AHPtools
was primarily developed as an aid to
research.
The functions in the AHPtools package are:
In the following sections are some illustrative use cases of the
AHPtools
package. The Consistency Ratio (CR) based, extant
consistency measure has been shown by several researchers to have
serious limitations. AHPtools
provides a compendium of
functions that could help research on consistency in the AHP.
# if (!requireNamespace("AHPtools", quietly = TRUE)) {
# install.packages("AHPtools")
# }
library(AHPtools)
#devtools::load_all(".")
suppressWarnings(suppressMessages(library(tidyverse)))
suppressWarnings(suppressMessages(library(dplyr)))
suppressWarnings(suppressMessages(library(knitr)))
suppressWarnings(suppressMessages(library(kableExtra)))
library(knitr)
library(kableExtra)
library(tidyverse)
library(dplyr)
The list of functions with the function signatures can be obtained:
Function_Name | Function_Signature |
---|---|
createRandomPCM | function (PCMsize) NULL |
sensitivity | function (PCM, typePCM = TRUE, granularityLow = TRUE) NULL |
viewAHPtree | function (ahp) NULL |
revisedConsistency | function (PCM, typePCM = TRUE) NULL |
AHPweights | function (ExcelPath, AHPsheet, PCMsheet) NULL |
createLogicalPCM | function (ord, prefVec = rep(NA, ord), granularityLow = TRUE) NULL |
createPCM | function (vec) NULL |
CR | function (PCM, typePCM = TRUE) NULL |
consEval | function (pcm) NULL |
improveCR | function (PCM, typePCM = TRUE) NULL |
The manual for any function, e.g. CR
, can be pulled out
from CRAN using the following R command.
The Consistency Ratio is used as a measure of consistency for a PCM in the AHP.
Random PCMs were used by Saaty to construct these thresholds, specifically to construct the Random Indices for various orders. This is one of the main reasons why consistency evaluation is biased and often flawed.
Needless to say Random PCMs are hardly consistent. The possibility of a PCM meeting the CR consistency threshold reduces with increase in the order of the PCM. Testing the CR consistency thresholds on simulated logical PCMs is demonstrated in this Research Nugget. Logical PCMs such as are assigned by human users are more likely to be representative candidates for benchmarking PCM consistency thresholds.
Consistency Ratios of simulated logical PCMs
and
simulated random PCMs
for various orders are compared. In
the following code snippet, 100 PCMs of each order for both the
categories are simulated and the CR values are captured.
runs <- rep(3:12, each=100)
R <- unlist(lapply(runs, function(x) CR(createRandomPCM(x))$CR))
L <- lapply(runs, function(x) CR(createLogicalPCM(x)))
Lcr <- unlist(lapply(L, function(x) x$CR))
Lcons <- unlist(lapply(L, function(x) x$CRconsistent))
exp1DF <- data.frame(Order=runs,Random.PCM=R,Logical.PCM=Lcr,Inconsistent=!Lcons)
We can get the order-wise mean CRs for the two categories of PCM, as follows.
suppressPackageStartupMessages(library(dplyr))
summaryExp1 <- exp1DF %>% group_by(Order) %>%
summarise(Random.PCM=mean(Random.PCM), Logical.PCM=mean(Logical.PCM)*100,
"Logical PCMs"=mean(Inconsistent))
summaryExp1$Random.PCM <- round(summaryExp1$Random.PCM,3)
summaryExp1$Logical.PCM <- round(summaryExp1$Logical.PCM,3)
summaryExp1$`Logical PCMs` <- round(summaryExp1$`Logical PCMs`,2)
kable(summaryExp1) %>%
add_header_above(c(" "=1, "Average CR of 100"=2, "% Inconsistent"=1), line=FALSE)
Order | Random.PCM | Logical.PCM | Logical PCMs |
---|---|---|---|
3 | 0.927 | 4.158 | 0.31 |
4 | 1.023 | 3.769 | 0.08 |
5 | 1.033 | 3.759 | 0.04 |
6 | 1.039 | 3.542 | 0.02 |
7 | 0.995 | 3.467 | 0.01 |
8 | 0.980 | 3.747 | 0.00 |
9 | 1.020 | 3.769 | 0.00 |
10 | 1.001 | 3.830 | 0.00 |
11 | 1.003 | 3.313 | 0.00 |
12 | 0.996 | 3.788 | 0.00 |
It is obvious that for logically constructed PCMs
the CR
threshold is far too liberal and would possibly result in large number
of false positives - i.e. inconsistent PCMs being adjudged to be
consistent.
Corroborating this is the empirical observation of the proportion of logical PCMs for each order that are CR inconsistent. The numbers are less than 5% for orders greater than 4. For order 3 however, 31% of PCMs are CR inconsistent.
Given that the logically constructed PCMs are representative of expert assigned PCMs, this highlights a possible gap in consistency evaluation using the CR method.
Given the criticality of consistency there have been several methods to enhance the consistency of a PCM. One of the earliest of these was due to Harker.
The function improveCR
takes any PCM as input and using
Harker’s method repeatedly, attempts to bring the CR to an acceptable
value.
This research nugget highlights a drawback in forcing the CR to a value that is deemed consistent. The principal eigenvector of a PCM indicates the relative importance of the alternatives. Improving consistency by focusing on errant pairwise ratios in isolation often changes the overall preferences for the alternatives.
The code in this section illustrates this by simulating 5 random PCMs
of order 7 each, and then using improveCR
to bring the CR
to an acceptable consistency level.
We also show, for each random PCM, the original CR value and the number of inversions or preference rank changes among the resultant principal eigenvector elements of the original PCM and the improved PCM. First we write a function to compute the number of inversions.
count_inversions <- function(x, y) {
rx <- rank(x)
ry <- rank(y)
n <- length(rx)
inversions <- 0
for (i in 1:(n - 1)) {
for (j in (i + 1):n) {
# Check if pair order differs between the two vectors
if ((rx[i] - rx[j]) * (ry[i] - ry[j]) < 0) {
inversions <- inversions + 1
}
}
}
return(inversions)
}
set.seed(93)
ind <- type <- ConsistencyRatio <- inverts <- vecRanks <- c()
oCR <- iCR <- cinv <- c()
for (i in 1:10) {
p1 <- createRandomPCM(9)
imp <- improveCR(p1)
ind <- c(ind, i)
#type <- c(type, c("Original", "Improved", ""))
oCR <- c(oCR, round(imp$CR.original,4))
iCR <- c(iCR, round(imp$suggestedCR,4))
spcm <- abs(Re(eigen(imp$suggestedPCM)$vectors[,1]))
opcm <- abs(Re(eigen(p1)$vectors[,1]))
cinv <- c(cinv, count_inversions(opcm, spcm))
# cat(c("Iteration", i, ": ", ConsistencyRatio, "Inversions=", count_inversions(opcm, spcm), "\n"))
# inverts <- c(inverts, c(imp$inversions, " ", ""))
# vecRanks <- c(vecRanks, c(imp$oriRank, imp$impRank, ""))
}
df <- data.frame(ind, oCR, iCR, cinv)
kable(df, linesep=FALSE) %>% kable_styling(position = "center")
ind | oCR | iCR | cinv |
---|---|---|---|
1 | 0.7806 | 0.0089 | 2 |
2 | 1.1202 | 0.0039 | 7 |
3 | 0.7915 | 0.0062 | 1 |
4 | 0.8970 | 0.0089 | 0 |
5 | 1.0941 | 0.0091 | 2 |
6 | 1.1280 | 0.0096 | 0 |
7 | 1.0388 | 0.0117 | 0 |
8 | 1.0681 | 0.0124 | 2 |
9 | 0.9050 | 0.0108 | 0 |
10 | 0.8999 | 0.0124 | 2 |
It is seen that while the CR value has been drastically improved, the ranks for the alternatives have changed as indicated by the number of inversions in 3 out of the 5 simulated random PCMs.
The function sensitivity
perturbs the upper triangular
elements of a PCM to a random selection from the set of the nearest five
elements in the Fundamental Scale. The lower triangular elements are
reciprocals of the corresponding upper triangular elements. 500
perturbations of a PCM are made.
The rank of the alternatives as in the principal eigen vector of the original PCM denotes the aggregated preference order for the alternatives.
The rank of the alternatives in each of the 500 perturbed PCMs is rank correlated with that of the original input PCM. The average of the 500 rank correlations is an indicator of the stability / sensitivity of the input PCM.
The nearer this measure is to 1 the less sensitive is the PCM, indicating greater consistency. Conversely, the lesser this measure is the greater are the indications of inconsistency.
As a proof-of-concept of the claim in the last paragraph, we shall
simulate 10 each random
and logical
PCMs each
of orders 5, 7 and 9. Random PCMs are expected to be less consistent
than logical PCMs. This implies that the average sensitivity score
should be lower for random PCMs
.
runs <- rep(c(5,7,9), each=20)
R <- unlist(lapply(runs, function(x) sensitivity(createRandomPCM(x))))
L <- unlist(lapply(runs, function(x) sensitivity(createLogicalPCM(x))))
exp2DF <- data.frame(Order=runs,Random.PCM=R,Logical.PCM=L)
exp2DF %>% group_by(Order) %>% summarise(R=mean(Random.PCM), L=mean(Logical.PCM), Rs=sd(Random.PCM), Ls=sd(Logical.PCM))
#> # A tibble: 3 × 5
#> Order R L Rs Ls
#> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 5 0.847 0.737 0.0976 0.283
#> 2 7 0.870 0.733 0.0539 0.157
#> 3 9 0.857 0.726 0.0853 0.137
In this function all different triads of elements of a PCM are
chosen. Triads are subsets of 3 elements chosen from the n
alternatives of an order-n PCM. A triad reversal is said to occur if any
two elements of the eigen vector of an order-3 PCM show a reversal in
preference when compared to the corresponding elements of the entire
eigen vector.
An example to illustrate this follows.
pcmVec <- c(1/3,1/2,1/8, 1,1/6, 3)
pcm <- createPCM(pcmVec)
colnames(pcm) <- rownames(pcm) <- c('a1', 'a2', 'a3', 'a4')
pcm
#> a1 a2 a3 a4
#> a1 1 0.3333333 0.5000000 0.1250000
#> a2 3 1.0000000 1.0000000 0.1666667
#> a3 2 1.0000000 1.0000000 3.0000000
#> a4 8 6.0000000 0.3333333 1.0000000
The eigen vector of this PCM is ( 0.1231476, 0.2772163, 0.6444789, 0.701878 ).
The triad reversals for this PCM can be obtained as follows.
#trdf <- triadReversal(pcm)
trdf <- consEval(pcm)$triadsData
trdf
#> triadE1 triadE2 triadE3 prefRev pcmWeightE1 pcmWeightE2 triadWeightE1
#> 1 a2 a3 a1 2.661258 0.2772163 0.6444789 0.7238135
#> 2 a3 a4 a1 1.427076 0.6444789 0.7018780 0.7832404
#> 3 a3 a4 a2 1.246666 0.6444789 0.7018780 0.7238135
#> triadWeightE2
#> 1 0.6323093
#> 2 0.5977243
#> 3 0.6323093
This PCM has 3 triad reversals indicated by the 3 rows displayed above.
Let us examine the first row displayed. It shows that the triad PCM constructed by taking the row-columns corresponding to a2, a3, a1 has a preference reversal of 2.661258.
The following code segment displays this PCM.
pcm3Vec <- c(1,3, 2)
pcm3 <- createPCM(pcm3Vec)
colnames(pcm3) <- rownames(pcm3) <- c('a2', 'a3', 'a1')
pcm3
#> a2 a3 a1
#> a2 1.0000000 1.0 3
#> a3 1.0000000 1.0 2
#> a1 0.3333333 0.5 1
The eigen vector of this triad PCM is ( 0.7238135, 0.6323093, 0.2761865 ).
The corresponding principal eigenvector elements for a2, a3 from the original order-4 PCM are 0.2772163, 0.6444789. For a2 and a3 the triadE1 and triadE2 elements have values 0.2772163 and 0.6444789.
The corresponding values for a2, a3 for the triad PCM are 0.7238135, 0.6323093.
There is a preference reversal in this triad because the preference ratio for a2:a3, is more than 1 for one of the comparisons and less for the other.
As per the eigen vector of the entire order-4 PCM, the preference ratio of a2:a3 is 0.2772163 / 0.6444789 = 0.4301402, indicating a higher preference for a3 compared to a2.
When we consider the triad a2, a3, a1 - row 1 of the data displayed - we see that the preference for a2 vis-a-vis a3, is 0.7238135 / 0.6323093 = 1.1447143, indicating a higher preference for a2 compared to a3.
This reversal in preference is counter-intuitive and indicative of inconsistency. The measure of intensity of the reversal is taken to be
as in the first row of the triad reversals data displayed.
Similarly, considering all possible triads - there are \(\binom{4}{3}\) triads possible - we see that there are three preference reversals, of values
Triads form the smallest units prone to inconsistency - a tuple is
never inconsistent because transverse elements in a pairwise comparison
matrix are reciprocals of one another. For an order n
PCM
there are \(\binom{n}{3}\) possible
triads, and for each triad there are \(\binom{3}{2}=3\) pairs which can have
preference reversals.
Thus, for an order-n
PCM the maximum possible number of
preference reversals is \(3 \times
\binom{n}{3}\).
In this example we see that there are 3 preference reversals out of a
maximum possible of \(3 \times
\binom{4}{3}\) = 12, with a maximum intensity of 2.661259 between
a2
and a3
considering a1
as the
third element in the triad.
The AHPtools
package also has a function
triadConsistency
which summarises the results of observed
reversals for any given PCM.
consEval(pcm)
#> $logitConsistency
#> [1] 2.261657e-09
#>
#> $prop3Rev
#> [1] 0.25
#>
#> $max3Rev
#> [1] 2.661258
#>
#> $triadsData
#> triadE1 triadE2 triadE3 prefRev pcmWeightE1 pcmWeightE2 triadWeightE1
#> 1 a2 a3 a1 2.661258 0.2772163 0.6444789 0.7238135
#> 2 a3 a4 a1 1.427076 0.6444789 0.7018780 0.7832404
#> 3 a3 a4 a2 1.246666 0.6444789 0.7018780 0.7238135
#> triadWeightE2
#> 1 0.6323093
#> 2 0.5977243
#> 3 0.6323093
The $triadsData
is a list of all preference reversals
observed and we have already seen this as an output of
triadReversal(pcm)
. We have already observed that there are
3 preference reversals out of a maximum possible 12. This gives the
value of consEval(pcm)$prop3Rev
.
Similarly, the maximum preference reversal is given by
consEval(pcm)$max3Rev
and is 2.661258 as we have already
seen.
The consEval(pcm)$logitConsistency
value is an indicator
of the consistency of the PCM. On a scale of 0 to 1, this value
indicates that the PCM is highly likely to be inconsistent.