# Case Study 3 – Six Sigma DMAIC Project Portfolio Selection

- Open the workbook
**DMAIC Project Selection - Optimization**. DiscoverSim Input Distributions will simulate the variability in the cost savings and probability of success for each project. We will specify cost savings as a Triangular Distribution for each project in cells**G14**to**G22**. The probability of success will be specified at cells**K14**to**K22**using the Bernoulli (Yes/No) Distribution. - DiscoverSim Input Controls (Discrete) will be used to “select” the project (0,1) and specified in cells
**O14**to**O22**. - Click on cell
**G14**to specify the “Cost Savings” Input Distribution for Project_A. Select**DiscoverSim**>**Input Distribution**: - Select
**Triangular Distribution**. - Click the input name cell reference and specify cell
**F14**containing the input**Name**“Project_A_CS” (denoting “Project A Cost Savings”). After specifying a cell reference, the dropdown symbol changes from to . - Click the
**Minimum**parameter cell reference and specify cell**C14**containing the minimum parameter value = 100 ($K). - Click the
**Mode**parameter cell reference and specify cell**D14**containing the mode (most likely) parameter value = 125 ($K). - Click the
**Maximum**parameter cell reference and specify cell**E14**containing the maximum parameter value = 150 ($K). - Click
**Update Chart**to view the**Triangular Distribution**as shown: - Click
**OK**. Hover the cursor on cell**G14**to view the DiscoverSim graphical comment showing the distribution and parameter values: - Click on cell
**G14**. Click the**DiscoverSim Copy Cell**menu button (**Do not use Excel’s Copy – it will not work!**). - Select cells
**G15:G22**. Now click the**DiscoverSim Paste Cell**menu button (**Do not use Excel’s Paste – it will not work!**). - Review the input comments in cells
**G15**to**G22**. - Click on cell
**K14**to specify the “Probability of Success” Input Distribution for Project_A. Select**DiscoverSim**>**Input Distribution**: - Select
**Discrete**and**Bernoulli Distribution**. Bernoulli is also sometimes referred to as the “Yes/No” distribution. - Click the input
**Name**cell reference and specify cell**J14**containing the input name “Project_A_Pr” (denoting “Project A Probability of Success”). - Click the
**Event Probability**parameter cell reference and specify cell**I14**containing the probability parameter value = 0.9. - Click
**Update Chart**to view the Bernoulli distribution as shown: - Click
**OK**. Hover the cursor on cell**K14**to view the DiscoverSim graphical comment showing the distribution and parameter values: - Click on cell
**K14**. Click the**DiscoverSim Copy Cell**menu button (**Do not use Excel’s Copy – it will not work!**). - Select cells
**K15:K22**. Now click the**DiscoverSim Paste Cell**menu button (**Do not use Excel’s Paste – it will not work!**). - Review the input comments in cells
**K15**to**K22**. - Now we will add Discrete Input Controls (also referred to as “Decision Variables”). These will be used to “select” the project (0 denotes project not selected, 1 denotes project selected). DiscoverSim will use stochastic global optimization to determine which projects to select and which not to select in order to maximize total cost savings. A constraint will also be added to ensure that the total number of resources is <= 20.
- Click on cell
**O14**. Select**Control**: - Click input
**Name**cell reference and specify cell**N14**containing the input control name “Project_A_Select”. - Select
**Discrete**(this input control setting will consider only integer values). Set the**Max**value to 1 as shown: - Click
**OK**. Hover the cursor on cell**O14**to view the comment displaying the Input Control settings: - Click on cell
**O14**. Click the**DiscoverSim Copy Cell**menu button (**Do not use Excel’s Copy – it will not work**!). - Select cells
**O15:O22**. Click the**DiscoverSim Paste Cell**menu button (**Do not use Excel’s Paste – it will not work!**). - Review the input control comments in cells
**O15**to**O22**. - Now we will add the constraint. Click on cell
**O26**. Select**Constraint**: - Enter the Constraint
**Name**as “Resource_Constraint”. Enter**O24**in the “Left Hand Side” (**LHS**). Select <= . Enter 20 in the “Right Hand Side” (**RHS**). - Click
**OK**. Review the comment at cell**O26**. - Now we will specify the output. Click on cell
**Q24**. The cell contains the Excel formula for the sum of expected savings as the sum of: Cost Savings * Probability of Success (if project is selected) or 0 (if project is not selected). - Select
**DiscoverSim**>**Output Response**: - Enter the output
**Name**as “Total_Savings”. Enter the Lower Specification Limit (**LSL**) as 1000. The**Weight**and**Output Goal**settings are used only for multiple response optimization, so do not need to be modified in this example. - Hover the cursor on cell
**Q24**to view the DiscoverSim Output information. - Now we are ready to perform the optimization to select projects. Select
**DiscoverSim**>**Run Optimization**: - Select “Maximize” for
**Optimization Goa**l and “Mean’ for**Statistic**. Select**Seed Value**and enter “123”, in order to replicate the optimization results given below. Change**Replication**s to 1000 to reduce the optimization time. All other settings will be default as shown: - Click
**Run**. This optimization will take approximately 5 to 6 minutes. - The final optimal parameter values are given as:
- The constraint equation evaluated at this optimum has been satisfied (amount violated = 0):
- You are prompted to paste the optimal values into the spreadsheet:
- Select
**DiscoverSim**>**Run Simulation**: - Select
**Seed Value**and enter “123” as shown, in order to replicate the simulation results given below. - The DiscoverSim Output Report displays a histogram, descriptive statistics and process capability indices:
- Now we will rerun the optimization to select projects, but this time, we will maximize the process capability index Ppl. (Ppl is used since we only have a lower specification limit). This will simultaneously maximize the mean and minimize the standard deviation:
- Reset the project selection by changing the Input Control values to 0:
- Select
**DiscoverSim**>**Run Optimization** - Select “Maximize” for
**Optimization Goal**, “Weighted Sum” for**Multiple Output Metric**and “PpL” 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: - The final optimal parameter values are given as:
- You are prompted to paste the optimal values into the spreadsheet:
- Select
**DiscoverSim**>**Run Simulation**: - Select
**Seed Value**and enter “123” as shown, in order to replicate the simulation results given below. - The DiscoverSim Output Report displays a histogram, descriptive statistics and process capability indices:
- Clearly, Project E is the key project to consider in this analysis. The impact of this project can be confirmed with a sensitivity analysis.
- Set all of the Input Control values to 1 as shown:
- Select
**DiscoverSim**>**Run Simulation**: - Click
**Report Options/Sensitivity Analysis**. Check**Sensitivity Regression Analysis**,**Sensitivity Charts - Correlation Coefficients**and**Regression Coefficients**. Select**Seed Value**and enter “123” as shown, in order to replicate the simulation results given below. - Click on the
**Sensitivity Correlations**sheet: - Click on the
**Sensitivity Regression**sheet to view the R-Square percent contribution to variation:

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Note: Cell **O24** contains the Excel formula: =SUMPRODUCT(O14:O22,M14:M22). This sums the product of resources*selected(0,1), in other words the sum of selected project resources.

The cell display is “TRUE” since the initial value for
**O24** is zero which obviously satisfies the constraint.

Click **OK**.

Note: Maximize Mean may not produce the lowest possible standard deviation, so we will consider Maximize Ppl later.

We will use the **Hybrid** optimization method which requires more time to compute, but is very powerful to solve complex optimization problems.

See Appendix: Optimization Methods for more details.

Projects E, F, H and I have been selected giving a predicted mean cost savings of 2004 ($K). Intuitively, these would be the projects one would select given that they have the highest expected values per resource. However, a simple “pick the winner” approach fails to take into account the variability and likelihood of meeting the minimum total cost saving requirement of 1000 $K.

Project E requires 10 resources, F requires 2, H requires 5 and I requires 3 for a total of 20, satisfying our constraint that the total resources be less than or equal to 20.

**Tip**: If the constraint condition shows an unacceptable violation amount, the constraint should be edited, increase penalty from the default of 100 and rerun the optimization.

Click **Yes**. This sets the Input Control values to the optimum values as shown:

Click **Run**.

From the histogram and capability report we see that the expected mean total savings are good (2011 $K), but the standard deviation is very large (1080 $K). Looking at the Actual Performance report we see that there is an 18.9% chance of failing to meet the management requirement of 1000 $K.

Ppl = (Mean – LSL)/(3*StDev).

An equivalent alternative would be to minimize Calculated DPM.

**Note:** A nonparametric alternative statistic would be %Ppl (percentile Ppl):

%Ppl = (Median – LSL)/(Median - 0.135 Percentile).

This is robust to the assumption of normality. Another robust statistic would be Actual DPM. With either of these we would recommend increasing the number of replications to 10,000 to improve the precision of the statistic, but this would require significantly more computation time for optimization.

Click **Run**. This optimization will take approximately 5 to 6 minutes.

Projects A, C, F, G, H, and I have been selected giving a predicted Ppl of 0.62.

Note: Project E, previously selected to maximize the mean, was not selected when the goal was to maximize the process capability, Ppl.

The constraint requirement that the total number of resources be <= 20 is satisfied with this selection.

Click **Yes**. This sets the Input Control values to the optimum values as shown:

Click **Run**.

Note that the mean total savings have been reduced somewhat from 2011 to 1925 $K, however, the standard deviation has been reduced from 1080 to 484 $K and the chance of failing to meet the management requirement of 1000 $K has been reduced from 18.9% to 4.0% (actual performance).

Considering the two selection scenarios, one can present the risks to management and make the project selection decision to go either way: take the risk and maximize the mean with potential larger savings, or choose the safer projects that result in less variation and significantly improve the chance of meeting a minimum savings of 1 million.

Click **Run**

The sensitivity charts confirm that Project E is the dominant “X” factor. This is a high impact project, but is also a difficult project with a probability of success of only 0.5 and a large variation (min = 2000, max = 2500). If possible, this project might be split into smaller, lower risk (higher probability of success) projects with less variation.

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