Case Study 4 – Catapult Variation Reduction
- Open the workbook Catapult Model. DiscoverSim Input Distributions will simulate the variability in the catapult input factor X’s. We will specify the Normal Distribution for each X in cells D30 to D33. The output, Distance, will be specified at cell D36 using the formula given in the Introduction.
- The Input Distribution parameters are given in the following table:
- Click on cell D30 to specify the Input Distribution for X1, Spring Constant. Select DiscoverSim > Input Distribution:
- We will use the selected default Normal Distribution.
- Click input Name cell reference and specify cell F30 containing the input name “Spring_Constant”. After specifying a cell reference, the dropdown symbol changes from to .
- Click the Mean parameter cell reference and specify cell I30 containing the mean parameter value = 47.3.
- Click the StdDev parameter cell reference and specify cell K30 containing the Standard Deviation parameter value = 0.1.
- Click Update Chart to view the Normal Distribution as shown:
- Click OK. Hover the cursor on cell D30 to view the DiscoverSim graphical comment showing the distribution and parameter values:
- Click on cell D30. Click the DiscoverSim Copy Cell menu button (Do not use Excel’s Copy – it will not work!).
- Select cells D31:D33. Now click the DiscoverSim Paste Cell menu button (Do not use Excel’s Paste – it will not work!).
- Review the input comments in cells D31 to D33:
- Click on cell D36. Note that the cell contains the Excel formula for Catapult Firing Distance. Excel range names are used rather than cell addresses to simplify interpretation.
- Select DiscoverSim > Output Response:
- Enter the output Name as “Distance”. Enter the Lower Specification Limit (LSL) as 47.5, Target as 50, and Upper Specification Limit (USL) as 52.5.
- Hover the cursor on cell D36 to view the DiscoverSim Output information.
- Select DiscoverSim > Run Simulation:
- Click Report Options/Sensitivity Analysis. Check Sensitivity Charts and Correlation Coefficients. Select Seed Value and enter “123” as shown, in order to replicate the simulation results given below.
- The DiscoverSim Output Report shows a histogram, descriptive statistics and process capability indices:
- Click on the Sensitivity Correlations sheet:
- In order to perform parameter optimization, we will need to add Input Controls in cells I30 to I33. Select Sheet1. The Input Control parameters (min/max) are also shown below:
- Click on cell I30. Select Control:
- Click Input Name cell reference and specify cell M30 containing the input control name “Spring_Constant_Control”.
- Click the Min value cell reference and specify cell N30 containing the minimum optimization boundary value = 30.
- Click the Max value cell reference and specify cell O30 containing the maximum optimization boundary value = 70:
- Click OK. Hover the cursor on cell I30 to view the comment displaying the input control settings:
- Click on cell I30. Click the DiscoverSim Copy Cell menu button (Do not use Excel’s Copy – it will not work!).
- Select cells I31:I33. Click the DiscoverSim Paste Cell menu button (Do not use Excel’s Paste – it will not work!).
- Review the Input Control comments in cells I31 to I33.
- The completed model is shown below:
- Now we are ready to perform the optimization. Select DiscoverSim > Run Optimization:
- Select “Maximize” for Optimization Goal, “Weighted Sum” for Multiple Output Metric and “Cpm” for Statistic. Select Seed Value and enter “123”, in order to replicate the optimization results given below. All other settings will be default as shown:
- Click Run. This optimization will take approximately 2 minutes to complete.
- The final optimal parameter values are given as:
- You are prompted to paste the optimal values into the spreadsheet:
- Select Run Simulation. Click Report Options/Sensitivity Analysis. Check Sensitivity Charts and Correlation Coefficients. Select Seed Value and enter “123”, in order to replicate the simulation results given below Click Run.
- The resulting simulation report confirms the predicted process improvement:
- We will continue our improvement efforts, now focusing on reducing the variation of the inputs. Click on the Sensitivity Correlations (2) sheet:
- Select Sheet1, and change the standard deviation for Pull Distance from 0.1 to 0.01 (K31), and standard deviation for Mass from .01 to .001 (K32) as shown.
- Select Run Simulation. Click Run. The resulting simulation report shows the further process improvement:
From the histogram and capability report we see that the catapult firing distance is not capable of meeting the specification requirements. The variation is unacceptably large. Approximately 36% of the shots fired would fail, so we must improve the process to reduce the variation.
Note: The output distribution is not normal (skewed left), even though the inputs are normal. This is due to the non-linear transfer function.
Launch Angle is the dominant input factor affecting Distance, followed by Pull Distance. At this point we could put procedures in place to reduce the standard deviation of these input factors, but before we do that we will run optimization to determine if the process can be made robust to the input variation.
These controls will vary the nominal mean values for Spring Constant, Pull Distance and Launch Angle. Mass is kept fixed at 0.5. The standard deviation values will also be kept fixed for this optimization.
Maximize Cpm is used here because it incorporates a penalty for mean deviation from target. We want the distance mean to be on target with minimal variation.
We will use the Hybrid optimization method which requires more time to compute, but is very powerful to solve complex optimization problems.
The predicted Cpm value is 0.49, which is an improvement over the baseline value of 0.3.
Click Yes. This replaces the nominal Input Control values to the optimum values.
Note that the expected values displayed in the Input Distributions are not automatically updated, but the new referenced Mean values will be used in simulation. To manually update the expected values and chart comments, click on each input cell, select Input Distribution and click OK.
The standard deviation has been reduced from 2.75 to 1.74. The % Total (out-of-spec) has been decreased from 36.0% to 15.3%. This was achieved simply by shifting the mean input values so that the catapult transmitted variation is reduced. The key variable here is the launch angle mean shifting from 30 degrees to 45 degrees as illustrated in the simplified diagram below:
Note: the illustrated improvement here is more dramatic than what was actually realized in the optimization due to the influence of the other input variables.
Pull Distance and Mass are now the dominant input factors affecting Distance. Note: Launch Angle now has the least influence due to the minimized transmitted variation at 45 degrees.
Of course it is one thing to type in new standard deviation values to see what the results will be in simulation, but it is another to implement such a change. This requires updates to standard operating procedures, training of operators, tightening of component tolerances and likely increased cost. Robust parameter optimization, as demonstrated above, is easy and cheap. Reducing the variation of inputs can be difficult and expensive.
The standard deviation has been further reduced from 1.74 to 0.47. The % Total (out-of-spec) has decreased from 15.3% to 0.2%
Note: The distribution is not normal (skewed left) due to the non-linearity in the transfer function.
This is not yet a Six Sigma capable process, but it has been dramatically improved. Additional process capability improvement could be realized by optimizing %Ppk (Percentile based Ppk is best for non-normal distributions) rather than Cpm. This would shift the mean to the right of the target, to compensate for the skew left distribution, but the trade-off of mean being on target versus reduction in defects below the lower specification would need to be considered.
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