In real life, DMs frequently disagree when expressing their ideas in assessment. Fuzzy sets (Zadeh, 1965) can only depict fuzziness of agreement but not reflect the disagreement of DMs, while IFSs (Atanassov, 1986) are an appropriate tool for describing DMs’ agreement and disagreement evaluations. In the recent decades, IFSs have been widely used in managing fuzziness in DMs’ assessments in complex socioeconomic situations (Kahraman et al., 2016).

TOPSIS developed by Hwang and Yoon (1981) is one of the widely recognized multi-criteria decision-making (MCDM) methods (Zavadskas et al., 2016). TOPSIS is established on the basis of the principle that the optimal solution should be closest to the positive ideal point and farthest to the negative ideal point (Aouadni et al., 2017; Dwivedi et al., 2018; Opricovic and Tzeng, 2004). Recently, numerous researchers have extended TOPSIS to solve intuitionistic fuzzy MCDM problems (Chen et al., 2016; Shen et al., 2018; Wan et al., 2015; Yue, 2014). Furthermore, TOPSIS has been integrated with other methods. Li and Wu (2016) proposed an improved interval-valued intuitionistic fuzzy TOPSIS integrated with a cumulative interval score function and applied it to manage MCDM problems with unknown weight information. Aloini et al. (2014) developed a peer-based modification to intuitionistic fuzzy MCGDM with TOPSIS and used it in packaging machine selection. Buyukozkan and Guleryuz (2016) established an integrated intuitionistic fuzzy MCGDM approach associated with Analytical Hierarchy Process (AHP) and TOPSIS and employed it to select a suitable product development partner.

The BWM proposed by Rezaei (2015) is a comparison-based method that establishes specific comparisons between items. For a comparison issue that contains n items, firstly, the best and worst items are determined, and the important degree of the best item to the worst one is then evaluated. Secondly, comparisons between the remaining n − 2 items and the best and worst items are needed. Lastly, a mathematical program model is constructed to derive the important values of items. In comparison with the traditional AHP (Saaty, 1980) and Analytical Network Process (ANP) (Saaty, 1996), BWM only requires to conduct 2 n − 3 comparisons. Statistical finding shows that BWM requires less comparison data but results in more consistent comparisons (Rezaei, 2016). To extend BWM to an uncertain environment, Mou et al. (2016) developed an intuitionistic fuzzy multiplicative BWM and applied it to MCGDM. Li et al. (2018) proposed a BWM method and used it in MCDM with probabilistic hesitant fuzzy information. Moreover, BWM was applied to supplier segmentation (Rezaei et al., 2015), water security sustainability evaluation (Nie et al., 2018a), failure mode and effects analysis (Nie et al., 2018b; Tian et al., 2018a) and performance evaluation of smart bike-sharing programs (Tian et al., 2018b).

Numerous researchers have studied the criteria and decision models involved in the process of selecting a suitable supplier (Chai et al., 2013). An increasing number of enterprises incorporate the green concept into the supplier chain management to comply with the trend of sustainable development. Numerous studies have focused on green supplier selection problems that allow for a set of conventional and environmental criteria (Beske et al., 2014; Govindan et al., 2015). In this regard, Govindan et al. (2015) found that environmental management system (EMS) was the most significant and comprehensive environmental criterion in the process of evaluating enterprises’ environmental performance and operation efficiency. Banaeian et al. (2015) identified EMS as a green criterion and financial, delivery and service and qualitative as the primary conventional criteria associated with a set of sub-criteria. These criteria were applied to select a green supplier in the food industry.

Recently, large numbers of green supplier evaluation and selection approaches have been developed, ranging from a single method to hybrid methods that are integrated with multiple techniques (Govindan et al., 2015). Dobos and Vorosmarty (2014) used Data Envelopment Analysis (DEA) as an evaluation tool. Dou et al. (2014) and Hashemi et al. (2015) combined ANP and Grey Relational Analysis (GRA) and applied them to evaluate green supplier development programs. Kuo et al. (2015) employed DEMATEL (DEcision MAking Trial and Evaluation Laboratory) associated with ANP to determine criterion value and VIKOR (Vlsekriterijumska optimizacija I KOmpromisno Resenje) to evaluate the environmental performance of suppliers in the electronic industry. Banaeian et al. (2014) proposed a hybrid model using Delphi and DEA. Tsui et al. (2015) developed a hybrid MCDM method using DEMATEL, ANP and PROMETHEE (Preference Ranking Organization METHod for Enrichment Evaluations) for green supplier selection. Freeman and Chen (2015) presented a comprehensive framework integrated with AHP, entropy and TOPSIS. Vahidi et al. (2018) developed a novel bi-objective two-stage mixed possibilistic-stochastic programming model to manage green supplier selection.

However, in green supplier evaluation and selection processes, some criteria are often precisely unknown, especially for environmental factors, such as easy recycling and reuse capability. Under this environment, fuzzy set theory can be regarded as an effective tool for addressing uncertainty. Kannan et al. employed a fuzzy TOPSIS method to select green suppliers for a Brazilian electronics company (Kannan et al., 2014). They also developed a fuzzy axiomatic design to select green suppliers for a Singapore plastic manufacturing company (Kannan et al., 2015). Akman (2015) combined the fuzzy c-means clustering and fuzzy VIKOR methods. Awasthi and Kannan (2016) incorporated the fuzzy Nominal Group Technique (NGT) with fuzzy VIKOR methods for green supplier development. Wang and Chan (2013) integrated the fuzzy AHP with fuzzy TOPSIS to support green supply chain management. Furthermore, fuzzy hybrid MCDM methods with multiple techniques, such as fuzzy ANP, DEMATEL and TOPSIS (Buyukozkan and Cifci, 2012), fuzzy AHP, Additive Ratio Assessment (ARAS) and Multi-Segment Goal Programming (MSGP) (Liao et al., 2016), and fuzzy TOPSIS, VIKOR and GRA (Banaeian et al., 2018) were developed. Interval type-2 fuzzy MCGDM methods based on Weighted Aggregated Sum Product Assessment (WASPAS) (Ghorabaee et al., 2016) and TODIM (An Acronym in Portuguese of Interactive and MCDM) (Qin et al., 2017), and probability hesitant fuzzy QUALIFLEX (QUALItative FLEXible Multiple Criteria Method) (Li and Wang, 2017) were proposed to manage green supplier selection. Table 1 presents a summary of preceding literature on green supplier selection.

As shown in Table 1, fuzzy set theory equipped with MCGDM methods has been applied to solve green supplier selection problems. However, minimal attention has been paid to intuitionistic fuzzy environment to address multi-criteria green supplier evaluation and selection problems. Furthermore, AHP and ANP methods are often employed to obtain criterion weights in green supplier selection. However, they may be tedious and complex in calculation to achieve consistent comparisons. The MCGDM methods with TOPSIS are used to rank green suppliers. The decision process is insufficiently cautious because DMs with different knowledge backgrounds are assigned with the same weights with respect to different criteria.

For an IFS { ⟨ x , μ A ( x ) , ν A ( x ) ⟩ | x ∈ X . }, the ordered tuple components ⟨ μ A ( x ) , ν A ( x ) ⟩ are described as intuitionistic fuzzy numbers (IFNs). Any IFN ⟨ μ , ν ⟩ must satisfy the conditions μ , ν ∈ [ 0 , 1 ] and 0 ⩽ μ + ν ⩽ 1.

The Euclidean distance between two intuitionistic fuzzy matrices can be defined as follows: Definition 6.

(See Yue, 2014.) Let A k = ( ⟨ μ i j k , ν i j k ⟩ ) m × n ( k = 1 , 2 ) be two intuitionistic fuzzy matrices, where the elements in A k are IFNs. Then, the distance between A 1 and A 2 is defined as follows: (3) d ( A 1 , A 2 ) = 1 2 m n ∑ i = 1 m ∑ j = 1 n ( ( μ i j 1 − μ i j 2 ) 2 + ( ν i j 1 − ν i j 2 ) 2 + ( π i j 1 − π i j 2 ) 2 ) , where π i j 1 = 1 − μ i j 1 − ν i j 1 and π i j 2 = 1 − μ i j 2 − ν i j 2 ( i = 1 , 2 , … , m; j = 1 , 2 , … , n).

This subsection presents an integrated methodology for green supplier selection on the basis of BWM and improved TOPSIS. Figure 1 shows the flowchart of the proposed approach.

For convenience, let M = { 1 , 2 , … , m }, N = { 1 , 2 , … , n } and T = { 1 , 2 , … , t }. The MCGDM problem concerned is described as follows.

Let A = { a 1 , a 2 , … , a m } be a set of m alternatives, C = { c 1 , c 2 , … , c n } be a set of n criteria, and E = { e 1 , e 2 , … , e t } be a set of t DMs. Assume that w j ( j ∈ N ) is the weight value of c j , where w j ⩾ 0 and ∑ j = 1 n w j = 1. Suppose that r i j k ( i ∈ M, j ∈ N, k ∈ T) represents the rating of alternative a i with respect to criterion c j provided by DM e k . Thus, the individual decision matrix provided by DM e k can be expressed as R k = ( r i j k ) m × n . Denote the weight vector of criteria by w = ( w 1 , w 2 , … , w n ), and the weight vector of DMs by ω = ( ω 1 , ω 2 , … , ω t ).

The focus is how to rank alternatives on the basis of individual decision matrices R k ( k ∈ T ) associated with weight information. To solve this type of MCGDM problem, a methodology integrated with intuitionistic fuzzy TOPSIS and BWM is introduced. The main steps of the proposed approach are briefly presented as follows.

In accordance with the principle of BWM developed by Rezaei (2015, 2016), DMs firstly select the best (e.g. most important and desirable) and the worst (e.g. least important and desirable) criteria.

Secondly, DMs determine the preferences of the best criterion over all the other criteria by using a number from 1 to 9 (1 means equally important and 9 signifies extremely important). The result is presented as a ‘best-to-others (BO)’ vector as follows: U B = ( u B 1 , u B 2 , … , u B n ) , where u B j indicates the preference of the best criterion B over criterion j, and u B B = 1.

Thirdly, DMs determine the preferences of the other criteria over the worst criterion by using a number from 1 to 9 (1 means equally important and 9 signifies extremely important). The result is presented as an ‘others-to worst (OW)’ vector as follows: V W = ( v 1 W , v 2 W , … , v n W ) T , where v j W indicates the preference of criterion j over the worst criterion W, and v W W = 1.

Lastly, establish a mathematical model and derive the optimal weights ( w 1 ∗ , w 2 ∗ , … , w n ∗ ). For each pair of w B / w j and w j / w W , the optimal weight should satisfy the conditions w B / w j = u B j and w j / w W = v j W . To meet these requirements, the maximum absolute differences | w B w j − u B j | and | w j w W − v j W | for all j should be minimized. Thus, the following model can be constructed by considering the sum condition and non-negativity of weights. (4) min max j { | w B w j − u B j | , | w j w W − v j W | } 108 p t ] s.t. w j ⩾ 0 , forall j ∑ j = 1 n w j = 1 .

Here, Model (4) can be transformed into the following linear programming model: (5) min ξ s.t. | w B w j − u B j | ⩽ ξ , for all j | w j w W − v j W | ⩽ ξ , for all j w j ⩾ 0 , for all j ∑ j = 1 n w j = 1 .

The optimal weights ( w 1 ∗ , w 2 ∗ , … , w n ∗ ) and consistency index ξ ∗ can be derived by solving Model (5). Furthermore, calculating the consistency level of comparisons is required. Rezaei (2015) defined the consistency index as follows.

As every DM is skilled in only some specific fields, it is more appropriate to allocate different weight values of each DM on different criteria.

For each criterion c j , the criterion value expressed by DM e k is transformed into an IFN vector r j k = ( r 1 j k , r 2 j k , … , r m j k ). Let ω j k be the weight of e k with respect to c j . To determine the criterion weight ω j k , two aspects should be considered simultaneously. One aspect is the closeness coefficient that captures the similarity between the individual decision matrix provided by DM e k and the collective one by the group of DMs. The other aspect is the proximity degree that measures the proximity between the individual decision matrix provided by DM e k and those matrices provided by all other DMs.

In accordance with the previous analysis, ω j k ( j ∈ N; k ∈ T) can be derived from two aspects. On one hand, an improved TOPSIS method inspired by the idea of TOPSIS (Hwang and Yoon, 1981; Wang et al., 2017; Yue, 2014) is developed to calculate the closeness coefficient. On the other hand, the proximity degree can be calculated on the basis of distance measure.

(1) Calculate the closeness coefficient on the basis of the improved TOPSIS.

The PID vector r j ∗ on criterion c j is defined as the arithmetic average of all individual decision vectors r j k ( k ∈ T ) on the basis of Eq. (1), that is, r j ∗ = ( r 1 j ∗ , r 2 j ∗ , … , r m j ∗ ), where (7) r i j ∗ = ⟨ μ i j ∗ , ν i j ∗ ⟩ = IFOWA ( r i j 1 , r i j 2 , … , r i j t ) = ⟨ 1 − ∏ k = 1 t ( 1 − μ i j σ ( k ) ) ϖ k , ∏ k = 1 t ( ν i j σ ( k ) ) ϖ k ⟩ ( i ∈ M , j ∈ N ) , where ϖ k is the associated weight value of the IFOWA operator, and its value can be determined in accordance with the normal distribution-based method (Xu, 2005).

The NID vectors consist of the individual negative ideal decision (INID) vector, left individual negative ideal decision (LINID) vector and right individual negative ideal decision (RINID) vector. The INID, LINID and RINID vectors on criterion c j are denoted by r j c = ( r 1 j c , r 2 j c , … , r m j c ), r j l − = ( r 1 j l − , r 2 j l − , … , r m j l − ) and r j r − = ( r 1 j r − , r 2 j r − , … , r m j r − ), respectively. In accordance with the complement operation in Definition 2, (8) r i j c = ⟨ μ i j c , ν i j c ⟩ , where μ i j c = ν i j ∗ = ∏ k = 1 t ( ν i j σ ( k ) ) ϖ k and ν i j c = μ i j ∗ = 1 − ∏ k = 1 t ( 1 − μ i j σ ( k ) ) ϖ k , (9) r i j l − = ⟨ μ i j l − , ν i j l − ⟩ , where μ i j l − = min k ∈ T { μ i j k } and ν i j l − = max k ∈ T { ν i j k } , (10) r i j r − = ⟨ μ i j r − , ν i j r − ⟩ , where μ i j r − = max k ∈ T { μ i j k } and ν i j r − = min k ∈ T { ν i j k } .

Subsequently, an extended closeness coefficient of each individual decision vector r j k with respect to the ideal decision vectors, including r j ∗ , r j c , r j l − and r j r − , is defined as follows: (11) φ j k = d ( r j k , r j c ) + d ( r j k , r j l − ) + d ( r j k , r j r − ) d ( r j k , r j ∗ ) + d ( r j k , r j c ) + d ( r j k , r j l − ) + d ( r j k , r j r − ) ( j ∈ N , k ∈ T ) .

(2) Calculate the average proximity degree on the basis of distance measure.

The proximity degree between r i j k and r i j l is denoted by γ i j l k and can be calculated as: (12) γ i j l k = 1 − d ( r i j l , r i j k ) , where d ( r i j l , r i j k ) is the distance between r i j k and r i j l on the basis of Eq. (2).

Furthermore, on the basis of Eq. (12), the average proximity degree η j k between DM e k and all the other DMs e l ( l ∈ T, l ≠ k) on criterion c j can be calculated as follows: (13) η j k = 1 m ( t − 1 ) ∑ l = 1 , l ≠ k t ∑ i = 1 m γ i j l k = 1 − 1 m ( t − 1 ) ∑ l = 1 , l ≠ k t ∑ i = 1 m d ( r i j l , r i j k ) , where d ( r i j l , r i j k ) is the distance between r i j k and r i j l on the basis of Eq. (2).

(3) Derive the weights of DMs with respect to different criteria.

To comprehensively consider the closeness coefficient and proximity degree, a control parameter θ ( 0 ⩽ θ ⩽ 1 ) is employed to construct the unified criterion weight λ j k of DM e k on criterion c j , as shown as follows: (14) λ j k = θ φ j k + ( 1 − θ ) η j k .

The unified criterion weight λ j k can tradeoff the closeness efficient versus the proximity degree by altering the values of parameter θ. Particularly, λ j k will only depend on the closeness efficient if θ = 1, and it will only depend on the proximity degree if θ = 0. Without loss of generality, a default control parameter θ = 0.5 can be set in practical application.

The unified criterion weights λ j k ( k ∈ T ) are normalized, and the weight ω j k of DM e k with respect to criterion c j can be obtained as follows: (15) ω j k = λ j k ∑ l = 1 t λ j l ( j ∈ N , k ∈ T ) .

Let R ˆ k = ( r ˆ i j k ) m × n be the weighted individual decision matrix. Then, the following result can be obtained: (16) R ˆ k = ( r ˆ i j k ) m × n = ( ω j k r i j k ) m × n = ( ⟨ μ ˆ i j k , ν ˆ i j k ⟩ ) m × n ( k ∈ T ) , where μ ˆ i j k = 1 − ( 1 − μ i j k ) ω j k and ν ˆ i j k = ( ν i j k ) ω j k on the basis of the operations in Definition 2. ω j k denotes the obtained weight of DM e k using Eq. (15).

In the following, the target is to rank all the alternatives on the basis of the improved TOPSIS method.

(1) Obtain the group decision matrix with respect to criteria.

For each alternative a i ( i ∈ M ), the weighted individual decision matrix in Eq. (16) is transformed into a group decision matrix of DMs with respect to the following criteria: (17) H i = ( h k j i ) t × n = ( ⟨ μ ˆ k j i , ν ˆ k j i ⟩ ) t × n ( i ∈ M ) , where the element h k j i in H i is the same as the element r ˆ i j k in R ˆ k in Eq. (16). Similar to the individual decision matrix R k , the matrix H i is called the alternative decision matrix. For each criterion c j , the weighted criterion values of alternative a i expressed by all the DMs e k ( k ∈ T ) are denoted as an IFN vector h j i = ( h 1 j i , h 2 j i , … , h t j i ).

(2) Determine the alternatives’ PID vector h j ∗ and the NID vectors h j c and h j − on criterion c j .

Similar to the procedures in Step 4, let h j ∗ = ( h 1 j ∗ , h 2 j ∗ , … , h t j ∗ ) denote the alternatives’ PID vector. The alternatives’ PID vector should be the best decision of all H i ( i ∈ M ) in Eq. (17). The elements in the alternatives’ PID vector can be calculated as follows: (18) h k j ∗ = ⟨ μ ˆ k j ∗ , ν ˆ k j ∗ ⟩ , where μ ˆ t j ∗ = max i ∈ M { μ ˆ k j i } and ν ˆ k j ∗ = min i ∈ M { ν ˆ k j i } .

Similar to the individual NID decision vectors, the alternatives’ NID vector should have maximum separation from the alternatives’ PID vector h j ∗ . It can naturally consider the complement ( h j ∗ ) c of h j ∗ , which shows the maximum separation from h j ∗ . Let h j c = ( h 1 j c , h 2 j c , … , h t j c ) denote the complement ( h j ∗ ) c of h j ∗ , where (19) h k j c = ⟨ μ ˆ k j c , ν ˆ k j c ⟩ , where μ ˆ k j c = ν ˆ k j ∗ = min i ∈ M { ν ˆ k j i } and ν ˆ k j c = μ ˆ k j ∗ = max i ∈ M { μ ˆ k j i } .

Moreover, the following alternatives’ decision vector also shows the maximum separation from the alternatives’ PID vector h j ∗ . Let h j − = ( h 1 j − , h 2 j − , … , h t j − ) denote one of the alternatives’ NID vectors, where (20) h k j − = ⟨ μ ˆ k j − , ν ˆ k j − ⟩ , where μ ˆ k j − = min i ∈ M { μ ˆ k j i } and ν ˆ k j − = max i ∈ M { ν ˆ k j i } .

(3) Calculate the TOPSIS-based index CI k j i and the comprehensive TOPSIS-based index CI ( a i ).

The distances between each alternative’s decision value h k j i and h k j ∗ , h k j c and h k j − are calculated on the basis of Eq. (2) and are denoted as d ( h k j i , h k j ∗ ), d ( h k j i , h k j c ) and d ( h k j i , h k j − ), respectively.

Furthermore, an improved TOPSIS-based index is developed to measure the discrimination of h k j i with respect to h k j ∗ , h k j c and h k j − and is defined as: (21) CI k j i = d ( h k j i , h k j c ) + d ( h k j i , h k j − ) d ( h k j i , h k j ∗ ) + d ( h k j i , h k j c ) + d ( h k j i , h k j − ) ( i ∈ M , j ∈ N , k ∈ T ) .

The improved TOPSIS-based index CI k j i can be employed to evaluate the performance of alternative a i with respect to criterion c j . By coupling the criterion weight w j k in terms of DM e k , the comprehensive TOPSIS-based index CI ( a i ) of the characteristics for alternative a i is expressed as follows: (22) CI ( a i ) = 1 t ∑ k = 1 t ∑ j = 1 n ( w j k CI k j i ) = 1 t ∑ k = 1 t ∑ j = 1 n ( w j k d ( h k j i , h k j c ) + d ( h k j i , h k j − ) d ( h k j i , h k j ∗ ) + d ( h k j i , h k j c ) + d ( h k j i , h k j − ) ) .

Significantly, the closer the alternatives’ decision value h k j i is to alternatives’ PID value  h k j ∗ , and the farther h k j i is from the alternatives’ NID values h k j c and h k j − , the closer the CI ( a i ) is to 1. Thus, the comprehensive TOPSIS-based index CI ( a i ) can be used to rank the preference order of all alternatives. A larger CI ( a i ) indicates a better alternative  a i .

Agriculture plays an important role in China as China consumes a large number of agriculture products. Agri-food production significantly contributes to the consumption of resources and presents remarkable environmental impacts. Company ABC, located in East China, is one of the leading manufacturers of processed vegetable, edible vegetable oils and condiments in China. With 26 large manufacturing facilities, company ABC has paid a major contribution to the economy and growth in the food sector. Recently, China’s government has paid considerable attention on sustainable development, which can push company ABC to incorporate the green concept into its management and administration. Company ABC is certified by ISO 14000 and uses the related guidelines to perform environmental duties, including encouraging its suppliers to improve their environmental practices and performance continuously. Company ABC needs to complete a supplier selection analysis. Under these circumstances, a decision committee consisting of three members, namely, the chief executive officer ( e 1 ), the chief marketing manager ( e 2 ) and an environmental expert ( e 3 ), has been formed to determine the optimal supplier among four possible alternatives. Experts use the BWM to obtain the subjective weights of criteria. Moreover, they assess the performance of each potential green supplier in terms of criteria. The collected opinions of experts are expressed in linguistic terms, which will then be transformed into IFNs.

The detailed procedures for evaluating and selecting the most appropriate green supplier are shown as follows.

Step 1: Define the overall goal, criteria, sub-criteria and associated alternatives for decision-making problems, and then establish the hierarchy of the considered problem.

A conventional and green supplier evaluation standard is identified on the basis of an extensive review of green supplier evaluation literature in the agri-food industry (Banaeian et al., 2018, 2015, 2014; Beske et al., 2014; Borghi et al., 2014; Brodt et al., 2013). This standard contains four main criteria associated with 15 sub-criteria, as shown in Fig. 2.

Financial ( c 1 ):

Capital and financial power of supplier company ( c 11 ), proposed raw material price ( c 12 ) and transportation cost to the geographical location (availability) ( c 13 ).

Delivery and service ( c 2 ):

Communication system (willingness to trade, attitude, acceptance of procedures and flexibility) ( c 21 ), on time delivery (lead time) ( c 22 ), after sales service (police, quality assurance and damage ratings) ( c 23 ) and production capacity ( c 24 ).

Qualitative ( c 3 ):

Quality (suppliers’ ability to access quality characteristics) ( c 31 ), operational control (reporting, quality control, inventory control and research and development) ( c 32 ), expert labour, technical capabilities and facilities ( c 33 ) and business experience and position among competitors ( c 34 ).

EMS ( c 4 ):

Environmental prerequisite (environmental staff training) ( c 41 ), environmental planning (program to reduce environmental impacts and green research and development) ( c 42 ), environmentally friendly material (low waste: easy recycling and reuse capability) ( c 43 ) and environmentally friendly technology (emission of pollutant: CO 2 equivalent and VOC, BOD and COD contents and etc.) ( c 44 ).

Step 2: Design and select the evaluation scale of IFS.

In the decision process, DMs use the 9 scale linguistic terms to evaluate the performance of suppliers in terms of each criterion. Furthermore, the linguistic evaluation values are transformed into IFNs (see Table 2 for linguistic terms used for the comparative rating of suppliers). Table 4 shows the linguistic evaluation information provided by DMs.

Step 3: Determine the weight vectors of criteria and sub-criteria.

In accordance with the principle of BWM developed by Rezaei (2015, 2016), DMs initially select the best and worst criteria. Then, the DMs determine the preferences of the best criterion over all the other criteria and the preferences of the other criteria over the worst criterion by using a number from 1 to 9 (1 means equally important and 9 signifies extremely important). The BO vector U B k and the OW vector V W k provided by e k ( k = 1 , 2 , 3 ) are shown in Table 5.

For the main criteria ( c 1 – c 4 ), the BO vector (1, 3, 1, 5) and the OW vector (5, 2, 4, 1) T of e 1 are used as examples. By incorporating the elements of the vectors into Model (5), then Model (23) can be established. (23) min ξ 1 s.t. w 1 w 4 − 5 ⩽ ξ 1 , w 1 w 2 − 3 ⩽ ξ 1 , w 1 w 3 − 1 ⩽ ξ 1 , w 2 w 4 − 2 ⩽ ξ 1 , w 3 w 4 − 4 ⩽ ξ 1 , w j ⩾ 0 , ∑ j = 1 4 w j = 1 .