Design and Analyze

Multiple Response Surface Optimization

Video 10. Multiple Response Optimization Explained with Example using Minitab Response Surface Methodology RSM

Video 10 provides the analysis and explanation of the results for the multiple response surface optimization.

The design of multiple response surfaces follows the exact same procedure, except for adding multiple response variables, such as y1, y2, y3, y4, and so on. Often, multiple responses (dependent variables, or y-variables) are already there and easier to collect and more economical, considering the amount of information gained through the additional responses. Experimental conditions or experimental units are the most costly deal in any design of experiment. Once the experimental conditions are set, collecting multiple responses are justified rather than not doing it.

Example Problem

There are many situations in which contradictory targets are needed to be achieved, such as better quality with reduced cost. For example, let’s look at the following bicycle riding speed and the heart rate responses. As we try to increase the speed of riding while keeping the heart rate as low as possible. Assume that we are optimizing the speed and the heart rate with respect to the pedaling speed and the tire pressure. Therefore, two independent variables (factors or predictors) and two dependent variables (responses) are used in a study of bicycle riding efficiency optimization. The descriptions of the variables are provided below.

  1. Independent Variable 1, X1: Pedaling Speed in Revolution Per Minute, RPM
  2. Independent Variable 2, X2: Tire Pressure in Pounds Per Square Inch, PSI
  3. Dependent Variable 1, Y1: The Average Speed of the Bicycle in Miles per Hour, MPH
  4. Dependent Variable 2, Y2: The Heart Rate in Beat Per Minute, BPM

Bicycles with multiple gears are used in the study so that any pedaling speed could be selected without increasing the effort or for any desired effort levels.

Some initial research on the variables are provided below.

  1. Heart rate (BPM)
    1. increases with the pedaling speed.
    2. may increase if the tire pressure is too low or too high due to too much rolling resistance or too bumpy rides, respectively.
  2. The average bicycle speed (MPH)
    1. is related with the pedaling speed in RPM. The average cycling speed (MPH) is maximum for an optimum level of pedaling speed (RPM) (neither too low nor too high).
    2. is dependent on the tire pressure. The average speed is maximum for an optimum level of tire pressure, while too low or too high pressure will reduce the speed by increasing the rolling resistances or bumpy rides, respectively.

The response surface methodology design and collected data is provided in Table 9. Assume that all riding conditions, including the test roads, riding distance, fitness level, weight, bicycle, and the riding equipment, for all 13 test subjects are kept in very similar to reduce any bias in the study. All experimental units are assumed to be identical twins, but independent.

Table 9

Multiple Response Optimization for Bicycling Efficiency

Analysis Results Explained

The analysis output results are provided in Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, and Table 10.

RSM Regression Analysis Output Explained

In Figure 20 and Figure 21, the quadratic model fitted by the pedaling speed and the tire pressure as the predictor variables are observed to be significant with respect to both responses, including the heart rate and the average speed. Moreover, the model shows excellent r-square values. However, while the response surface model prediction for the heart rate does not show any lack-of-fit, the prediction of average speed shows some lack-of-fit. The average speed prediction model shows a moderate prediction r-square value, indicating that more investigation is necessary for the average speed prediction.

The relative effects are visualized using the Pareto Chart in Figure 22. Any parameter that crosses the dashed line is observed to be statistically significant. Only the quadratic terms of the pedaling speed and the tire pressure are large and significant in predicting the average speed (top graph in Figure 22). Although both linear and quadratic terms for the tire pressure are observed to be significant, a relatively larger effect is observed from the linear term of the pedaling speed in predicting the heart rate (bottom graph in Figure 22).

Figure 20. Multiple Response Surface Regression: Speed versus Pedaling and Pressure

Figure 21. Multiple Response Surface Regression: HR versus Pedaling, Pressure

Figure 22. Multiple Response Surface Comparison of Effects, Top (Average Speed), Bottom (Heart Rate)

Individual Responses and Contour Plots Explained

The individual response surfaces and contour plots for both responses, including heart rate and the average speed, are visualized in Figure 23. The average speed (top two graphs) is significantly affected by almost equality by the tire pressure and the pedaling RPM. However, the average speed is slightly more affected by the pedaling speed than the tire pressure (top left graph in Figure 23). While the heart rate stays stable with respect to the tire pressure, it increases with respect to the pedaling speed (bottom two graphs). No significant interactions are observed between the tire pressure and the pedaling speed with respect to either heart rate or average speed.

Figure 23. Multiple Response Surfaces and Contour Plots for Individual Responses of the Average Speed (top two graphs) and the Heart Rate (bottom tow graphs)

Overlaid Contour Plot Explained

Figure 24 shows the overlaid contour plot using the heart rate between 149 and 155 BPM and the speed between 18 and 20 miles per hour. These selections for the responses involve some trial and error procedure to find the best overlaid plot that could be useful to draw some reasonable conclusions on the multiple response surface optimization. For example, in Figure 24, the maximum of speed 20 miles per hour can be achieved when the heart rate response is observed between 149 and 155 beats per minute.

Figure 24. Multiple Response Surface Overlaid Contour Plot to Optimize both the Riding Speed and the Heart Rate Together

To achieve these optimum responses for both heart rate and average speed, the tire pressure must be set between 95 and 105 psi (approximated from the plot) and the pedaling speed must be set between 88 and 92 revolutions per minute. While the Figure 24 provides the visual look at the approximated optimization, the detailed optimization results can be found in Table 10.

The multiple trial and error run in optimizing both the heart rate and average speed are provided in Table 10. The optimization result shows that target heart rate could be close to 152 beats per minute to achieve the maximum average speed. This optimization is shown in Figure 25.

Table 10

Simultaneous Multiple Response Surface Optimization

Composite Desirability

The composite desireability measures the overall predictability for both responses by the predictor parameters. While minimizing the heart rate is desirable when maximizing the average speed, the composite desirability of the model is much lower (72%) for this ideal situation. The composite desirability is observed to be highest (98%) for the target heart rate 152 beats per minute if the average speed is set to maximize.

Figure 25. Simultaneous Multiple Response Surface Optimization

Figure 26 shows the final overlaid contour plot of pedaling speed (rpm) vs tire pressure (psi) in maximizing the speed while keeping the heart at a target level.

Figure 26. Multiple Response Surface Overlaid Contour Plot to Optimize both the Riding Speed and the Heart Rate Together