Application of Fuzzy Set Theory in the Scheduling of a Tandem Cold-Rolling Mill

1. Application of Fuzzy Set Theory in the Scheduling of a Tandem Cold-Rolling Mill By U.S. Dixit & P.M. Dixit Department of Mechanical Engineering Indian Institute of Technology Ben Naseath Sep 12, 2005

2. References • Dixit, U. S., and Dixit, P. M., 1996, "A finite-element analysis of flat rolling and application of fuzzy set theory," Int. J. Mach. Tools Manufact., 36, pp. 947–969. first citation in article • Avitzur, B., 1962, "Pass reduction schedule for optimum production of a hot strip mill," Iron Steel Eng., Dec., pp. 104–114. first citation in article • Bryant, G. F., and Spooner, P. D., 1973, "On-line adoption of tandem mill schedules," Automation of Tandem Mills, Bryant, G. F., ed., The Iron and Steel Institute, London. first citation in article • Bryant, G. F., Halliday, J. M., and Spooner, P. D., 1973, "Optimal scheduling of a tandem cold-rolling mill," Automation of Tandem Mills, Bryant, G. F., ed., The Iron and Steel Institute, London. first citation in article • Zadeh, L. A., 1965, "Fuzzy Sets," Inf. Control., 8, pp. 338–353. first citation in article • Kaufmann, A., and Gupta, M. M., 1985, Introduction of Fuzzy Arithmetic: Theory and Applications, Van Nostrand Reinhold Company Inc., New York. first citation in article • Klier, G. J., and Folger, T. A., 1993, Fuzzy Sets, Uncertainty and Information, Prentice-Hall of India Private Limited, New Delhi. first citation in article • Dixit, U. S., and Dixit, P. M., 1997, "A study on residual stresses in rolling," Int. J. Mach. Tools Manuf., 37, pp. 837–853. first citation in article • Zhu, Y. D., and Avitzur, B., 1988, "Criteria for the prevention of split ends," ASME J. Eng. Ind., 110, pp. 162–172. first citation in article • Avitzur, B., Van Tyne, C. J., and Turczyn, S., 1988, "The prevention of central bursts during rolling," ASME J. Eng. Ind., 110, pp. 173–178. first citation in article • Wanheim, T., and Bay, N., 1978, "A model for friction in metal forming processes," Ann. CIRP, 27, pp. 189–194. first citation in article • Fletcher, R., 1981, Practical Methods of Optimization, Vol. 2, Constrained Optimization, Wiley, New York and Toronto. first citation in article • Valliappan, S., and Pham, T. D., 1993, "Fuzzy finite element analysis of a foundation on an elastic soil medium," Int. J. Numer. Anal. Methods Geomech., 17, pp. 771–789. first citation in article • Valliappan, S., and Pham, T. D., 1995, "Elasto-plastic finite element analysis with fuzzy parameters," Int. J. Numer. Methods Eng., 38, pp. 531–548. [Inspec]first citation in article • Zadeh, L. A., 1976, "A fuzzy-algorithmic approach to the definition of complex or imprecise concepts," Int. J. Man-Mach. Stud., 8, pp. 249–291. [Inspec]first citation in article • De Luca, A., and Termini, A., 1972, "A Definition of Nonprobabilistic Entropy in the Setting of Fuzzy Set Theory," Inf. Control., 20, pp. 301–312. [Inspec]first citation in article

3. Introduction

4. Introduction • Optimum Reduction Schedule • Correct output gage • Satisfactory shape • Surface finish • Literature • Sparse • See paper for a few references

5. Introduction • Current Practice based on • Past experience • Trial and error • Rules of thumb • Future • Computer based

6. Statement of the Problem • Objective of a scheduling problem • Set up a tandem cold rolling mill • Optimum reduction schedule • Proper • Interstand Pressure • Rolling speeds • Forces • Pressure • Minimum Power

7. Statement of the Problem

8. Statement of the Problem • Objective Function • Minimization of specific power • Constraints • Strip Tension • Upper limit - tearing limit = 1/3 yield stress • Lower Limit - enough to keep form buckling • For simplicity TL = 0

9. Constraints cont. • Residual Stress • Limit used to maintain good shape (bend, warp) • Neglected • Not effected by change in reduction with coefficient of friction and radius fixed. • Power • Dependent on motor • Roll Force • Neglected • Satisfied by power constarint

10. Constraint cont. • Velocity • Not considered in present work • Alligatoring • Burst • Controlled by Alligatoring

11. Optimization Problem • Minimize • Neglect Hydrostatic Stess and assume that interstand pressure is zero

12. Optimization Cont. • Kinematic • Material Behavior Levy-Mises coefficient Strain Strain Rate

13. Optimization Cont. • Continuity and Momentum • Velocity Relationship • He then says that he solved the FEM with the Wanheim and Bay method and that you can see Ref 1

14. Optimization with Fuzzy Parameters • Fuzzy Parameters • Yield Stress • Hardening parameters b and n • Coefficient of friction • These do not posses a fixed value. • They have a range of values

15. Fuzzy Parameters • Membership Grade • 0 for most least common • 1 for most common • Have either a linear or nonlinear value

16. Fuzzy Parameters • By using Fuzzy parameters the Power usage is also Fuzzy

17. Reliability of Schedule Design • With such a fuzzy range of parameters • How can one decide what they should use? • You use a Second term called Reliabilty

18. Reliability of Schedule Design • It is based on two terms • Possibility index • Reliability

19. Examples • In the examples we see that if we just look at power saving then all of the reduction should be done in the first pass. • Why don’t we use this? • It is not reliable? • How do we decide?

20. Decision Procedure • Assign the specific value a percentage value with 1 being the lowest possible power and 10 % more 0.5 • Then find the lesser of the power and reliability

21. Decision Procedure

22. Conclusion • We find that not only minimizing power is important but we must also be reliable.

23. Discussion