Desirability Function Example Desirability Functions Trade-off problems between various responses can be solved by using desirability functions. Desirability functions are a mathematical means of conducting trade-offs between two or more responses from a designed experiment. Fortunately software exists to help with the various desirability function calculations and graphs. DOE Wisdom is one such software package. When you are defining a response, DOE Wisdom will ask you to choose one of the following desirability functions: Increasing A higher response value is desirable. Decreasing A lower response value is desirable. Tent A nominal response value is desirable. The steps required for desirability functions are as follows: 1. Plan the experiment in the normal manner. 2. For each response, decide if the desired response is increasing, decreasing, or a tent. If increasing, determine the minimum acceptable value and maximum value beyond which no further improvement would be obtained. If decreasing, determine the minimum value below which no further improvement would be obtained and the maximum acceptable value. If a tent, determine the nominal, minimum, and maximum desired values. 3. For each response, determine the importance weight. This is a value between 1 and 10. The higher the number, the more important that particular response is. 4. Conduct the experiment. 5. Analyze the data using: Where: w 1, w 2,... w i are the weights specified for each response d n is the response converted to desirability units for each run. 6. Determine the settings for factors which maximize the desirability. The desirability
function ranges between zero and one. A desirability of zero represents a response value that is unacceptable. A desirability of one represents an optimal response value. For optimal results, the D(composite) value should be as close to 1 as possible. Desirability Function Example So far we have discussed some steps and calculations needed for conducting experiments using desirability functions. Now let s walk through an example that will help clarify this technique. Background Although the data has been modified slightly (upon the request of the customer), this example is based on an experiment conducted by Marcus Pozzetta of Caroba Plastics. Caroba Plastics is a custom injection molder in Englewood, CO. They provide molding services for customers in medical, high-tech, and consumer markets including three components of an oral syringe. The focus of this study was a syringe barrel for an oral medicine application. The syringe barrel is injection molded out of Amoco Homopolymer Polypropylene 7234. The amount of bow in the syringe is an area of concern. It must be within tolerance to ensure repeatability during the printing of the graduation scale on the syringe barrel. Bowing of a long cylindrical, injection molded part can be difficult to control due to the position and size of gate location. Improper position or size can cause an unbalanced material flow which can result in nonuniform wall stock. The lack of uniform wall stock can cause shrink differentials from one side to the other. This results in bowing. The mold for the syringe barrel is a one cavity mold. The molding machine used for the designed experiment was a 1996 Nissei 120 ton injection molding machine with a NC9000G Open Loop Controller. Experiment Objective After experiencing considerable difficulty with bowing, Caroba decided to perform a designed experiment on the syringe barrel. The objective of the experiment was to reduce the amount of bowing. Factor Settings and Design Matrix Factors studied in the experiment were:
FACTORS UNITS LOW HIGH Cure Time Seconds 20 30 Screw rpm rpm 50 70 First Stage Pressure psi 45 50 A Taguchi L8 design was used. Figure 3.1 shows the design matrix for this experiment. DESIGN MATRIX Run Cure Time Screw rpm 1 st Stage Pressure 1 20 50 45 2 20 50 50 3 20 70 45 4 20 70 50 5 30 50 45 6 30 50 50 7 30 70 45 8 30 70 50 Figure 3.1 Barrel temperature and melt temperatures for the molding machine were set to the material manufacturer s recommended settings. Responses The responses studied in the experiment were: Bow Wall Stock For this experiment the desirability function for the responses were set as follows: Bow Desirability = Decreasing Weight = 10 Minimum = 0.022 Maximum = 0.028
Stock Desirability = Tent Weight = 5 Minimum = 0.001 Target = 0.003 Maximum = 0.009 Data Five shots were conducted for each run of the orthogonal array but only the average is shown. The mold was a single cavity. The following data was obtained: Run Cure Screw 1 st Stage Bow Stock Time rpm Pressure 1 20 50 45 0.031 0.005 2 20 50 50 0.022 0.006 3 20 70 45 0.032 0.004 4 20 70 50 0.023 0.004 5 30 50 45 0.030 0.006 6 30 50 50 0.040 0.006 7 30 70 45 0.042 0.004 8 30 70 50 0.043 0.003 Main Effects Plots The statistical analysis for bow showed Cure Time and the Cure Time First Stage Pressure interaction to be statistically significant. The Main Effects Plot for bow is shown in Figure 3.2.
Figure 3.2 Notice that the big hitters for bow are: Cure Time The interaction between Cure Time and First Stage Pressure The statistical analysis for stock showed Screw rpm to be statistically significant. The Main Effects Plot for stock is shown in Figure 3.3. Figure 3.3
The Main Effects Plot graphically shows what the statistical analysis indicated. Notice that the big hitter for Stock is: Screw rpm Software In this example, we have an interaction that is important. This can make things really confusing if you are trying to only use a Main Effects Plot to determine where to set your factors. It gets even more complicated when you are trying to optimize more than one response. Here is the beauty of having outstanding DOE software. DOE Wisdom supports a hit a target feature. The hit a target feature allows the user to enter the target they are trying to hit and the software will automatically give the factor settings necessary to hit that target. We left all the factors and interactions in our prediction equation and used this feature. Since we defined a desirability function for each response, we will ask the software to maximize our D(composite) number. Figure 3.4 shows the results: Figure 3.4 In this case, we want to minimize the amount of bow and target the stock at 0.003. Using the hit the target feature with the Desirability Function within DOE Wisdom, we found that we could achieve a bow of 0.022 and a stock of 0.0052 by setting the factors as follows: Cure Time 20 Screw rpm 58 1 st Stage Pressure 50 These settings predict that we would achieve excellent Bow and acceptable Stock values.
Response Surface Graph Graphically, we can see this on the 3-D Response Surface Plot shown in Figure 3.5. Figure 3.5 Response Surface Plots provide the ability to graphically see, on a three dimensional plot, those factor settings that give a specific response. Contour Plot Now that the software has predicted where we need to set our factors to minimize bow and to obtain acceptable stock values, let s take a look at the Contour Plot. Figure 3.6 shows the Contour Plot for bow and Figure 3.7 shows the Contour Plot for stock.
Figure 3.6 Figure 3.7
Notice that the Contour Plot in Figure 3.7 would indicate that to minimize stock, you would want to set Cure Time to 30 and Screw rpm to 70, yet our software indicated we should set Cure Time to 20 and Screw rpm to 58 based upon trying to optimize TWO responses. Remember when we defined our responses? At that time we said that the Bow was more important and had a weight of 10; whereas, the Stock was less important and had a weight of only 5. This is where the desirability function is helpful. The software was able to optimize BOTH responses given the different weights we assigned to each response. Experiment Conclusion Before the designed experiment was conducted, it was believed that by simply increasing cure time and using the mold as a shrink fixture the bowing could be reduced. The amount of bow experienced before the designed experiment was ranging between 0.030" to 0.045". After some analysis by the printer, it was decided that bow less than 0.028" would produce acceptable parts. A bow of less than 0.022" would be ideal. As anticipated, the results indicated that cure time was the factor with the greatest influence over bow. The Pareto Chart in Figure 3.8 graphically demonstrates this. Figure 3.8. This, however, was the only thing our initial beliefs and the results had in common. Lower bow figures were actually found when the cure time was set at the low level
of 20 and not the high level of 30. The beauty of experimental design is that it can predict results of factor settings not captured in the initial eight runs. DOE Wisdom s target feature uses the prediction equation to determine the best factor settings to minimize bow and to generate parts with acceptable stock values. The following settings were determined to be optimal: Cure Time 20 Screw rpm 58 1 st Stage Pressure 50 The Contour Plots agreed with these settings. Although these exact settings were never actually tested during the designed experiment, the software was able to predict that by setting the factors to these levels, a bow of approximately 0.022" could be achieved. Confirmation Runs It is very important to run confirmation runs before reaching conclusions about the validity of the experiment. We recommend that 25 or more confirmation runs are done. For this experiment, the process was set to the above values. In order to ensure process stability, the molding machine was allowed to run for 30 minutes before samples were taken for measurement. After process stability was reached, 25 consecutive shots were measured to confirm the predictions from the experiment. All parts were acceptable for Bow and Stock! Desirability Function Summary Desirability functions are excellent to use when developing products in which many properties must be balanced. Although desirability functions have been used for many years, the implementation was extremely tedious to do by hand. Fortunately, easy-to-use software is now available so that injection molders may take full advantage of this valuable tool.