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The statistical analysis of the full factorial DOE results in the determination of the coefficients for a design space equation that relates all the control factors to the response factors. This equation includes interaction effects between control factors. This equation can then be used by designers to solve for the best overall system performance.

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Quick reference

DOE Functional Equation

One of the types of analysis that is created by a DOE study is the design space equation.  This equation describes mathematically how all the control factors impact the response variable – including interaction effects between factors.

When to use

The design space equation can be calculated once all the experimental runs are completed and the data recorded.  The design space equation is most commonly used when the goal of the DOE study includes the desire to optimize performance.

Instructions

The design space equation is calculated using the experimental run values.  The calculation consists of combining those values based upon the run configuration of the control factors.  The design space equation is relatively simple mathematically, requiring only simple algebra for the calculations.  However, if there are many terms it can become challenging to keep everything organized. 

The form of the equation is:

Where the β terms are coefficients that are associated with the appropriate control factors.

  1. In order to complete the calculations an experimental run configuration matrix that shows the configuration of each control factor for each run using the +1 and -1 convention is required.  This is normally completed using the Yates method.  An example of a very simple two factor  matrix is shown below.

  1. In addition, for each factor a formula is create that can be used to convert the actual values of the high and low levels into a value of +1 or -1.  For instance, in the example we will use, one factor is Cure Time.  The low level is 2 minutes and the high level is 10 minutes.  If we take the average of those two as the starting point and the range from the high value to the average as our normalizing factor we can create a normalizing equation.

           Cure Time Factor = (Cure Time – 6 minutes) / 4 minutes

          Where the average of 2 minutes and 10 minutes is 6 minutes and the range from 10 to 6 is 4.

          In our example that we use the second factor is pressure and the low level is 30 psi while the high level is 80 psi.  This gives us a normalizing equation for this factor of:

          Pressure Factor = (Pressure – 50psi) / 30 psi

  1. For our example, the experiments were run to determine the bond strength of a bonding process and the results are shown is matrix representing the control factor configuration.

  1. With this data, the first step in creating the design space equation is to determine the effect the factors have on the result.  For main effects (meaning only on factor) this is done by averaging all the test data values when the factor is high and subtracting the average of all the test data values when the factor is low.  So in our example:

          Cure time effect = (80 + 66) / 2 – (59 + 35) / 2 = 26 lbs

          Pressure effect = (80 + 59) / 2 – (66 + 35) / 2 = 19 lbs

  1. Interaction effects are calculated differently.  For each experimental run, the normalized factor level of +1 or -1 is multiplied together.  The interaction effect is calculated by taking the average of the data value when the product is positive (+1 * +1 or -1 * -1) and subtract the average of the data value when the product is negative (+1 * -1).

          Interaction effect for cure time and pressure = (80 + 35) / 2 – (66 + 59) / 2 = -5 lbs

  1. While these are the effects, they are not the β coefficients.  Recall that span represented by the low value to the high value using normalized factors is 2 (-1 to +1).  The coefficients represent the slope of the effect on the final value.  Slope is often described as the “rise over the run.”  Our effect value is the rise and the run value is span of the factors – which is 2.  So to get the actual coefficients, the effect for each factor and interaction must be divided by 2.

          βCure time = 26 / 2 = 13

          βPressure = 19 / 2 =9.5

          βCure time, Pressure = -5 / 2 = - 2.5 

  1. To create the design space equation, we also need the β0 constant.  This is found by averaging all the data points, which is called the grand mean. 

          Β0 = (80 + 66 + 59 + 35) / 4 = 60

  1. Our design space equation then becomes:

          Y = 60 + 13*Cure time Factor + 9.5*Pressure Factor  - 2.5*Cure Time Factor*Pressure Factor

  1. However, in the normal operation, settings are not conducted in factor units but rather in the real machine, equipment, or product units.  So the factor units must be replaced with factor unit equations discussed in step 2.

         Y + 60 + (13*(Cure time – 6) / 4) + (9.5 *(Pressure – 50) / 30) + (-2.5 *(Cure time – 6)/4 * (Pressure- 50)/30)

  1. Multiply and combine terms.  The final equation is units used by operators is:

          Y = 18.404 + (4.294 * Cure time) + ( .442 * Pressure) – (.021 * Cure time * Pressure)

Hints & tips

  • DOE statistical software will do this calculation for you.
  • When the study is large, the math is easy (it's just algebra) but the work can be time consuming to do the calculation on all the terms.
  • Be sure to include the scaling equations that convert the factor values from -1 or + 1 to real world settings.
  • Once the equation is known, you can set one or two control factors at values that are easy for operators establish and then vary the other factors to reach an optimal Y value performnce
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  • 00:04 Hello, I'm Ray Sheen and in this lesson,
  • 00:07 we'll explore the design space equation and how that's created using the DOE data.
  • 00:13 So first, let me explain what I mean by the design space equation.
  • 00:18 The design space equation shows how the control factors relate to each other and
  • 00:23 determine the value of the response factor.
  • 00:26 The full factorial DOE study will generate an equation that includes all
  • 00:30 the control factors and all the interaction effects between those factors.
  • 00:35 Here is the form of the design space equation, Y, which is the response
  • 00:39 variable equals beta 0, which is a number that shifts the value up and
  • 00:44 down to calibrate the answer, plus beta 1 times the first factor plus beta
  • 00:48 2 times the second factor plus beta 3 times the third control factor, and so on.
  • 00:54 In addition, we have beta 12, which is interaction effect, beta 13,
  • 00:59 another interaction, beta 14, and then beta 23 and 24.
  • 01:03 And we continue to add these until all the interaction effects have been included.
  • 01:08 The DOE analysis takes the data points and from those,
  • 01:11 it will derive the value of each of the beta terms.
  • 01:14 These terms calculate the slope of the line between the averages of the data
  • 01:18 points.
  • 01:19 One more item,
  • 01:20 recall that we had called the two levels of the control factors +1 and -1.
  • 01:26 Well, we'll have to eventually normalize our beta values to account for
  • 01:30 the real levels but more on that in a later slide.
  • 01:34 Let's go through the process for creating the design space equation.
  • 01:38 I'll illustrate this with a very simple two factor example.
  • 01:41 If you have many factors, you will use the same approach but
  • 01:44 it just requires more time and more math.
  • 01:48 At the time of planning the DOE study, you set the values that you would use for
  • 01:52 the +1 high value and the -1 low value for each factor.
  • 01:56 In this simple example, there would be just four experimental test runs.
  • 02:00 So we create the four samples.
  • 02:03 And when we run the four tests, we get four response values, one for
  • 02:07 each test configuration.
  • 02:10 Those results are used to calculate the beta factors in the formula.
  • 02:14 For the magnitude of the control factor effect, average the values
  • 02:19 when the factor is high and average the values when the factor is low.
  • 02:24 Then subtract the low average from the high average.
  • 02:28 For interaction effects, multiply the factor values for each configuration of
  • 02:32 those control factors, then average those values that have a positive product.
  • 02:37 And subtract the average of those values with a negative product.
  • 02:41 You're probably already confused so let's use an example to illustrate.
  • 02:46 So in our example, we want to find the design space equation for
  • 02:49 understanding the bond strength when gluing two materials together.
  • 02:53 The goal is to have a bond strength that is greater than 60 lbs.
  • 02:57 The study analyzes two control factors, which are pressure
  • 03:01 holding the two items together and the duration of the cure time.
  • 03:05 During the design phase of the study, the cure time levels were set at 2 minutes and
  • 03:09 10 minutes.
  • 03:11 And the pressure levels were set at 20 pounds per square inch and
  • 03:14 80 pounds per square inch.
  • 03:16 So our basic equation is Y or pull strength equals some constant beta
  • 03:22 0 plus the beta factor for cure time times the cure time.
  • 03:27 Then add the beta factor for pressure times the pressure.
  • 03:31 And finally, add the beta factor for the interaction between cure time and
  • 03:35 pressure times the cure time and times the pressure.
  • 03:40 Let's look at how those beta factors will be calculated.
  • 03:43 The cure time effect is the average of the high cure time values,
  • 03:47 which is 80 plus 66 divided by 2 to get the average minus the average
  • 03:53 of the low cure time values, which is 59 plus 35 divided by 2 for the average.
  • 03:59 The result of that calculation is 26 pounds.
  • 04:02 For the pressure effect, you need to determine the average of the high pressure
  • 04:07 values, which is 80 plus 59 divided by 2 to get the average minus the average of
  • 04:13 the low pressure values, which is 66 plus 35 divided by 2 for the average.
  • 04:18 The result of that calculation is 19 pounds.
  • 04:22 Finally, for the interaction effect, we determine the product of the + and- 1s.
  • 04:26 We first determine which runs have a positive interaction product.
  • 04:32 Those runs are the runs where both factors are both plus 1 or minus 1.
  • 04:37 If the factors are a +1 and a -1, then the product is negative.
  • 04:43 So let's take the average of the positive factor values.
  • 04:47 That means 80 plus 35 divided by 2 for the average minus the average of
  • 04:52 the negative factor values, which means 66 plus 59 divided by two for the average.
  • 04:58 The result of that calculation is -5 lbs.
  • 05:03 The largest value for the coefficient is 26 pounds.
  • 05:06 So cure time is the most significant effect.
  • 05:10 But we're not quite done yet with our calculations.
  • 05:13 Remember, the basic formula has a +1 or -1 for control factor values.
  • 05:18 To make the formula useful, we need to convert those initial results
  • 05:22 into units for the factors of minutes for time and psi for pressure.
  • 05:28 Let's start with the easy coefficient beta 0.
  • 05:31 To determine that value, we calculate the grand mean.
  • 05:35 That means we take the average of all the values, so add all 4 and divide by 4.
  • 05:41 The result of that beta 0 is 60 pounds.
  • 05:45 Next, to calculate the beta for
  • 05:46 each of the control factors, we have to make an adjustment.
  • 05:50 The control factors are the rise over the run.
  • 05:54 The rise is the difference between the average of the high control
  • 05:57 factor values and the average of the low control factor
  • 05:59 values that we calculated on the previous slide.
  • 06:02 The run is from -1 to +1 or a total of 2.
  • 06:08 Therefore, we take the magnitude of those effects we calculated on the previous
  • 06:12 slide and divide each of them by 2.
  • 06:15 That means that the beta for cure time is 26 divided by 2 or 13.
  • 06:21 The beta for pressure is 19 divided by 2 or 9.5.
  • 06:25 And the beta for the interaction is -5 divided by 2 or minus 2.5.
  • 06:32 But we're still not done.
  • 06:33 The control factors in the equation go from -1 to +1.
  • 06:38 But in reality, they go from 2 minutes to 10 minutes and 20 psi to 80 psi.
  • 06:44 So we need to scale these correctly.
  • 06:47 The factors we've just calculated are called the coded units.
  • 06:50 Now we need to calculate the uncoded units.
  • 06:54 The easiest way to do that is to add the high and low value for
  • 06:57 each factor divided by 2 to get the midpoint.
  • 07:00 The midpoint for time is 6 minutes and the midpoint for pressure is 50 psi.
  • 07:06 We then also determine the range to the midpoint
  • 07:10 by subtracting the midpoint from the high value.
  • 07:13 The range for the time is 4 minutes, and the range for the pressure is 30 psi.
  • 07:19 We also then determine the range to the midpoint
  • 07:22 by subtracting the midpoint from the high value.
  • 07:25 The range for time is 4 minutes and the range for pressure is 30 psi.
  • 07:30 So in order to make our time factor equal -1 and +1,
  • 07:35 we take the actual time minus the midpoint and
  • 07:38 divide by the range, which is time minus 6 divided by 4.
  • 07:43 At the high value of 10, this formula gives us a +1.
  • 07:46 At the low value of 2, it gives us a -1.
  • 07:50 This equation reduces to time divided by 4 minus 1.5.
  • 07:55 In the same way, the pressure formula becomes pressure minus 50 divided by 30.
  • 08:00 And that reduces to pressure over 30 minus 1.67.
  • 08:06 And now we can put all those together.
  • 08:09 The formula starts as Y equals 60 plus 13 times T for
  • 08:14 time over 4 minus 1.5 and
  • 08:17 then add 9.5 times pressure over 30 minus 1.67.
  • 08:23 And finally, minus 2.5 times time over 4 minus
  • 08:28 1.5 times pressure over 30 minus 1.67.
  • 08:33 I won't bore you with all the math to multiply things out and combine the terms.
  • 08:38 The final result is the equation pull strength response factor, or
  • 08:44 Y equals 18.404 plus 4.294 times time in minutes plus
  • 08:50 0.442 times pressure in psi minus .021 times time times pressure.
  • 08:58 To master the design space equation is simple algebra, but
  • 09:02 it can get complex due to many terms.
  • 09:05 You can do the calculations manually in Excel or with a calculator.
  • 09:09 Or you can let a statistical application like Minitab do the math for you

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