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

1. Dixit, U. S., and Dixit, P. M., 1996, "A
   finite-element analysis of flat rolling and
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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

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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

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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

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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

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Constraint cont.

• Velocity
• Not considered in present work
• Alligatoring
• Burst
• Controlled by Alligatoring

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Optimization Problem

• Minimize
• Neglect Hydrostatic Stess and assume that
  interstand pressure is zero

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Optimization Cont.

• Kinematic
• Material Behavior
• Levy-Mises coefficient
• Strain
• Strain Rate

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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

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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

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Fuzzy Parameters

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

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Fuzzy Parameters

• By using Fuzzy parameters the Power usage is also
  Fuzzy

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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

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Reliability of Schedule Design

• It is based on two terms
• Possibility index
• Reliability

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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 dont we use this?
• It is not reliable?
• How do we decide?

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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

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Decision Procedure
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Conclusion

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

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Discussion