In this section, we are going to examine applications of BWM that have been conducted in different fields. Rezaei et al. (2015) presented an integrated approach with the BWM, which includes willingness and capabilities as two features for classifying and evaluating the suppliers. This integrated approach helps companies divide their managerial resources effectively. In another research, Rezaei et al. (2016) discussed that choosing suppliers is a strategic decision that significantly influences the competitive advantage of a company. Due to the importance of selecting the suppliers and the sensitivity of this issue, they presented a novel method which aimed to select the suppliers during three phases, namely pre-selection, selection, and aggregation. They used BWM for initial screening. As for the decision-making phase, they made use of an MCDM method and the final decision was made at the aggregation phase based on the material price and the required annual quantity.

Scholars have applied the original BWM into different fields. Gupta and Barua (2016) conducted a study to identify the factors affecting technological innovations in India. Making use of the BWM, they presented a procedure for selecting the best empowerers. The primary purpose of their study was to identify vital enablers in the field of technological innovation in India. Rezaei et al. (2017) considered three alternatives related to transportation: through unit load devices, mixed unit load devices, and loose freight in trucks. To find the optimal freight bundling configuration, they considered three key performance indicators including cost, quality, and loading time. They used the BWM for finding the most suitable type of bundling configuration in the surface transportation. Their proposed framework facilitated risk assessment process. Torabi et al. (2016) used the BWM to present an enhanced risk assessment method within the business continuity management system framework. Sadaghiani et al. (2017) used the BWM to assess the importance of different types of energy such as oil and gas industries by sending questionnaires to academic experts on sustainable supply chain management. Their research helped companies to develop strategies for identifying external forces. The treatment of urban sewage sludge for reducing the threats of environmental pollution and its negative influences on human health is of high significance. Accordingly, Ren et al. (2017) presented a general framework for selecting the proper technology for the treatment of urban sewage. In this framework, the BWM was used to determine the weights of the criteria. Salimi and Rezaei (2018) measured the R&D performance using the BWM. The SERVQUAL model is designed to evaluate the service quality of a baggage handling system. Rezaei et al. (2018) collected the list of criteria for the SERVQUAL model and utilized the BWM to determine the weights of the criteria.

For some situations, the uncertain information of experts is easy to access and the uncertain BWMs have attracted the attention of scholars (Mi et al., 2019). Making some changes to the steps of the BWM, Rezaei (2016) presented a linear model, by which the weight of each criterion was obtained in intervals and the criteria were prioritized after comparing the obtained intervals. Using the fuzzy approach and triangular fuzzy numbers, Guo and Zhao (2017) presented the fuzzy BWM (FBWM). They believed that the FBWM obtained the suitable ranking of alternatives and that it has more comparison consistency than the BWM. In addition, Hafezalkotob and Hafezalkotob (2017) presented a fuzzy approach which combined individual and group alternatives, and used the FBWM in their approach in which triangular fuzzy numbers were employed. Gupta et al. (2017) presented an accurate and comprehensive approach for identifying various factors of managing energy efficiency in India. They believed that identifying the barriers is not enough and other proper solutions would also be presented. To rank the existing barriers, they used the BWM in their research. Mou et al. (2016) used intuitionistic fuzzy multiplicative preference relations for ranking criteria or alternatives and it can be considered as a tool for combining with other MCDM methods. It should be noted that their method did not consider the case that the consistency degree is not acceptable. Aboutorab et al. (2018) proposed the ZBWM by utilizing the Z-number approach to consider the uncertainty of the input data in MCDM problems. Their method has lower inconsistency ratio compared with the BWM.

Mou et al. (2017) presented a graph-based group decision-making approach for intuitionistic fuzzy BWM. Moreover, they presented three numerical examples to show the efficiency of the proposed method. Mi and Liao (2019) enabled BWM to accept hesitant numbers as input and Liao et al. (2019) fused the hesitant linguistic information in BWM. Mi et al. (2019) conducted a survey of the BWM related publications between 2015 and 2019. Hafezalkotob et al. (2019) proposed the interval MULTIMOORA and group interval BWM.

Since there is no research item on the combination of the grey system theory and BWM, the current study attempts to present the GBWM. As mentioned, the grey theory has features such as reality and vague aspects, and can be employed with an incomplete data, therefore, it can be a useful approach to solve decision-making problems (Zavadskas et al., 2009). The result calculated by the GBWM is more reliable than the one calculated by the fuzzy BWM because the GBWM has a smaller inconsistency ratio compared with the fuzzy BWM. We transform the weight determination model in the GBWM into linear models which can be solved efficiently and own high reliability. It should be noted that the rankings derived by the FBWM and GBWM are the same in the comparative analysis. Moreover, after defuzzification, the weights obtained by the FBWM fall within the weights that are obtained from the GBWM. Hence, the proposed method has appropriate performance in calculating weights. The calculation of grey numbers is simple compared to other uncertainty approaches. In the next section, the basic concepts of grey system theory are examined.

In this section, the related preliminaries of grey numbers are reviewed.

A grey number (Liu et al., 2012) is expressed as ⊗A∈[A_,A‾],A_<A‾.

To denote the central point of grey numbers, the “Kernel” of grey numbers (Guo et al., 2017) is proposed as: (2) ⊗Aˆ=12(A_+A‾).

For two grey numbers ⊗A and ⊗B , the arithmetic operations can be defined (Mahmoudi and Feylizadeh, 2018; Turskis and Zavadskas, 2010) as: (3) ⊗A+⊗B=[A_+B_,A‾+B‾], (4) ⊗A−⊗B=[A_−B‾,A‾−B_], (5) ⊗A.⊗B=[Min{A_B_,A‾B‾,A‾B_,A_B‾},Max{A_B_,A‾B‾,A‾B_,A_B‾}], (6) ⊗A⊗B=⊗A.⊗B−1=[Min{A_B_,A_B‾,A‾B_,A‾B‾},Max{A_B_,A_B‾,A‾B_,A‾B‾}].

The length of ⊗A is calculated by: (7) L(⊗A)=A‾−A_.

To compare the grey numbers, the greyness degree of ⊗A was calculated as [32]: (8) g0(⊗A)=μ(⊗A)/μ(Ω), where Ω represents the background of grey numbers and μ is the length of the background of grey numbers. For two grey numbers ⊗A=[A_,A‾] and ⊗B=[B_,B‾] , if ⊗Aˆ<⊗Bˆ , then ⊗A<G⊗B; if ⊗Aˆ=⊗Bˆ , then if go(⊗A)=go(⊗B) , then ⊗A=G⊗B ; if go(⊗A)<go(⊗B) , then ⊗A>G⊗B _{G}}\otimes B$]]>.

The grey possibility degree for numbers ⊗A=[A_,A‾] and ⊗B=[B_,B‾] are calculated by (Li et al. (2007), Zare et al. (2018)): (9) P{⊗A⩽⊗B}=Max{0,L(⊗A)+L(⊗B)−Max(0,A‾−B_)}L(⊗A)+L(⊗B).

Different methods have been presented for solving GLP models. Huang et al. (1995) presented a method for solving grey mixed-integer linear programming. Their model was suitable for grey models with the same sign in lower and upper bounds of grey numbers. After that, Li (2007) proposed another method to solve GLP problems named “Covered Solution”. The disadvantages of the method were complex calculations and it sometimes fails to meet stop conditions. Hajiagha et al. (2012) proposed a method to solve the GLP problem by using a multi-objective concept, yet their method presented the wrong solution for GLP problems as proved by Mahmoudi et al. (2018a). Li et al. (2014) proposed a method based on the concept of Covered Solution method, yet it had some problems, similarly as Li (2007)’s method. Nasseri et al. (2016) presented a new method using the primal simplex algorithm to solve GLP problems, but their method could solve the GLP problems just with the grey objective function. Liu et al. (2009) presented a positioned programming for solving GLP models. This method truly enjoys simplicity and covers all uncertainties in grey numbers. Moreover, this method can present crisp values based on ρ, β and δ parameters that are determined by the decision maker.

The current study employs the positioned programming method to solve GLP problems in the form of Model (3) (Liu et al., 2009; Mahmoudi et al., 2018b). (3) MaxS=C(⊗)Xs.t.A(⊗)X⩽b(⊗),X⩾0. Such that: (10) C(⊗)=[C1(⊗),C2(⊗),…,Cn(⊗)]T, (11) b(⊗)=[b1(⊗),b2(⊗),…,bm(⊗)]T, (12) A(⊗)=a11(⊗)a11(⊗)…a1n(⊗)a21(⊗)a11(⊗)…a2n(⊗)…………am1(⊗)a11(⊗)…amn(⊗). It should be noted that A(⊗) is the grey consumption matrix, C(⊗) is the grey price vector, b(⊗) is the grey constraint vector, and X is the problem decision vector. The parameters employed in Eqs. (10) to (12) are defined in Eqs. (13) to (15): (13) cj(⊗)∈[c_j,c‾j],c_j⩾0,j=1,2,…,n, (14) bi(⊗)∈[b_i,b‾i],b_i⩾0,i=1,2,…,m, (15) aij(⊗)∈[a_ij,a‾ij],a_ij⩾0,i=1,2,…,n,j=1,2,…,n.

To solve the GLP model, it ought to be whitened first.

This theorem has been proven in Liu et al. (2009).

In this section, the GBWM is presented. The eight steps of this method are outlined in detail as follows:

Step 1: Determine the criteria set {c1,c2,c3,…,cn} by decision-makers.

Step 2: Each decision-maker determines the best and the worst criteria. If there are k experts, then k best criteria and the k worst criteria would exist: {Bp1,Bp2,Bp3,…,Bpk} and {Wp1,Wp2,Wp3,…,Wpk} .

Step 3: At this step, each decision-maker determines the degrees of preference of the best criterion to the other criteria using the linguistic variables presented in Table 3. Equation (20) expresses best-to-others (BO) vectors of experts. (20) ⊗ABp1=(⊗aB1p1,⊗aB2p1,⊗aB3p1⋯⊗aBnp1),……⊗ABpk=(⊗aB1pk,⊗aB2pk,⊗aB3pk⋯⊗aBnpk).

In Eq. (20), ⊗ABp1 represents the opinion of the first decision-maker determining the degree of preference of the best criterion to criteria 1 to n.

Step 4: In Eq. (21), ⊗AWpk represents the viewpoint of the kth decision-maker determining the degree of preference of criteria 1 to n to the worst criterion. Equation (21) expresses others-to-worst (OW) vectors. (21) ⊗AWp1=(⊗a1Wp1,⊗a2Wp1,⊗a3Wp1⋯⊗anWp1)T,……⊗AWpk=(⊗a1Wpk,⊗a2Wpk,⊗a3Wpk⋯⊗anWpk)T.

Step 5: At this stage, the degree of optimal weight for each criterion is determined. Since the inputs of the problem are considered in grey numbers, Model (2) is converted into a grey model. For this purpose, we consider Model (5): (5) minξs.t.|WBWj−aBj|⩽ξ,for allj,|WjWw−ajw|⩽ξ,for allj,∑jWj=1,Wj⩾0,for allj.

According to the features of absolute value, Model (5) is equivalent to Model (6): (6) minξs.t.WBWj−aBj⩽ξ,for allj,−WBWj+aBj⩽ξ,for allj,WjWw−ajw⩽ξ,for allj,−WjWw+ajw⩽ξ,for allj,∑jWj=1,Wj⩾0,for allj.

To cross multiply the constraints of Model (6), Model (7) is ultimately obtained: (7) minξs.t.WB−WjaBj⩽Wjξ,for allj,−WB+WjaBj⩽Wjξ,for allj,Wj−Wwajw⩽Wwξ,for allj,−Wj+Wwajw⩽Wwξ,for allj,∑jWj=1,Wj⩾0,for allj.

Due to the multiplication of variables ξ and Wj , Model (7) is a non-linear model. Since solving grey non-linear models involves high levels of complexity, the model should be converted into a linear model. Also, since two continuous variables cause the non-linearity in Model (7), the McCormick method (Hijazi et al., 2017; McCormick, 1976) can be used for linearization. The steps are as follows: (22) If∅1=x1x2andx1∈[x1L,x1U],x2∈[x2L,x2U].

In Eq. (22), variables x1 and x2 are continuous and have specific upper and lower limits. Also, variable ∅1 has been considered as the product of multiplying the variables x1 and x2 . By considering Eq. (22) and adding four constraints mentioned in Eqs. (23) to (26), the linearization operation for variables x1 and x2 is undertaken. (23) ∅1⩾x1Lx2+x2Lx1−x1Lx2L, (24) ∅1⩾x1Ux2+x2Ux1−x1Ux2U, (25) ∅1⩽x1Lx2+x2Ux1−x1Lx2U, (26) ∅1⩽x1Ux2+x2Lx1−x1Ux2L.

In this section, the nonlinear Model (7) is converted into a linear model. Using Eq. (22), we first have the assumptions mentioned in Eq. (27): (27) ∅1=Wjξ,∅2=Wwξ,Wj∈[0,1],Ww∈[0,1].

The assumptions mentioned in Eq. (27) are not sufficient for linearization and the range of variable ξ should be determined in this regard. Based on Eq. (1), for variable ξ, we have (28) ξ=CI×CR.

The value of CI is determined based on Table 1 and the value of CR is always within the interval of [0,1] . In real contexts, the decision-maker may want the inconsistency rate not to be greater than the specified value of A. Therefore, the range of CR is considered as [0,A] . Equation (29) is thus formed: (29) ξ=[0,CI×A]CI⩾0,0⩽A⩽1.

Based on Eqs. (27) and (29) and the constraints from Eq. (23) to Eq. (26), the nonlinear Model (7) is changed into the grey linear Model (8): (8) min⊗ξs.t.⊗WB−⊗Wj⊗aBj⩽⊗∅1,for allj,−⊗WB+⊗Wj⊗aBj⩽⊗∅1,for allj,⊗∅1⩾0,⊗∅1⩾⊗ξ+CI.A.⊗Wj−CI.A,⊗∅1⩽CI.A.⊗Wj,⊗∅1⩽⊗ξ,⊗Wj−⊗Ww⊗ajw⩽⊗∅2,for allj,−⊗Wj+⊗Ww⊗ajw⩽⊗∅2,for allj,⊗∅2⩾0,⊗∅2⩾⊗ξ+CI.A.⊗,Ww−CI.A,⊗∅2⩽CI.A.⊗Ww,⊗∅2⩽⊗ξ,Wj‾−Wj_⩾ε,for allj,∑j⊗Wj=[0.8,1.2],⊗Wj⩾0,for allj,0⩽A⩽1,CI⩾0.

To find the optimal grey weights, the grey linear Model (8) can be formed and then solved using on positioned programming approach.

Step 6: Based on the opinion of each decision-maker, a weight is assigned to each criterion. To integrate the viewpoint of the experts with regard to each criterion, the grey geometric mean relation as defined in Eq. (30) is used. Since the opinions of decision-makers are of different significances, the weights ( Wk ) for each expert are determined using the linguistic variables presented in Table 4. (30) ⊗Wj=(⊗W1jp1⊗W1·⊗W2jp2⊗W2⋯⊗Wkjpk⊗Wk)1∑Wk.

Step 7. At this stage, the obtained weights are normalized by Eq. (31) (Dey and Chakraborty, 2016): (31) ⊗Wj∗=(Wj_12[∑j=1nWj_+∑j=1nWj‾],Wj‾12[∑j=1nWj_+∑j=1nWj‾]).

Step 8. To sort the grey interval numbers obtained for the weights of criteria, the order relation can be used. Another method to compare the weights of the criteria is grey possibility degree as shown in Eq. (9). To do so, the matrix of grey possibility degree is formed as: GPij=AB⋯NAB⋮NP(⊗A⊗⩽A)P(⊗B⩽⊗A)⋯P(⊗N⩽A⊗)P(⊗A⩽⊗B)P(⊗B⩽⊗B)…P(⊗N⩽⊗B)⋮⋮⋱⋮P(⊗A⩽⊗N)P(⊗B⩽⊗N)⋯P(⊗N⩽⊗N).

Ultimately, we have the following results: Pij=AB⋯NAB⋮NPAAPBA…PNAPABPBB…PNB⋮⋮⋱⋮PANPBN…PNN,

where Dij=1P(i⩽j)>0.5i,j=A,…,N,0P(i⩽j)⩽0.5i,j=A,…,N. 0.5\hspace{2.5pt}i,j=A,\dots ,N,\\ {} 0\hspace{1em}& P(i\leqslant j)\leqslant 0.5\hspace{2.5pt}i,j=A,\dots ,N.\end{array}\right.$]]>

By the sum of the horizontal components of the matrix Pij , the scores of criteria are obtained. Based on these scores, the criteria are prioritized. Finally, based on Eq. (1), the consistency ratio can be calculated while the consistency index has been shown in Table 5. For the first time, the consistency ratios for the GBWM are calculated in the current paper. To calculate CI-GBWM, we have employed the following equation (Rezaei, 2015). ξ2−(1+2aBW)ξ+(aBW2−aBW)=0whereaBW=1,3/2,5/2,7/2,9/2.

In this section, the collected data for an MCDM problem about purchasing a car is described. Different criteria may be considered for purchasing a car. Accordingly, three experts are consulted for a group decision making. The question is which criterion is the most important and how can we find optimal weights for the criteria. The solution is using GBWM for group decision making and under uncertainty conditions.

Step 1: Based on the opinions of the experts, quality, price, comfort, safety, and style are the most important criteria for purchasing a car.

Step 2: Each expert determines the best and the worst criteria as: {Pricep1,Pricep2,Qualityp3},{Stylep1,Safetyp2,Comfortp3}.

Step 3: The decision-makers determine the preference degrees of the best criterion to the other criteria using the linguistic variables presented in Table 6.

Step 4. The preference of each criterion to the worst criterion is determined by the experts as shown in Table 7.

Step 5. Based on the information collected through Steps 3 and 4 and using the data presented in Table 3, the linguistic variables are converted into grey numbers. Then, based on Model (8), Model (9) is formed for the first decision-maker as follows: (9) min⊗ξ,s.t.⊗W1−[23,32]⊗W2⩽⊗∅1,−⊗W1+[23,32]⊗W2⩽⊗∅1,⊗∅1⩾0,⊗∅1⩾⊗ξ+5×1×⊗W2−5×1,⊗∅1⩽5×1×⊗W2,⊗∅1⩽⊗ξ,⊗W1−[52,72]⊗W3⩽⊗∅2,−⊗W1+[52,72]⊗W3⩽⊗∅2,⊗∅2⩾0,⊗∅2⩾⊗ξ+5×1×⊗W3−5×1,⊗∅2⩽5×1×⊗W3,⊗∅2⩽⊗ξ,⊗W1−[52,72]⊗W4⩽⊗∅3,−⊗W1+[52,72]⊗W4⩽⊗∅3,⊗∅3⩾0,⊗∅3⩾⊗ξ+5×1×⊗W4−5×1,⊗∅3⩽5×1×⊗W4,⊗∅3⩽⊗ξ,⊗W1−[72,92]⊗W5⩽⊗∅4, −⊗W1+[72,92]⊗W5⩽⊗∅4,⊗∅4⩾0,⊗∅4⩾⊗ξ+5×1×⊗W5−5×1,⊗∅4⩽5×1×⊗W5,⊗∅4⩽⊗ξ,⊗W1−[72,92]⊗W5⩽⊗∅5,−⊗W1+[72,92]⊗W5⩽⊗∅5,⊗∅5⩾0,⊗∅5⩾⊗ξ+5×1×⊗W5−5×1,⊗∅5⩽5×1×⊗W5,⊗∅5⩽⊗ξ,⊗W2−[52,72]⊗W5⩽⊗∅5,−⊗W2+[52,72]⊗W5⩽⊗∅5,⊗W3−[32,52]⊗W5⩽⊗∅5,−⊗W3+[32,52]⊗W5⩽⊗∅5,⊗W4−[32,52]⊗W5⩽⊗∅5,−⊗W4+[32,52]⊗W5⩽⊗∅5,W1‾−W1_⩾0.001,W2‾−W2_⩾0.001,W3‾−W3_⩾0.001,W4‾−W4_⩾0.001,W5‾−W5_⩾0.001,⊗W1+⊗W2+⊗W3+⊗W4+⊗W5=[0.8,1.2],⊗W1⩾0,⊗W2⩾0,⊗W3⩾0,⊗W4⩾0,⊗W5⩾0.

After solving Model (9) for this problem using on positioned programming approach, the weight values of the criteria for the 1st decision-maker are obtained as shown by P1 in Table 8. After formulating the model for the second and third decision-makers and solving them, columns P2 and P3 pertaining to the opinions of the 2nd and 3rd decision-makers are also obtained.

The opinions of the experts regarding each criterion have been compared in Fig. 1. According to Fig. 1, ‘price’ has the highest score by the first two decision-makers, which seems rational considering the fact that it has also been selected as the ‘best’ criterion. Also, the 3rd decision-maker has selected ‘quality’ as the best criterion. Similarly, the worst criterion for each decision-maker can be observed in Fig. 1. Overall, Fig. 1 demonstrates the differences between the viewpoints of experts with regard to each criterion.

Step 6. Using Eq. (30) and according to the data presented in Table 8, the weight of each criterion is presented in Table 9.

Step 7. The normalized weights are presented in Table 10. Also, Fig. 2 represents the final obtained weights.

Step 8. Based on the order relation of grey numbers, the criteria are prioritized as: Price>Quality>Comfort>Style>Safety. \mathrm{Quality}>\mathrm{Comfort}>\mathrm{Style}>\mathrm{Safety}.\]]]>

Another method which yields a relatively similar result is the formation of a matrix of grey possibility degree which is concluded as Eq. (32). (32) GPij=0.500.641.001.001.000.350.501.001.001.000.000.000.500.520.510.000.000.470.500.480.000.000.480.510.50, (33) Pij=0.001.001.001.001.000.000.001.001.001.000.000.000.001.001.000.000.000.000.000.000.000.000.001.000.0043201

Based on the sum of the horizontal components of the matrix in Eq. (33), the prioritization of the criteria is concluded as Price>Quality>Comfort>Style>Safety \mathrm{Quality}>\mathrm{Comfort}>\mathrm{Style}>\mathrm{Safety}$]]>.

In this section, we aim to analyse the sensitivity of the example solved in the previous section. A primary reason for undertaking sensitivity analysis is to investigate the changes of output parameters resulting from changes in the input data. Since the current study makes use of the positioned programming approach, the parameter β is considered as constant in the sensitivity analysis. The sensitivity of the consistency ratio is calculated according to the changes of the parameters ρ and δ. In other words, we are going to examine the impact of parameters ρ and δ on consistency ratio.

As is depicted in Fig. 3, the parameter ρ in the positioned programming is indifferent to changes in the consistency ratio. The parameter δ has a direct impact on the consistency ratio, such that any increase in this parameter would result in an increase in the consistency ratio. Fig. 3 demonstrates the degree of sensitivity of the 1st decision-maker.

It can be seen from Fig. 4 that the consistency ratio has a direct relationship with the parameter δ in the positioned programming. The consistency ratio is indeterminate to the parameter ρ. Fig. 4 is based on the viewpoint of the 2nd decision-maker.

According to Fig. 5, the consistency ratio has a direct correlation with the parameter δ in the positioned programming. If a need arises, the definitive response could be achieved by determining the parameters ρ, β and δ based on the positioned programming method. The lower the parameter δ is and the higher the parameters ρ and β are, the lower the inconsistency response ratio will be achieved. Determining suitable parameters for positioned programming is a challenge as well. Finding the optimal values for these parameters is not an easy job and it depends on different factors. In future researches, Scholar can provide a methodology for finding optimal values. This is another benefit of GBWM that can present a crisp solution based on expert’s needs in different conditions.

In this section, we solve the numerical example by the FBWM (Guo and Zhao, 2017) for each expert to analyse the weights of criteria and the consistency index compared with the GBWM proposed in this paper. We tabulate the detailed computation results of the three experts in Tables 11–13. It should be noted that the results of GBWM have been obtained based on ideal and critical solutions using on position programming method while we have (ρ,β,δ)=(0,0,1) and (ρ,β,δ)=(1,1,0) . These two situations include the highest and lowest consistency ratio for the problem.

Based on the information given in Tables 11, 12, 13, we can obtain some characteristics of the GBWM compared with the FBWM.

It is interesting to mention that the linguistic variables for both GBWM and FBWM are same, but GBWM employs a grey linear model as a core model and that is why the results are more reliable than of FBWM method. On the other hand, employing grey system theory can contribute to decrease the volume of calculations. Therefore, it is clear why current research suggests using GBMW method. Moreover, GBWM method can provide a crisp solution for decision-maker if there is a need. Decision-makers should just provide suitable values for the parameters in a grey linear model and this is another advantage of GBMW method. In conclusion, the GBWM is valid in deducing the weights of criteria and advantageous in keeping reliable results and requiring less computational complexity.