Abstract
Multi-objective evolutionary algorithm (MOEA) is an efficient tool for solving different problems in engineering and various other fields. This chapter deals with an approach used to establish input–output relationships of a process utilizing the concepts of multi-objective optimization and cluster-wise regression analysis. At first, an initial Pareto-front is obtained for a given process using a multi-objective optimization technique. Then, these Pareto-optimal solutions are applied to train a neuro-fuzzy system (NFS). The training of the NFS is implemented using a meta-heuristic optimization algorithm. Now, for generating a modified Pareto-front, the trained NFS is used in MOEA for evaluating the objective function values. In this way, a new set of trade-off solutions is formed. These modified Pareto-optimal solutions are then clustered using a clustering algorithm. Cluster-wise regression analysis is then carried out to determine input–output relationships of the process. These relationships are found to be superior in terms of precision to that of the equations obtained using conventional statistical regression analysis on the experimental data. To validate the performance of the developed method, an engineering problem, related to the electron beam welding (EBW) of SS 304, is selected and its input–output relationships have been established.
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Appendices
Appendices
Appendix 1 collected experimental data
Sl. no. | Power (W) | Speed (mm/min) | Depth of penetration (mm) | Bead width (mm) |
|---|---|---|---|---|
1 | 3200 | 1800 | 2.73 | 4.82 |
2 | 3200 | 1500 | 3.27 | 5.36 |
3 | 4000 | 1800 | 3.43 | 4.46 |
4 | 3200 | 1200 | 4 | 3.4 |
5 | 4000 | 1500 | 4.13 | 4.4 |
6 | 4800 | 1800 | 3.41 | 5.54 |
7 | 5600 | 1800 | 4.55 | 3.4 |
8 | 4800 | 1500 | 4.5 | 3.5 |
9 | 4000 | 1200 | 4.6 | 4.7 |
10 | 3200 | 900 | 3.9 | 6.92 |
11 | 5600 | 1500 | 4.8 | 5.1 |
12 | 4800 | 1200 | 5.29 | 5.31 |
13 | 4000 | 900 | 5.69 | 4.99 |
14 | 5600 | 1200 | 5.8 | 5.5 |
15 | 4800 | 900 | 7.15 | 5 |
16 | 5600 | 900 | 8.2 | 5.6 |
Appendix 2 Experimental data collected for testing the performance of the developed approach
Sl. no. | Power (W) | Speed (mm/min) | Depth of penetration (mm) | Bead width (mm) |
|---|---|---|---|---|
1 | 4800 | 1650 | 4.37 | 3.38 |
2 | 4800 | 1325 | 5.03 | 3.87 |
3 | 5200 | 1650 | 4.39 | 3.86 |
4 | 5200 | 1325 | 5.27 | 4.51 |
5 | 5600 | 1000 | 7.91 | 5.28 |
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Das, A.K., Das, D., Pratihar, D.K. (2018). Multi-Objective Optimization and Cluster-Wise Regression Analysis to Establish Input–Output Relationships of a Process. In: Mandal, J., Mukhopadhyay, S., Dutta, P. (eds) Multi-Objective Optimization. Springer, Singapore. https://doi.org/10.1007/978-981-13-1471-1_14
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