Two-level Factorial and Fractional Factorial Designs in Blocks of Size Two

YUYUN JESSIE YANG and NORMAN R.DRAPER

Journal of Quality Technology ,35 , p294 ,2003

報告者 :梁凱傑

Introduction

IN many experimental situations, it is desirable to group sets of experimental runs together in blocks.

The block size is governed by many considerations, and represents, in most experiments, the number of runs that can be made without worrying (much) about variation caused by factors not being studied specifically in the experiment.

Often, a block is some natural interval of time (e.g., a week, a day, or a work shift), of space (an oven, a greenhouse, a work bench, or a reactor), of personnel (a research worker or a research team), and so on.

Consider a product that can be made in different ways by varying a set of input factors, each with two levels.

In making boots, for example, variable factors that may be considered are the type of leather in the uppers, stiffness of the leather uppers, type of sole/heel cushioning, type of insoles, thickness of insoles, flexibility of sole, padded or thin tongue, overall weight of boot, Velcro or laced closure, and so forth.

Thus, in making boots, one could perform a two level factorial design that employed every combination of such levels or perhaps a subset of these combinations. If each boot of every pair were made to the same specification and if the boots were worn and used by testers, the boot results would beperfectly confounded with the testers.

Introduction

The Two-Factor, Design22

The Three Factor, Design32

The Four Factor, Design42

The Five Factor, Design52

Stage 1 :(1, 1, 1, 1, 1) and (0, 0, 0, 0, 0, 0, 0, 0, 0, 0), total 5Stage 2 :(1, 1, 2, 2, 2) and (0, 1, 1, 1, 1, 1, 1, 0, 0, 0), total 14Stage 3 :(1, 2, 2, 3, 3) and (1, 1, 2, 2, 2, 1, 1, 1, 1, 0), total 23Stage 4 :(2, 3, 3, 3, 4) and (1, 1, 3, 2, 2, 2, 1, 2, 1, 1), total 31.

Stage 1 :(1, 1, 1, 0, 0) and (0, 0, 1, 1, 0, 1, 1, 1, 1, 0), total 9Stage 2 :(2, 2, 1, 1, 1) and (0, 1, 1, 1, 1, 1, 1, 2, 2, 0), total 17Stage 3 :(3, 2, 2, 2, 1) and (1, 1, 1, 2, 2, 2, 1, 2, 3, 1), total 26.

1 2 3 4

1 2 3 4

1 2 3 4

Type 3 : B = 4, B = 5 , B = 13, B = 23;

Type 4 : B = 3, B = 15, B = 25, B = 45;

Type 3 : B = 2, B = 5 , B = 14, B = 34.

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

Type 5 : B = 15, B = 25, B = 35, B = 45;

Type 3 : B = 1 , B = 2 , B = 35, B = 45;

Type 3 : B = 1 , B = 3 , B = 25, B = 45;

Type 4 : B = 4 , B = 15, B = 25, B = 45

Example1

Consider a design defined by I = 1234,

If we are prepared to assume that "> or =3fi=0", the labels reduce to 1, 2, 3, 12+34, 13+24, 23+14, and 4.

To split apart the 12+34, 13+24, and 23+14 combinations, we need to use the design defined by I = -1234.

This addition results in a design of 48 runs.

4 12IV

4 12IV

Example2

Consider the design defined by I = 12345. The 32 possible estimates are confounded in 16 pairs

I + 12345, 1 + 2345, 2 + 1345, 3 + 1245,4 + 1235, 5 + 1234, 12 + 345, 13 + 245,14 + 235, 15 + 234, 23 + 145, 24 + 135,25 + 124, 34 + 125, 35 + 124, 45 + 123.

If we are prepared to assume that "> or =3fi=0"

5 12V

We can use any of the groups of three arrangements shown in Table 9 (taken from Table 8) to estimate all the main effects and 2fis of the projected four factors.

These designs, divided into blocks of size two, each require a total of 64 runs.

If we use designs, divided into blocks of size two, we need three sets, comprising 96 runs in total,

52

ConclusionThere are numerous ways to divide factorial and fractional factorial designs into blocks of size two, and the various possibilities achieve various objectives in terms of the estimation of effects and interactions.

Here, we have assumed that only main effects and 2fis are of interest,

and that the blocking is done by conventional methods.

This leads to reductions in the numbers of runs that are needed, and provides choices that depend on the experimenter's requirements.