 42 (2015) 2198–2204 2199

For the comparative study we selected WGM–AIP, weighted arith-metic mean method (WAM), and some recently presented modelsin addition to WGM–AIJ (Hosseinian et al., 2012; Huang et al.,2009; Regan, Colyvan, & Markovchick-Nicholls, 2006; Sun &Greenberg, 2006). These models were selected because they areeasy to understand and could be attractive for many applications.

The three new measures, the satisfactory (SAT) index (Huanget al., 2009) and the fitting performance (FP) index (Hosseinianet al., 2012) were employed in the evaluation study, whichcompared seven group AHP methods in a theoretical example.Additionally, a case study that compared the criteria for selectingbuilding construction method and material for an industrial typeof building was performed. In the study, three groups of stakehold-ers were included in the decision making. To aggregate thestakeholders’ judgments we suggest utilizing AIJ within the groupsfirst and then AIP between the groups. In the paper we proposedtwo new stage group approaches, namely WGM–WAM andWGM–LW-AHP. Seven known group AHP methods and two newlyproposed were applied in the case study for deriving group prior-ities. The results of nine group AHP methods were compared withfive evaluation measures: GED, GMV, WD, SAT index, and FP index.

The next section offered a brief description of group AHP meth-ods applied in the study. Further, we proposed the measures forevaluating the group AHP methods. The theoretical part of paperwas followed by the theoretical example and a case study. Finally,some conclusions were provided.

2. Revision of group AHP prioritization methods

Let n be the number of criteria (or alternatives) and m the num-ber of DMs. The standard AHP 1–9 scale (Saaty, 1980) was used forthe judgments of each DM, which were written in the comparison

matrices AðkÞ ¼ ðakijÞn�n

; k ¼ 1; . . . ;m: If the DMs’ opinions were notequally important, the relative importance weight of kth DM’s opin-ion was denoted by ak, for k = 1, . . . ,m, with ak > 0 and

Pmk¼1ak ¼ 1.

There are many methods for deriving priority vectors but in thisstudy we primarily used the eigenvector method (Saaty, 1980)

resulting in wk ¼ ðwk1; . . . ;wk

nÞT; k ¼ 1; . . . ;m as DMs’ priority vec-

tors. In the study we focused on the additive error structureaij ¼ wi

wjþ eij for inconsistent comparison matrix A and used additive

normalization conditionPn

i¼1wi ¼ 1 for all priority vectors for one ormore DMs (Sun & Greenberg, 2006). The consistency of judgments inthe comparison matrix A was measured by the consistency ratio

CRA ¼ kA;max�nðn�1ÞRIn

, where RIn was the average random consistency index.

A consistency ratio of less than 0.1 was considered acceptable.Of the AIJ methods WGM–AIJ is the only method that meets

several required axiomatic conditions, such as the reciprocalproperty (Aczél & Alsina, 1986). The individual judgmentsak

ij; k ¼ 1; . . . ;m were aggregated into a group judgment aWGMMij by

weighted geometric mean:

aWGMMij ¼

Ymk¼1

ðakijÞ

ak ð1Þ

The group priority vector was derived from the group compar-ison matrix AWGMM by the eigenvector method.

The AIP is a suitable method when a group is non-homogenousand consists of stakeholders from different fields. Both the WAMand WGM approaches can be used for the AIP. First, each DM k,k = 1, . . . ,m, applies for the eigenvector method for deriving the

priority vector wk ¼ ðwk1; . . . ;wk

nÞT

from its comparison matrix.The individual priority vectors are then synthesized into the group

priority vector w ¼ ðw1; . . . ;wnÞT using the weighted arithmeticmean (WAM) (2) or weighted geometric mean (WGM–AIP) (3):

wi ¼Xm

k¼1

akwki ; i ¼ 1; . . . ;n; ð2Þ

wi ¼Ymk¼1

ðwki Þ

ak; i ¼ 1; . . . ;n ð3Þ

2.1. LW-AHP model

The Lehrer–Wagner (LW) model (Lehrer & Wagner, 1981) wasadopted for the AHP by Regan et al. (2006). In this study, it wasassigned as the LW-AHP model. The base of this model was placedin the philosophy of negotiation (Regan et al., 2006) and used forthe AIP. The initial priority vectors 0wk ¼ ð0wk

1; . . . ; 0wknÞ

T

k = 1, . . . ,m were derived by the eigenvector method from DMs’comparison matrices. They were revised according to weights ofrespect, wij

s , which were based on the strength of the differencesbetween the priorities of DMs for each criterion (or alternative) s,s = 1, . . . ,n.

wijs ¼

1� j0wis � 0wj

sjPmj¼1ð1� j0wi

s � 0wjsjÞ; ð4Þ

The weights are gathered in the matrices of weights of respectWs ¼ ðwij

s Þm�m. Let 0Ps denote the vector of DMs’ priorities of thecriterion s: 0Ps ¼ ð0w1

s ; . . . ; 0wms Þ

T . The updated priorities of the cri-terion s after the first round of the aggregation result in1Ps ¼Ws

0Ps ¼ ð1w1s ; . . . ; 1wm

s ÞT . The process of aggregation was

repeated with the same weights of respect: rPs ¼ ðWsÞr 0Ps. As rapproaches infinity, the revised priorities of criterion s convergedtowards the final priority cws ¼ cw1

s ¼ . . . ¼ cwms , which was equal

for all DMs and where c was the number of iterations needed toreach the convergence.

2.2. GWLS model

Sun and Greenberg (2006) proposed a GWLS model for derivinggroup priorities:

minXm

k¼1

Xn

j¼1

Xn

i¼1

akðakijwj �wiÞ

2

subject to :Xn

i¼1

wi ¼ 1;

wi > 0; i ¼ 1; . . . ;n

ð5Þ

and proved that the solution of model (5) is given by

w ¼ C�1k; ð6Þ

where

C ¼ eA þ eAT �K; eA ¼ ð~aijÞn�n; ~aij ¼Xm

k¼1

akaðkÞij ;

aij ¼Xm

k¼1

ak aðkÞij

� �2þ 1

� �; gj ¼

Xn

i¼1

aij; K ¼ diagðg1;g2; . . . ;gnÞ

and C�1 ¼ ð�cijÞn�n; k ¼ k2;k2; . . . ;

k2

� �T

; k ¼ 2=Xn

i¼1

Xn

j¼1

�cij

!:

ð7Þ

2.3. PD&R model

Huang et al. (2009) proposed a group AHP model considering thedifferences of preference among criteria (or alternatives) and theranks of the criteria (or alternatives) for each DM. The priority vec-tor of DM k, k = 1, . . . ,m was originally derived by the logarithmicleast squares method (Crawford & Williams, 1985), but we used

2200 P. Grošelj et al. / Expert Systems with Applications 42 (2015) 2198–2204

the eigenvector method instead. The preferential differencebetween criterion i and criterion j of DM k is defined as:

hkij ¼ jwk

i �wkj j: ð8Þ

The preferential differences are used for the AIJ which are gath-ered in Adiff ¼ ðadiff

ij Þn�n:

adiffij ¼

Ymk¼1

ðakijÞ

hkij

! 1Pm

k¼1hkij for i; j ¼ 1; . . . ; n ð9Þ

The eigenvector method is employed for deriving the priority

vector wdiff from Adiff. The vector of adjusting weights wrank, consid-ering the preferential ranks, is defined as

wranki ¼ diPn

i¼1di; ð10Þ

where di denotes the sum of rank-adjusting factors for all DMs forthe criterion i, di ¼

Pmk¼1d

ki ¼

Pmk¼1

nrk

i, and rk

i the preferential rank

of the criterion i according to DM k.The final group priorities of the PD&R model are the normalized

products of wdiffi and wrank

i :

wi ¼wdiff

i wrankiPn

i¼1wdiffi wrank

i

; i ¼ 1; . . . ;n ð11Þ

The PD&R is an appealing model as it is the only model whichincludes the individual ranks of criteria.

2.4. LP-GW-AHP model

The approaches for linking the AHP and the data envelopmentanalysis are studied frequently. Thus we decided to includeLP-GW-AHP model (Hosseinian et al., 2012), which employedconcepts from the data envelopment analysis. The criteria (or alter-natives) i = 1, . . . ,n are viewed as decision making units. Criteria(alternatives) are viewed as outputs so the model has n outputs.The WGM is used for the AIJ. The group priorities w1, . . . ,wn werederived by solving the linear programming model (12):

max z subject to wi P z, i = 1, . . . ,n,Xn

j¼1

Ymk¼1

ðakijÞ

ak

!v j �wi ¼ 0; i ¼ 1; . . . ;n;

Xn

i¼1

wi ¼ 1

v i �1b

wi P 0; i ¼ 1; . . . ;n;

v i �1n

wi 6 0; i ¼ 1; . . . ;n;

wi P 0; v i P 0; i ¼ 1; . . . ;n;

ð12Þ

where b ¼min maxi1ri

Pnj¼1aWGMM

ij rj

� �;maxi

1ci

Pnj¼1aWGMM

ij cj

� �n oand

r1, . . . ,rn and c1, . . . ,cn are the row sums and the column sums of

group comparison matrix AWGMM, respectively.

3. Measures for evaluating group AHP methods

For comparisons of group AHP methods results, we need suit-able measures of evaluation. We found only two existing measuresfor evaluating different group AHP methods in the literature. TheFP index (Hosseinian et al., 2009), which is measured by the Euclid-ean distance:

FP ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1n2

Xn

i¼1

Xn

j¼1

aWGMMij �wi

wj

� �2vuut ð13Þ

The FP index prefers methods which employ AWGMM and doesnot shed light on the distance between the individual judgmentsor priorities and the final group priorities. The SAT index (Huanget al., 2009) composes differences between the priorities anddifferences between the ranks:

q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiYmk¼1

1n

Xn

i¼1

ðgki Þ

1ki

!m

vuut ; ð14Þ

where gki ¼

ðjwki�wi jÞ

�1Pn

i¼1ðjwk

i�wi jÞ

�1 and 1ki ¼ jrk

i � rij, i = 1, . . . ,n. A higher SAT

index indicates greater satisfaction of DMs with the group result.The drawbacks of the SAT index are that it is not a continuous func-tion and it is not defined if wk

i ¼ wi.We developed three new evaluation measures because of the

deficiencies of the existing measures. Two measures are general-ized from Euclidian distance and minimum violations from oneDM’s case (Srdjevic, 2005; Mikhailov, 2006):

GED ¼ 1m

Xm

k¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i¼1

Xn

j¼1

akij �

wi

wj

� �2vuut ð15Þ

The group Euclidian distance (GED) measures the average sumof distances between the judgments of DMs and the related ratiosof group priorities vector.

GMV ¼ 1m

Xm

k¼1

Xn�1

i¼1

Xn

j¼iþ1

Iij;

Iij ¼

1 if wi > wj and aij < 1;1 if wi < wj and aij > 1;0:5 if wi ¼ wj and aij–1;0:5 if wi–wj and aij ¼ 1;0 otherwise

8>>>>>><>>>>>>:ð16Þ

The group minimum violations (GMV) averages the violationsof each DM associated with the order reversals.

For evaluation of distances between DMs’ priorities and grouppriorities we proposed the distance between weights (WD):

WD ¼ 1m

Xm

k¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i¼1

ðwki �wiÞ

2

vuut ð17Þ

4. Theoretical example

Let us consider three equally important DMs comparing fourcriteria and providing comparison matrices A, B and C.

A ¼

1 5 3 115 1 1

517

13 5 1 1

2

1 7 2 1

2666437775; B ¼

1 1 2 21 1 7 612

17 1 1

12

16 1 1

2666437775; C ¼

1 12 2 7

2 1 5 912

15 1 9

17

19

19 1

2666437775ð18Þ

Their consistency ratios are 0.0385, 0.0674, and 0.0909, respec-tively, which indicates that all comparison matrices are acceptablyconsistent. The group priority vectors are obtained using theWGM–AIJ, WGM–AIP, WAM, LW-AHP, GWLS, PD&R, and theLP-GW-AHP methods. The results are presented in Table 1.The rankings of criteria for the WAM and LW-AHP are identicaland the priorities are similar. This similarity is due to the fact thatthe LW-AHP is a type of WAM with unfixed weights. The rankingsobtained by the other five methods are equal and the priorities ofthe WGM–AIJ, WGM–AIP and LP-GW-AHP are similar as they all

Table 1The group priorities and the ranking (r) of criteria, according to the WGM–AIJ, WGM–AIP, WAM, LW-AHP, GWLS, PD&R and LP-GW-AHP model.

Criteria WGM–AIJ r WGM–AIP r WAM r LW-AHP r GWLS r PD&R r LP-GW-AHP r

C1 0.38 1 0.38 1 0.31 2 0.31 2 0.63 1 0.45 1 0.38 1C2 0.30 2 0.30 2 0.36 1 0.37 1 0.15 2 0.42 2 0.30 2C3 0.18 3 0.18 3 0.15 4 0.15 4 0.15 3 0.07 3 0.18 3C4 0.14 4 0.14 4 0.17 3 0.16 3 0.07 4 0.06 4 0.14 4

P. Grošelj et al. / Expert Systems with Applications 42 (2015) 2198–2204 2201

use the geometric mean within the aggregation. The priorities ofthe GWLS and PD&R method greatly differ from the priorities ofother methods.

Table 2 shows the evaluation of the results from Table 1 accord-ing to the five presented evaluation measures: FP index (13), SATindex (14), group ED (15), group MV (16) and WD (17). The FP indexprefers the WGM–AIJ and LP-GW-AHP, which employ the geomet-ric mean of judgments. The SAT index indicates that the WAM andLW-AHP are the best methods. Group ED favors the LW-AHPmethod. Group MV is equal for all group AHP methods. The WD pre-fers the LW-AHP and WAM. The results showed that in our examplethe LW-AHP was the best group AHP method according to most cri-teria. The methods WGM–AIJ, WGM–AIP, WAM and LP-GW-AHPhad some advantages and weaknesses and their evaluation mea-sures placed them in the middle of the rankings. The priorities ofthe GWLS and PD&R methods were outstanding and the evaluationmeasures confirmed that other methods provided more suitablegroup priority vectors.

5. Case study

A study was conducted to select the most suitable constructiontype among solid wood construction, wood-frame construction,aerated concrete, brick construction, and steel construction, forindustrial buildings. The construction of prefabricated industrialwooden buildings today is supported by strong arguments. Innova-tions and improvements introduced in the early 1980s have helpedto promote wooden prefabricated buildings around the world.Further, in construction of industrial buildings, the followingadjustments are very important: (a) the transition from on-siteconstruction to industrial prefabrication, (b) the transition fromstick-building to modular construction, (c) an increased use of gluedlumber in construction, (d) the development of environmentallyfriendly solutions for wood protection, and (e) the shift from smallto large panel system (LPS) construction (Leskovar Zegarac &Premrov, 2012). The choice of the material is the most importantdecision and it has long-term consequences for the owner of thestructure (Johnson, 1990).

The goal of the case study was to select the most suitableconstruction type for industrial buildings. According to the criteriadifferent construction materials for industrial buildings were com-pared: solid wood construction, wood-frame construction, aeratedconcrete, brick construction, and steel construction. The research

Table 2The results of evaluation of group priorities according to the FP index, SAT index, GED, GM

WGM–AIJ WGM–AIP WAM

FP index 0.162 0.168 0.340Ranks 2 3 4SAT index 0.404 0.404 0.541Ranks 7 5 1GED 9.288 9.271 9.292Ranks 3 2 5GMV 1.833 1.833 1.833Ranks 1 1 1WD 0.286 0.286 0.255Ranks 4 3 2

was carried out by survey, which included experts from severalEuropean countries, such as architect engineers, construction engi-neers, and wood-technology engineers. The selection of the criteriawas conducted by the Delphi method (Gupta & Clarke, 1996). Inthe first round of the decision making process experts selectedeighteen most important construction criteria and grouped theminto five categories. In the second round nine most importantcriteria out of the eighteen were selected. They were classified intothree groups: (1) Mechanical and technical criteria (energy effi-ciency, load capacity, form and dimension planning limitations-FDP limitations, fire safety), (2) Economic criteria (constructioncosts, depreciation costs, construction time), and (3) Residentialcriteria (aesthetics, quality of living). These nine criteria wereincluded in the AHP decision tree (Fig. 1). The pairwise compari-sons of criteria were conducted by twenty-seven experts. Thehighly inconsistent comparison matrices with CR > 0.18 wererejected. The most inconsistent judgments in comparison matriceswith 0.1 < CR < 0.18 were adjusted. In the end, fifteen acceptedcomparison matrices had CR < 0.1.

Three groups of experts consisting of fivewood-technologyengineers, four architect engineers, and six construction engineers,with different respects to the constructing criteria, took part in theresearch. Since more homogenous judgments within the groupsthan between the groups were expected, we suggested two newmethods: the WGM–WAM and WGM–LW-AHP, which were usedfor the AIJ within each group of engineers and for the AIP of thethree expert groups. In both methods we used the WGM–AIJ forderiving the priority vectors of each group of experts. In theWGM–WAM method, the WAM was applied and in the WGM–LW-AHP method, the LW-AHP was applied for aggregating threegroup vectors into the final priority vector. The results of all eightgroup methods are presented in Table 3. All methods put firesafety, load capacity and energy efficiency first, and aesthetics last.Despite that, the PD&R method stands out with high priorities.

An evaluation of all nine methods with the same five measuresused in the theoretical example was conducted. The results arepresented in Table 4. The measures evaluated the methods veryheterogeneously. This indicated that no group AHP method is‘‘the best’’. Nevertheless, the methods with the highest evaluationsaccording to most of the measures were WGM–WAM and WGM–LW-AHP and they evaluated better than the unmodified WAMand LW-AHP methods, respectively. This signified that it was use-ful to create smaller homogenous groups of DMs. The evaluation of

V and WD and the rankings of the group AHP methods.

LW-AHP GWLS PD&R LP-GW-AHP

0.352 1.789 2.220 0.1625 6 7 10.539 0.406 0.413 0.4042 4 3 69.220 11.210 10.279 9.2911 7 6 41.833 1.833 1.833 1.8331 1 1 10.251 0.471 0.305 0.2861 7 6 5

Fig. 1. The AHP decision tree with criteria and alternatives in the construction of industrial buildings.

Table 3The group priorities and the rankings (r) of construction criteria, according to the WGM–AIJ, WGM–AIP, WAM, WGM–WAM, LW-AHP, WGM–LW-AHP, GWLS, PD&R and LP-GW-AHP model.

WGM–AIJ r WGM–AIP r WAM r WGM–WAM r LW-AHP r WGM–LW-AHP r GWLS r PD&R r LP-GW-AHP r

FDP limitations 0.08 6 0.08 6 0.09 5 0.08 6 0.09 5 0.08 6 0.07 6 0.04 5 0.08 6Quality of living 0.06 8 0.06 8 0.06 8 0.06 8 0.06 8 0.06 8 0.06 8 0.02 7 0.06 8Construction costs 0.09 5 0.09 5 0.08 6 0.09 5 0.08 6 0.09 5 0.08 4 0.03 6 0.09 5Construction time 0.06 7 0.06 7 0.06 7 0.06 7 0.06 7 0.06 7 0.06 7 0.02 8 0.06 7Depreciation costs 0.09 4 0.09 4 0.09 4 0.09 4 0.09 4 0.09 4 0.08 5 0.05 4 0.09 4Fire safety 0.24 1 0.24 1 0.23 1 0.24 1 0.23 1 0.24 1 0.26 1 0.39 1 0.24 1Aesthetics 0.04 9 0.04 9 0.04 9 0.04 9 0.04 9 0.04 9 0.05 9 0.01 9 0.04 9Load capacity 0.22 2 0.22 2 0.22 2 0.22 2 0.22 2 0.22 2 0.22 2 0.35 2 0.22 2Energy efficiency 0.13 3 0.13 3 0.12 3 0.13 3 0.12 3 0.13 3 0.11 3 0.09 3 0.13 3

2202 P. Grošelj et al. / Expert Systems with Applications 42 (2015) 2198–2204

Table 4The results of evaluation of group priorities of construction criteria according to the FP index, SAT index, GED, GMV and WD and the rankings of the group AHP methods.

WGM–AIJ WGM–AIP WAM WGM–WAM LW-AHP WGM–LW-AHP GWLS PD&R LP-GW-AHP

FP index 0.364 0.371 0.341 0.337 0.343 0.338 0.413 6.526 0.361Ranks 6 7 3 1 4 2 8 9 5SAT index 0.278 0.274 0.274 0.323 0.274 0.320 0.294 0.165 0.279Ranks 5 6 7 1 8 2 3 9 4GED 19.163 19.168 19.544 19.234 19.406 19.231 19.467 57.142 19.166Ranks 1 3 8 5 6 4 7 9 2GMV 9.433 9.433 9.500 9.433 9.500 9.433 9.567 9.367 9.433Ranks 2 2 7 2 7 2 9 1 2WD 0.215 0.215 0.215 0.215 0.215 0.215 0.217 0.287 0.215Ranks 5 2 7 4 1 3 8 9 6

Table 5The priorities of alternatives for construction of industrial buildings regarding the construction criteria.

Construction type FDPlimitations

Quality ofliving

Constructioncosts

Constructiontime

Depreciationcosts

Firesafety

Aesthetics Loadcapacity

Energyefficiency

Steel construction 0.23 0.11 0.20 0.29 0.11 0.20 0.05 0.20 0.20Solid wood

construction0.22 0.26 0.12 0.24 0.07 0.20 0.38 0.20 0.20

Wood – frame 0.22 0.26 0.27 0.29 0.16 0.20 0.38 0.20 0.20Aerated concrete 0.22 0.13 0.18 0.12 0.52 0.20 0.04 0.20 0.20Brick construction 0.12 0.24 0.23 0.06 0.14 0.20 0.15 0.20 0.20

the PD&R method was the least favorable, followed by the GWLSmethod.

Five different types of construction materials were assessedseparately for each of the nine key criteria of construction. Theweighting coefficients of the construction costs criterion wereselected on the basis of average costs per square meter of selectedwall types by chosen manufacturers. Depreciation costs wereassessed in relation to the service life of the material and construc-tion costs. The criterion of form and dimension planning

limitations was estimated on the basis of indicators such asfunctionality, span possibility, multistory construction, systemsolutions and surface efficiency. Factors such as prefabricationlevel, drying, transport, knowledge and experience in using the ele-ments affected the estimation of the construction time criterion.The quality of living was assessed on the basis of health andpsychological factors. The weighting coefficients for the aestheticscriterion were selected on the basis of the survey. For three out ofnine criteria (load capacity, fire safety, and energy efficiency) the

Table 6The final priorities and ranks (r) of different types of construction for industrial buildings.

WGM–AIJ r WGM–AIP r WAM r WGM–WAM r LW-AHP r WGM–LW-AHP r GWLS r PD&R r LP-GW-AHP r

Steel frame 0.190 4 0.190 4 0.188 4 0.189 4 0.189 4 0.189 4 0.188 4 0.195 4 0.190 4Solid wood 0.195 3 0.195 3 0.197 3 0.196 3 0.197 3 0.196 3 0.199 3 0.196 3 0.195 3Wood frame 0.219 1 0.219 1 0.221 1 0.220 1 0.220 1 0.220 1 0.222 1 0.206 2 0.219 1Concrete 0.214 2 0.215 2 0.214 2 0.213 2 0.214 2 0.213 2 0.208 2 0.212 1 0.214 2Brick 0.183 5 0.182 5 0.181 5 0.183 5 0.181 5 0.183 5 0.183 5 0.192 5 0.183 5

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parameters were set as the construction standards. With regard tothe fact that with proper planning requirements of limited stan-dards for each of these materials can be met, all construction typeswere ascribed the same weighting coefficients for the criteria ofload capacity, fire safety, and energy efficiency. The results are pre-sented in Table 5.

The final priorities for the types of construction given in Table 6were calculated as the weighted sums of priorities from Table 5,where the weights in the weighted sums are the priorities fromTable 3. The results of all group methods, except PD&R, indicatedthat wood frame construction was the most appropriate for indus-trial building. Concrete construction was placed second, followedby solid wood, steel frame and brick construction as the last. Letus comment that the differences between the priorities of differenttypes of construction are small. So even though brick constructionwas ranked last, it did not mean that brick construction is com-pletely inappropriate for industrial building.

The findings of the study serve as a starting point for investors,civil engineers, architects and the others involved in buildingprocess to understand the importance of the selection criteria,however, further studies on a larger scale are needed to confirmthese observations. On the methodological part, the group resultswere compared with the five evaluation measures GED, GMV,WD, SAT index, FP index. The results of the evaluation showed thatin the case of a non-homogenous group of decision makers, divid-ing the group into smaller, homogenous groups and applying thenewly developed WGM–WAM or WGM–LW-AHP methods pro-duced better results. We do not recommend the PD&R and GWLSmethods until their regularity is more thoroughly investigated.

6. Conclusions and further work

In order to evaluate group AHP methods we developed threenew measures: group Euclidean distance, group minimum viola-tions, and distance between weights for evaluating group AHPmethods. We applied these new measures in addition to the exist-ing SAT index and FP index measures in a theoretical example andin a case study of industrial building construction. The aim of thetheoretical example was evaluation of seven carefully selectedAHP group methods techniques, which are appealing for use inmany applications. Our comparison indicated that the LW-AHPmodel performed the best of the selected AHP group methods.Therefore, we recommend applying the LW-AHP model to real-world group decision making scenarios.

The goal of the case study was to select the most suitableconstruction material for industrial buildings based on ninemechanical, technical, economic, and residential criteria. Threegroups of engineers took part in the survey. We presumed thatjudgments will be more homogenous within each group ofengineers than between groups. We proposed two new two stagegroup approaches, WGM–WAM and WGM–LW-AHP to aggregateindividual judgments into group priority. We applied seven groupAHP methods from the theoretical example and two newapproaches to evaluate five different types of construction materi-als in the case study. The results showed that wood frameconstruction was the most appropriate for industrial buildings.

The advantages of wood as a construction material with lowerembodied global warming potential, and embodied carbon posi-tively associated with well-being, aesthetic and eco-friendliness,and realistic end-of-life disposal options (Praznik, Butala, &Zbašnik-Senegacnik, 2014).

The AHP group methods have been only evaluated in twoexamples, which is the main limitation of our study. Thereforesome open issues remain, which should be studied in the future.For each method its robustness and the stability of the solutionshould be studied. In addition, theoretical analysis of evaluationmeasures should be investigated and the correlation betweenthem examined. The results of evaluation depend on the definitionof the evaluation measure. The best result was achieved by themethod that minimizes this measure. In this way the evaluationmeasure can become a group method.

Acknowledgements

Manja Kuzman would like to acknowledge the SlovenianResearch Agency for financial support within the frame of theprogram P4-0015 and Ministry of Education, Science and SportRS in the frame of the WoodWisdom-Net+ Project W3B WoodBelieve.

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