You can create blocked or unblocked Box-Behnken designs. The illustration below shows a three-factor Box-Behnken design. Points on the diagram represent the experimental runs that are performed:

You may want to use Box-Behnken designs when performing non-sequential experiments. That is, you are only planning to perform the experiment once. These designs allow efficient estimation of the first- and second-order coefficients. Because Box-Behnken designs have fewer design points , they are less expensive to run than central composite designs with the same number of factors.

Box-Behnken designs can also prove useful if you know the safe operating zone for your process. Central composite designs usually have axial points outside the "cube" (unless you specify an athat is less than or equal to one). These points may not be in the region of interest, or may be impossible to run because they are beyond safe operating limits. Box-Behnken designs do not have axial points, thus, you can be sure that all design points fall within your safe operating zone. Box-Behnken designs also ensure that all factors are never set at their high levels simultaneously.

 

Run (DOE)

Each experimental condition or factor level combination at which responses are measured. Typically, each run corresponds to a row in the worksheet and results in one or more response measurements, or observations. For example, you conduct a full factorial design with two factors, each with two levels. Your experiment has four runs:

Run

Factor1

Factor2

Response

1 -1 -1 112 1 -1 123 -1 1 104 1 1 9

Note When conducting an experiment, the run order should be randomized.

Each run corresponds to a design point, and the entire set of runs is the design. Multiple executions of the same experimental conditions are considered separate runs and are called replicates.

Design points, response surface design

Include factorial or "cube" points, axial points (also called star points), and center points for the most commonly used response surface design — a central composite design. Points on the diagrams below represent the experimental runs for a central composite design with two factors:

The points in the cube portion of the design have factor levels, which are coded -1 and +1.

The points in the axial, or star, portion of the design are at (+0) (, 0) (0, ) (0, ).

Here, the "cube" and axial portions, along with the center point, are shown. The design center is at (0,0).

The effects that can be estimated depend on the point type:

   Cube points - allow for the estimation of linear and interaction effects, but not curvature. (These points are comparable to a the corner points of a 2K factorial design.)

   Center points - add center points to check for curvature, but not individual quadratic terms. (The point in the middle of the cube represents the center points for both the cube and the axial blocks.)

   Axial points - add axial points, in addition to center points, to estimate quadratic terms. (The points joined by dotted lines indicate points outside or on the surface of the cube.)

Block

A group of experimental runs conducted under relatively homogeneous conditions. Although every measurement should be taken under consistent experimental conditions (other than those that are being varied as part of the experiment), this is not always possible. Use blocks in experimental design and analysis to minimize bias and error variance due to nuisance factors. For example, you want to test the quality of a new printing press. However, press setup takes several hours and can only be done four times a day. Because the design of the experiment requires at least eight runs, you need at least two days to test the press. You should account for any differences in conditions between days by using "day" as a blocking variable. To distinguish

between any block effect (incidental differences between days) and effects due to the experimental factors (temperature, humidity, and press operator), you must account for the block (day) in the experimental design. You should randomize run order within blocks.

 

Replicates

Multiple experimental runs with the same factor settings (levels). Replicates are subject to the same sources of variability, independently of one another. You can replicate combinations of factor levels, groups of factor level combinations, or entire designs.

In experimental design, replicate measurements are taken from identical but different experimental runs. This is in contrast to repeats, which are simply repeated observations at the same settings. You can use replicates to estimate the variance (experimental error) caused by slightly different experimental conditions. The experimental error serves as a benchmark to determine whether observed differences in the data are statistically different. To make sure all the experimental variability is observed and quantified, replicates should be randomized to cover the entire range of experimental conditions. If the number of runs is too large to be completed under steady state conditions, you may block on replicates. Blocking allows you to estimate the block effects independently of the experimental error.

For example, if you have three factors with two levels each and you test all combinations of factor levels (full factorial design), one replicate of the entire design would consist of 8 runs (23). You can choose to run the design once or have multiple replicates.

Your experimental design includes the number of replicates you should run. Considerations for replicates:

    Screening designs to reduce a large set of factors usually don't use multiple replicates.

    If you are trying to create a prediction model, multiple replicates may increase the precision of your model.

    If you have more data, you may be able to detect smaller effects or have greater power to detect an effect of fixed size.

    Your resources may dictate the number of replicates you can run. For example, if your experiment is extremely costly, you may be able to run it only once.

 

Blocking a Box-Behnken Design main topic  

When the number of runs is too large to be completed under steady state conditions, you need to be concerned with the error that may be introduced into the experiment. Running an experiment in blocks allows you to separately and independently estimate the block effects (or different experimental conditions) from the factor effects. For example, blocks might be days, suppliers, batches of raw material, machine operators, or manufacturing shift.

For a Box-Behnken design, the number of ways to block a design depends on the number of factors. All of the blocked designs have orthogonal blocks. When you are creating a design, Minitab displays the appropriate choices. A design with:

    Three factors cannot be blocked

    Four factors can be run in three blocks

    Five, six, seven, or ten factors can be run in two blocks

    Nine factors can be run in five or ten blocks

If you add replicates to your design, you can also block on replicates. How this works depends on whether you have blocks in your design.

   If your design does not have blocks, Minitab places each set of replicates in separate blocks.

   If your design includes blocks, Minitab replicates the existing blocking scheme. The points in each existing block are replicated to form new blocks. The number of blocks in your design will equal the number of original blocks multiplied by the number of replicates. The number of runs in each block stays the same.

   If your design includes blocks but you do not block on replicates, Minitab replicates the points within each block. The total number of runs in the each block equals the number of original runs times the number of replicates. The total number of blocks in the design stays the same

SL NO

RUN Order

Coded Values

Blocks Natural Values

MRR TWRX1 X2 X3 I (A) ton (μs) toff(μs)

1 28 0 0 0 1 14 79 7 40.94631 0.132872 16 -1 -1 0 1 8 60 7 22.53602 0.073403 27 0 +1 +1 1 14 98 9 38.83285 0.086464 20 -1 0 -1 1 8 79 5 19.05476 0.046895 8 +1 0 +1 1 20 79 9 60.69574 0.19060

6 26 0 -1 +1 1 14 60 9 45.13658 0.194807 7 -1 0 +1 1 8 79 9 21.46137 0.070698 17 +1 -1 0 1 20 60 7 66.15446 0.275019 15 0 0 0 1 14 79 7 41.22875 0.1819710 12 0 +1 +1 1 14 98 9 38.05354 0.2448411 9 0 -1 -1 1 14 60 5 41.79545 0.0877112 10 0 +1 -1 1 14 98 5 34.75443 0.1166613 29 0 0 0 1 14 79 7 40.76725 0.2280814 24 0 -1 -1 1 14 60 5 40.41885 0.3590515 30 0 0 0 1 14 79 7 41.56889 0.0451916 5 -1 0 -1 1 8 79 5 19.27846 0.1065017 14 0 0 0 1 14 79 7 38.64282 0.1661618 22 -1 0 +1 1 8 79 9 20.14924 0.0935319 18 -1 +1 0 1 8 98 7 14.52333 0.0324620 23 +1 0 +1 1 20 79 9 62.05969 0.1354221 25 0 +1 -1 1 14 98 5 35.87051 0.0768422 3 -1 +1 0 1 8 98 7 16.27185 0.0716423 4 +1 +1 0 1 20 98 7 51.26461 0.1625324 21 +1 0 -1 1 20 79 5 62.52854 0.0624725 11 0 -1 +1 1 14 60 9 42.44533 0.0460226 6 +1 0 -1 1 20 79 5 57.10004 0.1969527 1 -1 -1 0 1 8 60 7 21.37529 0.0238628 13 0 0 0 1 14 79 7 48.34334 0.0418529 19 +1 +1 0 1 20 98 7 56.76232 0.1192630 2 +1 -1 0 1 20 60 7 57.69092 0.18380

Work piece- EN24 Tool electrode- Copper & Brass

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RSM- Box-Behnken Design approach (BBD)

1. MRR2. TWR