# Hesitant Fuzzy Linguistic Term and TOPSIS to Assess Lean Performance
## 4 Numerical Example  
This section introduces a real-life example, which was applied in an automotive company based in Ciudad Juarez, Chihuahua, Mexico. The company works under an LM methodology and focuses on minimizing operational waste; thus, managers are particularly interested in assessing the real impact of the LM methodology. To this end, a group of experts first assessed the company's LM implementation improvement metrics. Simultaneously, we described the set of criteria and the KPIs depicted like alternatives as follows: \(C_{1}\): Defects, \(C_{2}\): Productivity, \(C_{3}\): Lead time, \(C_{4}\): Customer, \(C_{5}\): Demand satisfaction, \(C_{6}\): Cycle time, \(C_{7}\): Tack time, \(C_{8}\): Effectiveness, \(C_{9}\): Levels of inventory \(C_{10}\): Suppliers. Additionally, during the evaluation of lean projects, nineteen alternatives to be considered are summarized: A1: Sales, A2: Markeshare, A3: Maintenance, A4: OEE, A5: On-time delivery, A6: 5,S, A7: KAIZEN, A8: Bottleneck removal, A9: Cross-functional work force, A10: Focused factory production, A11: JIT/continuous flow production, A12: Lot size reductions, A13: Maintenance optimization, A14: Process capability measurements, A15: Kanban, A16: Quick changeover, A17: Total quality management, A18: Self-directed work teams, A19: Safety improvement programs.  
\[\omega_{i}=\left\{0.238,0.164,0.139,0.109,0.089,0.070,0.064,0.041,0.051,0.035 \right\}^{T}.\]  
**Step 4.** Compute the positive-ideal solution vector \((\dot{A}^{+})\) and Negative-ideal solution vector\((\dot{A}^{-})\):  
\[\dot{A}^{+}=\left(\{50,52\},\{56,58\},\{50,52\},\{57,58\},\{57,58\},\{57,58\},\{5 7,58\},\{57,58\},\{57,58\},\{56,58\}\right),\]  
\[\dot{A}^{-}=\left(\{57,58\},\{50,52\},\{57,58\},\{50,51\},\{50,51\},\{50,51\}, \{50,51\},\{50,51\},\{50,51\},\{50,51\}\right).\]  
**Step 5.** Construct positive ideal distance matrix \((HPiS^{+})\) and negative ideal distance matrix \((HNiS^{-})\), which are denoted as follows:  
\[HPiS^{+}=\left(\begin{array}{ccccccccccccc}1.902&+&0.996&+&0.278&+&0.531&+&0. 709&+&0.632&+&0.551&+&0.338&0.154&+&0.280\\ 0.475&+&0.657&+&0.804&+&0.652&+&0.956&+&0.772&+&0.639&+&0.205&+&0.103&+&0.175\\ 0.951&+&0.986&+&0.973&+&0.678&+&0.621&+&0.691&+&0.639&+&0.123&+&0.257&+&0.3 86\\ 1.902&+&0.657&+&112&+&0.455&+&0.532&+&0.421&+&0.447&+&0.207&+&0.3 80&+&0.140\\ 0.951&+&1.150&+&0.556&+&1.136&+&0.621&+&0.140&+&0.447&+&0.328&+&0.3 00&+&0.140\\ 1.427&+&1.150&+&1.251&+&0.899&+&0.355&+&0.562&+&0.064&+&0.164&+&0.154&+&0.0 70\\ 0.951&+&1.315&+&0.844&+&1.807&+&0.621&+&0.491&+&0.319&+&0.246&+&0.411&+&0.175 \\ 0.476&+&0.657&+&1.251&+&0.305&+&0.443&+&0.702&+&0.270&+&0.246&+&0.3 00&+&0.280\\ 2.651&+&0.657&+&1.390&+&0.652&+&0.532&+&0.491&+&0.511&+&0.369&+&0.306&+&0.2 00\\ 1.902&+&0.164&+&1.112&+&1.807&+&0.709&+&0.421&+&0.511&+&0.123&+&0.011&+&0.2 00\\ 1.189&+&1.150&+&0.695&+&0.217&+&0.266&+&0.421&+&0.256&+&0.082&+&0.257&+&0.1 05\\ 1.664&+&0.657&+&1.251&+&0.425&+&0.355&+&0.140&+&0.31&+&0.33&+&0.3 65&6.406\\ 2.140&+&0.657&+&0.556&+&0.578&+&0.532&+&0.632&+&0.319&+&0.123&+&0.514&+&0.0 05\\ 1.189&+&0.986&+&0.695&+&0.543&+&0.709&+&0.562&+&0.447&+&0.287&+&0.257&+&0.1 75&5.851\\ 2.615&+&0.493&+&1.529&+&0.543&+&1.153&+&0.772&+&0.73&+&0.082&+&0.3 00&+&0.315\\ 2.140&+&1.643&+&1.300&+&0.540&+&0.621&+&0.632&+&0.757&+&0.256&+&0.411&+&0.2 01\\ 2.651&+&0.657&+&0.695&+&0.682&+&0.696&+&0.421&+&0.256&+&0.257&+&0.245&+&0.246&+&0.2 04\\ 0.713&+&0.657&+&0.834&+&0.581&+&0.976&+&0.351&+&0.447&+&0.246&+&0.617&+&0.4 02\\ 1.427&+&1.429&+&1.251&+&1.304&+&0.709&+&0.842&+&0.639&+&0.246&+&0.514&+&0.140\\ \end{array}\right)
\begin{tabular}{c} \((1.189\\ 1.664\\ 2.615\\ 1.199\\ 1.199\\ 2.140\\ 2.  
Table 6 depict the hesitant relative closeness index called \(HSRC_{i}\)  
Step 7 Ranking of the alternatives.  
\[A_{11}>A_{14}\succ A_{5}\succ A_{2}\succ A_{8}\succ A_{10}\succ A_{18}\succ A_{ 6}\succ A_{3}\succ A_{4}\succ A_{7}\succ A_{12}\succ A_{13}\succ A_{1}\succ A_{ 7}\succ A_{9}\succ A_{19}\succ A_{18}.\]