# Fuzzy MCDM Methodology for Analysis of Fibre Laser Cutting Process
## 3 Results and Discussion  
### Ranking of Alternatives and Analysis of Results  
The ranking results of alternative fibre laser cutting conditions after applying the above-discussed fuzzy MCDM methodology are given in Figure 3.  
The results from Figure 3 indicate that alternative A10 stands out as the most favourable combination of fibre laser cutting parameters (\(f=1.5\) mm, \(v=3.6\) m/min, and \(p=0.75\) bar). One needs to note that this cutting condition is favourable if one considers the spent energy to cut a unit length, i.e., linear specific energy, which is a useful indicator of laser cutting efficiency and eco-friendliness [32]. Moreover, attaining less interaction time means less heat input, heat affected zone (HAZ) and mechanical stress to the workpiece material being cut. It must be noted that the ARAS-F method identified alternative A6 as the best trade-off solution. On the other hand, alternative A2 represents the least preferable combination of fibre laser cutting parameters (\(f=2\) mm, \(v=1.8\) m/min, and \(p=0.85\) bar) by all three fuzzy MCDM methods. This lowest ranking can be explained given that shifting the focus position away from the upper sheet surface widens kerf width [43], and in addition, a decreasing cutting speed has a significant effect on the resulting material removal rate. The results from Figure 3 also clearly indicate the complexity of the laser cutting process, variable effect of parameter inputs and existence of parameter interactions to a greater or lesser extent. Namely, alternative A1 (_f_ = 1 mm, \(v\) = 1.8 m/min, and \(p\) = 0.85 bar) drops from 10 to 13 ranking position of alternative A2 (_f_ = 2 mm, \(v\) = 1.8 m/min, and \(p\) = 0.85 bar) because of changing focusing position, which results in the following phenomena: kerf width on a straight cut increased by 31%, kerf width on a curved cut increased by 27% and surface roughness (_R\({}_{z}\)_) increased from 15.84_ \(\upmu\)m to 22.49_ \(\upmu\)m, while the surface productivity remained unchanged. On the other hand, alternative A3 (_f_ = 1 mm, \(v\) = 3.6 m/min, and \(p\) = 0.85 bar) rises from eight to four ranking position of alternative A4 (_f_ = 2 mm, \(v\) = 3.6 m/min, and \(p\) = 0.85 bar) because of changing focusing position, which results in the following phenomena: kerf width on a straight cut increased by 42%, kerf width on a curved cut increased by 29% and surface roughness (_R\({}_{z}\)_) significantly decreased from 40.55_ \(\upmu\)m to 5.51_ \(\upmu\)m, while the surface productivity remained unchanged. This observation indicates that the effect of focus position on cut quality characteristics should be considered through the interaction with the cutting speed, which can be explained by changing interaction time, laser energy absorption of the material, formation and behaviour of the melt front, melt flow and ejection, as well as cut kerf geometry characteristics at different combinations of cutting speed and focus position. Actually, the existence of a change in the effects of one parameter by the variation of another parameter has been previously observed in fibre laser cutting of aluminium [44], stainless steel [45], aluminium alloy [46] and carbon fibre reinforced polymer composites [47]. The previous observations clearly indicate that laser cutting parameters may have opposite effects on process performance characteristics due to the existence of interaction effects of laser cutting parameters to some extent, which is consequently reflected in the final ranking of alternatives. The cut surface patterns obtained under the most preferable cutting condition and the least preferable cutting condition are given in Figure 4. As can be observed, there is a minimal difference between the upper (laser dominated) and the lower (melt-flow controlled) cut surface, indicating \"striation free\" fibre laser cutting of mild steel [48]. Figure 3: Ranking of alternatives, i.e., different fibre laser cutting conditions.