The MAUT decision models are defined with aid of utility functions \(u_1,\ldots,u_n\) which are evaluated over indexes \(x_1,\ldots,x_n\) and those utilities are aggregated considering additional weights \(w_1,\ldots,w_n\), the whole final utility is given by the sum
\[u(x_1,\ldots,x_n) = \sum_{1\leq i \leq n} w_i u_i\ ( x_i )\]
With mau you can build and test decision models based in Multi Attribute Utility Theory (MAUT). The utilities of any level of the decision model can be easily evaluated.
To install mau you can proceed in the following way making use of the devtools library
library( devtools )
install_github( "pedroguarderas/mau" )The utility functions for a MAUT model could be defined in a practical format when those are are piecewise defined like constant risk averse functions, in such case it is only necessary to define the parameters of the function for each part of the domain of definition. This is because, the constant risk averse functions are of the form \(u(x) = a \cdot x + b\) or \(u(x) = a \cdot e^{b \cdot x} + c\).
File format for the piecewise definition of utilities, is specified
as follows.
>Header
>
>Function name
>min1 max1 a1 b1 c1
>min2 max2 a2 b2 c2
>min3 max3 a3 b3 c3
>…
>Function name
>min1 max1 a1 b1 c1
>min2 max2 a2 b2 c2
>min3 max3 a3 b3 c3
>…
If \(c_i\) is \(0\) then the utility is linear, otherwise is an exponential function. For example:
library( mau )
file <- system.file("extdata", "utilities.txt", package = "mau" )
lines <- readLines( file )
for ( i in 1:length( lines ) ) {
cat( lines[i], '\n' )
}
#> Utilities
#>
#> Project
#> 1 2 1.5 -0.5 0
#> 2 3 1.5 -0.5 0
#>
#> Self implementation
#> 1 2 1.5 -0.5 0
#> 2 3 1.5 -0.5 0
#>
#> External and local relations
#> 1 10 1 0 0
#> 0 1 0 1 0
#>
#> Scope of capabilities
#> 6 15 1 0 0
#> 0 6 1.225 -1.225 0.2824In the sources below is developed a complete example of a decision
model, the package mau is employed to load utilities
defined in the file utilities.txt, provided in the package
itself, automatically the script with utilities is built and saved in
the local working directory, after that with eval_utilities
every function is evaluated over the columns of the index table, the
names for utilities were previously standardized with
stand_string. With another file tree.csv the
decision tree associated to the MAUT model is built and every weight and
relative weight assigned with the make_decision_tree
function, in addition the whole model with utilities of every criteria
is obtained with compute_model. The simulation of
constrained weights is made with sim_const_weights, the
result could be employed for a sensitivity test of the decision model
under a variation of weights.
# Loading packages --------------------------------------------------------------------------------
library( mau )
library( data.table )
library( igraph )
library( ggplot2 )
# Table of indexes --------------------------------------------------------------------------------
index <- data.table( cod = paste( 'A', 1:10, sep = '' ),
i1 = c( 0.34, 1, 1, 1, 1, 0.2, 0.7, 0.5, 0.11, 0.8 ),
i2 = c( 0.5, 0.5, 1, 0.5, 0.3, 0.1, 0.4, 0.13, 1, 0.74 ),
i3 = c( 0.5, 1.0, 0.75, 0.25, 0.1, 0.38, 0.57, 0.97, 0.3, 0.76 ),
i4 = c( 0, 0.26, 0.67, 0.74, 0.84, 0.85, 0.74, 0.65, 0.37, 0.92 ) )
# Loading utilities -------------------------------------------------------------------------------
file <- system.file("extdata", "utilities.txt", package = "mau" )
lines <- 17
skip <- 2
encoding <- 'utf-8'
functions <- read_utilities( file, lines, skip, encoding )
# script <- 'utilities.R'
# write( functions[[ 2 ]], script )
functions <- functions[[ 1 ]]
# Index positions ---------------------------------------------------------------------------------
columns <- c( 2, 3, 4, 5 )
# Function names
functions <- sapply( c( 'Project',
'Self implementation',
'External and local relations',
'Scope of capabilities' ),
FUN = stand_string )
names( functions ) <- NULL
# Evaluation of utilities -------------------------------------------------------------------------
utilities <- eval_utilities( index, columns, functions )
# Tree creation -----------------------------------------------------------------------------------
file <- system.file("extdata", "tree.csv", package = "mau" )
tree.data <- read_tree( file, skip = 0, nrow = 8 )
tree <- make_decision_tree( tree.data )
# Compute the decision model ----------------------------------------------------------------------
weights <- tree.data[ !is.na( weight ) ]$weight
model <- compute_model( tree, utilities, weights )
# Weights simulation ------------------------------------------------------------------------------
n <- 200
alpha <- c( 0.2, 0.5, 0.1, 0.2 )
constraints <- list( list( c(1,2), 0.7 ),
list( c(3,4), 0.3 ) )
S <- sim_const_weights( n, utilities, alpha, constraints )
plot.S <- plot_sim_weight( S$simulation, title = 'Simulations', xlab = 'ID', ylab = 'Utility' )
plot( plot.S )