# Case Study 5 – Robust New Product Design

**Spring Force Simulation and Optimization with DiscoverSim**

- Open the workbook
**Shutoff Valve Spring Force**. DiscoverSim Input Distributions will simulate the variability in feature tolerance. We will use the Uniform Distribution, since we do not have historical data to estimate a best fit distribution, and will assume an equal probability for a feature to have a minimum tolerance or maximum tolerance value. Input Distributions with Uniform Distributions (Random) will be specified in cells**I15**to**I22**. Nominal (midpoint) values are specified in cells**H15**to**H22**with initial values determined by engineering best estimate. Input Controls will be applied here for optimization. The sum of Nominal + Random are used to calculate the X1 to X8 variables and given in cells**J15**to**J22**. The output, Spring Rate, will be specified at cell**J27**using the formula given above. - The Input Distribution parameters are given in the following table:
- Click on cell
**I15**to specify the Input Distribution for X1, Lip Height. Select**DiscoverSim**>**Input Distribution**: - Select
**Uniform Distribution**. - Click input
**Name**cell reference and specify cell**L15**containing the input name “X1_Dist”. After specifying a cell reference, the dropdown symbol changes from to . - Click the
**Minimum**parameter cell reference and specify cell**M15**containing the minimum parameter value = -0.1. - Click the
**Maximum**parameter cell reference and specify cell**N15**containing the maximum parameter value = 0.1. - Click
**Update Chart**to view the**Uniform Distribution**as shown: - Click
**OK.**Hover the cursor on cell**I15**to view the DiscoverSim graphical comment showing the distribution and parameter values: - Click on cell
**I15.**Click the**DiscoverSim Copy Cell**menu button (**Do not use Excel’s Copy – it will not work!**). - Select cells
**I16:I22**. Click the**DiscoverSim Paste Cell**menu button (**Do not use Excel’s Paste – it will not work!**). - Review the input comments in cells
**I16**to**I22**: - Click on cell
**J27**. - Select
**DiscoverSim**>**Output Response**: - Enter the output
**Name**as “Spring_Force”. Enter the Lower Specification Limit (**LSL**) as 20,**Targe**t as 22, and Upper Specification Limit (**USL**) as 24. - Hover the cursor on cell
**J27**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: - Select
**Sheet1**, click on cell**M19**. Change the**Uniform Minimum**value from -0.5 to -0.25. Click on cell**N19**. Change the**Uniform Maximum**value from 0.5 to 0.25. - DiscoverSim references to cells are dynamic, so this change will take effect on the next simulation run, but the input comment will need to be refreshed manually. Click on cell
**I19**(Input X5_Dist). Select**Input Distribution**. Click**OK**. Review the revised input comment in cell**I19**: - Rerun the simulation without sensitivity charts:
- In order to run optimization, we will need to add Input Controls (also known as Decision Variables). Select
**Sheet1**. The Input Control parameters are given in the table: - Click on cell
**H15**. Select**Control**: - Click input
**Name**cell reference and specify cell**P15**containing the input control name “X1_Control”. - Click the
**Min**value cell reference and specify cell**Q15**containing the minimum optimization boundary value = 2. - Click the
**Max**value cell reference and specify cell**R15**containing the maximum optimization boundary value = 5: - Click
**OK**. Hover the cursor on cell**H15**to view the comment displaying the input control settings: - Click on cell
**H15**. Click the**DiscoverSim Copy Cell**menu button (**Do not use Excel’s Copy – it will not work!**). - Select cells
**H16:H22**. Click the**DiscoverSim Paste Cell**menu button (**Do not use Excel’s Paste – it will not work!**). - Review the input control comments in cells
**H16**to**H22**. - Now we will add the Constraints. Click on cell
**U16**. Select**Constraint**:

- Type
**X2_Control – X1_Control**in the “Left Hand Side”**(LHS**). Select “Greater Than >”. Enter 0.2 in the “Right Hand Side” (**RHS**). - Click
**OK**. Review the comment at cell**U16**. - Repeat steps 35 and 36 with constraints at cells
**U18**and**U22**as shown: - 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. Set**Replications**to 1000 to reduce the optimization time. All other settings will be default as shown: - Click
**Run**. This hybrid optimization will take approximately 2 minutes. - Scroll to the top of the optimization report window. The initial method in Hybrid, DIRECT global optimization, found a solution that resulted in a very good Cpm value of 3.05:
- The second method in Hybrid, Genetic Algorithm global optimization, used the above optimal values as a starting point and found a solution that resulted in an astonishing Cpm value of 7.38!
- The final method in Hybrid, Sequential Quadratic Programming local optimization, used the above optimal values as a starting point and improved the solution slightly to a Cpm value of 7.39.
- The constraint equations evaluated at this optimum are satisfied:
- You are prompted to paste the optimal values into the spreadsheet:
- Select
**Run Simulation**. Click**Run**. - The resulting simulation report confirms the predicted dramatic improvement:
- Recall the histogram for the initial simulation run:
- In summary, DiscoverSim was used to dramatically improve the Spring Force performance as follows:
- Mean centered from 21.6 to 22.0
- Standard Deviation reduced from 1.24 to 0.09, more than a ten-fold reduction!
- Ppk increased from 0.43 to 7.29, Cpm increased from 0.51 to 7.29!
- Actual % Total (out of spec) reduced from 12.5% to 0%!
- The benefits do not stop here. Since the design is now so robust, we can review the input tolerances to see if there is a cost saving opportunity by widening the feature tolerances, and re-running the simulation to study the impact.
- Finally the results predicted here should be validated with physical prototypes before proceeding to finalize the design parameters. Remember “All models are wrong, some are useful!” George Box.

Cell** I16**

Cell **I17**

Cell **I18**

Cell **I19**

Cell **I20**

Cell **I21**

Cell **I22**

Note: The cell contains the Excel formula for Spring Force: =_X7 + _R * (_X5 - _L). Excel range names _X7, _R, _X5 and _L are used rather than cell addresses to simplify interpretation.

Click **OK.**

Click** Run**.

From the histogram and capability report we see that the Spring Force is not capable of meeting the specification requirements. The process mean is off target and the variation due to feature tolerance is unacceptably large. Approximately 12.5% of the shutoff valves would fail, so we must improve this design to center the mean and reduce the variation.

X5 (Initial Compression) is the dominant input factor affecting Spring Force, followed by X4 (Spring Gap Depth). The tolerance on Initial Compression is +/- 0.5. Discussing this with our supplier it turns out that the tolerance for this spring feature can be tightened to +/- 0.25 without increasing the cost.

This Sensitivity Chart uses the Spearman Rank Correlation, and the results may be positive or negative. If you wish to view R-Squared percent contribution to variation, rerun the simulation with
**Sensitivity Regression Analysis** and **Sensitivity Charts**,
**Regression Coefficients** checked.

Tightening the tolerance of X5, Initial Compression resulted in an improvement with the Spring Force standard deviation reduced from 1.24 to 0.86, actual percent out of specification from 12.5 to 4.0% and Ppk increased from .43 to .62. However a Six Sigma design should have a minimum Ppk value of 1.5.

At this point one could use Excel’s Solver to further improve the process by finding the nominal values that center the mean, but that still would not achieve the desired quality level.

DiscoverSim’s Stochastic Global Optimization will not only find the optimum X settings that result in the best mean spring force value, it will also look for a solution that will reduce the standard deviation. Stochastic optimization looks for a minimum or maximum that is robust to variation in X, thus reducing the transmitted variation in Y. This is referred to as “Robust Parameter Design” in Design For Six Sigma (DFSS).

The controls will vary the nominal X values in **H15** to
**H22**. The nominal values are added to the random input distribution (**J15** to
**J22**) as discussed above in Step 1.

Constraints will also be added as shown so that there can be no overlap due to tolerance. For example if X2, Height, was at the low end of its tolerance and X1, Lip Height, was at the high end of its tolerance, it is possible that an overlap would leave no place for the spring to be seated.

The cell display is “TRUE” since the initial value for X2_Control (5) is greater than the initial value for X1_Control (3).

Maximize Cpm is used here, rather than Ppk, because it incorporates a penalty for mean deviation from target. We want the spring force mean to be on target with minimal variation. Note, however, that this solution may not produce the lowest possible standard deviation. We will use the
**Hybrid** optimization method which requires more time to compute, but is very powerful to solve complex optimization problems.

**Note**: The constraint cell values in Excel will all appear as TRUE at the optimum.

Click **Yes**. This replaces the nominal input control values to the optimum values.

The slight discrepancy in predicted Cpm (Optimization 7.39 versus Simulation 7.29 is due to the difference in number of replications for Optimization (1000) versus Simulation (10,000).

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