Problem Analysis for 2 Blocks:

The effect(s) that would confound with the blocks, we confound effect ABC because in doing so contrast of the confounded effect is equal to the contrast of the block

The propose a design:

block 1: (1), ab, ac, bc

block2: a,b,c,abc

Problem Analysis for 4 Blocks:

The confound the effects AB, AC, BC with the blocks

The proposed design is shown below:

Block 1: (1), abc

Block 2: a, bc

Block 3: b, ac

Block 4: c, ab

library(DoE.base)
## Loading required package: grid
## Loading required package: conf.design
## Registered S3 method overwritten by 'DoE.base':
##   method           from       
##   factorize.factor conf.design
## 
## Attaching package: 'DoE.base'
## The following objects are masked from 'package:stats':
## 
##     aov, lm
## The following object is masked from 'package:graphics':
## 
##     plot.design
## The following object is masked from 'package:base':
## 
##     lengths
tool_life <- c(22, 32, 35, 55, 44, 40, 60, 39)

A<-c(-1, 1,-1,1,-1,  1, -1,  1)
B<-c(-1,-1, 1,1,-1, -1,  1,  1)
C<-c(-1,-1,-1,-1,1,  1,  1,  1)

dat<-data.frame(tool_life, A, B, C)
model = lm(tool_life~A*B*C, data=dat)
halfnormal(model)

## no significant effects

Analysis of Half Normal Plot

The half-normal plot shows that most effects lie close to the reference line, indicating they are likely due to random noise. However, three points fall far to the right and rise noticeably above the line, identifying them as statistically significant: the A*C interaction, and the main effects of B and C. These effects are much larger than the remaining factors, which cluster near the line and therefore appear insignificant. The blocking effect (ABC) also lies near the noise region, suggesting that the block was not significant in the experiment.