# Fuzzy MCDM Methodology for Analysis of Fibre Laser Cutting Process
## 2 Materials and Methods
### Fuzzy MCDM Methodology  
In recent years, the application of different MCDM approaches has become increasingly significant when taking a number of aspects and different laser cutting performances into consideration [27]. To acknowledge a certain degree of uncertainty and vagueness in the decision-making process [34], this study proposes a fuzzy MCDM methodology (Figure 2). Figure 1: Experimental setup. The algorithm is implemented in six major steps:  
Step 1. For the initial step, five experts (\(k=1\div 5\)) in the field of laser cutting evaluated four criteria using a regular analytic hierarchy process (AHP) and [PERSON]'s 1-9 scale [35]. The AHP can be implemented in a few simple steps, and in this case, it was used only for the determination of the criteria weights. The results are five criteria weights vectors \(w_{k}=\{w_{k1},w_{k2},w_{k3},w_{k4}\}\). The aggregated fuzzy values of criteria weights \(\left(w_{j}^{l},w_{j}^{m},w_{j}^{u}\right)\) for each criterion (\(j=1\div 4\)) are obtained using the minimal, arithmetic mean and maximal value of the corresponding crisp values \(w_{kj}\) as shown by Equation (1). \[\widetilde{w}_{j}=\left(w_{j}^{l},w_{j}^{m},w_{j}^{u}\right)=\left(\underset {k}{min}\left(w_{kj}\right),\;\frac{\sum_{k=1}^{5}w_{kj}}{5},\underset{k}{max} \left(w_{kj}\right)\right) \tag{1}\]  
The result is the fuzzy criteria weights vector \(\widetilde{\mathrm{w}}=\{\widetilde{w}_{1},\widetilde{w}_{2},\widetilde{w}_{3 },\widetilde{w}_{4}\}=\{(0.10,0.14,0.20)\)\((0.05,0.14,0.32)\)\((0.17,0.38,0.53)\)\((0.17,0.35,0.58)\}\). Step 2. In the second step, a fuzzy decision matrix is obtained according to experimental data. In the case of the first three criteria (\(C_{1}\)--kerrid width on a straight cut [mm], \(C_{2}\)--kerrid width on a curved cut [mm], and \(C_{3}\)--surface roughness [um]), similarly to the  
Figure 2: A complete algorithm of the proposed fuzzy MCDM methodology. Step 4. In the fourth step, the results obtained using different fuzzy MCDM methods are compared using [PERSON]'s tau-b and [PERSON]'s rho correlation coefficient tests. Step 5. The uncertainty in solving Decision-Making Problems (DMPs) with Multi-Criteria Decision-Making (MCDM) methods is often significantly influenced by the weighting of criteria. Thus, assessing how variations in criteria weights affect the ranking results produced by the chosen MCDM methods is crucial [40; 41; 42]. In this research, the Monte Carlo simulation was employed, generating 1000 different scenarios of criteria weights for two cases: Case 1 with 10% variations, and Case 2 with 25% variations of original weights. Each value \(w_{ki}\) in five criteria weights vectors \(w_{k}=\{w_{k1},w_{k2},w_{k3},w_{k4}\}\), obtained by selected experts (\(k=1\div 5\)), has been randomly varied from interval \([w_{kiu},w_{kil}]\), where \(w_{kiu}\) and \(w_{kil}\) represent the upper and the lower boundary of each original weight \(w_{ki}\) (\(i=1\div 4\), number of criteria) defined as \(\pm 10\%\) (Case 1) or \(\pm 25\%\) (Case 2) of original \(w_{ki}\), as shown in Equation (3):  
\[ \begin{array}{l}\text{{Case 1}}:\qquad\begin{array}{l}w_{kiu}=1.1\cdot w_{ki} \qquad\begin{array}{l}w_{kil}=0.9\cdot w_{ki}\\ w_{kil}=0.75\cdot w_{ki}\end{array}\end{array}\end{array}  \tag{3}\]  
For each scenario, the aggregation procedure described in Step 1 of the proposed methodology incorporating Equation (1) is applied, so 1000 fuzzy criteria weights vectors \(\widetilde{\text{w}}=\left\{\widetilde{w}_{1},\widetilde{w}_{2},\widetilde{w }_{3},\widetilde{w}_{4}\right\}\) are obtained (for both cases). In the next step for thus-defined fuzzy criteria weights vectors and for all selected fuzzy MCDM methods, complete rankings of the alternatives were calculated. The changes of alternative ranks relative to the first solution (obtained with original weights) are monitored and sums of all rank changes are calculated. Step 6.