# Blockchain Integrated IoT for Food Supply Chain: A Grey Based Delphi-DEMATEL Approach
## 3 Methodology
### Grey DEMATEL  
A grey DEMATEL approach combines grey theory and DEMATEL to address the incomplete information, subjectivities, and biases in expert input. Furthermore, this method can improve the accuracy of the observations [61; 73]. Following is the stepwise procedure for grey DEMATEL [15; 74]:  
Step 1: Construct the initial direct-relation matrix  
Develop an initial direct relation matrix of CSFs (c = {ci | i = 1, 2,..., n}) of B-IoT adoption in FSC, utilizing the five-point grey linguistic scale (refer to Table 2), from \"No Influence\" to \"Very High influence\" by k experts through pair-wise comparisons. \begin{table}
\begin{tabular}{c c} \hline
**Linguistic Scale** & **Grey Number** \\ \hline Very low important (VL) & [0,1] \\ Low important (L) & [1; 2] \\ Medium important (M) & [2; 3] \\ High important (H) & [3; 4] \\ Very high important (VH) & [4; 5] \\ \hline \end{tabular}
\end{table} 
Table 1: Linguistic scale and their associated grey number. \begin{table}
\begin{tabular}{c c} \hline
**Linguistics Term** & **Grey Number** \\ \hline No influence (No) & [0,0] \\ Very low influence (VL) & [0,1] \\ Low influence (L) & [1,2] \\ High influence (H) & [2; 3] \\ Very high influence (VH) & [3; 4] \\ \hline \end{tabular}
\end{table} 
Table 2: Linguistics terms and their corresponding grey scales. The linguistic term is substituted with the corresponding grey number in order to convert the obtained initial direct relationship matrix into a grey initial direct relationship matrix. As K-number of experts provided their responses, the K grey direct relationship matrix \(X^{1}\), \(X^{2}\), \(X^{3}\)..., \(X^{K}\) is obtained. The direct-relation grey matrix is represented as follows by Equation (8):  
\[X^{k}=\left[ \begin{array}{cccccc}0&\otimes x_{12k}&\otimes x_{13k}&\cdots& \otimes x_{1(n-1)k}&\otimes x_{1 nk}\\ \otimes x_{21k}&0&\otimes x_{23k}&\cdots&\otimes x_{2(n-1)k}&\otimes x_{2 nk}\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ \otimes x_{(n-1)1k}&\otimes x_{(n-1)2k}&\otimes x_{(n-1)3k}&\cdots&0&\otimes x _{(n-1)nk}\\ \otimes x_{n1k}&\otimes x_{n2k}&\otimes x_{n3k}&\ldots&\otimes x_{n(n-1)k}&0 \end{array} \right] \tag{8}\]  
The element of [X], \(\otimes x_{ijk}\) = (\(\underline{\otimes}x\)\({}_{ijk}\), \(\overline{\otimes}x_{ijk}\)) shows the influence of CSF '\(i\)' on CSF '\(j\)' by the \(k\)th expert. The \(\underline{\otimes}x\)\({}_{ijk}\) represents the lower and \(\overline{\otimes}x_{ijk}\) the upper limit of grey values. Step 3: Develop the overall grey relation matrix  
In order to formulate the overall grey relation matrix, we combine all grey direct-relation matrices by using Equation (9). \[\otimes\mathrm{X}=\sum\
olimits_{i=0}^{K}(\frac{\sum\underline{\otimes}x\)\({}_{ijk}}{K},\frac{\sum\overline{\otimes}x_{ijk}}{K}) \tag{9}\]  
Step 4: Express the normalised grey direct-relation matrix  
In order to transform the grey relation matrix into a normalized grey direct-relation matrix N, Equations (4)-(6) are used. \[\otimes\mathrm{s}=[\underline{\mathrm{s}},\overline{\mathrm{s}}]=\frac{1}{ \underset{0\leq i\leq n}{max}}\frac{1}{\underset{j=0}{max}}\ i,\ j=1,2,3 \ldots\ldots\ldots n \tag{10}\]  
\[\mathrm{N}=\otimes\mathrm{s}\cdot\mathrm{X} \tag{11}\]  
\[\otimes n_{ij}=[\underline{\mathrm{s}},\ \otimes x_{ij},\overline{\mathrm{s}},\ \otimes x_{ij}] \tag{12}\]  
Step 5: Calculate the total relation matrix  
The total relation matrix \"T\" is obtained from the normalized grey direct-relation matrix by using Equation (13). \[\otimes\mathrm{T}=\otimes\mathrm{N}.(\otimes\mathrm{I}-\otimes\mathrm{N})^{-1} \tag{13}\]  
where \"\(\otimes\)I\" is the grey identity matrix.