Response Surface (pt 2)¶

Setting the Optimization Criteria¶

The program allows you to set criteria for all variables, including factors and propagation of error (POE). (We will detail POE later.) The program restricts factor ranges to factorial levels (plus one to minus one in coded values) — the region for which this experimental design provides the most precise predictions. Response limits default to observed extremes. In this case, you should leave the settings for time, temperature, and catalyst factors alone, but you will need to make some changes to the response criteria.

Now you get to the crucial phase of numerical optimization: assigning “Optimization Parameters.” The program uses five possibilities as a “Goal” to construct desirability indices (di):

• Maximize,

• Minimize,

• Target->,

• In range,

• Equal to -> (factors only).

Desirabilities range from zero to one for any given response. The program combines individual desirabilities into a single number and then searches for the greatest overall desirability. A value of one represents the ideal case. A zero indicates that one or more responses fall outside desirable limits. The program uses an optimization method developed by Derringer and Suich, described by Myers, Montgomery and Anderson-Cook in Response Surface Methodology, 3rd edition, John Wiley and Sons, New York, 2009.

For this tutorial case study, assume you need to increase conversion. Click Conversion and set its Goal to maximize. As shown below, set Lower Limit to 80 (the lowest acceptable value), and Upper Limit to 100, the theoretical high.

Conversion criteria settings¶

You must provide both these thresholds so the desirability equation works properly. By default, thresholds will be set at the observed response range, in this case 51 to 97. By increasing the upper end for desirability to 100, we put in a ‘stretch’ for the maximization goal. Otherwise we may come up short of the potential optimum.

Now click the second response, Activity. Set its Goal to a target-> of 63. Enter Lower Limits and Upper Limits of 60 and 66, respectively. These limits indicate that it is most desirable to achieve the targeted value of 63, but values in the range of 60-66 are acceptable. Values outside that range are not acceptable.

Activity criteria settings¶

The above settings create the following desirability functions:

1. Conversion:

• if less than 80%, desirability (di) equals zero

• from 80 to 100%, di ramps up from zero to one

• if over 100%, di equals one

2. Activity:

• if less than 60, di equals zero

• from 60 to 63, di ramps up from zero to one

• from 63 to 66, di ramps back down to zero

• if greater than 66, di equals zero

Changing Desirability Weights and the (Relative) Importance of Variables¶

You can select additional parameters called “weights” for each response. Weights give added emphasis to upper or lower bounds or emphasize target values. With a weight of 1, di varies from 0 to 1 in linear fashion. Weights greater than 1 (maximum weight is 10) give more emphasis to goals. Weights less than 1 (minimum weight is 0.1) give less emphasis to goals.

Note

Weights can be quickly changed by clicking and dragging the handles (squares ▫) on desirability ramps. Try pulling the square on the left down and the square on the right up as shown below.

Weights change by grabbing handles with mouse¶

This might reflect a situation where your customer says they want the targeted value (63), but if it must be missed due to a trade-off necessary for other specifications, it would be better to err to the high side. Before moving on from here, reenter the Lower and Upper Weights at their default values of 1 and 1; respectively. This straightens them to their original ‘tent’ shape.

“Importance” is a tool for changing relative priorities to achieve goals you establish for some or all variables. If you want to emphasize one over the rest, set its importance higher. Stat-Ease 360 offers five levels of importance ranging from 1 plus (+) to 5 plus (+++++). For this study, leave the Importance field at +++, a medium setting. By leaving all importance criteria at their defaults, no goals are favored over others.

Now click the Options button to see what you can control for the numerical optimization.

Optimization Options dialog box¶

Click OK to close Optimization Options.

Running the optimization¶

Start the optimization by clicking the Solutions tab. It defaults to the Ramps view so you get a good visual on the best factor settings and the desirability of the predicted responses.

Numerical Optimization Ramps view for Solutions (Your results may differ)¶

The program randomly picks a set of conditions from which to start its search for desirable results – your results may differ. Multiple cycles improve the odds of finding multiple local optimums, some of which are higher in desirability than others. The program then sorts the results from most desirable to least. Due to random starting conditions, your results are likely to be slightly different from those in the report above.

The Solutions Toolbar provides three views of the same optimization. Click Report.

Report on numerical optimization¶

Now select the Bar Graph view from the Solutions Toolbar.

Solution to multiple response optimization — desirability bar graph¶

The bar graph shows how well each variable satisfies the criteria: values near one are good.

Optimization Graphs¶

Select the Graphs tab to view a contour graph of the overall desirability and all of your responses. On the Factors Tool palette, right-click C:Catalyst. Make it the X2 axis. Temperature then becomes a constant factor at 90 degrees (this level is picked automatically by the selected solution #1).

You can also use the droplist on the Graphs toolbar to look at bigger versions of each plot, such as the desirability plot below.

Desirability graph (after changing X2 axis to factor C)¶

The screen shot above is a graph displaying graduated colors – cool blue for lower desirability and warm yellow for higher.

The program sets a flag at the optimal point (as selected by the button bar at the top , number 1 is selected in the screenshot above). To view a response associated with the desirability, select the desired Response from its droplist. Take a look at the Conversion plot.

Conversion contour plot (with optimum flagged)

Note

Right-click over this graph and choose Graph Preferences. Then go to Surface Graphs and click Show 2D grid lines.

Show 2D grid lines option¶

Grid lines help locate the optimum, but for a more precise locator right-click the flag and Flag Size->Full to see the coordinates plus many more predicted outcome details. To get just what you want on the flag, right-click it again and select Edit Info.

Flag size toggled to see select detail¶

To look at the desirability surface in three dimensions, again select Desirability from the Graphs toolbar drop-down. Then, press 3D Surface. Next, right-click the graph and select Set rotation and change horizontal control to 170. Press your Tab key or click the graph. What a spectacular view!

3D desirability plot¶

Now you can see there’s a ridge where desirability can be maintained at a high level over a range of catalyst levels. In other words, the solution is relatively robust to factor C.

Note

Right-click over your graph to bring up Graph preferences. Via the Surface Graphs tab change the 3D graph shading option to Wire Frame.

3D graph shading changed to wire frame¶

One way or another, please show your colleagues what Stat-Ease 360 does for pointing out the most desirable process factor combinations. We’d like that! The best way to show what you’ve accomplished is not on paper, but rather by demonstrating it on your computer screen or by projecting your output to larger audiences. In this case, you’d best shift back to the default colors and other display schemes. Do this by pressing the Default button on Surface Graphs and any other Graph Preference screens you experimented on.

P.S. Stat-Ease 360 offers a very high Graph resolution option. Try this if you like, but you may find that the processing time taken to render this, particularly while rotating the 3D graph, can be a bit bothersome. This, of course, depends on the speed of your computer and the graphics card capability.

Now move on to graphical optimization—this may the best way to convey the outcome of an RSM experiment by displaying the “sweet spot” for process optimization.

Confirmation¶

After finding the optimum settings based on your RSM models, the next step is to confirm that they actually work. To do this, click the Confirmation node (left side of your screen).

Confirmation predictions set to solution #1¶

Look at the 95% prediction interval (“PI low” to “PI high”) for the Activity response. This tells you what to expect for an individual (n = 1) confirmation test on this product attribute. You might be surprised at the level of variability, but it will help you manage expectations. (Note: block effects, in this case day-by-day, cannot be accounted for in the prediction.)

Of course you would not convince many people by doing only one confirmation run. Doing several would be better. For example, let’s say that the experimenters do three confirmatory tests. Click the Confirmation Location #1 tab and next to Runs enter 3 then press tab. You will now have 3 rows under Response Data, for Activity enter 62, 63 and 64.

Entering confirmation run results¶

Notice that the prediction interval (PI) narrows as n increases. Does the Data Mean (63) fall within this range? If so, the model is confirmed. If not, it will turn red.

Confirmation results are not within the interval for activity¶

Save the Data to a File¶

Now that you’ve invested all this time into setting up the optimization for this design, it would be prudent to save your work. If you are not worn out yet, you will need this file in Part 3 of this series of tutorials.