PRODUCTION SYSTEMS ENGINEERING

Chapter 5: Continuous Improvement of Bernoulli Lines

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

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

Motivation: Continuous improvement is universally recognized as the

most important managerial tool. Currently, continuous improvement projects have no theory

for their design and are, typically, carried out based on experience, common sense, and sometimes, discrete event simulations. As a result, quite often predicted improvements do not materialize.

The purpose of this chapter is to present analytical tools for designing continuous improvement projects with predictable results.

In this framework, both constrained and unconstrained improvabilities are addressed.

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OUTLINE

1. Constrained Improvability 2. Unconstrained Improvability 3. Measurement-Based Management 4. Case Studies 5. Summary

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1.1 Resource constraints and definitions Constraints:

Buffer capacity (BC) constraint

Workforce (WF) constraint

1. Constrained Improvability

Definition:

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A Bernoulli line is improvable w.r.t. BC if there exist s.t. and

improvable w.r.t. WF if there exist s.t. and

improvable w.r.t. BC and WF simultaneously if the exist and s.t.

and

1.1 Resource constraints and definitions (cont.)

1.2 Improvability w.r.t. WF

1.2.1 Necessary and sufficient conditions

Theorem: A Bernoulli line is unimprovable w.r.t. WF if and only if Corollary: Under the above condition, i.e.,

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WF-Improvability Indicator: A Bernoulli line is practically unimprovable w.r.t. WF if each buffer is close to being half full.

“Common sense” justification of this indicator:

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1.2.2 WF improvability indicator and continuous improvement procedure

WF-Continuous Improvement Procedure: (1) Evaluate WIPi

(2) Determine the buffer for which |WIPi - Ni/2| is maximum (3) Re-allocate workforce (4) Return to (1) (5) Continuous until max|WIPi - Ni/2| is sufficiently small

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1.2.2 WF improvability indicator and continuous improvement procedure (cont)

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1.2.3 Example

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1.2.4 PSE Toolbox

Theorem: If , then

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1.2.5 Unimprovable allocation of WF

Theorem: The unimprovable , is given by

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1.2.5 Unimprovable allocation of WF (cont)

Corollary: If Ni := N, i = 1,…, M, unimprovable is a “flat” inverted bowl:

In most cases, this allocation does not give a substantially larger than the uniform allocation.

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1.2.5 Unimprovable allocation of WF (cont)

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1.2.6 PSE Toolbox

1.3 Improvability w.r.t. WF and BC simultaneously

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1.3.1 Necessary and sufficient conditions

Theorem: A Bernoulli line is unimprovable w.r.t. WF and BC simultaneously if and only if

Corollary: Under this condition, BLi = STi, i = 2,…, M - 1, BL1 = STM. In addition, and

Denote Theorem: The unimprovable sequence and are given by

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1.3.2 Unimprovable allocation of pi and Ni

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1.3.3 PSE Toolbox

1.4 Improvability w.r.t. BC

1.4.1 Necessary and sufficient conditions Theorem: A Bernoulli line is unimprovable w.r.t. BC if and only if is maximized over all sequences such that

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1.4.2 Practical equivalence statement

Numerical Fact: The production rate under the above allocation is practically always the same as that for allocation that minimize

over all sequences such that “Common sense” justification of this fact:

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BC-Improvability Indicator: A Bernoulli line is practically unimprovable w.r.t. BC if the average occupancy of each buffer is as close to the average availability of its downstream buffer as possible.

BC-Continuous Improvement Procedure: (1) Evaluate WIPi

(2) Determine the buffer with the largest |WIPi – (Ni+1 – WIPi+1 )| (3) Re-allocate Ni and Ni+1 (4) Return to (1) (5) Continuous until arriving at a limit cycle and select the best allocation

from the limit cycle

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1.4.3 BC improvability indicator and continuous improvement procedure

M = 11; pi = 0.8, i ≠ 6; p6 = 0.6. Total buffer capacity 24. BC-Continuous Improvement Procedure results in N1 = 1, N2 = N3 = N4 = 2, N5 = 5, N6 = 4, N7 = N8 = N9 = N10 = 2. Resulting throughput . Goldratt’s allocation Ni = 1, i ≠ 5, N5 = 17. Resulting throughput (27% less). Improved Goldratt’s allocation Ni = 1, i ≠ 5,6 N5 = N6 =8. Resulting throughput (16% less).

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1.4.4 Example

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1.4.5 PSE Toolbox

2. Unconstrained Improvability

2.1 Bottleneck machine Definition: mi, i ϵ {1,…, M}, is the bottleneck machine (BN-m) if

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The worst machine is not necessarily the BN-m. The machine with the largest WIP is not necessarily the BN-m.

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2.1 Bottleneck machine (cont)

Theorem: In WF-unimprovable Bernoulli lines, Thus, in a WF-unimprovable system, BN-m is necessarily the worst

machine.

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2.1 Bottleneck machine (cont)

Definition: bi, i = 1,…, M - 1, is the bottleneck buffer (BN-b) if

The smallest capacity buffer is not necessarily the BN-b.

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2.2 Bottleneck buffer

2.3 Bottlenecks in two-machine lines

Theorem: if and only if ; if and only if ;

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Illustration

STi 0 0.0215

BLi 0.1215 0

2.3 Bottlenecks in two-machine lines (cont.)

2.4 Bottlenecks in M > 2-machine lines

2.4.1 Arrow assignment rule: BLi > STi+1 → arrow from mi to mi+1

BLi < STi+1 → arrow from mi+1 to mi

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STi 0 0.01 0.39 0.37 0.27 BLi 0.41 0.20 0.27 0.01 0

A single machine with no emanating arrows is the BN-m.

Among multiple machines with no emanating arrows, the one with the largest severity is the Primary BN (PBN-m):

BN-b is the buffer in front of the BN-m if it is more often starved than blocked or right after the BN-m if it is more often blocked than starved

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2.4.2 Bottleneck Indicator

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2.4.3 Illustrations

(a) Single bottleneck case

(b) Multiple bottleneck case

5000 five-machine lines

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Statistical experiment

2.4.4 Numerical justification

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Results (using calculation data)

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Results (using simulation data)

Hypothesis: implies implies

Lemma: For any , there exists N*, such that if then

Theorem: Under the Hypothesis, BN-m is downstream of mj if and upstream of mj if .

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2.4.5 Analytical justification

(1) Determine BLi and STi. (2) Identify BN-m and BN-b. (3) Increase pi of the BN-m. (4) If impossible, increase N of the BN-b. (5) Return to (1).

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2.4.6 BN-Continuous Improvement Procedure

The most effective way for continuous improvement!

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2.4.7 PSE Toolbox

2.5 Buffer potency

2.5.1 Definition

The buffering is weakly potent if BN-m is the worst one in the system; potent if it is weakly potent and PR ≈ 0.95pi(BN-m); strongly potent if it is potent and is the smallest

possible to ensure the desired production rate.

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Analysis of production losses

Observation: Out of roughly 240 parts/h lost, 80 parts/h are due to machines, while about 160 parts/h are due to MHS.

Thus, ensuring MHS potency is an important resource of production systems improvements.

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2.5.2 Illustration: Potency of automotive ignition module assembly system

3. Measurement-Based Management

3.1 Purpose and structure Designing continuous improvement project requires relatively

detailed information concerning the machines and buffers. A simpler method is needed for daily managerial duties. MBM is such a method. It consists of the following:

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3.2 Simplified block diagrams

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3.3 Measurement of blockages and starvations

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3.4 Identifying bottlenecks

3.5 Managerial decisions

Based on the previous steps, develop and plan activities to eliminate the bottlenecks identified.

Continue in the same manner in the never-ending process of continuous improvement.

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4. Case Studies

4.1 Automotive ignition coil processing system

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4.1.1 Bernoulli model

Constraints:

Unimprovable allocations

Observation: PR is still low, i.e., constrained improvability does not solve the problem.

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4.1.2 Constrained improvability

Increasing BN-b by 1 and p9-10 by 10% lead to ( ). Observation: PR is still low.

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4.1.3 Unconstrained improvability

Increasing b5 by 1 leads to ( parts/hour), which has been viewed as acceptable performance

Returning to exponential model (using B-exp transformation):

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4.1.3 Unconstrained improvability (cont)

4.2 Automotive paint shop production system

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4.2.1 Bernoulli model

Constraints:

Unimprovable allocation:

After some adjustment, ( = 57.45 parts/hour). Observation: PR is relatively low.

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4.2.2 Constrained improvability

Assuming that there are no starvations for skids, ( jobs/hour).

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4.2.3 Unconstrained improvability

Increasing efficiency of m3 by 4% results in

() )

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4.2.3 Unconstrained improvability (cont)

5. Summary

Improvements can be accomplished in either constrained or unconstrained scenarios.

WF-unimprovable system: buffers close to being half full. WF- and BC-unimprovable system: blockages and starvations of

all “internal” machines are the same. BC-unimprovable system: each machine is blocked and starved

with almost equal frequencies. BN-m and BN-b can be identified using the arrow-based rule. Buffer is potent if the worst machine is indeed the BN.

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