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Ch. 9 : Managing Flow Variability
Chapter 9: Managing Flow Variability
Sections 9.3.4 to End
Team 10
Alex Ichiroku
Vivian Ramos
Hamid Orandi
(and…Shehzad Khan)
Ch. 9 : Managing Flow Variability
9.3.4 Control Charts

Statistical process control involves setting a “range of acceptable variations” in the
performance of the process, around its mean.

If the observed values are within this range:
– Accept the variations as “normal”
– Don’t make any adjustments to the process

If the observed values are outside this range:
– The process is out of control
– Need to investigate what’s causing the problems – the assignable cause
Ch. 9 : Managing Flow Variability
9.3.4 Control Charts … Continued

Let  be the expected value of the performance
Set up a “control band” around 
» UCL = Upper Control Limit
» LCL = Lower Control Limit

Calculate the standard deviation, 

Decide how “tightly” we want to monitor and control the process
The smaller the value of “z”, the tighter the control
Ch. 9 : Managing Flow Variability
9.3.4 Control Charts … Continued

The Upper and Lower Control Bands:
LCL =  - z
Process Control Chart:
UCL =  + z
Ch. 9 : Managing Flow Variability
9.3.4 Control Charts … Continued

If observed data within the control band:
Performance variability is normal

If observed data outside the control band:
Process is “out of control”

Data Misinterpretation
Type I error, : Process is “in control”, but data outside the Control Band
Type II error, : Process is “out of control”, but data inside the Control Band
Ch. 9 : Managing Flow Variability
9.3.4 Control Charts … Continued

Optimal Degree of Control
Depends on 2 things:
1. How much variability in the performance measure we consider acceptable
2. How frequently we monitor the process performance.
Optimal frequency of monitoring is a balance between the costs and benefits
The value of “z” determines how tightly we control the process
LCL =  - z
UCL =  + z
A small value of “z”: Narrower Control Band -- Tighter Control
Ch. 9 : Managing Flow Variability
9.3.4 Control Charts … Continued

Optimal Degree of Control
If we set ‘z’ to be too small:


We’ll end up doing unnecessary investigation
Incur additional costs
If we set ‘z’ to be too large:


We’ll accept a lot more variations as normal
We wouldn’t look for problems in the process – less costly
Ch. 9 : Managing Flow Variability
9.3.4 Control Charts … Continued

Optimal Degree of Control
In practice, a value of z = 3 is used

99.73% of all measurements will fall within the “normal” range
Ch. 9 : Managing Flow Variability
9.3.4 Control Charts … Continued

Average and Variation Control Charts
Monitor process performance by taking random samples
For each Sample:
Calculate the average value, A1, A2….AN
 Calculate the variance of each sample, V1, V2….VN

Sample Averages:

Normally distributed
 Mean of A
 Standard Deviation of A
Ch. 9 : Managing Flow Variability
9.3.4 Control Charts … Continued

Average and Variation Control Charts
A = /n
(n = sample size)
LCL =  - z/n
and
UCL =  + z/n
Take it one step further:
Estimate  by the overall average of all the sample averages, A
A = (A1+ A2+…+AN) / N
(N = # of samples)
Also estimate  by the standard deviation of all N x n observations, S
Ch. 9 : Managing Flow Variability
9.3.4 Control Charts … Continued

Average and Variation Control Charts
Our New, Improved equations for Upper and Lower Control Limits are:
LCL = A - zs/n
and
UCL = A + zs/n
We can do the same calculations with the Sample Variances:
CalculateV -- the average variance of the sample variances
V = (V1+ V2+…+VN) / N
(N = # of samples)
Also calculate SV -- the standard deviation of the variances
Ch. 9 : Managing Flow Variability
9.3.4 Control Charts … Continued

Average and Variation Control Charts
The New equations for Variance Control Limits are:
LCL = V - z sV and
UCL = V + z sV
If observed variations fall within this range:
Process Variability is stable
If not:
Need to investigate the cause of abnormal variations
Ch. 9 : Managing Flow Variability
9.3.4 Control Charts … Continued

Average and Variation Control Charts
Garage Door Example revisited…
Ex: A1 = (81 + 73 + 85 + 90 + 80) / 5 = 81.8 kg
Ex: V1 = (90 - 73) = 17 kg
Std. Dev. of Door Weights:
Std. Dev. of Sample Variances:
s = 4.2 kg
sV = 3.5 kg
Ch. 9 : Managing Flow Variability
9.3.4 Control Charts … Continued

Average and Variation Control Charts
Average Weights of Garage Door Samples:
Ch. 9 : Managing Flow Variability
9.3.4 Control Charts … Continued
Average and Variation Control Charts
Let z = 3
Sample Averages
UCL = A + zs/n = 82.5 + 3 (4.2) / 5 = 88.13
LCL = A - zs/n = 82.5 – 3 (4.2) / 5 = 76.87
Average Weight Control Chart
90
88
Average Wt. (Kg)

UCL = 88.13
86
84
82
`
80
78
LCL = 76.87
76
74
1
2
3
4
5
6
7
8
9
10
11
Days
12
13
14
15
16
17
18
19
20
Ch. 9 : Managing Flow Variability
9.3.4 Control Charts … Continued
Average and Variation Control Charts
Let z = 3
Sample Variances
UCL = V + z sV = 10.1 + 3 (3.5) = 20.6
LCL = V - zs sV = 10.1 – 3 (3.5) = - 0.4
Variance Control Chart
Variance (range) of Wt.
(Kg)

25
UCL = 20.6
20
15
10
5
LCL = 0
0
1
2
3
4
5
6
7
8
9
10 11
Days
12 13 14
15 16 17
18 19 20
Ch. 9 : Managing Flow Variability
9.3.4 Control Charts … Continued

Extensions
Continuous Variables:
 Garage Door Weights
 Processing Costs
 Customer Waiting Time
Use Normal distribution
Discrete Variables:
 Number of Customer Complaints
 Whether a Flow Unit is Defective
 Number of Defects per Flow Unit Produced
Use Binomial or Poisson distribution
Ch. 9 : Managing Flow Variability
9.3.5 Cause-Effect Diagrams

Cause-Effect Diagrams
Sample
Plot
Abnormal
Observations
Control Charts
Variability !!
Now what?!!
Brainstorm Session!!
Answer 5 “WHY” Questions !
Ch. 9 : Managing Flow Variability
9.3.5 Cause-Effect Diagrams … Continued

Why…? Why…? Why…? (+2)
Our famous “Garage Door” Example:
1. Why are these doors so heavy?
Because the Sheet Metal was too ‘thick’.
2. Why was the sheet metal too thick?
Because the rollers at the steel mill were set
incorrectly.
3. Why were the rollers set incorrectly?
Because the supplier is not able to meet our
specifications.
4. Why did we select this supplier?
Because our Project Supervisor was too busy
getting the product out – didn’t have time to
research other vendors.
5. Why did he get himself in this situation?
Because he gets paid by meeting the
production quotas.
Ch. 9 : Managing Flow Variability
9.3.5 Cause-Effect Diagrams … Continued

Fishbone Diagram
Ch. 9 : Managing Flow Variability
9.3.6 Scatter Plots

The Thickness of the Sheet Metals
 Change Settings on Rollers
 Measure the Weight of the Garage Doors
 Determine Relationship between the two
Roller Settings & Garage Door Weights
Plot the results on a graph:
Door Weight (Kg)
Scatter Plot
20
15
10
5
0
0
1
2
3
4
5
6
7
8
9
Roller Setting (m m )
10 11 12 13
Ch. 9 : Managing Flow Variability
9.4 Process Capability




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
Ease of external product measures (door operations and durability) and internal
measures (door weight)
Product specification limits vs. process control limits
Individual units, NOT sample averages - must meet customer specifications.
Once process is in control, then the estimates of μ (82.5kg) and σ (4.2k) are reliable.
Hence we can estimate the process capabilities.
Process capabilities - the ability of the process to meet customer specifications
Three measures of process capabilities:
– 9.4.1 Fraction of Output within Specifications
– 9.4.2 Process Capability Ratios (Cpk and Cp)
– 9.4.3 Six-Sigma Capability
Ch. 9 : Managing Flow Variability
9.4.1 Fraction of Output within Specifications

The fraction of the process output that meets customer specifications.
 We can compute this fraction by:
- Actual observation (see Histogram, Fig 9.3)
- Using theoretical probability distribution
Ex. 9.7:
- US: 85kg; LS: 75 kg (the range of performance variation that customer is
willing to accept)
See figure 9.3 Histogram: In an observation of 100 samples, the process is 74%
capable of meeting customer requirements, and 26% defectives!!!
OR:
– Let W (door weight): normal random variable with mean = 82.5 kg and
standard deviation at 4.2 kg,
Then the proportion of door falling within the specified limits is:
Prob (75 ≤ W ≤ 85) = Prob (W ≤ 85) - Prob (W ≤ 75)
Ch. 9 : Managing Flow Variability
9.4.1 Fraction of Output within Specifications cont…

Let Z = standard normal variable with μ = 0 and σ = 1, we can use the standard
normal table in Appendix II to compute:
AT US:
Prob (W≤ 85) in terms of:
Z = (W-μ)/ σ
As Prob [Z≤ (85-82.5)/4.2] = Prob (Z≤.5952) = .724 (see Appendix II)
(In Excel: Prob (W ≤ 85) = NORMDIST (85,82.5,4.2,True) = .724158)
AT LS:
Prob (W ≤ 75)
= Prob (Z≤ (75-82.5)/4.2) = Prob (Z ≤ -1.79) = .0367 in Appendix II
(In Excel: Prob (W ≤ 75) = NORMDIST(75,82.5,4.2,true) = .037073)
THEN:
Prob (75≤W≤85)
= .724 - .0367 = .6873
Ch. 9 : Managing Flow Variability
9.4.1 Fraction of Output within Specifications cont…
SO with normal approximation, the process is capable of producing 69% of doors
within the specifications, or delivering 31% defective doors!!!
Specifications refer to INDIVIDUAL doors, not AVERAGES.
We cannot comfort customer that there is a 30% chance that they’ll get doors that is
either TOO LIGHT or TOO HEAVY!!!
Ch. 9 : Managing Flow Variability
9.4.2 Process Capability Ratios (C pk and Cp)






2nd measure of process capability that is easier to compute is the process capability
ratio (Cpk)
If the mean is 3σ above the LS (or below the US), there is very little chance of a
product falling below LS (or above US).
So we use:
(US- μ)/3σ
(.1984 as calculated later)
and (μ -LS)/3σ
(.5952 as calculated later)
as measures of how well process output would fall within our specifications.
The higher the value, the more capable the process is in meeting specifications.
OR take the smaller of the two ratios [aka (US- μ)/3σ =.1984] and define a single
measure of process capabilities as:
Cpk = min[(US-μ/)3σ, (μ -LS)/3σ] (.1984, as calculated later)
Ch. 9 : Managing Flow Variability
9.4.2 Process Capability Ratios (C pk and Cp)

Cpk of 1+- represents a capable process
 Not too high (or too low)
 Lower values = only better than expected quality
Ex: processing cost, delivery time delay, or # of error per transaction process
 If the process is properly centered
– Cpk is then either:
(US- μ)/3σ or (μ -LS)/3σ
As both are equal for a centered process.
Ch. 9 : Managing Flow Variability
9.4.2 Process Capability Ratios (C pk and Cp) cont…

Therefore, for a correctly centered process, we may simply define the process
capability ratio as:
– Cp = (US-LS)/6σ
(.3968, as calculated later)
Numerator = voice of the customer / denominator = the voice of the process
 Recall: with normal distribution:
Most process output is 99.73% falls within +-3σ from the μ.
 Consequently, 6σ is sometimes referred to as the natural tolerance of the process.
Ex: 9.8
Cpk = min[(US- μ)/3σ , (μ -LS)/3σ ]
= min {(85-82.5)/(3)(4.2)], (82.5-75)/(3)(4.2)]}
= min {.1984, .5952}
=.1984
Ch. 9 : Managing Flow Variability
9.4.2 Process Capability Ratios (C pk and Cp)



If the process is correctly centered at μ = 80kg (between 75 and 85kg), we
compute the process capability ratio as
Cp = (US-LS)/6σ
= (85-75)/[(6)(4.2)]
= .3968
NOTE: Cpk = .1984 (or Cp = .3968) does not mean that the process is capable of
meeting customer requirements by 19.84% (or 39.68%), of the time. It’s about
69%.
Defects are counted in parts per million (ppm) or ppb, and the process is assumed
to be properly centered. IN THIS CASE, If we like no more than 100 defects per
million (.01% defectives), we SHOULD HAVE the probability distribution of door
weighs so closely concentrated around the mean that the standard deviation is 1.282
kg, or Cp=1.3 (see Table 9.4)
Test: σ = (85-75)/(6)(1.282)] = 1.300kg
Ch. 9 : Managing Flow Variability
Table 9.4
Table 9.4 Relationship Between Process Capability Ratio and Proportion Defective
Defects (ppm)
10000
1000
100
10
1
2 ppb
Cp
0.86
1
1.3
1.47
1.63
2
Ch. 9 : Managing Flow Variability
9.4.3 Six-Sigma Capability
The 3rd process capability
 Known as Sigma measure, which is computed as
 S = min[(US- μ /σ), (μ -LS)/σ] (= min(.5152,1.7857) = .5152 to be calculated later)
 S-Sigma process
If process is correctly centered at the middle of the specifications,
S = [(US-LS)/2σ]
Ex: 9.9
Currently the sigma capability of door making process is
S=min(85-82.5)/[(2)(4.2)] = .5952
By centering the process correctly, its sigma capability increases to
S=min(85-75)/[(2)(4.2)] = 1.19
THUS, with a 3σ that is correctly centered, the US and LS are 3σ away from the
mean, which corresponds to Cp=1, and 99.73% of the output will meet the
specifications.

Ch. 9 : Managing Flow Variability
9.4.3 Six-Sigma Capability cont…

SIMILARLY, a correctly centered six-sigma process has a standard deviation so
small that the US and LS limits are 6σ from the mean each.
 Extraordinary high degree of precision.
Corresponds to Cp=2 or 2 defective units per billion produced!!! (see Table 9.5)

In order for door making process to be a six-sigma process, its standard deviation
must be:
σ = (85-75)/(2)(6)] = .833kg
 Adjusting for Mean Shifts
Allowing for a shift in the mean of +-1.5 standard deviation from the center of
specifications.
Allowing for this shift, a six-sigma process amounts to producing an average of 3.4
defective units per million. (see table 9.5)
Ch. 9 : Managing Flow Variability
Table 9.5
Table 9.5 Fraction Defective and Sigma Measure
Sigma S
3
4
5
Capability Ratio Cp
1
1.33
1.667
Defects (ppm)
66810
6210
233
6
2
3.4
Ch. 9 : Managing Flow Variability
9.4.3 Six-Sigma Capability cont…

Why Six-Sigma?
– See table 9.5
– Improvement in process capabilities from a 3-sigma to 4-sigma = 10-fold
reduction in the fraction defective (66810 to 6210 defects)
– While 4-sigma to 5-sigma = 30-fold improvement (6210 to 232 defects)
– While 5-sigma to 6-sigma = 70-fold improvement (232 to 3.4 defects, per
million!!!).

Average companies deliver about 4-sigma quality, where best-in-class companies
aim for six-sigma.
Ch. 9 : Managing Flow Variability
9.4.3 Six-Sigma Capability cont…

Why High Standards?
– The overall quality of the entire product/process that requires ALL of them to
work satisfactorily will be significantly lower.
Ex:
If product contains 100 parts and each part is 99% reliable, the chance that the
product (all its parts) will work is only (.99)100 = .366, or 36.6%!!!
- Also, costs associated with each defects may be high
- Expectations keep rising
Ch. 9 : Managing Flow Variability
9.4.3 Six-Sigma Capability cont…

Safety capability
- We may also express process capabilities in terms of the desired margin [(US-LS)zσ] as safety capability
- It represents an allowance planned for variability in supply and/or demand
- Greater process capability means less variability
- If process output is closely clustered around its mean, most of the output will fall
within the specifications
- Higher capability thus means less chance of producing defectives
- Higher capability = robustness
Ch. 9 : Managing Flow Variability
9.4.4 Capability and Control

So in Ex. 9.7: the production process is not performing well in terms of MEETING
THE CUSTOMER SPECIFICATIONS. Only 69% meets output specifications!!!
(See 9.4.1: Fraction of Output within Specifications)

Yet in example 9.6, “the process was in control!!!”, or WITHIN US & LS LIMITS.

Meeting customer specifics: indicates internal stability and statistical predictability
of the process performance.
In control (aka within LS and US range): ability to meet external customer’s
requirements.
 Observation of a process in control ensures that the resulting estimates of the
process mean and standard deviation are reliable so that our measurement of the
process capability is accurate.

Ch. 9 : Managing Flow Variability
9.5 Process Capability Improvement

Shift the process mean
 Reduce the variability
 Both
Ch. 9 : Managing Flow Variability
9.5.1 Mean Shift

Examine where the current process mean lies in comparison to the specification
range (i.e. closer to the LS or the US)
 Alter the process to bring the process mean to the center of the specification range
in order to increase the proportion of outputs that fall within specification
Ch. 9 : Managing Flow Variability
Ex 9.10

MBPF garage doors (currently)
-specification range: 75 to 85 kgs
-process mean: 82.5 kgs
-proportion of output falling within specifications: .6873
 The process mean of 82.5 kgs was very close to the US of 85 kgs (i.e. too
thick/heavy)
 To lower the process mean towards the center of the specification range the supplier
could change the thickness setting on their rolling machine
Ch. 9 : Managing Flow Variability
Ex 9.10 Continued



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


Center of the specification range: (75 + 85)/2 = 80 kgs
New process mean: 80 kgs
If the door weight (W) is a normal random variable, then the proportion of doors
falling within specifications is: Prob (75 =< W =< 85)
Prob (W =< 85) – Prob (W =< 75)
Z = (weight – process mean)/standard deviation
Z = (85 – 80)/4.2 = 1.19
Z = (75 – 80)/4.2 = -1.19
Ch. 9 : Managing Flow Variability
Ex 9.10 Continued

[from table A2.1 on page 319]
Z = 1.19
Z = -1.19
(1 - .8830)
.8830
.1170
Prob (W =< 85) – Prob (W =< 75) =
.8830 - .1170 = .7660
 By shifting the process mean from 82.5 kgs to 80 kgs, the proportion of garage
doors that falls within specifications increases from .6873 to .7660

Ch. 9 : Managing Flow Variability
9.5.2 Variability Reduction

Measured by standard deviation
 A higher standard deviation value means higher variability amongst outputs
 Lowering the standard deviation value would ultimately lead to a greater proportion
of output that falls within the specification range
Ch. 9 : Managing Flow Variability
9.5.2 Variability Reduction Continued

Possible causes for the variability MBPF experienced are:
-old equipment
-poorly maintained equipment
-improperly trained employees
 Investments to correct these problems would decrease variability however doing so
is usually time consuming and requires a lot of effort
Ch. 9 : Managing Flow Variability
Ex 9.11






Assume investments are made to decrease the standard deviation from 4.2 to 2.5 kgs
The proportion of doors falling within specifications:
Prob (75 =< W =< 85)
Prob (W =< 85) – Prob (W =< 75)
Z = (weight – process mean)/standard deviation
Z = (85 – 80)/2.5 = 2.0
Z = (75 – 80)/2.5 = -2.0
Ch. 9 : Managing Flow Variability
Ex 9.11 Continued

[from table A2.1 on page 319]
Z = 2.0
Z = -2.0
(1 - .9772)
.9772
.0228
Prob (W =< 85) – Prob (W =< 75) =
.9772 - .0228 = .9544
 By shifting the standard deviation from 4.2 kgs to 2.5 kgs and the process mean
from 82.5 kgs to 80 kgs, the proportion of garage doors that falls within
specifications increases from .6873 to .9544

Ch. 9 : Managing Flow Variability
9.5.3 Effect of Process Improvement on Process Control

Changing the process mean or variability requires re-calculating the control limits
 This is required because changing the process mean or variability will also change
what is considered abnormal variability and when to look for an assignable cause
Ch. 9 : Managing Flow Variability
9.6 Product and Process Design

Reducing the variability from product and process design
-simplification
-standardization
-mistake proofing
Ch. 9 : Managing Flow Variability
Simplification

Reduce the number of parts (or stages) in a product (or process)
-less chance of confusion and error
 Use interchangeable parts and a modular design
-simplifies materials handling and inventory control
 Eliminate non-value adding steps
-reduces the opportunity for making mistakes
Ch. 9 : Managing Flow Variability
Standardization

Use standard parts and procedures
-reduces operator discretion, ambiguity, and opportunity for making mistakes
Ch. 9 : Managing Flow Variability
Mistake Proofing

Designing a product/process to eliminate the chance of human error
-ex. color coding parts to make assembly easier
-ex. designing parts that need to be connected with perfect symmetry or with
obvious asymmetry to prevent assembly errors
Ch. 9 : Managing Flow Variability
9.6.2 Robust Design

Designing the product in a way so its actual performance will not be affected by
variability in the production process or the customer’s operating environment
 The designer must identify a combination of design parameters that protect the
product from the process related and environment related factors that determine
product performance
Ch. 9 : Managing Flow Variability
QUESTIONS
???