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This example shows how to improve the performance
of an engine cooling fan through a Design for Six Sigma approach using
Define, Measure, Analyze, Improve, and Control (DMAIC). The initial
fan does not circulate enough air through the radiator to keep the
engine cool during difficult conditions. First the example shows how
to design an experiment to investigate the effect of three performance
factors: fan distance from the radiator, blade-tip clearance, and
blade pitch angle. It then shows how to estimate optimum values for
each factor, resulting in a design that produces airflows beyond the
goal of 875 ft^{3} per minute using test data.
Finally it shows how to use simulations to verify that the new design
produces airflow according to the specifications in more than 99.999%
of the fans manufactured. This example uses MATLAB^{®}, Statistics and Machine Learning Toolbox™,
and Optimization Toolbox™.

This example addresses an engine cooling fan design that is
unable to pull enough air through the radiator to keep the engine
cool during difficult conditions, such as stop-and-go traffic or hot
weather). Suppose you estimate that you need airflow of at least 875
ft^{3}/min to keep the engine cool during
difficult conditions. You need to evaluate the current design and
develop an alternative design that can achieve the target airflow.

Load the sample data.

load(fullfile(matlabroot,'help/toolbox/stats/examples','OriginalFan.mat'))

The data consists of 10,000 measurements (historical production data) of the existing cooling fan performance.

Plot the data to analyze the current fan's performance.

plot(originalfan) xlabel('Observation') ylabel('Max Airflow (ft^3/min)') title('Historical Production Data')

The data is centered around 842 ft^{3}/min
and most values fall within the range of about 8 ft^{3}/min.
The plot does not tell much about the underlying distribution of data,
however. Plot the histogram and fit a normal distribution to the data.

figure() histfit(originalfan) % Plot histogram with normal distribution fit format shortg xlabel('Airflow (ft^3/min)') ylabel('Frequency (counts)') title('Airflow Histogram')

pd = fitdist(originalfan,'normal') % Fit normal distribution to data

pd = NormalDistribution Normal distribution mu = 841.652 [841.616, 841.689] sigma = 1.8768 [1.85114, 1.90318]

`fitdist`

fits a normal
distribution to data and estimates the parameters from data. The estimate
for the mean airflow speed is 841.652 ft^{3}/min,
and the 95% confidence interval for the mean airflow speed is (841.616,
841.689). This estimate makes it clear that the current fan is not
close to the required 875 ft^{3}/min. There
is need to improve the fan design to achieve the target airflow.

Evaluate the factors that affect cooling fan performance using
design of experiments (DOE). The response is the cooling fan airflow
rate (ft^{3}/min). Suppose that the factors
that you can modify and control are:

Distance from radiator

Pitch angle

Blade tip clearance

In general, fluid systems have nonlinear behavior. Therefore, use a response surface design to estimate any nonlinear interactions among the factors. Generate the experimental runs for a Box-Behnken design in coded (normalized) variables [-1, 0, +1].

CodedValue = bbdesign(3)

CodedValue = -1 -1 0 -1 1 0 1 -1 0 1 1 0 -1 0 -1 -1 0 1 1 0 -1 1 0 1 0 -1 -1 0 -1 1 0 1 -1 0 1 1 0 0 0 0 0 0 0 0 0

The first column is for the distance from radiator, the second column is for the pitch angle, and the third column is for the blade tip clearance. Suppose you want to test the effects of the variables at the following minimum and maximum values.

Distance from radiator: 1 to 1.5 inches

Pitch
angle: 15 to 35 degrees

Blade tip clearance:
1 to 2 inches

Randomize the order of the runs, convert the coded design values to real-world units, and perform the experiment in the order specified.

runorder = randperm(15); % Random permutation of the runs bounds = [1 1.5;15 35;1 2]; % Min and max values for each factor RealValue = zeros(size(CodedValue)); for i = 1:size(CodedValue,2) % Convert coded values to real-world units zmax = max(CodedValue(:,i)); zmin = min(CodedValue(:,i)); RealValue(:,i) = interp1([zmin zmax],bounds(i,:),CodedValue(:,i)); end

Suppose that at the end of the experiments, you collect the following response
values in the variable `TestResult`

.

TestResult = [837 864 829 856 880 879 872 874 834 833 860 859 874 876 875]';

Display the design values and the response.

disp({'Run Number','Distance','Pitch','Clearance','Airflow'}) disp(sortrows([runorder' RealValue TestResult]))

'Run Number' 'Distance' 'Pitch' 'Clearance' 'Airflow' 1 1.5 35 1.5 856 2 1.25 25 1.5 876 3 1.5 25 1 872 4 1.25 25 1.5 875 5 1 35 1.5 864 6 1.25 25 1.5 874 7 1.25 15 2 833 8 1.5 15 1.5 829 9 1.25 15 1 834 10 1 15 1.5 837 11 1.5 25 2 874 12 1 25 1 880 13 1.25 35 1 860 14 1 25 2 879 15 1.25 35 2 859

Save the design values and the response in a `table`

.

Expmt = table(runorder', CodedValue(:,1), CodedValue(:,2), CodedValue(:,3), ... TestResult,'VariableNames',{'RunNumber','D','P','C','Airflow'});

`D`

stands for `Distance`

, `P`

stands
for `Pitch`

, and C stands for `Clearance`

.
Based on the experimental test results, the airflow rate is sensitive
to the changing factors values. Also, four experimental runs meet
or exceed the target airflow rate of 875 ft^{3}/min
(runs 2, 4,12, and 14). However, it is not clear which, if any, of
these runs is the optimal one. In addition, it is not obvious how
robust the design is to variation in the factors. Create a model based
on the current experimental data and use the model to estimate the
optimal factor settings.

The Box-Behnken design enables you to test for nonlinear (quadratic) effects. The form of the quadratic model is:

$$\begin{array}{l}AF\text{=}{\beta}_{0}\text{+}{\beta}_{1}\text{*}Distance\text{+}{\beta}_{2}\text{*}Pitch\text{+}{\beta}_{3}\text{*}Clearance\text{+}{\beta}_{4}\text{*}Distance\text{*}Pitch\\ \text{+}{\beta}_{5}*Distance*Clearance+{\beta}_{6}*Pitch*Clearance+{\beta}_{7}*Distanc{e}^{2}\\ +{\beta}_{8}*Pitc{h}^{2}+{\beta}_{9}*Clearanc{e}^{2},\end{array}$$

where *AF* is
the airflow rate and *B _{i}* is
the coefficient for the term

`fitlm`

function
from Statistics and Machine Learning Toolbox.`mdl = fitlm(Expmt,'Airflow~D*P*C-D:P:C+D^2+P^2+C^2');`

Display the magnitudes of the coefficients (for normalized values) in a bar chart.

figure() h = bar(mdl.Coefficients.Estimate(2:10)); set(h,'facecolor',[0.8 0.8 0.9]) legend('Coefficient') set(gcf,'units','normalized','position',[0.05 0.4 0.35 0.4]) set(gca,'xticklabel',mdl.CoefficientNames(2:10)) ylabel('Airflow (ft^3/min)') xlabel('Normalized Coefficient') title('Quadratic Model Coefficients')

The bar chart shows that *Pitch* and
*Pitch*^{2} are dominant factors.
You can look at the relationship between multiple input variables and one output
variable by generating a response surface plot. Use `plotSlice`

to generate response
surface plots for the model `mdl`

interactively.

plotSlice(mdl)

The plot shows the nonlinear relationship of airflow with pitch. Move the blue
dashed lines around and see the effect the different factors have on airflow.
Although you can use `plotSlice`

to
determine the optimum factor settings, you can also use Optimization Toolbox to automate the task.

Find the optimal factor settings using the constrained
optimization function `fmincon`

.

Write the objective function.

`f = @(x) -x2fx(x,'quadratic')*mdl.Coefficients.Estimate;`

The objective function is a quadratic response surface
fit to the data. Minimizing the negative airflow using `fmincon`

is
the same as maximizing the original objective function. The constraints
are the upper and lower limits tested (in coded values). Set the initial
starting point to be the center of the design of the experimental
test matrix.

lb = [-1 -1 -1]; % Lower bound ub = [1 1 1]; % Upper bound x0 = [0 0 0]; % Starting point [optfactors,fval] = fmincon(f,x0,[],[],[],[],lb,ub,[]); % Invoke the solver

Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the default value of the function tolerance, and constraints are satisfied to within the default value of the constraint tolerance.

Convert the results to a maximization problem and real-world units.

maxval = -fval; maxloc = (optfactors + 1)'; bounds = [1 1.5;15 35;1 2]; maxloc=bounds(:,1)+maxloc .* ((bounds(:,2) - bounds(:,1))/2); disp('Optimal Values:') disp({'Distance','Pitch','Clearance','Airflow'}) disp([maxloc' maxval])

Optimal Values: 'Distance' 'Pitch' 'Clearance' 'Airflow' 1 27.275 1 882.26

The optimization result suggests placing the new fan one inch from the radiator, with a one-inch clearance between the tips of the fan blades and the shroud.

Because pitch angle has such a significant effect on airflow, perform additional analysis to verify that a 27.3 degree pitch angle is optimal.

load(fullfile(matlabroot,'help/toolbox/stats/examples','AirflowData.mat')) tbl = table(pitch,airflow); mdl2 = fitlm(tbl,'airflow~pitch^2'); mdl2.Rsquared.Ordinary

ans = 0.99632

The results show that a quadratic model explains the effect of pitch on the airflow well.

Plot the pitch angle against airflow and impose the fitted model.

figure() plot(pitch,airflow,'.r') hold on ylim([840 885]) line(pitch,mdl2.Fitted,'color','b') title('Fitted Model and Data') xlabel('Pitch angle (degrees)') ylabel('Airflow (ft^3/min)') legend('Test data','Quadratic model','Location','se') hold off

Find the pitch value that corresponds to the maximum airflow.

pitch(find(airflow==max(airflow)))

ans = 27

The additional analysis confirms that a 27.3 degree pitch angle is optimal.

The improved cooling fan design meets the airflow requirements. You also have a model that approximates the fan performance well based on the factors you can modify in the design. Ensure that the fan performance is robust to variability in manufacturing and installation by performing a sensitivity analysis.

Suppose that, based on historical experience, the manufacturing uncertainty is as follows.

Factor | Real Values | Coded Values |
---|---|---|

Distance from radiator | 1.00 +/- 0.05 inch | 1.00 +/- 0.20 inch |

Blade pitch angle | 27.3 +/- 0.25 degrees | 0.227 +/- 0.028 degrees |

Blade tip clearance | 1.00 +/- 0.125 inch | -1.00 +/- 0.25 inch |

Verify that these variations in factors will enable to maintain
a robust design around the target airflow. The philosophy of Six Sigma
targets a defect rate of no more than 3.4 per 1,000,000 fans. That
is, the fans must hit the 875 ft^{3}/min target
99.999% of the time.

You can verify the design using Monte Carlo simulation. Generate 10,000 random numbers for three factors with the specified tolerance. First, set the state of the random number generators so results are consistent across different runs.

`rng('default')`

Perform the Monte Carlo simulation. Include a noise variable
that is proportional to the noise in the fitted model, `mdl`

(that
is, the RMS error of the model). Because the model coefficients are
in coded variables, you must generate `dist`

, `pitch`

,
and `clearance`

using the coded definition.

dist = random('normal',optfactors(1),0.20,[10000 1]); pitch = random('normal',optfactors(2),0.028,[10000 1]); clearance = random('normal',optfactors(3),0.25,[10000 1]); noise = random('normal',0,mdl2.RMSE,[10000 1]);

Calculate airflow for 10,000 random factor combinations using the model.

```
simfactor = [dist pitch clearance];
X = x2fx(simfactor,'quadratic');
```

Add noise to the model (the variation in the data that the model did not account for).

simflow = X*mdl.Coefficients.Estimate+noise;

Evaluate the variation in the model's predicted airflow using a histogram. To estimate the mean and standard deviation, fit a normal distribution to data.

pd = fitdist(simflow,'normal'); histfit(simflow) hold on text(pd.mu+2,300,['Mean: ' num2str(round(pd.mu))]) text(pd.mu+2,280,['Standard deviation: ' num2str(round(pd.sigma))]) hold off xlabel('Airflow (ft^3/min)') ylabel('Frequency') title('Monte Carlo Simulation Results')

The results look promising. The average airflow is 882 ft^{3}/min
and appears to be better than 875 ft^{3}/min
for most of the data.

Determine the probability that the airflow is at 875 ft^{3}/min
or below.

```
format long
pfail = cdf(pd,875)
pass = (1-pfail)*100
```

pfail = 1.509289008603141e-07 pass = 99.999984907109919

The design appears to achieve at least 875 ft^{3}/min
of airflow 99.999% of the time.

Use the simulation results to estimate the process capability.

S = capability(simflow,[875.0 890]) pass = (1-S.Pl)*100

S = mu: 8.822982645666709e+02 sigma: 1.424806876923940 P: 0.999999816749816 Pl: 1.509289008603141e-07 Pu: 3.232128339675335e-08 Cp: 1.754623760237126 Cpl: 1.707427788957002 Cpu: 1.801819731517250 Cpk: 1.707427788957002 pass = 99.9999849071099

The `Cp`

value is 1.75. A process is considered
high quality when `Cp`

is greater than or equal to
1.6. The `Cpk`

is similar to the `Cp`

value,
which indicates that the process is centered. Now implement this design.
Monitor it to verify the design process and to ensure that the cooling
fan delivers high-quality performance.

You can monitor and evaluate the manufacturing and installation process of the new fan using control charts. Evaluate the first 30 days of production of the new cooling fan. Initially, five cooling fans per day were produced. First, load the sample data from the new process.

load(fullfile(matlabroot,'help/toolbox/stats/examples','spcdata.mat'))

Plot the *X*-bar and *S* charts.

figure() controlchart(spcflow,'chart',{'xbar','s'}) % Reshape the data into daily sets xlabel('Day')

According to the results, the manufacturing process is in statistical control, as indicated by the absence of violations of control limits or nonrandom patterns in the data over time. You can also run a capability analysis on the data to evaluate the process.

[row,col] = size(spcflow); S2 = capability(reshape(spcflow,row*col,1),[875.0 890]) pass = (1-S.Pl)*100

S2 = mu: 8.821061141685465e+02 sigma: 1.423887508874697 P: 0.999999684316149 Pl: 3.008932155898586e-07 Pu: 1.479063578225176e-08 Cp: 1.755756676295137 Cpl: 1.663547652525458 Cpu: 1.847965700064817 Cpk: 1.663547652525458 pass = 99.9999699106784

The `Cp`

value of 1.755 is very similar to
the estimated value of 1.73. The `Cpk`

value of 1.66
is smaller than the `Cp`

value. However, only a `Cpk`

value
less than 1.33, which indicates that the process shifted significantly
toward one of the process limits, is a concern. The process is well
within the limits and it achieves the target airflow (875 ft^{3}/min)
more than 99.999% of the time.