The grey system theory, proposed in 1980s by Julong (
1989), is a mathematical concept which has a widespread application in MCDM. It is considered as a highly effective method for encountering uncertainty problems associated with unknown and incomplete information (Liu and Lin,
2010). Generally, the information pertaining to the preferences of decision-makers for certain criteria and the reasons for such preferences are expressed based on the qualitative judgments of decision-makers. Also, in practice, the judgments of decision-makers are often uncertain and thus cannot be represented by accurate numerical values. The grey theory is one of the concepts used for studying uncertainty and incompleteness. This theory has been used in the mathematical analysis of incomplete information system (Chalekaee
et al.,
2019; Mahmoudi and Feylizadeh,
2018). The importance degrees of criteria in a decision-making process can be expressed by numerical intervals. These numerical intervals would include uncertain information. In other words, the accurate values of grey numbers are unknown, but the interval which covers a value is almost known (Liu
et al.,
2017). Since we will compare the GBWM with the FBWM in the current study, the grey system theory and fuzzy set theory (Zadeh,
1965) are compared with each other from different aspects in Table
2 (Mahmoudi
et al.,
2019).
3.2 Grey Linear Programming (GLP)
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).
Such that:
It should be noted that
$A(\otimes )$ is the grey consumption matrix,
$C(\otimes )$ is the grey price vector,
$b(\otimes )$ 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):
To solve the GLP model, it ought to be whitened first.
Definition 1.
If the values of
${\delta _{ij}}$,
${\beta _{j}}$ and
${\rho _{j}}$ for
$i=1\dots m$ and
$j=1\dots n$ fall within the closed interval
$[0,1]$, the white values of the grey parameters are defined by Eqs. (
16) to (
18).
After the whitening stage, we are left with Model (
4):
If
$(\rho ,\beta ,\delta )=(0,0,1)$, we have the lowest value after solving model (
4). It can be displayed by
$\operatorname{Max}\underline{s}$. On the other hand, if
$(\rho ,\beta ,\delta )=(1,1,0)$, we have the highest value after solving model (
4), represented by
$\operatorname{Max}\overline{s}$.
Theorem 1.
Eq. (
19)
holds true for different values of δ, ρ and β within the interval $[0,1]$.
This theorem has been proven in Liu
et al. (
2009).
3.3 Grey Best Worst Method (GBWM)
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 $\{{c_{1}},{c_{2}},\hspace{2.5pt}{c_{3}},\dots ,{c_{n}}\}$ 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: $\{{B^{p1}},{B^{p2}},{B^{p3}},\dots ,{B^{pk}}\}$ and $\{{W^{p1}},{W^{p2}},{W^{p3}},\dots ,{W^{pk}}\}$.
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.
Table 3
Linguistic variables of decision-makers.
Linguistic variable |
Value |
Equally Important (EI) |
$[1,1]$ |
Weakly Important (WI) |
$\big[\frac{2}{3},\frac{3}{2}\big]$ |
Fairly Important (FI) |
$\big[\frac{3}{2},\frac{5}{2}\big]$ |
Very Important (VI) |
$\big[\frac{5}{2},\frac{7}{2}\big]$ |
Absolutely Important (AI) |
$\big[\frac{7}{2},\frac{9}{2}\big]$ |
In Eq. (
20),
$\otimes {A_{B}^{p1}}$ 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),
$\otimes {A_{W}^{pk}}$ 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.
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):
According to the features of absolute value, Model (
5) is equivalent to Model (
6):
To cross multiply the constraints of Model (
6), Model (
7) is ultimately obtained:
Due to the multiplication of variables
ξ and
${W_{j}}$, 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:
In Eq. (
22), variables
${x_{1}}$ and
${x_{2}}$ are continuous and have specific upper and lower limits. Also, variable
${\varnothing _{1}}$ has been considered as the product of multiplying the variables
${x_{1}}$ and
${x_{2}}$. By considering Eq. (
22) and adding four constraints mentioned in Eqs. (
23) to (
26), the linearization operation for variables
${x_{1}}$ and
${x_{2}}$ is undertaken.
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):
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
The value of
$\mathit{CI}$ is determined based on Table
1 and the value of
$\mathit{CR}$ 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
$\mathit{CR}$ is considered as
$[0,A]$. Equation (
29) is thus formed:
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):
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 (
${W_{k}}$) for each expert are determined using the linguistic variables presented in Table
4.
Step 7. At this stage, the obtained weights are normalized by Eq. (
31) (Dey and Chakraborty,
2016):
Table 4
Grey linguistic variables for determining the significance of each expert.
Very low |
Low |
Medium low |
Medium |
Medium-high |
High |
Very high |
[$1,2$] |
[$2,3$] |
[$3,4$] |
[$4,5$] |
[$5,6$] |
[$6,7$] |
[$7,8$] |
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:
Ultimately, we have the following results:
where ${D_{ij}}=\left\{\begin{array}{l@{\hskip4.0pt}l}1\hspace{1em}& P(i\leqslant j)>0.5\hspace{2.5pt}i,j=A,\dots ,N,\\ {} 0\hspace{1em}& P(i\leqslant j)\leqslant 0.5\hspace{2.5pt}i,j=A,\dots ,N.\end{array}\right.$
Table 5
Consistency Index for linguistic grey numbers.
Linguistic terms |
Equally Important (EI) |
Weakly Important (WI) |
Fairly Important (FI) |
Very Important (VI) |
Absolutely Important (AI) |
$\otimes {a_{BW}}$ |
$[1,1]$ |
$[\frac{2}{3},\frac{3}{2}]$ |
$[\frac{3}{2},\frac{5}{2}]$ |
$[\frac{5}{2},\frac{7}{2}]$ |
$[\frac{7}{2},\frac{9}{2}]$ |
CI-GBWM |
0.00 |
0.20 |
0.71 |
1.31 |
1.96 |
By the sum of the horizontal components of the matrix
${P_{ij}}$, 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).