# Hesitant Fuzzy Linguistic Term and TOPSIS to Assess Lean Performance
## 3 Materials and Methods  
This section presents the material and method used in the investigation. We introduce the procedure of HFLTs and TOPSIS for the Lean Improvement Assessment. At the same time, in order to explain the proposed method, Figure 1 shows the flowcharts of the different steps about it.  
_Hesitant Fuzzy Linguistic Term and TOPSIS to Assess Lean Performance_  
In this section, we introduce an algorithm through a Hesitant Fuzzy Linguistic Term and TOPSIS in order to be applied to Lean Improvement Assessment. The method is described in the following steps:  
**Step 1**. Determine the hesitant fuzzy decision matrix called \(Y^{l}\)=\([\rho^{l}_{ij}]_{mxn}\) for the MCDM problem. Appraisal the alternative with respect to DM preferences and the criteria.
**Step 2**. Calculate the aggregated decision matrix called \(Z\). This process requires the aggregation of the preferences of the DMs \((Y^{1},Y^{2},\ldots,Y^{k})\) through Equations (14) and (15).  
Then, \(Z=[z_{ij}]\), where \(z_{ij}=[_{\xi p_{ij}},\xi_{q_{ij}}]\),  
Figure 1: Flowcharts of the algorithms to assess the Lean Manufacturing Performance.  
\[\xi_{p_{ij}}=min\left\{\min_{l=1}^{k}(max\,\rho_{ij}^{l}),\max_{l=1}^{k}(min\,\rho_ {ij}^{l})\right\}, \tag{14}\]  
\[\xi_{q_{ij}}=max\left\{\min_{l=1}^{k}(max\,\rho_{ij}^{l}),\,\max_{l=1}^{k}(min\, \rho_{ij}^{l})\right\}. \tag{15}\]  
**Step 3**. Determine the importance or preference about criteria called vector \(\omega_{j}\), for the MCDM problem via Analytic Hierarchy Process (AHP) method proposed by [33]. Appraise the criteria with respect to DM preferences.
**Step 4**. Compute the positive-ideal solution vector \((A^{+})\) and Negative-ideal solution vector \((A^{-})\). At this mode, the evaluation of alternative \(A_{i}\) by mean of criterion \(C_{j}\) is symbolized as \(z_{ij}\) using an aggregated matrix \(Z\). Thus, \(\theta_{B}\) depicts a set of benefit criteria and represents the greater preference of the criterion \(C_{j}\) and \(\theta_{C}\) depicts a set of cost criteria and describes the smaller preference of the criterion \(C_{j}\):  
\[\dot{A}^{+}=\left[\left(\max_{l=1}^{k}\left(\max_{i}\,\rho_{ij}^{l}\right) \right)\right]j\in\theta_{B},\,\left(\min_{l=1}^{k}\left(\min_{i}\,\rho_{ij}^ {l}\right)\right)|j\in\theta_{C}\right],\textit{where }i=1,\ldots,m. \tag{16}\]  
Then, \(A^{+}=\left[\hat{R}_{1}^{+},\hat{R}_{2}^{+}\ldots,\hat{R}_{n}^{+}\right];\, \hat{R}_{j}^{+}=\left[\xi_{p_{ij}},\xi_{q_{ij}}\right](j=1,\ldots n)\),  
\[\dot{A}^{-}=\left[\left(\min_{l=1}^{k}\left(\min_{i}\,\rho_{ij}^{l}\right) \right)\right]j\in\theta_{B},\,\left(\max_{l=1}^{k}\left(\max_{i}\,\rho_{ij}^ {l}\right)\right)|j\in\theta_{C}\right],\textit{where }i=1,\ldots,m, \tag{17}\]  
thus \(A^{-}=\left[\hat{R}_{1}^{-},\hat{R}_{2}^{-}\ldots,\hat{R}_{n}^{-}\right];\, \hat{R}_{j}^{-}=\left[\xi_{p_{ij}},\xi_{q_{ij}}\right](j=1,\ldots n)\).
**Step 5**. Construct positive ideal distance matrix \((HPIS^{+})\) and negative ideal distance matrix \((HNIS^{-})\), which are denoted as follows:  
\[HPIS^{+}=\left(\begin{array}{cccccc}\omega_{1}d(z_{11},R_{1}^{+})&+&\ldots& +&\omega_{n}d(z_{1n},R_{n}^{+})\\ \omega_{2}d(z_{21},R_{1}^{+})&+&\ldots&+&\omega_{1}d(z_{2n},R_{n}^{+})\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \omega_{j}d(z_{m1},R_{1}^{+})&+&\ldots&+&\omega_{j}d(z_{mn},R_{2}^{+})\end{array} \right), \tag{18}\]  
\[HNIS^{-}=\left(\begin{array}{cccccc}\omega_{1}d(z_{11},R_{1}^{-})&+&\ldots& +&\omega_{n}d(z_{1n},R_{n}^{-})\\ \omega_{2}d(z_{21},R_{1}^{-})&+&\ldots&+&\omega_{n}d(z_{2n},R_{n}^{-})\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \omega_{j}d(z_{m1},R_{1}^{-})&+&\ldots&+&\omega_{n}d(z_{mn},R_{n}^{-})\end{array} \right). \tag{19}\]  
**Step 6**. Calculate the relative closeness \((HSRC_{i})\) of each alternative to the ideal solution as follows:  
\[HSRC_{i}=\frac{HNIS^{-}}{HNIS^{-}+HPIS^{+}}, \tag{20}\]  
where  
\[HPIS^{+}=\sum_{j=1}^{n}\omega_{j}d(z_{ij},R_{j}^{+}),\]  
and  
\[HNIS^{-}=\sum_{j=1}^{n}\,\omega_{j}d(z_{ij},R_{j}^{-}).\]