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Box-Behnken How-To

This walkthrough builds the three factor Box-Behnken design and shows what makes it different from a central composite: no run sets every factor to an extreme at once.

Nothing here needs data. Every number below is determined by the design.

Steps

  1. Start the builder

    QXL DOE New > Create Design. Choose Box-Behnken and press Next.

  2. Choose the design

    The first page is a list of the seven designs that exist, each written as its factor count and its run count. Choose 3 Factors, 15 runs and press Next.

  3. Name the factors

    On Enter Factor Names and Coding:

    Factor Low High
    Temp 180 220
    Press 40 60
    Time 10 20

    There is no Categorical column on this page for this design. Every factor in a Box-Behnken design is quantitative, because the design works by holding factors at a middle setting and a categorical factor has no middle. Press Next.

  4. Define the output

    Name one output, Yield, and press Next.

  5. Leave the centre points alone

    The Center points page shows Extra number of center points: at 0. The design already carries three, and this box adds to them rather than replacing them. Below it the two totals read Total number of center points: 3 and Total number of runs: 15. Press Finish.

What you get

Fifteen runs: twelve design points and three centre points. In coded units:

Press Copy for Excel and paste the block into an empty part of the worksheet, beside your design sheet, to compare it row by row rather than by eye.

Run Temp Press Time
1 -1 -1 0
2 -1 +1 0
3 +1 -1 0
4 +1 +1 0
5 -1 0 -1
6 -1 0 +1
7 +1 0 -1
8 +1 0 +1
9 0 -1 -1
10 0 -1 +1
11 0 +1 -1
12 0 +1 +1
13 0 0 0
14 0 0 0
15 0 0 0

Read the structure off the table

The twelve design points come in three blocks of four, and each block moves one pair of factors while holding the third at its middle:

Runs Moving Held at 0
1 to 4 Temp and Press Time
5 to 8 Temp and Time Press
9 to 12 Press and Time Temp

Each block is the four corners of a two-factor square, so the design is built entirely from the edges of the cube plus its centre.

No run sits at a corner of the cube. Look for a row with no zero in it and there is none. That is the property to take away: a Box-Behnken design never combines the extremes of every factor at once, so it suits a process where running everything at its limit together is expensive, unsafe or impossible.

Every column is balanced. Add any column and it comes to zero: four runs at -1, four at +1 and seven at 0.

Three levels per factor is what makes a curved model estimable. A two-level design can only fit a straight line in each factor; the middle setting here is what lets a squared term be estimated.

Things to try next

  • Type 6 into Extra number of center points: and watch the totals below the box. They become Total number of center points: 9 and Total number of runs: 21, because the box adds to the three the design already carries rather than setting the count.
  • Build the four factor design instead, which is 4 Factors, 27 runs. The same all-pairs rule produces six blocks of four rather than three, one per pair of factors.
  • Compare it with a central composite on the same three factors. The CCD reaches outside the cube along each axis at a distance of alpha; the Box-Behnken never leaves the edges.

See Also