Path of Steepest Ascent/Descent

Locked lesson.

Upgrade

About this lesson

Some DOE analyses will indicate that the optimal performance of the system would occur when control factors are set beyond the bounds of the study. When this occurs, it is best to shift the study to the likely region of optimal performance and then determine the best control factor settings. Following the path of steepest ascent or descent will ensure that the new analysis is conducted in a region with maximum or minimum performance.

Quick reference

Path of Steepest Ascent/Descent

The path of steepest ascent or descent can shift the DOE analysis into a zone of control factor inputs that result in a more favourable response variable performance.  

When to use

When conducting a DOE for new design, there is often a performance target for the response variable.  If the DOE responses are not in the region of the targeted performance and the DOE analysis shows that the performance zone is not near a minimum or maximum, the path of steepest ascent or descent will shift the DOE into a better performance zone.

Instructions

Often a DOE analysis has response factor performance that does not meet the desired objective.  Even if the objective was to be “faster,” “cheaper,” or more “precise” than existing designs, the DOE may not be performed in a region of control factors that reaches the optimal point.  This is indicated in several ways.  If the factorial plots have lines that are converging, but have not crossed yet, the optimal points is outside the DOE zone of analysis.  Also, if the DOE had center points, the analysis will calculate a curvature value and a P value associated with that.  If the P value is below 0.05, you are not near an optimal point and you should conduct a path of steepest ascent of descent. If the P value is greater than 0.05, this technique will not provide any significant improvement in performance.

The purpose of this technique is to make sure that when changing the zone of control factors for a DOE, you are moving in a direction that will lead to improved performance.  If the goal of the response factor was to reach a minimum, you follow the path of steepest descent.  If it was to reach a maximum, follow the path of steepest ascent.   That path is defined by the relationship between the coefficients in the coded design space equation, 

To determine how much to change the control factors, you first must determine the magnitude of a “coded” unit for each control factor.  Recall that the Yates matrix uses +1 and -1 levels.  The difference between those two is 2 coded units.  The uncoded magnitude of a coded unit for a control factor is then the difference between the high value and the low value, divided by 2.   This is the uncoded magnitude of one coded unit.  Calculate this value for each control factor.

Now select one of the factors to be your baseline factor.  Normally this is the factor with the largest coefficient in the coded design space equation.  However, if you have a control factor that is very difficult to control or that only moves in discrete units, use that as the baseline factor.   The new control factor settings will be to increase or decrease the baseline factor by the uncoded value of one coded unit.  Then use the ratio of the coefficients in the coded design space equation of the all the other factors to the baseline factor to determine how much to move the other factors.  For instance, if the baseline factor coefficient was 5 and another control factor coefficient was 3, the ratio is 3/5 or 60%.  Change the second factor to increase or decrease it by 60% of the uncoded value of one coded unit for that factor.

With these increments determined, add (or subtract) them from the center point of your control factor in the original DOE.  Do an experiment with these new control factor settings. The response variable result should be either higher (path of ascent) or lower (path of descent) than the average of the DOE analysis.  Continue to increment all of the control factors in the same manner until the response variable no longer increases (path of ascent) or decreases (path of descent).   You are now near an optimal point on that path.  Conduct a DOE at that point setting your high and low factor values above and below the most recent values.  If the DOE analysis at that point still has a curvature P value that is below 0.05, continue along a new path that would be based upon the coefficients from the new coded design space equation generated by the DOE study that was just completed.   Once your response has met the desired level, or you no longer have a curvature P value below 0.05, you have reached the zone of peak performance.

ANOVA and DOE

The DOE methodology relies on ANOVA statistical analysis.  In this analysis there are several important statistical measures that can be used to evaluate the DOE mathematical model.  We have mentioned several times the P value and significance.  The ANOVA analysis will calculate the P value.  The implication of the P value is to determine if the item being analysed is varying due to random chance or if there is a real statistically significant relationship.  The P value is associated with each of the terms and their coefficient.  A low P value (below 0.05) indicates that the factor and coefficient have a statistically significant effect on the response variable.

The other statistical measure that comes from the ANOVA is degrees of freedom.  The degrees of freedom are a way of expressing how many factors the model can use to try to optimize the mathematical model to fit the real-world data.  Generally, the more degrees of freedom available for model fit, the better the model.

A DOE ANOVA will have one less degree of freedom than they have runs in the analysis.  So if there are 32 runs, there are 31 degrees of freedom.  However, that does not mean that all of those will help to reduce errors in the model.  Some of them are used in part of the basic model analysis.  Every factor in the design space equation will require one degree of freedom.  If there are no replicates or center points and no insignificant factors excluded, all the degrees of freedom are need just to create the design space equation.  If replicates or center points are added, the number of runs goes up and that increases the number of degrees of freedom.  However, if center points are included, one of the degrees of freedom will be used for the curvature analysis.  All of the unused degrees of freedom will benefit the accuracy of the model and reduce errors.  

Hints & tips

  • Be sure you are clear when working with uncoded and coded units.  The real world is uncoded, the DOE Yates matrix is coded.   All of these calculations should be with coded units.
  • Don’t forget to consider the sign when looking at the ratios between factors.  If the baseline factor is increasing and a coefficient ratio for another control factor is negative, that factor should decreases by the appropriate percentage of the uncoded unit.
  • Only the quantitative factors are changed when following a path of steepest ascent or descent.  Set the qualitative at the appropriate best case level.  However, one conducting the new DOE at the optimal point, include those factors.  
Login to download
  • 00:05 Hi, I'm Ray Sheen.
  • 00:06 Sometimes, the results of your DOE will indicate that the best performance
  • 00:11 will likely occur when the control factors are in a different region than what
  • 00:15 you use with your analysis.
  • 00:18 When that's the case, you should follow the path of steepest ascent or
  • 00:22 descent to improve the performance.
  • 00:24 So let's look at what we need to do to get the optimum level of performance.
  • 00:30 When the results of the DOE are not an optimal point, the Sath of steepest Ascent
  • 00:34 is a technique that guides the analysis to the optimal point.
  • 00:38 The way to know if you're close is to check the curvature P value.
  • 00:42 Now, you only get a curvature analysis if your DOE included center points.
  • 00:47 When the P value for curvature is greater than 0.05, then you are near
  • 00:52 an optimal point, and there is no benefit in trying the path of steepest ascent.
  • 00:57 But if the curvature P value is less than 0.05, then you should use this technique.
  • 01:03 This technique works with coded units, not the normal real world units.
  • 01:08 Coded units refer back to the+1 and -1 value for the Yates matrix.
  • 01:12 And the design space equation will be the one with coded coefficients,
  • 01:17 not real world coefficients.
  • 01:19 The way this works is that each of the control factor values is increased or
  • 01:24 decreased in a proportional manner, depending upon
  • 01:28 whether you want to increase the response variable or decrease it.
  • 01:32 Start by selecting one of the factors as your base factor.
  • 01:35 This is usually selected, based upon which factor has the largest coded coefficient.
  • 01:41 Although you may use a different method, such as the factor that is most difficult
  • 01:46 to set and control or factor that can only change in discrete levels.
  • 01:50 Also, just to be clear, we're talking about quantitative factors,
  • 01:54 not qualitative.
  • 01:56 So take the center point of the base factor and increment it by one coded unit
  • 02:01 in the direction you want to go, either ascending or descending.
  • 02:05 What do I mean by coded unit?
  • 02:07 Well, remember that when we set up the DOE, we were using +1 and
  • 02:12 -1 in the Yates table.
  • 02:14 The plus and minus 1 are the coded units.
  • 02:16 So to determine a coded unit value, take the difference between your high and
  • 02:22 low value for the base control factor.
  • 02:25 That represents a span of two coded units.
  • 02:28 One coded unit is half of that.
  • 02:31 Shift the baseline factor center point by one coded unit.
  • 02:36 Then shift the center point for the other factors by a fraction of a coded unit.
  • 02:41 That fraction is based upon the coefficients
  • 02:44 of the coded design space equation, not the real world equation.
  • 02:49 Don't worry, Minitab will give you those values.
  • 02:52 So if the baseline coefficient was 5, and another factors coefficient was 3, then
  • 02:57 change the second factor by three-fifths or 60% of a coded unit for that factor.
  • 03:04 Now, do a test run with all of these new control factors and
  • 03:07 determine the response.
  • 03:10 So what do we mean by a path?
  • 03:12 Continue to do this incrementing of the control factors by the same ratio of
  • 03:16 coded units until the response variable no longer increases or
  • 03:21 decreases, depending on which path you're going down.
  • 03:24 In this new region,
  • 03:25 conduct a new DOE to determine the design space equation in that zone.
  • 03:30 This path will move the model performance on an efficient
  • 03:33 line that is perpendicular to the contour lines of the Y response variable.
  • 03:39 At the point where the response is no longer changing, conduct the new DOE.
  • 03:43 You will probably need to do a new screening fractional factorial study.
  • 03:48 If that is the optimal point, you can continue with the refining and
  • 03:52 optimizing studies.
  • 03:54 However, if the curvature P value for this new DOE screening study is still low,
  • 04:00 and by that I mean below 0.05, then you're still not at an optimal point.
  • 04:05 But you may have gone as far in that direction as you should go.
  • 04:08 Now, start a new path based upon the coefficients from the new coded
  • 04:13 design space equation.
  • 04:15 Essentially, we're doing an old fat type of study, but
  • 04:18 we are changing multiple factors in a controlled ratio manner.
  • 04:23 Once you get to the point with a high curvature, you can complete the DOE and
  • 04:27 optimize for your performance.
  • 04:30 One more thing to address is the use of the ANOVA analysis in DOE.
  • 04:35 The heart of the DOE analysis is the ANOVA statistical analysis methodology.
  • 04:40 There are several statistical measures that come from the ANOVA analysis, and
  • 04:43 let's look at just a couple of those.
  • 04:45 The first thing to look at is the P value coefficient for each of the terms.
  • 04:49 The P value is a statistical term that looks at the statistical significance
  • 04:53 of a calculated relationship.
  • 04:55 If the value is low, meaning below 0.05,
  • 04:58 then that means the relationship is significant.
  • 05:02 If the value is high, relationship is due as much to random effects as to any
  • 05:06 significant statistical effect between the factors.
  • 05:10 So we want to be certain to include coefficients with a low P value.
  • 05:15 Now let's talk about the coefficients for a minute.
  • 05:17 Remember, we set up our DOE with +1 and
  • 05:20 -1units in your Yates model of sample configuration.
  • 05:25 The +1 and -1 are referred to as coded units and simplify the DOE design.
  • 05:30 But the real world uses real world units.
  • 05:33 In most cases, the ANOVA will decode the coefficients so
  • 05:36 that they can be used in the design space equation with normal or uncoded units.
  • 05:43 The other statistical measure to consider is the degrees of freedom.
  • 05:46 They're calculated by ANOVA.
  • 05:48 The degrees of freedom are a way of determining how good the fit
  • 05:52 of the model is to the actual real world data.
  • 05:55 The number of degrees of freedom in this study are the number of runs minus 1.
  • 06:00 So a 32 run study has 31 degrees of freedom.
  • 06:05 But some of those degrees of freedom are used up to account for
  • 06:08 the different factors and the effects in the model.
  • 06:12 Every term in your design space equation uses a degree of freedom.
  • 06:17 This is one of the reasons to get rid of the insignificant terms and
  • 06:20 allow additional degrees of freedom for improving the fit of the model.
  • 06:25 If you have included center points,
  • 06:27 then an additional degree of freedom is used for curvature analysis.
  • 06:31 So you can see increasing the number of runs, and therefore degrees of freedom.
  • 06:35 Or decreasing the number of terms which constrain a degree of freedom will leave
  • 06:40 more degrees available for improving the fit of the model.
  • 06:44 Following the path of steepest ascent or descent, and
  • 06:47 understanding the significance of the ANOVA statistics will
  • 06:51 help you to improve the application of your DOE.

Lesson notes are only available for subscribers.

PMI, PMP, CAPM and PMBOK are registered marks of the Project Management Institute, Inc.

Gift this course