Introduction
A recurring challenge of organisations attempting to implement Design for Six Sigma (DFSS) is to develop useful transfer functions (models) relating higher level outputs to lower level inputs. Building these functions can be incredibly complex tasks. Some are built, based upon first principles of science and engineering, and others are built using mathematical approximations. In this paper, we will see how a designed experiment approach combined with simulation can be used to efficiently provide insights into the engineering design of a classic device, the trebuchet, and for a more modern product.
The Trebuchet Example
In the 21st century, the super weapons of choice are nuclear. In the 12th century, mechanical siege weapons were greatly feared. Complex mechanical firing mechanisms emerged in this time frame, and one of the most intriguing was the trebuchet (pronounced treh buh shay). In checking the internet, you will find an extensive list of sites for trebuchet hobbyists who are engaged in building various sizes of these devices (from small ones for tossing tennis balls to ones large enough to toss wrecked cars).
Unlike the catapult (a popular device used by many in Six Sigma training), the trebuchet is highly nonlinear. Attempts to fit the device with a simple linear or quadratic equation over a specific range of interest can provide very poor predictive ability. We will use both statistical analysis and Monte Carlo simulation software tools to create a useful transfer function and assess potential variation in the trebuchet.
The statistical analysis software of choice for this activity is JMP software (version 5.1). JMP, a product of SAS Institute, is specifically geared toward the scientist or engineer who wants to perform a multitude of statistical tasks, including the setup and analysis of powerful designed experiments. Once we have generated the mathematical approximation in JMP, we will enter the equation into Microsoft Excel and use Crystal Ball simulation software to perform Monte Carlo analysis of the equation.
Wintreb drove the engineering nerd within us to build our own real life table-top trebuchet. One of our early engineering units is displayed in Figure 1.
Figure 1: The Trebuchet
Monte Carlo simulation is a random sampling method used to test the effects of variation or uncertainty. Credit for inventing the Monte Carlo method in 1946 goes to Stanislaw Ulam, a mathematician who worked for John von Neumann on the Manhattan Project during World War II. Ulam is primarily known for designing the hydrogen bomb with Edward Teller in 1951.
Creating the Experimental Design
For the designed experiment for the trebuchet, we looked to a computer simulation program, Wintreb for inspiration. Developed by the US Army (West Point), Wintreb (Figure 1) is simple way to view the effects of changing variables on the throwing distance achieved by the trebuchet. The simulation allows for as many as eight variables to be changed.
By clicking on the Go button, we can view either an animated fling of the projectile (Figure 2) or a trajectory plot that will provide maximum projectile height and projectile flight distance (Figure 3).
Figure 2 – An animated fling of the projectile
Figure 3: A trajectory plot that will provide maximum projectile height and projectile flight distance
Applying JMP to the Trebuchet Experimental Design
Our first step in creating a flexible trebuchet design was to collect experimental data from our actual table top trebuchet. Using the power of JMP, we utilised the custom design feature under the DOE option in the main menu to generate an efficient family of experimental trials. As evidenced in the results (a portion of which are shown in Table 1), JMP adroitly laid out a design with five levels for each of the two control factors. We chose this particular design because it allows for the estimation of factors effects beyond a quadratic model (cubic terms).
Table 1: JMP design table for Trebuchet
A big advantage of JMP is its ability to allow for design types with more than just two or three levels.
In the above table, the factor Rel bar allows us to vary the projectile release angle, and S Length allows us to vary the length of the projectile sling. The response dist is the actual flight distance for the projectile obtained for each run of the experiment. We then used JMP to create a standard regression output table (Table 2). The JMP command we used was analyze>fit model> with personality of standard least squares.
Table 2: Standard regression output table from JMP
Simulating the Transfer Function
We moved the transfer function from JMP to Microsoft Excel using the basic protocol for entering equations and tested it through Monte Carlo analysis with Crystal Ball. JMP’s strength lies in the analysis of data as opposed to simulation. Crystal Ball was specifically created to perform simulations to test the effects of variable inputs (or causal factors) on an output (a transfer function or other Excel formula).
Note that it is essential to verify that a transfer function actually provides useful predictions over the region of interest. We do this by performing predictions (with JMP in this case) and actually running these predicted best trials. For the following discussion, we will assume this step has been completed.
Figure 4: Microsoft Excel model of transform function
Click here for larger image
Suppose our goal is to hit a target value of 199 inches with minimal variation. To represent the variation, we use Crystal Ball to define an input probability distribution shape, mean and standard deviation for Rel Bar and for S Length (Figures 5 and 6). In this case, we initially assume Normal distributions for each of the inputs with a standard deviation set at 10% of the mean (based upon our actual prototype unit design and engineering experience we determined this was a reasonable conservative assumption. If we were to build additional units we could use actual data to determine the best distribution shape and variability).
Figure 5: Rel Bar distribution
Figure 6: S Length distribution
>We then define the distance (the transform function) as a Crystal Ball forecast and simulate the distance 1000 times (Figure 7). For each trial, Crystal Ball enters a random number into the two inputs based on their underlying Normal distributions. Once the new numbers are entered into the inputs, Excel recalculates and creates a new distance. After 1000 such trials, you have 1000 distance measurements that you can statistically analyse. Notice that the mean appears to be very near 198, but individual shots could vary from about 187 to 207 inches.
Figure 7. Crystal Ball forecast chart showing simulation results
Another highly useful simulation output is the sensitivity chart (Figure 8). This chart indicates which variables have the greatest impact on variation in the response distance. In our example, reducing variation in the factor Rel Bar would have the greatest impact, so this is where we would focus our design efforts.
Figure 8. Crystal Ball sensitivity chart
Thermoplastic Injection Molding of LEGO Toys
In this example, we will conduct a designed experiment so as to examine the manufacturing process for the LEGO building block (Figure 9). Suppose LEGO blocks are being injectionmolded in a single cavity tool, one part at a time. The response, or output, we are interested in is the outside length of the part. The only manufacturing factors we will examine are Fill Speed (two and five inches per second), and Hold Pressure (4600 and 6500 psi. plastic). The two and five inches per second for Fill Speed are just different set points, commonly referred to as levels.
Figure 9: LEGO building block
Obviously, the manufacturer would like to fill as fast as possible, for a shorter cycle time. We are studying pressure because thermoplastic material at high temperature and pressure is highly compressible, and so pressure can have great impact on several characteristics of the molded part, including dimensions. JMP was used to setup a relevant factorial design. Four shots (four parts in this case) were made at each run with relevant values recorded in the following JMP table (Table 3).
Table 3. Factorial design in JMP for LEGO building block
The statistical analysis from the JMP design is shown in Table 4:
Table 4: Parameter estimates for LEGO building block
Using JMP, we determined a part length of 57.3 (specification nominal) could be obtained if we set fill speed at 5 in/sec and hold pressure at 6150 psi.
As with the prior trebuchet example, we then use the length transfer function from JMP and enter this formula into Excel (Figure 10)
Figure 10. Excel model of length of LEGO brick
We assumed the input variations for both factors, fill speed (Figure 11) and hold pressure (Figure 12), were uniformly distributed (distribution type and anticipated variation were determined from historical data from a 3 month production period).
Figure 11. Distribution for Fill Speed
Figure 12. Distribution of hold pressure
We then tested the effects of variability on the transform function with Crystal Ball. After 1000 simulation trials (Figure 13), the predicted values from the Crystal Ball simulation suggest the simulated variability in overall LEGO block length will be from 57.29 to 57.31. Since the length tolerances were 57.1 to 57.5, the variation predicted through the simulation appears to be well within accepted tolerances.
Figure 13: Forecast of length of LEGO building block
Conclusion
Experimental design approaches can provide useful mathematical approximations of complex phenomena, including non-linear equations. With the statistical tool JMP, you can set up experiments and model even a highly non-linear mechanism. Given the transfer function from JMP, Crystal Ball in Excel then simulates known variations in factor inputs and so predicts output variation around the transform function. If the initial variation in the output is too great, Crystal Ball pinpoints which key inputs are driving the variation so that we can most efficiently reduce output variation to an acceptable level.
NOTE:
‘Crystal Ball’ is a registered trademark of Decisioneering, Inc. All Rights Reserved. Microsoft is a registered trademark of Microsoft Corporation in the U.S. and other countries.
SAS and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS Institute Inc.
LEGO is a trademark of the LEGO Group
