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

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Introduction

Optimum Reduction Schedule – Correct output gage – Satisfactory shape – Surface finish Literature – Sparse – See paper for a few references

Introduction Current Practice based on – Past experience – Trial and error – Rules of thumb Future – Computer based

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

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 T L = 0

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

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

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

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

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

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

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

Fuzzy Parameters By using Fuzzy parameters the Power usage is also Fuzzy

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

Reliability of Schedule Design It is based on two terms – Possibility index – Reliability

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?

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

Decision Procedure

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

Discussion