PRODUCTION SYSTEMS ENGINEERING

Chapter 14: Design of Lean Serial Lines with Continuous Models of Machine Reliability

Instructors: J. Li, S.M. Meerkov, and L. Zhang

Copyright © 2008-2012 J. Li, S.M. Meerkov, and L Zhang

Motivation The same as in Bernoulli lines: the need for analytical

methods for selecting the smallest buffering, which is necessary and sufficient to ensure the desired efficiency of a production line.

The methods developed for Bernoulli lines are generalized here to serial lines with continuous time model of machine reliability.

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OUTLINE

1. Parametrization and Problem Formulation 2. Lean Buffering in Synchronous Lines with Identical Exponential machines 3. Lean Buffering in Synchronous Lines with Non-Identical Exponential Machines 4. Lean Buffering in Synchronous Lines with Non-Exponential Machines 5. Summary

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1. Parametrization and Problem Formulation

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Assume, for the moment, that

Parametrizations: Level of buffering – capacity of the buffer in units of downtime:

Line efficiency:

Definition: Lean level of buffering (kE) – the smallest level of buffering necessary and sufficient to ensure E.

Develop methods for calculating kE for exponential lines.

Investigate sensitivity of kE to type of up- and downtime distributions in non-exponential lines.

Based on this analysis, provide empirical formulas for calculating kE for non-exponential lines.

Generalize the results to systems with non-identical machines and buffers.

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Problems

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2. Lean Buffering in Synchronous Lines with Identical Exponential machines

2.1 Two-machine case In this case:

Theorem:

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2.1 Two-machine case (cont)

Observations: does not depend on Tup and Tdown (only Tup/Tdown). If E > e, the system must have a buffer. If E ≤ e, just-in-time is possible. → ∞ as E → 1.

Theorem:

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2.2 Three-machine case

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2.2 Three-machine case (cont)

Observations: Again, does not depend on Tup and Tdown. If , the system must have a buffer. If , just-in-time is possible. → ∞ as E → 1.

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2.3 M > 3-machine case

Theorem:

where

Approximation of Q:

Accuracy: ΔQ, ΔkE

≤ 5%.

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2.3 M > 3-machine case (cont)

Behavior of kE as a function of e

Behavior of kE as a function of M

Rule-of-thumb: For M ≥ 10

Q: Does buffer N = 1000 lean or not? A: It depends: If Tdown = 1000, the buffer is too lean (since kE = 1). If Tdown = 10, the buffer is very much not lean (since kE = 100).

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3. Lean Buffering in Synchronous Lines with Non-Identical Exponential Machines

3.1 Two-machine case Parametrization

Theorem:

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3.1 Two-machine case (cont)

Behavior of kE as functions of e and E

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3.1 Two-machine case (cont)

Observations:

depends on Tup and Tdown explicitly.

If E > min(e1, e2), the systems must have buffers.

If E ≤ min(e1, e2), JIT is possible.

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3.2 M > 2-machine case

Closed formula and recursive approaches to estimate ki,E, i = 1,…, M − 1, are developed (similar to those for Bernoulli lines)

Closed formula approaches: I. Local pair-wise approach: use the two-nonidentical-machine

formula for consecutive machines:

II. Global pair-wise approach: Use the two-nonidentical-machine formula for possible pairs of machines:

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Closed formula approaches

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III. Local upper bound approach: Substitute each pair of consecutive machines by

and use the two-identical-machines formula:

IV. Global upper bound approach: Substitute the line by an M-machine line with identical machines defined by

and use M-identical-machine formula:

Measures of quality of each approach

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Average level of buffering

where

Fraction of systems where Es is not reached:

Results

Observation: Local pair-wise selection of kE is not acceptable. Global pair-wise and local upper bound approaches are the best.

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Effect of M on approaches I-IV

Observation: Global pair-wise approach is the best for M ≤ 15. For M > 15, local upper bound approach is preferable.

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Recursive approaches:

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V. Full search approach:

VI. Bottleneck-based approach: Use Approach I, identify c-BN and increase buffers around this machine:

Performance metrics: As above + time ( )

Results

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Observations

Full search results in smallest buffering but longest time.

Approaches II-IV are practically instantaneous but lead to 2-10 larger buffering than approach V.

Approach VI provides a good trade off: 20% larger buffering than V but up to two order of magnitude faster.

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Illustration

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Illustration (cont)

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LLB estimates for Line 1 LLB estimates for Line 2

Illustration (cont)

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LLB estimates for Line 3 LLB estimates for Line 4

4. Lean Buffering in Synchronous Lines with Non-exponential Machines

4.1 Approach Select a representative set of up- and downtime distributions.

By simulations, evaluate PR and kE.

Investigate the sensitivity of kE to ftupand ftdown

Derive an empirical formula for kE.

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Identical reliability model

Mixed reliability model

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Systems analyzed

Assumptions

or and

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Systems analyzed (cont)

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Reliability models considered

4.2 Sensitivity of kE to machine reliability model

Case

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Observations: kE is monotonically increasing as a function of CV. kE for various reliability models are almost the same, i.e., kE is not

sensitive to ftupand ftdown (at most 10% difference) 14-32

Case

Case

What affects kE more: CVup or CVdown?

Formalization of this question: Inequality

implies that CVdown has a larger effect.

If the inequality is reversed, CVup has a larger effect.

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Thus, CVdown has a larger effect on kE than CVup. However, the difference is not too dramatic (within 30%).

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Results

4.3 Empirical bound for kE(M, E, e, CV)

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Case or

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Tightness of the empirical bound

Case

Define

Then

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Tightness

Generalization to non-identical machines

To account for Tup, i ≠ Tup, l , Tdown, i ≠ Tdown, l , use

To account for CVup, i ≠ CVup, l , CVdown, i ≠ CVdown, l , use

Then

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5. Summary

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Lean buffering in serial lines with continuous time models of machine reliability can be characterized in terms of two dimensionless parameters: the level of buffering (which quantifies buffer capacities in units of downtime) and the line efficiency (which quantifies the desired production rate in units of the largest production rate available in the system).

Closed formulas for the lean level of buffering (LLB) have been derived for exponential lines with identical machines.

For exponential lines with non-identical machines, closed formulas are available only for two- and three-machine cases. For longer lines, LLB approximations, based on either closed formulas or recursions, have been developed.

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Among the closed formulas approaches, the global pair-wise approach is recommended in lines with less than 15 machines; otherwise, the local upper bound approach is deemed the best. Among the recursive approaches, the bottleneck-based one is preferred.

For lines with non-exponential machines, it is shown that LLB is not sensitive to the shape of up- and downtime distribution and depends mainly on their coefficients of variation.

Based on this observation, empirical formulas are derived, which provide an upper bound on LLB as a function of up- and downtime CV’s, given that both of them are less than 1.