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International Journal of Systems Science: Operations &Logistics

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An intuitionistic fuzzy-grey superiority andinferiority ranking method for third-party reverselogistics provider selection

Madjid Tavana, Mohsen Zareinejad & Francisco J. Santos-Arteaga

To cite this article: Madjid Tavana, Mohsen Zareinejad & Francisco J. Santos-Arteaga (2016):An intuitionistic fuzzy-grey superiority and inferiority ranking method for third-party reverselogistics provider selection, International Journal of Systems Science: Operations & Logistics,DOI: 10.1080/23302674.2016.1256448

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INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE: OPERATIONS & LOGISTICS, http://dx.doi.org/./..

An intuitionistic fuzzy-grey superiority and inferiority ranking method forthird-party reverse logistics provider selection

Madjid Tavanaa,b, Mohsen Zareinejadc and Francisco J. Santos-Arteagad,e

aDistinguished Chair of Business Systems and Analytics, La Salle University, Philadelphia, PA, USA; bBusiness Information Systems Department,Faculty of Business Administration and Economics, University of Paderborn, Paderborn, Germany; cYoung Researchers and Elite Club, ShirazBranch, Islamic Azad University, Shiraz, Iran; dSchool of Economics and Management, Free University of Bolzano, Bolzano, Italy; eInstitutoComplutense de Estudios Internacionales, Universidad Complutense de Madrid, Madrid, Spain

ARTICLE HISTORYReceived February Accepted October

KEYWORDSThird-party reverse logistics;intuitionistic fuzzy set;TOPSIS; ANP; superiority andinferiority rankings; strategicreporting

ABSTRACTOrganisations often outsource reverse logistics (RL) to third-party RL providers (3PRLPs) to focus ontheir primary business and reduce costs. The existence of multiple criteria available for choosing a3PRLP, which are sometimes contradictory and yet related to each other, has led decision-makers toconsider the development of multi-criteria decision-making models. The purpose of this study is todevelop a hybrid model integrating the analytic network process (ANP) and the intuitionistic fuzzy-grey superiority and inferiority ranking (IFG-SIR) process to help an industrial production group selecta 3PRLP. The ANP method is used to analyse the relationships among the different selection criteriaand to obtain a weight indicating the relative importance of each criterion. The best 3PRLP is chosenusing the IFG-SIR process. The classical SIR technique requires a sufficient amount of data while rely-ingon the technique for orderpreferenceby similarity to ideal solutions and simple additiveweightedmethods. We use intuitionistic fuzzy sets to account for the subjectivity inherent to the potentiallystrategic opinions of the experts and of grey relation analysis to simplify the ranking process. Wepresent a real-world case study to exhibit the applicability and demonstrate the efficacy of the pro-posed model.

1. Introduction

Product quality, customer satisfaction, and presence incompetitive markets are among the fundamental require-ments for any organisation nowadays. Indeed, these char-acteristics have become organisational standards, leadingcompanies to search for other factors that influence thepurchasing decisions of potential customers. For this rea-son, reverse logistics (RL) has become a key factor con-tributing to the success of many organisations.

The increasing importance of RL has led many manu-facturers to design dismantling and reconstructionmeth-ods as a part of their sustainable development initiatives(Grenchus, Johnson, &McDonnell, 2001; Min, Jeung Ko,& SeongKo, 2006).Moreover, ecologically conscious cus-tomers tend to spend more on environmentally friendlyproducts, increasing the income of those companies thatuse RL to comply with the requirements of such con-sumers (Meade & Sarkis, 2002). In this regard, Senthil,Srirangacharyulu, and Ramesh (2014) and Zhu, Sarkis,and Lai (2008) emphasised the significance of the rela-tionship between the environment and RL.

CONTACT Madjid Tavana tavana@lasalle.edu

After acknowledging the importance of RL, manymanufacturers and retailers have outsourced their RLprocesses. Outsourcing RL services to third-party RLproviders (3PRLPs) helps companies entering new mar-kets and reduces their logistics costs considerably (Kan-nan, Pokharel, & Sasi Kumar, 2009). Therefore, the task ofevaluating and selecting a 3PRLP is fundamental, since itdetermines the overall performance of the company.

It should be emphasised that the implementation ofRL processes has important consequences that extendbeyond the domain of a given company. As illustratedin Table 1, this is particularly the case for developingcountries designing sustainable environmentally friendlypolicies while facing considerable internal barriers tothe adoption of RL processes. Brazil (Bouzon, Govin-dan, Taboada Rodriguez, & Campos, 2016; Guarnieri,Sobreiro, Nagano, & Serrano, 2015) and India (Prakash& Barua, 2015, 2016) constitute two recent examples ofdeveloping countrieswhoseRL adoption constraints havebeen analysed in the literature. In this case, strategic con-siderations regarding the reports and opinions received

© Informa UK Limited, trading as Taylor & Francis Group

2 M. TAVANA ET AL.

Table . Importance of RL processes for environmental and development policies.

Authors Case study

Vahabzadeh and Yusuff () Elicitation of expert judgments regarding environmental sound practices in RLsGuarnieri et al. () Brazilian National Policy of Solid Waste obliging companies to implement RLsBouzon et al. () Barriers to RL adoption in the Brazilian electrical-electronic industry sectorPrakash and Barua () Barriers to RL adoption in the Indian electronics industryPrakash and Barua () Analysis of the Indian electronics industry

from the experts when evaluating potential RLPs shouldbe included in the corresponding selection process.

The selection process of a 3PRLP generally encom-passes several decision criteria, various forms of uncer-tainty and different combined/integrated decision-making models.

� Multi-criteria decision-making (MCDM) methods,such as the analytic hierarchy process (AHP) and theanalytic network process (ANP), are frequently usedto model the complex relationships between the dif-ferent decision-making elements and criteria (Liao,1998).

� Due to the imprecisions inherent in real-worldinformation, people usually face some uncertaintywhen assigning their preference evaluations to thechoice objects being considered. In order to pro-vide a more comprehensive model of human per-ception and cognition of uncertainty, Antanassov(1986) extended Zadeh’s concept of fuzzy sets tothat of intuitionistic fuzzy sets (IFS). An IFS ischaracterised by a membership function, a non-membership function, and a hesitancy function,providing a more suitable framework to express theuncertainty inherent in the evaluations of experts(Wei, Wang, Lin, & Zhao, 2011; Xu & Liao, 2014).

� Since the selection of a 3PRLP involves both ambi-guity and strategic uncertainty, we will combine theANP and the intuitionistic fuzzy-grey superiorityand inferiority (IFG-SIR) models in order to min-imise their effect on the final decision. That is, wepropose a hybrid conceptual model that combinesthe ANP and the IFG-SIR methods to evaluate andselect 3PRLPs.

The contribution of the current paper is determined inpart by the chosen combination of the decision-makingmodels used to evaluate the set of 3PRLPs. Its main fea-ture consists of the intuitionistic fuzzy structure definedto account for the subjective and strategically biased eval-uations of the experts when rating the different 3PRLPsbased on the selected decision criteria. The main stepsdefining the decision theoretical structure introduced inthe current paper are

(1) Determine the main decision criteria (attributes)on which the selection process of the 3PRLPs willbe based. The opinions of different experts will beacquired in this stage. However, a crisp approachwill be followed to differentiate these experts fromthose providing direct evaluations of the RLPs inthe following stage.

(2) Given the decision criteria determined in the ini-tial stage, the set of RLPs is evaluated using theambiguous and strategically uncertain reports of asecond group of experts. In order to account forambiguous and potentially strategic reports, theevaluation process will be divided in a series ofsteps defined to smooth any opinion biases exhib-ited within this second group of experts. Thesesteps include(a) Implementing an intuitionistic fuzzy aggre-

gation technique to average the evaluationsreceived based on the subjective importanceor credibility assigned to each expert.

(b) The definition of superiority and inferiorityperformancematrices that will be used to gen-erate two different rankings so as to accountfor potential biases inherent to the opinions ofthe experts.

(c) Applying grey relation analysis (GRA) to theentries of both matrices in order to obtain arelative evaluation of each attribute for everyRLP.

(3) Combine the weights assigned to each deci-sion criterion through ANP with the evaluationsobtained via IFG-SIR for each potential RLP intoa hybrid decision-makingmodel that delivers botha superiority and an inferiority ranking.

(4) Weight the superiority and inferiority rankings inorder to obtain a unique final ranking of the dif-ferent RLPs.

The validity and applicability of the hybrid ANP-IFG-SIR model will be assessed systematically using a real-world example.

The remainder of the paper is organised as follows.The next section provides a review of the literature on theselection of 3PRLPs. In Section 3,we describe the solution

INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE: OPERATIONS & LOGISTICS 3

methodology and review some basic notions regardingIFSs as well as the ANP, SIR, andGRAmethods. The con-ceptual model is developed in Section 4. Section 5 con-tains a real-world study illustrating the performance ofthe proposed methodology. Section 6 analyses the resultsobtained and provides managerial insights. Conclusionsand directions for future research are given in Section 7.

2. Literature review

RL constitutes a vital business opportunity for companiesas well as a useful means of protecting the environment.As a result, an increasing number of researchers havefocused on this subject and more academics and expertshave beenmotivated to actualise the analysis of RL, bridgethe gaps across studies, and identify research limits. Theseefforts have led to the development of diverse qualitativeand quantitative models.

The initial studies on this topic focused on quan-titative models, while qualitative or non-mathematicalmodels were used to describe systems focusing on com-munication plans and the existing relationships betweeninternal and external factors. Examples of these modelswere proposed in studies performed on the applicationof information and communication technology and deci-sion support systems to RL (Daugherty, Richey, Genchev,& Chen, 2005; Dhanda & Hill, 2005; Wu & Liu, 2008).

Since the outsourcing of RL is an important challengeto organisations, it has drawn a considerable amount ofresearch attention. Companies outsource non-core activ-ities in relation to their ability to gain competitive advan-tage and the complexity of particular activities. Due to thecomplexity of RL, this process is generally outsourced toa third party called the 3PRLP. In this regard, cost reduc-tion and strategic decisions are considered to be the mostimportant factors leading to the outsourcing of RL ser-vices to 3PRLPs (Sahay & Mohan, 2006).

2.1. Initial developments and evolution of theliterature

The 3PLP industry has grown faster than the demandof companies for logistic services. Hence, as the mar-ket for logistic services evolves, 3PLPs have become thesubject of an increasingly significant field of research,where qualitative and MCDM techniques, such as AHP,ANP, data envelopment analysis (DEA), risk matrices,and the fuzzy comprehensive evaluation model, havebeen consistently applied to model the provision of RLservices. For instance, Bian and Yu (2006) applied theAHP to measure the capacity of various Asian countriesto implement RL services for internationalmanufacturersof electrical equipment, while Feng (2008) used the AHP

and DEA techniques to study the selection process of a3PRLP.

Researchers gradually realised that it is highly impor-tant to determine the relationships existing among thecriteria involved in the selection of a 3PLP. As a result, theinterpretive structuralmodelling (ISM) technique startedgaining increasing popularity in this field. For example,Kannan, Haq, Sasikumar, and Arunachalam (2008) useda hybrid AHP/ISMmodel to identify and analyse mutualeffects among the criteria defined to select third-partyproviders based on environmental conditions. Govindan,Palaniappan, Zhu, and Kannan (2012) conducted a sim-ilar research to analyse inter-criteria relationships whenselecting 3PRLPs.

In addition, due to the uncertainties inherent to theinput data, researchers inevitably incorporated fuzzy setsinto their analyses. For instance,Haq andKannan (2006a)suggested the use of the AHP technique and a fuzzy algo-rithm to select a third-party provider, whileHaq andKan-nan (2006b) combined the ISM method together withfuzzyAHP in order to evaluate different alternatives. Sim-ilarly, Kannan et al. (2009) employed the ISM techniquealong with fuzzy technique for order preference by sim-ilarity to ideal solutions (TOPSIS) to select and evalu-ate 3PRLPs. Recently, Senthil et al. (2014) evaluated andselected 3PRLPs using AHP, TOPSIS, and fuzzy sets.

Although the AHP and ISM models have been usedextensively in this field of research, they are subject toimportant limitations. For instance, the ISM techniqueincreases the ability of decision-makers to analyse theinteractions between the criteria involved in the evalu-ation of 3PRLPs, but lacks statistical validity (Govindanet al., 2012). In addition, the simplicity of the hierar-chical structure defining the AHP may deemphasise theinterdependencies existing among criteria (Roper-Lowe& Sharp, 1990). As a result, network models have beenintroduced to deal with such deficiencies.

Network models structure the interactions and theanalyses of interdependences in a logical and uniqueway (Saaty & Ozdemir, 2005). For instance, Tengda andXiangyang (2008) and Ravi, Shankar, and Tiwari (2005)used the ANP to study outsourcing and decision-makingin the application of RLs. These authors asserted that,since the ANP considers various dimensions of ana-lytic information, it is capable of identifying the criticaland necessary criteria required for every strategic deci-sion. However, given the difficulty of determining exactparameter values within a complex evaluation system,such systems must be divided into several subsections inorder to simplify the corresponding evaluation process(Liou, Tzeng, & Chang, 2007; Zareinejad, Javanmard, &Arak, 2013). We analyse the main consequences derivedfrom the structural complexity of the evaluation systemin the next section.

4 M. TAVANA ET AL.

Table . Current developments and trends in the literature and paper contribution.

Authors Attribute ranking Fuzzy ranking PLP selection Fuzzy selection Expert evaluation Fuzzy evaluation

Efendigil et al. () Fuzzy AHP√

Artificial neuralnetwork

Kannan () Fuzzy AHP√

Govindan andMurugesan ()

Fuzzy extent analysis√

Guarnieri et al. () Systematic literaturereview

Conceptualmethodologicalframework

Sharma and Kumar()

Quality functiondeployment

Taguchi loss function

Vahabzadeh andYusuff ()

Fuzzy VIKOR√

Bouzon et al. () Fuzzy Delphi AHP√

Govindan et al. () Grey DEMATEL√

Sahu et al. () Interval-valued fuzzynumbers

√

Khodaverdi andHashemi ()

AHP Grey relationalanalysis

√

Alkhatib et al. () Fuzzy DEMATEL√

Fuzzy TOPSIS√

Prakash and Barua()

Fuzzy AHP√

Fuzzy TOPSIS√

Prakash and Barua()

Fuzzy AHP√

Fuzzy TOPSIS√

Current paper ANP SIR-GRA√

IFS√

2.2. Current developments and trends in theliterature

In the latter years, the literature on RLs has experienceda substantial incremental attention given, among oth-ers, the positive environmental effects derived from theimplementation of RL services. Three recent surveys onthe selection of RLPs are provided by Agrawal, Singh,and Murtaza (2015), Govindan, Soleimani, and Kannan(2015), and Vahabzadeh and Yusuff (2015).

A fundamental feature of the RLs literature is the rela-tively small amount of 3PRLPs models developed follow-ing fuzzy approaches within MCDM environments. Thatis, the classification of the literature performed by Govin-dan et al. (2015) reveals a marginal amount of researchpapers dealing with 3PRLP selection problems from afuzzy-MCDM perspective, a total of three (Efendigil,Önüt, & Kongar, 2008; Govindan & Murugesan, 2011;Kannan, 2009).

We should emphasise the fact that the final choicemade is based on the opinions of different groups ofexperts that must be elicited along the evaluation pro-cess. As already stated, the ambiguity and imprecisioninherent to the reports provided by different experts havebeen repeatedly acknowledged in the literature. How-ever, despite this fact, the experts have not been evalu-ated in terms of either their credibility or the existence ofsubjective biased interests. In this regard, the maincontribution of the current paper has been explicitlyhighlighted in Table 2, where the defining characteristics

of the main models related to the current one have alsobeen summarised.

As emphasised at the end of the previous section,the initial research on the selection of 3PRLPs focusedon ranking the main attributes (decision criteria) withwhich a 3PRLP should be endowed using fuzzy eval-uation methods. Efendigil et al. (2008) implemented atwo-phase model based on fuzzy AHP to select the mostimportant attributes and an artificial neural network toclassify the different 3PRLPs. Kannan (2009) used fuzzyAHP to rank the main attributes determining the selec-tion of 3PRLPs, while Govindan and Murugesan (2011)applied fuzzy extent analysis to perform the same typeof task. Similarly to other alternative techniques such asVIKOR, TOPSIS, and ELECTRE, extent analysis requiresa substantial amount of numerical calculations, increas-ing the time required to make a decision.

Absent explicit fuzzy considerations regarding thereports of the experts, Guarnieri et al. (2015) performeda systematic literature review to propose a conceptualmethodological framework that helps decision-makersselecting 3PRLPs. Sharma and Kumar (2015) used qual-ity function deployment to relate the attributes to therequirements of the decision-makers, while the perfor-mance of the 3PLPs was evaluated through the weightedloss scores computed using a Taguchi loss function. Boththese papers acknowledge the two sources of expert infor-mation required to evaluate the potential providers. How-ever, they do not consider the strategic incentives of theexperts when evaluating the set of 3PRLPs, a particularlyimportant issue in developing countries or in those with a

INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE: OPERATIONS & LOGISTICS 5

weak institutional environment such as Brazil (Guarnieriet al., 2015).

The literature has focused on exploiting the proper-ties of the fuzzy attribute ranking techniques, those of thefuzzy 3PLP selection processes, or both of them.

� Within the first group of papers, Vahabzadeh, Asi-aei, and Zailani (2015) implemented a fuzzy VIKORapproach to measure and rank the effect that differ-ent green environmental factors have on the mainrecovery options in RLs.More importantly, they alsosuggested the design of a group decision-makingprocess based on fuzzy multi-attribute decision-making methods when considering the best recov-ery option.

Bouzon et al. (2016) collected the opinions of differ-ent experts in order to identify the main barriers to RLadoption using the fuzzy Delphi method. These barri-ers were then ranked by other set of experts through theAHP. Govindan, Khodaverdi, and Vafadarnikjoo (2016)used grey decision-making trial and evaluation labora-tory (DEMATEL) to determine the interdependent rela-tionships existing among different 3PLP selection crite-ria, an approach that can be considered as an alternativeto the ANP when ranking the set of attributes.

� Within the second group of papers, Sahu, Datta, andMahapatra (2015) defined a fuzzy environment toaccount for group decision-making processes wheredecision-makers issue subjective judgments. Kho-daverdi and Hashemi (2015) combined AHP - torank the selection criteria – and grey relational anal-ysis – to select 3PRLPs – in their hybrid modelbut did not consider evaluating the experts or theirreports.

� The literature has evolved towards scenarios wherethe uncertainty arising from the opinions of thedifferent groups of experts ranking the attributesand selecting the 3PRLP has been accounted forand integrated within the corresponding decision-making models. Alkhatib, Darlington, Yang, andNguyen (2015) used fuzzy DEMATEL to determinethe weights of logistics resources and fuzzy TOPSISto rank the alternatives.

Prakash and Barua (2015) combined fuzzy AHP – toweight barriers to the adoption of RL processes – withfuzzy TOPSIS – to deliver a ranking of RLs adoption solu-tions that can be implemented to overcome the barriers. Asimilar approach was followed by these authors (Prakash& Barua, 2016) to define the set of criteria consideredin the selection 3PRLPs. Note, however, that in all these

decision-makingmodels the subjectivemotivations of theexperts providing the reports remain unstudied.

3. The solutionmethodology

We propose a hybrid ANP-based model that introducesIFSs and grey relations within the standard SIR method.It should be noted that the SIR technique is superior to theTOPSIS one in terms of duration and mutual inferiorityof options (Memarzade, Alvandi, &Mavi 2011). TheANPmethod is used to obtain theweights of the evaluation cri-teria, while the proposed IFG-SIR approach is introducedto account for the uncertainty and the lack of precisionassociated with the input information received from theexperts evaluating potential 3PRLPs. More precisely

� The unique capabilities of the ANP will be prop-erly utilised to evaluate inter-criterion interactions(Zareinejad et al., 2013).

� As already stated, systems are generally ambiguousdue to the uncertainties inherent to their input infor-mation (Kumar, Bhatia, & Kaur, 2009). Therefore,we will make use of IFSs and GRA to address thepotential inadequacy of the input data received fromthe experts (Chen & Tzeng, 2004).

The resulting hybrid model provides a proper frame-work to analyse the existing correlations among the deci-sion criteria and allows us to obtain an ideal solution ina highly-complex, multi-criteria, decision-making envi-ronment. Figure 1 describes the set of steps composingthe proposed hybrid model.

3.1. Definition of criteria

A recent study conducted by Govindan et al. (2012) anal-ysed 35 decision criteria generally considered in the selec-tion process of 3PRLPs. However, even though this studyseemed to be sufficiently comprehensive, it did not con-sider the different stages of the product life cycle (PLC)incorporated byMeade and Sarkis (2002). Besides, poten-tial risks should not be overlooked, since every step ofthe PLC is associated with specific risks (Xia & Chen,2011). In this regard, the results of the literature reviewperformed on 3PRLP evaluation criteria are presented inTable 3.

3.2. ANPmethodology

The ANP method is an extended version of the AHP.It determines the relative significance of criteria andclusters in terms of the correlations and feedbacks

6 M. TAVANA ET AL.

Figure . Proposed framework.

INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE: OPERATIONS & LOGISTICS 7

Table . Evaluation criteria.

Criterion Sub-criterion References

IT applications (IT) (A) •Warehouse management(A)

Dowlatshahi (), Holguı’n-Veras (), Li, Wu, Lai, and Liu (), Govindan et al.(), Bun and Ishizuka (), and van den Berg and Zijm ()

• Order management (A)• Supply chain planning (A)• Shipment and tracking (A)• Freight payment (A)

Impact of use of PL (IUPL) (B) • Customer satisfaction (B) Boyson, Corsi, Dresner, and Rabinovich (), Jähn, Zimmermann, Fischer, andKäschel (), Govindan et al. (), and Lynch ()

• Profitability (B)• Frequent updating (B)• Employee morale (B)

Third-party logistics services • Inventory replenishment (C) Davis and Gaither (), Dowlatshahi (), Gunasekaran, Patel, and Tirtiroglu (),(PLS) (C) •Warehouse management

(C)Gupta and Bagchi (), Holguı’n-Veras (), Govindan et al. (), Bun and Ishizuka(), Kleinsorge, Schary, and Tanner (), and van den Berg and Zijm ()

• Shipment consolidation (C)• Direct transportationservices (C)

User satisfaction (US) (D) • Effective communication (D) Andersson and Norrman (), Boyson et al. (), Govindan et al. (),• Service improvement (D) Lynch (), and Mohr and Spekman ()• Cost saving (D)• Overall working relations(D)

RL functions (RLF) (E) • Collection (E) Dowlatshahi (), Cochran and Ramanujam (), Kaliampakos, Benardos, and• Packing (E) Mavrikos (), Govindan et al. (), Schwartz (), and van Dijck ()• Storage (E)• Sorting (E)• Transitional processing (E)• Delivery (E)

Organisational performance • Flexibility (F) Andersson and Norrman (), Boyson et al. (), Govindan et al. (), Kim, Park,criteria (OPC) (F) • Service (F) and Jeong (), Kim et al. (), Lynch (), and Stock, Greis, and Kasarda ()

• Time (F)• Cost (F)• Quality (F)

Organisational role (OR) (G) • Reclaim (G) Dowlatshahi (), Demir and Orhan (), Govindan et al. (), Meade and Sarkis• Recycle (G) (), and Schwartz ()• Remanufacture (G)• Reuse (G)• Disposal (G)

Product life cycle stages • Introduction (H) Mead and Sarkis ()(PLC) (H) • Growth (H)

•Maturity (H)• Decline (H)

existing among the different decision factors. It alsoextends the super matrix method (Saaty, 2001a). Moreimportantly, this model overcomes the structural limita-tions of the linear AHP since it uses networks regardlessof their levels. In fact, the criteria are included in clustersthat replace the levels. Moreover, the ANP model con-siders both inter-cluster (outer-dependence) and intra-cluster (inter-dependence) interactions and feedback.Hence, it can be used to analyse the whole set of poten-tial relationships arising among the decision factors. TheANPprocess is generally comprised of the following steps(Saaty, 2005):

(1) Construction of a network model: after identi-fying the main criteria, which are based on theobjective of the problem, the decision elementsare classified into clusters and distributed withina network structure. The network includes all

mutual relationships and feedback among bothelements and clusters.

(2) Performance of pair-wise comparisons: in thisstage, each of the sub-criteria composing the dif-ferent decision criteria is compared pair-wise.These pair-wise comparisons follow the sameintuition as those performed in the AHP method.

(3) Formation of the super matrix: the relative prior-ity eigenvectors obtained for each pair-wise com-parison matrix are placed in a larger partitionedmatrix. The resulting super matrix allows for theanalysis of the existing interdependences amongthe whole set of criteria composing the network.The function of this super matrix is similar to thatof a Markov chain process. Every ANP networkhas three super matrices associated with it, namedun-weighted, weighted, or normalised and limit.

(4) Analysis of the information: all calculationsregarding the super matrices are easily performed

8 M. TAVANA ET AL.

Figure . Sample super matrix.

using the super decisions software. The columnsof the limit super matrix, which are all identical,determine the ultimate priority of the whole setof criteria composing the network. As a result, theANP allows us to identify and select the criterionor alternative with the highest priority.

Figure 2 describes a super matrix that contains Nclusters. In particular, Ck refers to the kth cluster, com-posed by nk elements or criteria, which are denotedby ek1, ek2, ........, eknk . Each column (w ji) in this figuredescribes the relation and subsequent effect that the col-umn criterion has on the row criterion. That is, each col-umn is a relative priority vector derived from pair-wisecomparisons among the corresponding criteria using theAHP eigenvector method. If there are no relationshipsamong the elements of two clusters, the resulting matrixwill be a zero matrix (Saaty & Vargas, 2006).

The sum of the weights composing the columns ofthe primary or un-weighted super matrix may result in anumber greater than one. Thus, a weighted or normalisedsupermatrixmust be obtained bymultiplying the value ofeach criterion composing the un-weighted super matrixby the (eigenvector) weight assigned to its correspond-ing cluster. Finally, theweighted supermatrix is convertedinto a limit super matrix by raising it to the power 2k+1,with k being a sufficiently large number.

If more than one limit matrix is obtained, it is nec-essary to calculate the Cesaro sum in order to establisha priority ranking among the resulting super matrices(Chen&Chen, 2012). The Cesaro sum is given by the fol-lowing expression: LimN→∞( 1

N )∑N

k=1 wk, with w refer-ring to the weighted super matrix used to calculate thelimit one.

3.3. The classic SIR technique

The classic SIR technique was first introduced by Rebai(1993). This technique was based on the construction ofsuperiority and inferioritymatrices thatwere used to rankthe options available. Consider a decision-maker whomust rank m available options, denoted by Ai, with i =1, 2, . . . ,m, with respect to n cardinal criteria, denotedby g j, with j = 1, 2, . . . , n. Let g j(Ai) be a real valuedfunction describing the criteria value or performance ofthe ith alternative, Ai, with respect to the jth criterion,g j. These criteria values or performances are used to con-struct the decision matrixD as follows:

D =

⎡⎢⎢⎢⎣g1(A1) g2(A1) . . . gn(A1)

g1(A2) g2(A2) . . . gn(A2)...

......

...g1(Am) g2(Am) . . . gn(Am)

⎤⎥⎥⎥⎦ .

Assume now that f j is a generalised criterion adoptedby the decision-maker in order to compare options Aiand Ak with respect to the jth criterion g j. That is, f j isa monotonic function from the set of real numbers in theinterval [0, 1].

The intensity of preference or superiority of thealternative Ai relative to Ak based on the jth criterionis determined by the following relation Pj (Ai,Ak) =f j (g j(Ai) − g j(Ak)) (Xu, 2001).The superiority, S j(Ai), and the inferiority, I j(Ai),

indexes for alternative Ai with respect to the jth criterionare calculated using the following equations:

S j(Ai) =m∑k=1

Pj(Ai,Ak) and I j(Ai) =m∑k=1

Pj(Ak,Ai).

(1)Superiority and inferiority indexes are used in the con-

struction of the superiority (S = (S j(Ai))m×n ) and infe-riority (I = (I j(Ai))m×n ) matrices.

Finally, standardMCDMprocedures such as SAWandTOPSIS can be implemented on the S and I matrices inorder to aggregate the corresponding indexes into twodifferent global preference indexes, namely, the superior-ity (S-flow) ϕ>(·) and the inferiority (I-flow) ϕ<(·) flows.The S and I flows reflect the global degree of superiorityand inferiority of each alternative, respectively (Xu, 2001).

... SIR rankingFollowingXu (2001), the superiorityϕ>(·) and inferiorityϕ<(·) flows can be used to rank the set of alternatives.Note that the higher the value of the superiority flow, themore preferred is a given alternative. Similarly, the lowerthe value of the inferiority flow, the more preferred is agiven alternative.

INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE: OPERATIONS & LOGISTICS 9

Let P> and I> denote the strict preference and indif-ference relations relative to the superiority flow. A sim-ilar notation can be introduced for the inferiority flow.The superiority ranking (�> = {P>, I>}) of the alter-natives based on the descending order of ϕ>(·) istherefore given by AiP>Ak iff ϕ>(Ai) > ϕ>(Ak) andAiI>Ak iff ϕ> (Ai) = ϕ>(Ak). Similarly, the inferior-ity ranking (�< = (P<, I<)) of the alternatives based onthe descending order of ϕ<(·) is given by AiP<Ak iffϕ<(Ai) < ϕ<(Ak) and AiI<Ak iff ϕ< (Ai) = ϕ<(Ak).

3.4. Grey relation analysis (GRA)

The first researcher to theorise grey systems was Deng(1988). The theory of grey systems yields satisfactory out-puts using relatively little information and highly variablecriteria. Similarly to the theory of fuzzy systems, this the-ory also provides an effective mathematical tool to solveuncertain and ambiguous problems (Deng, 1988).

GRA is a part of the theory of grey systems that isgenerally used to solve problems with complex inter-factor and inter-variable relationships (Moran, Granada,Míguez, & Porteiro, 2006). In particular, GRA has beenwidely used in data processing and modelling activities,systems analysis, as well as to solve MCDM problems(Deng, 1988; Memarzade et al., 2011; Wei, 2011; ZareMehrjerdi, 2014). In a grey relation environment, a col-lection of non-functional models can be used to meetthe requirements of statistical methods for the analysis oflarge sample volumes (Chen & Tzeng, 2004).

Themain definitions commonly employed in GRA arepresented below.Definition 3.4.1 (Chen & Tzeng, 2004): Let X be a set ofgrey relation coefficients (GRCs) of determination, x0 ∈X the referential sequence, and xi ∈ X the comparativesequence, with x0( j) and xi( j) representing the numeri-cal values of x0 and xi at point j.Definition 3.4.2 (Chen&Tzeng, 2004):TheGRC,whichdetermines the proximity between xi( j) and x0( j), isgiven by

γ (x0( j), xi( j)) = �min+ζ �max�i j + ζ �max

;i = 1, ...,m j = 1, ..., n, (2)

where the sub-index i refers to the options available, jaccounts for the selection criteria,

�ij = |x0(j) − xi

(j)|

�min = min{�i j, i = 1, 2, . . . ,m; j = 1, 2 . . . , n

}�Max = Max

{�i j, i = 1, 2, . . . ,m; j = 1, 2 . . . , n

}

and ζ ∈ [0, 1] is the distinguished coefficient used toexpand or contract the range of the GRC. Note that thevalue of the GRC increases as xi( j) approaches x0( j).Definition 3.4.3 (Chen&Chen, 2012):Letw j denote theweight of the jth criterion. The grey relation grade (GRG)is given by

�(x0, xi) =n∑

j = 1

wjγ(x0

(j), xi

(j))

. (3)

The GRC relation describes the existing correla-tion between the target referential and the comparativesequence. The best alternative is the one endowed withthe highest GRG.

3.5. Intuitionistic fuzzy set (IFS)

Atanassov (1986) extended the fuzzy model developedby Zadeh (1965). Together with the degrees of mem-bership and non-membership defining the elements of afuzzy set, Atanassov introduced a degree of hesitancy asa main characteristic of the elements within the set. Thisdegree of hesitancy accounts for the fact that a decision-maker is not always able to determine definite member-ship degrees. Hence, an IFS is defined as follows.Definition 3.5.1 (Atanassov, 1986): Let X be a non-empty finite set. An IFS A in X is given by

A = {x, μA (x) , vA (x) |x ε X} , (4)

with μA, vA : X → [0, 1] denoting the degree of mem-bership and non-membership of x to A, respectively, and0 ≤ μA(x) + vA(x) ≤ 1, ∀ x ∈ X .

Szmidt and Kacprzyk (2000) introduced an intuition-istic index of x in A, defined by πA (x) = 1 − μA(x) −vA(x) for each IFS in X . This index determines the hesi-tancy degree of x to A. Given this index and the parame-ters composing it, an IFS performs better than a fuzzy onein uncertain situations.

... Measuring the intuitionistic fuzzy distanceFollowing Szmidt and Kacprzyk (2000) and Chai, Liu,and Xu (2012), we propose two different ranges based onthe μA(x), vA(x) and πA (x) parameters to measure thedistance between two IFSs. Assume that A and B are twoIFSs inX = {x1, x2, . . . , xn}, the following distances areproposed:

(1) The Hamming distance,

h (A, B) = 1/2n∑

i = 1

(|μA (xi) − μB (xi)| + |vA (xi)

− vB (xi) | + |πA (xi) − πB (xi)|). (5)

10 M. TAVANA ET AL.

(2) The Euclidean distance,

d(A,B) =√√√√1/2

n∑i=1

(μA(xi) − μB(xi))2+(νA(xi) − νB(xi))2 + (πA(xi) − πB(xi))2. (6)

Generally, decisions are made by a group of decision-makers, so the aggregation of opinions is an essentialcomponent of any decision-making method (Wei, Zhao,& Wang, 2012). Recently, Xu (2007a), Chai et al. (2012),Wei and Zhao (2012), and Zhao and Wei (2013) havedeveloped several operators (such as IFHA, IFHG; IFWA,IFWG; I-IFCA, I-IFCG; IFEHA, IFEHG) to aggregateopinions within an intuitionistic fuzzy space. These oper-ators can be used to aggregate the intuitionistic fuzzy val-ues obtained by a specific alternative in relation to eachcriterion. The result of these calculations is also an intu-itionistic fuzzy value that helps the decision-maker toselect the best alternative available.Definition 3.5.1.1: Let αi = (μαi, vαi ) (i = 1, . . . , n)

be a set of intuitionistic fuzzy values, then IFWA: �n →� is defined as follows (Xu, 2007b):

IFWAω = n⊕i=1

(ωiαi) =(1 −

n∏i=1

(1 − μαi )ωi,

n∏i=1

(vαi )ωi

).

(7)

Definition 3.5.1.2: Let αi = (μαi, vαi ) (i = 1, . . . , n)

be a set of intuitionistic fuzzy values, then IFWG: �n →� is defined as follows (Xu & Yager, 2006):

IFWGω = n⊕i=1

(αωii ) =

( n∏i=1

(μαi )ωi,1 −

n∏i=1

(1 − vαi )ωi

),

(8)

where ωi = (ω1, ω2, ..., ωn)T is the weight vector of

αi (i = 1, . . . , n) with ωi ∈ [0, 1] and∑n

i=1 ωi = 1.

3.6. The proposed IFG-SIRmethodology

The following inputs are required to implement themethod proposed:

Input 1: The weights assigned to the kth expert: wk =(µk, vk), with k = 1,…, L;

Input 2: The weights of the criteria:Wj, with j = 1,…,n;

Input 3: The value of the report provided by the kthexpert: d(k)

i j = (μ(k)i j , v (k)

i j ), with i = 1, …,m.Chai et al. (2012) developed the SIR method on the

basis of IFSs. In the proposed model, the weights of the

criteria correspond to those obtained after implementingthe ANP, from where they are directly imported.

Our model is defined in nine sequential stages. How-ever, before starting with the initial one, we determine thevalue of the intuitionistic weights, wk, associated to eachexpert. In this regard, it should be emphasised that thefirst six stages have been defined to account for the relia-bility of the experts and their reports.

Stage 1: The main objective of the initial stage is toascertain the significance of the personal decisions ofeach expert, ξk, with k= 1,…,L. To this end, we will com-pute the relative distance between the weight assignedto the kth expert, wk = (μk, υk), and the correspondingideal value,w+

k = (μ+k, υ

+k, π

+k), using Equations (5)

and (6). We will denote the distance functions defined inthese equations by Dk(·), when applied to the kth expert.

The positive-ideal intuitionistic fuzzy weight, w+k =

(μ+k, υ

+k, π

+k), and the negative-ideal intuitionistic

fuzzy weight, w−k = (μ−

k, υ−k, π

−k), are given by

w+ = (1, 0, 0), andw− = (0, 1, 0), respectively. The sig-nificance of the report provided by the kth expert is there-fore given by the relative closeness of his weight to thepositive-ideal one,

ξk(wk, w

+) = Dk(wk, w

−)/Dk

(wk, w

−)+Dk

(wk, w

−)(9)

Stage 2: The second stage consists of computing thegroup aggregated decision values, di j = (μi j, υi j), whichare based on the intuitionistic fuzzy aggregation opera-tors defined in Equations (7) and (8). According to Chaiet al. (2012), IFWA should be used when the aggregationof the decision values aims at preserving the subjectivejudgments of the experts, and therefore it is the criterionwe use,

di j = IFWAξk = ξ1 d(1)i j ⊕ . . . ⊕ ξLd(L)

i j

Stage 3: In this stage, the group aggregated decisionvalues, di j = (μi j, υi j), are transformed into the entriesdefining the performance decision matrix D. Therefore,g j( Si) behaves as an operational performance functionmapping relative ideal distances into the [0, 1] interval.The procedure is similar to the one described in the firststage and has therefore been omitted.

Stage 4: The IFG-SIR index and the correspondingsuperiority and inferiority matrices are constructed inthis stage. As described in Section 3.3, the degree of pref-erence Pj(Si, St ) reflecting the superiority of alternative Si

INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE: OPERATIONS & LOGISTICS 11

Figure . Network model.

relative to St based on the jth criterion is given by

Pj (Si, St ) = f j(g j(Si

) − g j (St )), withj = 1, . . . , n; i, t = 1, . . . ,m and i = t,

(10)

where f j is a monotonic function that can be determinedby each decision-maker.

The corresponding superiority (Su) and inferiority (I)indices based on the jth criterion will be defined as fol-lows:

Su j (Si) =m∑

i = 1

Pj (Si, St) (11)

I j (Si) =m∑

i = 1

Pj (St, Si) . (12)

Stage 5: In this stage, the maximum values obtained bythe alternatives for each criterion within the superioritymatrix and the minimum values obtained by the alter-natives for each criterion within the inferiority matrixwill be used to define superiority and inferiority referenceseries for each respective matrix.

Stage 6: In this stage, we use Equation (2) to calcu-late the GRC for each one of the elements composing thesuperiority and inferiority matrices.

Stage 7: The weights of the criteria obtained fromthe ANP are inputted into the GRA model, usingEquation (3) to calculate the GRG of each alternativewithin the superiority and inferiority matrices.

Stage 8: The alternatives are ranked based on theGRGs obtained in the previous stage for the superiority

(Su-GRG) and inferiority (I-GRG)matrices. In each case,the best alternative is the one whose GRG is closer to 1(Zareinejad, 2013).

Stage 9: In the final stage, the Su-GRG and the I-GRGrankings are combined. To this end, we define a problem-weighted index that behaves similarly to the V-weightedindex introduced in the VIKOR technique (Opricovic &Tzeng, 2004). The V index is designed to achieve maxi-mum group desirability and is usually equal to 0.5 (Opri-covic & Tzeng, 2004; Zareinejad, 2013). Thus, the finalweight assigned to each alternative is obtained using thefollowing equation:

�Sui,Ii (x0 j, xi j) = V × �Sui (x0 j, xi j)+ (

(1 −V ) × �Ii(x0 j, xi j)). (13)

The final ranking is determined by the �Sui, Ii (x0j, xij)value obtained by each alternative.

4. The conceptual model

Without understanding the interactions and relation-ships among the various criteria defining each alterna-tive, it is not possible to properly select a 3PRLP. There-fore, when preparing, designing, and providing a formalframework for our decision-making model, it should beacknowledged that RL services must focus on non-linearrelationships instead of linear ones. Consequently, a pow-erful decision-making method is required to account forthe opinions of experts and integrate abstract ideas withconcrete criteria. In order to gain a better understandingof the problem being analysed and the type of MCDMstructure required, the conceptual model is illustrated inFigure 3.

12 M. TAVANA ET AL.

Figure . General relations between criteria and sub-criteria in the ANP model.

This model describes the selection process in termsof inter-factor relationships, dependencies, and feedback.It applies holism and systems theory resulting from thecharacteristics of the ANP.

5. Case study

In this paper, we apply the proposed framework to Pipex1,amanufacturer of flexible and reinforced composite pipesin West Virginia. Pipex is the leading manufacturer ofpipes, joints, and composite tanks in West Virginia. Thecorrosion and damage of composite tanks and fiber-glass materials pose serious threats to the environment.Composites contain carbon fibres and are coated withchrome linings. Composites with hexavalent chromiumcoatings are known to be hazardous wastes. Therefore,since chromiummay be leached into the ground, it is notpossible to dispose of these coatings in the environment.

One of the main priorities of Pipex is the reduction ofthe costs derived from the collection and return of com-posites. Hence, one of the main objectives of the com-pany is the selection of a 3PRLP to handle its collectionand return operations. As a result of the conflicts aris-ing among the different selection indices, the company islooking for a systematic method to select 3PRLPs.

5.1. Inter-criterion dependencies

All of the relationships defined among the differentcriteria and sub-criteria determining the choice of an

alternative are based on the information retrieved fromseven half-matrix questionnaires. The resulting networkstructure illustrated in Figure 4 follows from the relationsdescribed in Figure 3 and the opinions of the experts.

5.2. Pair-wise comparisons and determination oflocal weight

After building the ANP network, experts perform pair-wise comparisons among all the sub-criteria on the basisof the nine-point scale of preference proposed by Saaty(2000).Wedistributed copies of the pair-wise comparisonquestionnaires to 10 logistic specialists, out of which onlyseven responded. The opinions of the experts were aggre-gated using the geometric mean method (Saaty, 2001a).

5.3. Formation of a super-matrix

The weights assigned to each sub-criterion are obtainedusing special vectors (Saaty, 1980), whose values form thecolumns of a larger matrix that reflects the effect of eachsub-criterion composing each cluster on all the other ele-ments of the system. The formation of a super matrixallows for the analysis of the correlations existing amongall the elements composing the ANP model. The pri-mary (or un-weighted) super matrix, which is describedin Figure 5, is composed of letters and zero values. Theletters represent the relationships shown in Figure 4. If an

INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE: OPERATIONS & LOGISTICS 13

Figure . Sub-matrices in the super matrix.

element does not influence another element, its relativeimpact priority is set equal to zero.

In order to provide more insight, a subset of the cal-culations of block H, i.e. those involving the introduc-tion stage (H1) of the PLC and the elements of cluster F,are presented in Figure 6. The calculations of the weightsassigned to each criterion are performed using the superdecisions software.

It should be highlighted that the final limit matrix(Saaty, 2001b) determines the weights associated to eachcriterion that will be used in the following stages of themodel.

5.4. Results of weight calculations

In this stage, the 15 sub-criteria with the highest relativeglobal significance are selected from the limit matrix forinclusion in the proposed IFG-SIR model. The weightsassociated to each one of these decision criteria are pre-sented in Table 4.

5.5. Deciding on the final ranking using IFG-SIR

Four qualified experts are selected to provide subjectiveevaluations of the five 3PRLPs available for selection by

Table . The final normalised weights of the criteria.

Initial values of the Values of the finalSub-criterion final weight normalised weight

A . .A . .A . .F . .F . .F . .F . .F . .E . .E . .D . .D . .B . .B . .B . .

the company. The experts are also evaluated and a weightwk is assigned to each one of them in order to estimate therelative significance of their decisions. The input infor-mation provided by the experts as well as the evaluationsused to describe their significance are presented in theform of verbal descriptions. The expressions employedtogether with their corresponding IFVs are presented inTable 5.

... Intuitionistic fuzzy evaluationmatricesIn this stage, we build the intuitionistic fuzzy matricesdescribing the linguistic evaluations of each 3PRLP pro-vided by the experts based on each of the selection crite-rion considered. These matrices are presented in Table 6.

... Measuring the significance of personaldecisionsThe following linguistic weights are ascribed to the opin-ions of the experts: (1) expert one: very important (VI);(2) expert two: extremely important (EI); (3) expert three:

Figure . Pair-wise comparison matrix of organisational performance criteria with respect to the introduction sub-criterion (H).

14 M. TAVANA ET AL.

altogether important (AI); (4) expert four: very, veryimportant (VVI).

The relative closeness of the above values to thepositive-ideal solution, w+ = (1, 0, 0), is calculatedusing Equation (9). The resulting significance levelsassigned to the subjective decisions made by each expertare given by

ξk = (ξ1, ξ2, ξ3, ξ4) = (0.7405, 1.000, 0.9000, 0.8314).

... Aggregation decision-makingThe group aggregated decision values, di j = (μi j, υi j),are obtained after applying the intuitionistic fuzzy aggre-gation operator defined in Equation (7). The correspond-ing intuitionistic decision-making matrix is presented inTable 7.

... Calculation of the performance functionmatrixThe group aggregated decision values are now trans-formed, using Equations (5) and (9), into the entriesdefining the performance (relative to the positive-idealsolution) decision matrix D. The resulting decisionmatrix, [g j( Si)], which is entirely composed by crispnumbers, is given by

g j(Si) =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0.9975 0.9986 0.9840 0.9905 0.9931 0.9912 0.9834 0.9862 0.9744 0.9987 0.9684 0.9929 0.9807 0.9409 0.9896

0.9755 0.9969 0.9641 0.9889 0.9894 0.9796 0.9774 0.9907 0.9954 0.9857 0.9971 0.9896 0.9969 0.9755 0.9866

0.9986 0.9894 0.9437 0.9992 0.9533 0.9755 0.9443 0.9969 0.9969 0.9906 0.9930 0.9842 0.9824 0.9248 0.9754

0.9971 0.9182 0.9814 0.9533 0.9960 0.9846 0.7471 0.9437 0.9931 0.9921 0.9858 0.9755 0.9537 0.9866 0.9473

0.9422 0.8994 0.9989 0.9951 0.9981 0.9889 0.9969 0.9712 0.9994 0.9973 0.9947 0.9987 0.9282 0.9547 0.9920

⎤⎥⎥⎥⎥⎥⎥⎥⎦

.

... Formation of superiority and inferioritymatricesThe degree of preference, Pj(Si, St ), is now calculatedfor each jth criterion using Equation (10). The resultingsuperiority and inferiority matrices follow after applyingthe superiority (Su) and inferiority (I) indices describedin Equations (11) and (12), respectively,

Table . Verbal expressions and their respective intuitionistic fuzzy number.

Description of the importance of experts Description of the option performance Intuitionistic fuzzy number

Extremely important EI Extremely positive EP (., .)Altogether important AI Altogether positive AP (., .)Very very Important VVI Very very positive VVP (., .)Very important VI Very positive VP (., .)Important I Positive P (., .)Medium M Medium M (., .)Less important LS Negative N (., .)Non-important NI Very negative VN (., .)Inconsiderable UC Extremely negative EN (., .)

Table . Expert opinions represented by verbal expressions.

Expert Expert Expert Expert

Sub-criterion S S S S S S S S S S S S S S S S S S S S

A P M P VP VVP VP VP P VVP VP VP P VP VP VP P VVP VP VVP VPA VP VP N VN AP VVP VVP N N VVP P P M VN VVP P VP VVP M VPA VP P VVP M M P VP VVP M P VVP VVP VP M VP VP VVP AP VP VPB P VP VVP VP AP VP AP VP VVP AP VP VVP AP VVP AP P VP VVP VP VVPB AP VVP AP AP VVP VVP VP VVP AP AP AP P M VP VP VVP VP P AP VVPB P VP P VVP VP P VVP VP VP P P VVP AP VP AP VP AP VP P VVPD AP VVP VP VVP AP AP VP VVP P VVP VP P M P VVP VP VVP VP P PD VP VVP P M M VP VVP P M M P VP VP P M VP AP VVP VP APE M P N P M M P M VP P P VP N VP M P VP VP VVP VVPE VP VP P N VVP VP VP P M VP VVP VVP VP P VP VP P VP VP VPF AP VVP VP VP M VP P AP VVP P AP P AP VV M VP P VP AP APF VP VVP P N M AP VVP VP M M AP VP VP M N VVP AP AP P MF P M M VP VVP VVP VP M P AP P P M P VVP VP P VP VVP PF VVP VP AP N AP VVP VP AP M VVP VP VP AP M VP P VP VP VVP VPF VP P N VVP VP VVP P M AP AP VVP P M VVP AP VP AP VVP M VP

INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE: OPERATIONS & LOGISTICS 15

Table . Aggregation decision-making matrix.

Sub-criterion S S S S S

A (.,.) (.,.) (.,.) (.,.) (.,.)A (.,.) (.,.) (.,.) (.,.) (.,.)A (.,.) (.,.) (.,.) (.,.) (.,.)B (.,.) (.,.) (.,.) (.,.) (.,.)B (.,.) (.,.) (.,.) (.,.) (.,.)B (.,.) (.,.) (.,.) (.,.) (.,.)D (.,.) (.,.) (.,.) (.,.) (.,.)D (.,.) (.,.) (.,.) (.,.) (.,.)E (.,.) (.,.) (.,.) (.,.) (.,.)E (.,.) (.,.) (.,.) (.,.) (.,.)F (.,.) (.,.) (.,.) (.,.) (.,.)F (.,.) (.,.) (.,.) (.,.) (.,.)F (.,.) (.,.) (.,.) (.,.) (.,.)F (.,.) (.,.) (.,.) (.,.) (.,.)F (.,.) (.,.) (.,.) (.,.) (.,.)

Su =

⎡⎢⎢⎢⎢⎣0.5500 1.6020 0.3548 0.2423 0.2743 0.0844 1.2710 0.3472 0.0000 0.0533 0.0000 0.0760 0.5594 0.0511 0.34220.1990 1.5718 0.0799 0.2239 0.2294 0.0034 1.1970 0.4344 0.0845 0.0000 0.1816 0.0448 1.0150 0.6977 0.29050.5721 1.4386 0.0000 0.3832 0.0000 0.0000 0.9995 0.5863 0.0996 0.0048 0.1243 0.0105 0.5968 0.0000 0.14610.5418 0.0682 0.3055 0.0000 0.3160 0.0214 0.0000 0.0000 0.0675 0.0086 0.0587 0.0000 0.1220 1.1061 0.00000.0000 0.0000 0.7743 0.3068 0.3503 0.0561 1.5339 0.1404 0.1298 0.0409 0.1455 0.1664 0.0000 0.2011 0.3900

⎤⎥⎥⎥⎥⎦

I =

⎡⎢⎢⎢⎢⎣0.0240 0.0000 0.0434 0.0192 0.0067 0.0000 0.0358 0.0267 0.3658 0.0000 0.4539 0.0067 0.0517 0.6135 0.00120.2826 0.0006 0.3494 0.0292 0.0264 0.0487 0.0804 0.0077 0.0036 0.0727 0.0000 0.0186 0.0000 0.0244 0.00760.0000 0.0280 1.0613 0.0000 1.1368 0.1032 0.8852 0.0000 0.0013 0.0225 0.0040 0.0620 0.0412 0.6168 0.11710.0005 2.0727 0.0608 0.3912 0.0009 0.0124 3.9989 1.2329 0.0118 0.0141 0.0513 0.2148 0.5991 0.0000 1.04211.5800 2.5810 0.0000 0.0034 0.0000 0.0011 0.0000 0.2410 0.0000 0.0004 0.0012 0.0000 1.6010 0.2671 0.0000

⎤⎥⎥⎥⎥⎦ .

Moreover, in the current stage we also define the refer-ence series for the superiority and inferiority matrices,

� Reference series for the superiority matrix (RS),

{max SU1(Ai),max SU2(Ai), ...,max SUn(Ai)}RS = {0.5721, 1.6020, 0.7743, 0.3832, 0.3503,0.0844, 1.5339, 0.5863, 0.1298, 0.0533,0.1816, 0.1664, 1.0150, 1.1061, 0.3900}

� Reference series for the inferiority matrix (RI),

{min I1(Ai),min I2(Ai), ...,min In(Ai)}RI = {0.0000, 0.0000, 0.0000, 0.0000,0.0000, 0.0000, 0.0000, 0.0000, 0.0000,0.0000, 0.0000, 0.0000, 0.0000, 0.0000,0.0000}

... Grey relation coefficient (GRC)The GRC is calculated for each of the entries com-posing the superiority and inferiority matrices usingEquation (2). A summary of the numerical resultsobtained is presented in Table 8. For instance, the[SuF1(S1)] coefficient for the F1 criterion represented inthe first column of the superiority matrix is calculated

as follows:

γSu = γ (x0( j),

xS1( j)) = 0 + ζ0.5721(0.5721 − 0.5500) + ζ0.5721

= 0.92828,

where ζ = 0.5.

... Grey relation grade (GRG)In this stage, we introduce the weights calculated for eachdecision criterion using the ANP into Equation (3) inorder to obtain the GRG of each available option. Table 9presents the results of these calculations.

... Comparison of rankingsGiven the fact that the best GRG is the one whose value iscloser to 1, the resulting superiority and inferiority rank-ings are

Su − GRG : 3PRLP − 1 � 3PRLP − 5 � 3PRLP− 3 � 3PRLP − 2 � 3PRLP − 4

I − GRG : 3PRLP − 2 � 3PRLP − 1 � 3PRLP− 3 � 3PRLP − 4 � 3PRLP − 5.

16 M. TAVANA ET AL.

Table . Grey relation coefficients of superiority and inferiority matrixes.

Superiority matrix Inferiority matrix

Criterion S S S S S S S S S S

A . . . . . . . .A . . . . . . . . A . . . . . . . .B . . . . . . . . B . . . . . . . .B . . . . . . . .D . . . . . . . . D . . . . . . . .E . . . . . . . .E . . . . . . . . F . . . . . . . .F . . . . . . . .F . . . . . . . . F . . . . . . . .F . . . . . . . .

Table . Grey relation grade for S and I matrices.

PRLP S-GRG Normal S-GRG I-GRG Normal I-GRG

PRLP- . . .PRLP- . . . PRLP- . . . .PRLP- . . . .PRLP- . . . .

3PRLP-1 3PRLP-5

3PRLP-2 3PRLP-3 3PRLP-4

Figure . Priorities of options.

Table . Final rankings.

PRLP V= V= V= . Norm. V= .

PRLP- . . . PRLP- . . . .PRLP- . . . .PRLP- . . . .PRLP- . . . .

The above results suggest that Options 1 and 2 are thebest choices. Figure 7 illustrates how the available optionscan be prioritised according to the two aforementionedrankings.

... Final decisive rankingFinally, the Su-GRG and I-GRG rankings are combinedusing Equation (13), and the final ranking is obtainedbased on the problem-weighted V indexes presented inTable 10.

3PRLP − 1 �� 3PRLP − 2 �� 3PRLP−3 �� 3PRLP − 5 �� 3PRLP − 4

6. Discussion andmanagerial insights

The main feature of the hybrid ANP-IFG-SIR model isits focus on the reliability of the reports received fromthe experts. That is, the evaluations of the initial groupof experts, who must determine the relative importanceof the attributes of the RLPs, are inherently impreciseand could be processed using a fuzzy version of the ANP.However, these experts do not have a direct incentiveto misrepresent the information provided. On the otherhand, the second group of experts evaluates directly theset of 3PRLPs based on the selection criteria obtainedfrom the ANP. The potential incentives of the expertswithin this second group to bias their evaluations mustbe accounted for by any decision-maker facing a strategicreporting scenario.

As a result, the first six stages of the current modelhave been explicitly defined to smooth potential biasesin the evaluations received from the second group ofexperts. This is why we have considered the credibility ofthe experts together with their evaluations from the verybeginning of the decision-making process. The aggrega-tion of intuitionistic fuzzy reports introduced in the firsttwo stages of the model weights down potential biasesusing the credibility of the experts. The decision-maker istherefore endowed with the resulting matrix of subjectivefuzzy aggregated evaluations describing the performanceof each alternative for all the selection criteria.

Given its advantages over the standard TOPSISapproach, we have applied the SIR technique to thematrix of aggregated evaluations and generated the supe-riority and inferiority matrices to further smooth thebiases in the evaluations received regarding the rela-tive performance of the potential 3PRLPs. It should alsobe emphasised that the amount of numerical computa-tions required increases considerably if a TOPSIS-type

INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE: OPERATIONS & LOGISTICS 17

approachwould have been followed, which hasmotivatedthe use of GRA to derive the relative evaluations of eachalternative.

Note that we have implemented two filters to accountfor potential biases in the subjective reports received. Thefirst one softens strategic evaluation peaks while the sec-ond classifies the alternatives based on their relative supe-riority and inferiority performances. After both filtershave been implemented, the decision-maker can derivethe correspondingGRGs and generate two different rank-ings that will be subjectively combined in order to makea final decision.

The main consequence of the model from a strategicmanagement perspective implies that managers shouldbe particularly careful when selecting the experts giventhe possibility of being subject to strategic manipulationwhen receiving the reports. Moreover, managers shouldhave a ranking method at their disposal that is both easyto compute and implement but sufficiently flexible so asto account for strategically biased evaluations. Clearly, thecapacity of a model to smooth the biased evaluations ofexperts depends on the relative number of experts report-ing strategically.

From an applied viewpoint, it should be highlightedthat the literature is focusing on the implementationof environmentally friendly RLs policies in develop-ing countries. For example, the Brazilian governmenthas recently enacted the National Policy of Solid Waste(Bouzon et al., 2016), encouraging companies to incorpo-rate RLs practices in their business processes (Guarnieriet al., 2015). A similar framework has been described forthe Indian electronics industry (Prakash & Barua, 2015,2016). However, both countries are constrained by weakinstitutional and regulatory systems that impose consid-erable barriers to the implementation of these policies,starting with the selection of adequate providers.

We conclude by emphasising that the results obtainedare not only sensitive to the subjective evaluations of theexperts but also to those of the decision-maker selectinga 3PRLP. Note, in particular, that the subjective prefer-ences of the decision-maker play a fundamental role intwo distinct stages of the model: when assigning credibil-ity scores to the experts and when deciding the weights ofthe superiority and inferiority rankings, V and (1 −V ),whose convex combination determines the final choicebeing made.

7. Conclusion

The importance of RL processes has been recentlystressed due to the existence of factors that directlyinfluence customers’ decisions when making a purchasein a competitive environment. Organisations generally

outsource RL to reduce costs, focus on their core busi-ness, and avoid the complexities associated with theseprocesses. Consequently, the selection of a third-party RLprovider is an essential element of the supply chain andorganisations require a proper decision-making modelto address this problem. At the same time, the litera-ture has also emphasised the important consequencesthat the implementation of RL processes has for devel-oping countries designing sustainable environmentallyfriendly policies while facing considerable internal bar-riers to the adoption of such processes. The proposedhybrid MCDM model obtained by combining the ANPand IFG-SIRmethods is used in this study to evaluate andselect the best 3PRLP in the presence of strategic uncer-tainty.

The present research includes a case study aimed atassessing the reliability of the proposed hybrid model.Among the numerous interrelated criteria involved inthe selection of a 3PRLP, flexibility, a sub-criterion oforganisational performance, has been shown to be themost important one, followed by services (a sub-criterionof organisational performance) and profitability (a sub-criterion of the impact of using 3PLs). Given these evalu-ations, the IFG-SIR part of the model was used to createthe final ranking of alternatives and select one of them.

The selection process is subject to a substantial amountof ambiguity and subjectivity that has to be dealt with bythe decision-making model, which must also provide arelatively simple computational approach that can be eas-ily understood and implemented by the managers. In thisregard, the use of any combination of techniques – otherthan IFG-SIR – designed to smooth the influence of sub-jective biases on the final decision should be encouragedso as to prevent the potential manipulation derived fromstrategic reporting.

The advantages of the proposed model are seven-fold. The ANP and IFG-SIR method: (1) applies to bothqualitative and quantitative structures; (2) handles inter-criterion mutual effects based on systems theory; (3)enhances group decision-making while accounting forthe credibility of the experts; (4) allows for a simultane-ous comparison of the superiority and inferiority evalua-tions of the alternatives; (5) does not require the demand-ing TOPSIS and SAW calculations performed in the clas-sic SIR method; (6) accounts for the lack of adequateinformation and inexperienced experts through GRA;(7) allows to manage ambiguous and strategically biasedinformation.

Note

1. The name has been changed to protect the anonymity ofthe manufacturing company.

18 M. TAVANA ET AL.

Disclosure statement

No potential conflict of interest was reported by the authors.

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