Background

In this case study, we are a compressor manufacturer in the process of developing a new type of compressor. Our project team was charged with developing the design for the compressor using Design for Six Sigma (DFSS) tools and techniques. As we worked through the DMADV (Define, Measure, Analyze, Design, Verify) process, we used simulation and optimization to provide project justification, lend insight into the critical drivers of quality, and help create a cost effective design that meets customer requirements.

For DFSS, critical benefits of simulation and optimization are the ability to prototype new products or processes without an appreciable investment of time or money, minimal defects, and sales driven through improved customer satisfaction.

Define

The first step in our Six Sigma process was to estimate the financial impact of this project. We started by developing a simple spreadsheet model (DFSS Case Study Define.xls) in Microsoft^{® }Excel to take into account the greatest risks in developing this new product. We considered three major areas of concern: technical feasibility, manufacturing capability, and market uptake. Once the model was complete, we used Crystal Ball^{® }to define the variability or uncertainty around project elements (as probability distributions) and simulate potential outcomes (Figure 1).

Figure 1. Forecast Chart for Product NPV

Using this model, we predicted a 72% overall probability of success. While the success rate for the product was not exceptional, the potential rewards of this project were great, compared to the relatively minor risk. The percentile values of the NPV forecast shown in Figure 2 illustrate that the likelihood of losing more than $1.9MM was only 10% while the likelihood of creating an NPV of at least $49MM was more than 70%. Based on this information, the team and process leader decided to proceed with the project. We could have also developed models to forecast the projected timeline for the project and the required resources in the Define phase.

Figure 2. Percentile Values for the NPV Forecast

Measure

In the measure phase, we needed to establish valid and reliable metrics to help monitor our progress towards the project goals. Our primary goal was to establish the Critical-to-Quality (CTQ) characteristics. Based on our customer feedback, we know the top CTQ is flow rate through the compressor.

We built a model (DFSS Case Study Measure.xls) to calculate the flow rate from the input parameters. We applied Crystal Ball assumptions to each of the input parameters to describe their variability in previous systems. The data for these parameters were available from similar compressor systems our company had already designed. We then ran a simulation and viewed the Sensitivity Chart (Figure 3) for the Flow Rate, which we have defined as a forecast.

Figure 3. Sensitivity Analysis of Flow Rate

The variability in the piston stroke length was the primary contributor to the variability in the mass flow rate. In order to produce compressors with the least variability in the flow rate (as desired by our customers), we had to minimize the variability in the piston stroke length of the compressor.

Analyze

Next, in the Analyze phase, we needed to determine our design options for the piston assembly. We developed a model (DFSS Case Study Analyze.xls) that calculated the piston stroke length using the pre-defined piston dimensions and user-defined Decision Variables for the nominal values of arm length and crank length. Decision variables are model inputs over which we have control and wish to optimize. Both simulation and optimization play an important role in the design analysis, and we needed to run an optimization on these two elements to ensure a proper design.

Since the crank angle producing the maximum stroke length depends on the arm length and crank length, we wrote a simple macro that invokes Solver prior to each simulation to find this crank angle based on the two decision variables. The macro was called before each simulation through a setting defined in the Crystal Ball run preferences.

We ran OptQuest^{®}, the optimization tool in Crystal Ball Professional Edition, with the objective of minimizing the mean error from the target piston stroke length. We used OptQuest’s Solution Analysis tool (Figure 4) to analyze multiple designs that resulted in a nominal stroke length very near 30.80. The optimal stroke length had been determined earlier in the design of the compressor.

Figure 4. Analysis of Near-Optimal Solutions in OptQuest

OptQuest quickly identified five solutions that closely approximate the desired stroke length of 30.80, and these are the solutions we compared in the Design phase.

Design

In this phase of development, we designed the product so that it met customer requirements. With five possible piston assembly solutions from the Analyze phase, our goal was to choose the optimum quality design that was also cost efficient.

The variation (uncertainty) in this design model was represented by the error in each of the five piston dimensions used in the calculation of piston stroke length. We defined these errors as normal assumptions. We then modified our compressor model to include cost functions and levels of machined quality for each of the five dimensions. Because we have control over the desired level of quality, we defined these five quality factors as decision variables.

We also calculated total cost to produce one million in specification parts based on these quality costs and the simulated DPMO. As we increased quality (and decreased variability), our machining costs increased, so we needed our optimization to determine the lowest possible cost we could expect for the required level of piston quality.

We ran OptQuest on the model (DFSS Case Study Design.xls), with the objective of minimizing the total cost to produce 1,000,000 in specification parts. OptQuest minimized cost by adjusting the nominal crank and arm lengths as well as the quality level of each of the five parameters that determine piston stroke length. The results are displayed below in Figure 5.

Figure 5. Optimization to Minimize Total Cost

The Performance Graph, shown in the bottom of Figure 5, is useful for gauging OptQuest’s progress. As the graph leveled off and remained unchanged, we deduced that OptQuest had converged on an optimal or near-optimal solution. OptQuest selected the third design solution with a total cost of $17 million, which met our cost expectations.

Verify

But did this final design meet customer requirements? The Verify phase required us to check that the customer needs were satisfied. We copied the optimum design parameters back into our Excel model and ran the simulation for 10,000 trials. We found that with these design parameters and specification limits, only about 84% of our production would be within the specification limits.

Figure 6. Certainty of achieving specification limits

While the economic analysis indicated that this was the most cost-effective solution, we were charged with designing a piston assembly that provides 95% certainty of a stroke length between 30.70 and 30.90. To solve this problem, we returned to OptQuest one last time.

With this in mind, we ran the optimization again, this time with a requirement of 95% certainty that the stroke length was within the specification limits and the objective of minimizing total cost to produce 1,000,000 in specification parts. The resulting optimization is shown in Figure 7.

Figure 7. Minimization of Costs Subject to 95% in Specification Requirement

This new optimization, which selected design solution 1, was successful. While the costs rose to $24 million, we ensured that 95% of the piston assemblies would produce a maximum stroke length within specifications. Based on this design, our company created a compressor with few defects and our customers were satisfied.

Conclusion

For more information on how Crystal Ball is used for Six Sigma, visit the Crystal Ball Web site (www.crystalball.com/sixsigma).