Analytic hierarchy process (AHP) is a decision-making method that decomposes the elements related to decision-making into levels such as objectives, quasi indicators and schemes, and carries out qualitative and quantitative analysis on the basis of sub levels. This paper describes the implementation process of R through an example.
summary
The basic idea of calculating index weight by analytic hierarchy process is to first establish an effective hierarchical index system, then compare the indexes to construct a judgment matrix, and then conduct digital processing and consistency test according to the judgment matrix to obtain the relative importance weight of each index.
example:
In the evaluation of regional macroeconomic benefits, three indicators are selected: capital profit and tax rate (x1), investment effect coefficient (x2) and labor productivity (x3). An expert believes that the capital profit and tax rate is extremely important than labor productivity, slightly more important than the investment effect coefficient, and the investment effect coefficient is more important than labor productivity. Try to determine the weights of the three evaluation indicators according to the expert's judgment.
| index | X1 | X2 | X3 |
|---|---|---|---|
| X1 | 1 | 3 | 9 |
| X2 | 1/3 | 1 | 5 |
| X3 | 1/9 | 1/5 | 1 |
tibble storage decision matrix
options(digits = 2) library(tidyverse) macro <- tibble(x1=c(1,1/3,1/9), x2=c(3,1,1/5), x3=c(9,5,1)) macro # A tibble: 3 x 3 # x1 x2 x3 # <dbl> <dbl> <dbl> # 1 1 3 9 # 2 0.333 1 5 # 3 0.111 0.2 1
Calculate row geometric average
That is, calculate the row data score, and then find the P-th root of the row product result, that is, the row geometric average.
# Add w variable macro %>% mutate(w = '^'(x1*x2*x3, 1/3)) -> macro macro # A tibble: 3 x 4 # x1 x2 x3 w # <dbl> <dbl> <dbl> <dbl> # 1 1 3 9 3 # 2 0.333 1 5 1.19 # 3 0.111 0.2 1 0.281
Normalizing w variables
The value of the w variable divided by the sum of the w column vectors.
# Define normalization function
unif <- function(x){
x / sum(x)
}
# Weights are calculated by normalization
macro %>% mutate_at(c("w"), .funs = std) -> macro
macro
# A tibble: 3 x 4
# x1 x2 x3 w
# <dbl> <dbl> <dbl> <dbl>
# 1 1 3 9 0.672
# 2 0.333 1 5 0.265
# 3 0.111 0.2 1 0.0629
The following is to test the weight of three variables
Consistency test
Consistency test ensures that the determination of the relative importance of each index should be coordinated and consistent, and there should be no contradiction.
The condition for judging the consistency of matrix B is that the maximum eigenvalue of matrix B is equal to the number of indexes.
The calculation process is as follows:
options(digits = 2) library(tidyverse) # Random consistency table ri_table <- c(0, 0, 0.58, 0.89, 1.12, 1.26, 1.36, 1.41, 1.46, 1.49, 1.52,1.54) b <- as.matrix(macro[,-4]) w <- as.matrix(macro[,4]) ## matrix product bw <- b %*% w ## Maximum characteristic root lmda <- 1/3 * sum(bw / wein) lmda ## Consistency index CI ci <- (lmda-length(bw)) / (length(bw) -1) ci ## Consistency ratio CR cr <- ci / ri_table[length(bw)] cr # [1] 0.025 # CR = 0.025 < 0.10, the consistency test is passed, and the weight of w above is reasonable # w # [1,] 0.672 # [2,] 0.265 # [3,] 0.063
CR = 0.025 < 0.10, the consistency test is passed, so the weight of w above is reasonable.
Finally, X1 (67%), X2 (27%) and X3 (6%) are calculated, and the sum of the three is equal to 1
Complete code
options(digits = 2)
library(tidyverse)
macro <- tibble(x1=c(1,1/3,1/9), x2=c(3,1,1/5), x3=c(9,5,1))
macro
# Define normalization function
unif <- function(x){
x / sum(x)
}
# Weights are calculated by normalization
macro %>% mutate_at(c("w"), .funs = std) -> macro
macro
# Random consistency table
ri_table <- c(0, 0, 0.58, 0.89, 1.12, 1.26, 1.36, 1.41, 1.46, 1.49, 1.52,1.54)
# Consistency test
b <- as.matrix(macro[,-4])
w <- as.matrix(macro[,4])
bw <- b %*% w
lmda <- 1/3 * sum(bw / wein)
lmda
ci <- (lmda-length(bw)) / (length(bw) -1)
ci
cr <- ci / ri_table[length(bw)]
cr