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 1. 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 articleInt. J. Mach. Tools Manufact., 36, pp. 947–969first citation in article 2. Avitzur, B., 1962, "Pass reduction schedule for optimum production of a hot strip mill," Iron Steel Eng., Dec., pp. 104–114. first citation in articlefirst citation in article 3. 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 articlefirst citation in article 4. 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 articlefirst citation in article 5. Zadeh, L. A., 1965, "Fuzzy Sets," Inf. Control., 8, pp. 338–353. first citation in articlefirst citation in article 6. Kaufmann, A., and Gupta, M. M., 1985, Introduction of Fuzzy Arithmetic: Theory and Applications, Van Nostrand Reinhold Company Inc., New York. first citation in articlefirst citation in article 7. Klier, G. J., and Folger, T. A., 1993, Fuzzy Sets, Uncertainty and Information, Prentice-Hall of India Private Limited, New Delhi. first citation in articlefirst citation in article 8. 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 articleInt. J. Mach. Tools Manuf., 37, pp. 837–853first citation in article 9. 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 articlefirst citation in article 10. 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 articlefirst citation in article 11. Wanheim, T., and Bay, N., 1978, "A model for friction in metal forming processes," Ann. CIRP, 27, pp. 189–194. first citation in articlefirst citation in article 12. Fletcher, R., 1981, Practical Methods of Optimization, Vol. 2, Constrained Optimization, Wiley, New York and Toronto. first citation in articlefirst citation in article 13. 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 articlefirst citation in article 14. 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 articleInt. J. Numer. Methods Eng., 38, pp. 531–548[Inspec]first citation in article 15. 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[Inspec]first citation in article 16. 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[Inspec]first citation in article

3. Introduction

4. 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. 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 T L = 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