Hesitant Fuzzy Linguistic Term and TOPSIS to Assess Lean Performance

: Manufacturing companies usually expect strategic improvements to focus on reducing both waste and variability in processes, whereas markets demand greater flexibility and low product costs. To deal with this issue, lean manufacturing (LM) emerged as a solution; however, it is often challenging to evaluate its true effect on corporate performance. This challenge can be overcome, nonetheless, by treating it as a multi-criteria problem using the Hesitant Fuzzy linguistic and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method. In fact, the hesitant fuzzy linguistic term sets (HFLTS) is vastly employed in decision-making problems. The main contribution of this work is a method to assess the performance of LM applications in the manufacturing industry using the hesitant fuzzy set and TOPSIS to deal with criteria and attitudes from decision makers regarding such LM applications. At the end of the paper, we present a reasonable study to analyze the obtained results.


Introduction
Lean manufacturing (LM) combines a wide rage of management practices, such as just in time (JIT), quality systems, work teams, cellular manufacturing, and supply chain management (SCM) in a whole system [1]. The LM method aims at saving costs by reducing waste in the manufacturing system, thereby dealing with economic aspects [2]. Nowadays, LM covers the multiple stages of a product's life cycle, from its development and manufacturing to its delivery [3]; however, LM is also a challenge amid mass production practices, especially as quality products, and customer satisfaction are prioritized, inventory, time to market and manufacturing space, and everything that adds no value to a product is systematically categorized as waste [4]. LM is often discussed with respect to key performance indicators (KPIs) [5,6]. In addition, Kan et al. [7] affirm the KPI parameters have an association with LM performance. In fact, research evidence has found that LM practices have a positive impact on operational performance [8,9], yet it is often challenging to assess company performance with respect to LM implementation [10,11] and, according to [12,13], it is an attractive and hot topic for exploration through multi-criteria decision-making (MCDM) methodologies.
Definition 5 (Distance [26,28,31,32]). Assuming that H 1 and H 2 are two HFLTS and env (H 1 S ) = ς α , ς β and env(H 2 where β represents a higher element from H 1 andβ depicts the maximum or higher element from H 2 . Thus, α denotes a low element from H 1 andα depicts the minimal or low element from H 2 : where ρ depicts # of the elements of the set ς and, I(ς i ) stands for the subfix of linguistic term ς i Definition 6 ([30]). Let ς = {ς 0 , . . . , ς τ } be a linguistic term set. A hesitant fuzzy linguistic term sets (HFLTS), H ς , is an ordered and finite subset of the consecutive linguistic terms of S.
In addition, an HFLTS can be used to elicit several linguistic values for a linguistic term, yet it is still not comparable to human thinking and reasoning processes. Thus, Rodríguez et al. [18] further presented a context-free grammar to describe linguistic terms that are more parallel to the human expressions and can be simply denoted by means of HFLTSs. In addition, according to [20], the grammar H ς is used to express the linguistic term and transformed to HFLEs by using

TOPSIS in Conventional Version
In this section, the conventional manner of TOPSIS is presented Step 1. Establish the final decisión matrix. Table 1 shows the set of the alternatives A i (A 1 , A 2 , . . . A m ) and C j (C 1 , C 2 . . . C n ) be a finite set of criteria involved in the MCDM problem.
Step 3. Construct the aggregate matrixR where i = 1, 2, . . . , m, j = 1, 2, . . . , n and w z represents the weight vector of the criteria C j (j = 1, . . . n) Step 4. Establish the vector ideal positive A + and the vector anti-ideal negative A − by means of Equations (9) and (10): where δ depicts the sets of benefit criteria and δ represents the sets of cost criteria.
Step 6. Ranking of the alternatives

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 =[ρ 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 , . . . , Y k ) through Equations (14) and (15).
Step 3. Determine the importance or preference about criteria called vector ω j , for the MCDM problem via Analytic Hierarchy Process (AHP) method proposed by [33]. Appraise the criteria with respect to DM preferences. 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, θ B depicts a set of benefit criteria and represents the greater preference of the criterion C j and θ C depicts a set of cost criteria and describes the smaller preference of the criterion C j : Then, Step 5. Construct positive ideal distance matrix (HPIS + ) and negative ideal distance matrix (HN IS − ), which are denoted as follows: Step 6. Calculate the relative closeness (HSRC i ) of each alternative to the ideal solution as follows: where and Step 7. Rank all the alternatives

Numerical Example
This section introduces a real-life example, which was applied in an automotive company based in Ciudad Juárez, 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.
Step 1. Determine the hesitant fuzzy decision matrix called ρ ij for the MCDM problem. Appraise the alternative with respect to DM preferences and the criteria. Establish the final decision matrix. Let Y l = [ρ l ij ] mxn be a fuzzy decision matrix for the MCDM problem, and the following notations are used to depict the considered problems. At the same time, the matrices (Tables 2 and 3) describe the preferences DM 1 ,DM 2 , DM 3 , DM 4 , DM 5 and DM 6 . Table 2. Decision matrix Y 1 with respect to decision makers 1, 2, and 3. Table 3. Decision matrix Y 2 with respect to decision makers 4, 5, and 6.
Step 2. Calculate the aggregated decision matrix called Z. This process requires the aggregation of the preferences of the DMs using the matrices (Y 1 andY 2 ) through Equations (14) and (15). Table 4 shows the hesitant aggregated matrix called Z. Step 3. Determine the importance or preference about criteria called vector ω j for the MCDM problem via the Analytic Hierarchy Process (AHP) method. Appraise the criteria with respect to DM preferences. Table 5 depict the preferences of the criteria in order to obtain the vector ω j .  {0.238, 0.164, 0.139, 0.109, 0.089, 0.070, 0.064, 0.041, 0.051, 0.035} T .
Step 5. Construct positive ideal distance matrix (HPIS + ) and negative ideal distance matrix (HN IS − ), which are denoted as follows: Step 6. Calculate the relative closeness (HSRC i ) of each alternative to the ideal solution as follows:  Step 7. Ranking of the alternatives.

Result Analysis and Discussions
The method proposed by [26] present a weakness to determine the position of the alternatives due to duplicate ranking of the closeness coefficients values. The information shown in Table 7 depicts a comparison that reports this kind of duplicate issue. However, there is the alternative A 11 as a best option identified by both analyses. Normally, the manufacturing company handles a high standard of the KPIs to monitor the best performances of LM. At this sense, our method offers the initiative to appraise the key performance indicators (KPIs). Table 8 introduces the correlation between the three methods by taking into account their results. As can be observed, there is a significant correspondence between our approach and the two MCDM approaches proposed by [26] and [28], respectively. Table 8. Correlation matrix.

Proposed by [26] Proposed by [28] Our Method
Proposed by [26] 1.000 0.820 0.677 Proposed by [28] 0.820 1.000 0.630 Our method 0.677 0.630 1.000 Similarly, Table 9 lists the residual covariances between the methods. On the other side, Table 10 lists the statistical parameters of the case studies. As can be observed, the mean and standard deviation values are similar in the three methods. In fact, the results can be interpreted with minimal error in the three case studies. Finally, Table 11 lists the internal consistency values as expressed by the Cronbach's alpha coefficient. Our study reported an overall Cronbach's alpha value of 0.9008, which is considerably higher than 0.7, the usual threshold. This confirms the reliability of the results, since higher values of Cronbach's alpha imply greater internal data consistency. To perform an error analysis on the ranking results, we employed a neural network. In this sense, Figure 2 indicates that almost 78 epochs are found below the minimal error. The results from the neural network indicate that the major contribution of the LM methodology is offered by JIT/continuous production flow. In this sense, a productivity bonus shares for the workers based on the top 10 metrics classified using the Hesitant Fuzzy Linguistic Term and the TOPSIS method. Similarly, we plan to develop a waste minimization project to take into account the ranking results obtained from the assessments. Additionally, a sensitivity analysis was planted, which implies the comparisons with other methods in order to check the stability of our application and the results are shown in Figure 3. Observing in Figure 3, we can notice the stability of the gained results. In addition, two different methods were applied and the ranking of the best position does not change. Finally, we demonstrated that there is a significant correspondence between our approach and the two approaches compared.

Conclusions
In this research, we propose an operative method for dealing with hesitant assessments in lean manufacturing problems. TOPSIS and HFLTS are a useful tool for managers who wish to assess the KPI's performance of the LM projects. In this research, we propose a multi-criteria decision-making method to find the desirable alternatives. Likewise, the results from our proposed can be used to design an action plan. Normally, developing cost minimization projects in a manufacturing environment is challenging, yet HFLTS and TOPSIS offer a systematic method for establishing priorities, thereby helping managers determine what key performance indicators (KPIs) have a low performance. Finally, the results represent a robust solution to deal with KPI assessments and provides visibility in terms of how lean manufacturing projects impact corporate performance. In addition, we present the use of AHP in order to determine the weights of criteria. There are some guidelines for future research where MCDM problems exist within the context of HSFLT situations-for example, evaluating the Lean Six Sigma projects, appraising performance of supply chains, among others. In addition, the consideration of the comparisons with other methods of MCDM.