## 3 Methodology
### Rough Analytic Hierarchy Process (AHP)  
The rough set theory was developed to handle the subjective judgment of the decision-makers when determining the boundary intervals, which are later integrated with the arithmetic operation to generally analyze complex and vague information. Thus, in this case, the consistency of preferences is measured while the judgment made by the decision-makers is managed. In addition, the rough set theory is utilized to solve and eliminate uncertainties, which include the vagueness of the judgment or the opinion of an expert who may not have experience in the related field. Furthermore, experts are asked to provide the scale of importance to evaluate the criteria for a particular problem. Therefore, the rough set theory evaluates the given criteria within a small set of data, lowering the uncertainty to provide the best decision. The following steps are followed to develop the rough AHP tool [21; 22; 23]:  
Step C1: Develop a multiple decision-making pairwise matrix formed by the expert's team working in the warehouse, the same matrix considered during the AHP process.  
Step C2: After determining the crisp value, obtain the rough numbers by following a series of equations.  
Consider \(U\) as the universe consisting of all the objects, and \(X\) is considered a random object from universe \(U\). The following assumption in Equation (2) is a specific set built with the decision-makers' preferences.  
\[\underline{Apr}\big{(}J_{q}\big{)}=R=(J_{1},J_{2},\ldots\ldots,J_{k})\]  
with the given condition:  
\[J_{1}<J_{2}<,\ \ldots\ldots,<J_{k} \tag{2}\]  
Then, for \(\forall\ X\ \epsilon\ U,\ J_{q}\ \epsilon\ R,\ 1\leq q\leq k\), the lower approximation \(\underline{Apr}\big{(}J_{q}\big{)}\) and upper approximation \(\overline{Apr}\big{(}J_{q}\big{)}\) are determined.  
Lower approximation:  
\[\underline{Apr}\big{(}J_{q}\big{)}=\text{U}\ \big{\{}X\ \epsilon\ U/R(X)\ \leq J_{q}\big{\}} \tag{3}\]  
Upper approximation:  
\[\overline{Apr}\big{(}J_{q}\big{)}=\text{U}\ \big{\{}X\ \epsilon\ U/R(X)\ \geq J_{q}\big{\}} \tag{4}\]  
Step C3: Define the random object \(X\) with the rough number \(R.N.\) with the lower limit \(\underline{Lim}\big{(}J_{q}\big{)}\) and upper limit \(\overline{Lim}\big{(}J_{q}\big{)}\):  
\[\underline{Lim}\big{(}J_{q}\big{)}=\frac{1}{M_{L}}\ \sum R(X)\bigg{|}\ X \ \epsilon\ \underline{Apr}\big{(}J_{q}\big{)} \tag{5}\]  
\[\overline{Lim}\big{(}J_{q}\big{)}=\frac{1}{M_{U}}\ \sum R(X)\bigg{|}\ X \ \epsilon\ \overline{Apr}\big{(}J_{q}\big{)} \tag{6}\]  
where \(M_{L}\) and \(M_{U}\) represent the sums of objects in the lower and upper object approximation of \(J_{q}\), respectively.  
Step C4: Create a rough comparison matrix consisting of rough numbers following Equation (7):  
\[R.N.\Big{(}x_{gh}^{\epsilon}\big{)}=[x_{gh}^{\epsilon L},x_{gh}^{\epsilon U}] \tag{7}\]where \(x_{gh}^{el}\) is the lower limit of the rough number \(R.N.\) (\(x_{gh}^{el}\)) and \(x_{gh}^{elU}\) is the upper limit of the rough number \(R.N.\) (\(x_{gh}^{el}\)). Then, the average rough number limits are taken as  
\[x_{gh}^{L}=\frac{x_{gh}^{1L}+x_{gh}^{2L}+\ldots+x_{gh}^{SL}}{S} \tag{8}\]  
\[x_{gh}^{U}=\frac{x_{gh}^{1U}+x_{gh}^{2U}+\ldots+x_{gh}^{SU}}{S} \tag{9}\]  
where \(x_{gh}^{L}\) and \(x_{gh}^{U}\) are the average lower and upper limits of rough number \(R.N.\) (\(x_{gh}^{e}\)).  
The rough comparison matrix is constructed in the same following step as  
\[M=\left[\begin{array}{cccc}[1,1]&[x_{12}^{L},\ x_{12}^{U}]&\ldots&[x_{1m}^{ L},\ x_{1m}^{U}]\\ [x_{21}^{L},\ x_{21}^{U}]&[1,1]&\ldots&[x_{2m}^{L},\ x_{2m}^{U}]\\ \vdots&\vdots&\ddots&\vdots\\ [x_{m1}^{L},\ x_{m2}^{U}]&[x_{m2}^{L},\ x_{m2}^{U}]&\ldots&[1,1]\end{array}\right] \tag{10}\]  
Step C5: Obtain the weight of each criterion by applying the power root for the weights obtained using Equation (11):  
\[W_{g}=\sqrt[n]{\prod_{h=1}^{m}x_{gh}^{L}},\ \sqrt[n]{\prod_{h=1}^{m}x_{gh}^{U}} \tag{11}\]  
Finally, the weights are normalized:  
\[\hat{W_{g}}=W_{g}/\max\left(W_{g}^{U}\right) \tag{12}\]