r-mau

Decision Models with Multi Attribute Utility Theory.

Introduction

The MAUT decision models are defined with aid of utility functions u₁, …, u_n which are evaluated over indexes x₁, …, x_n and those utilities are aggregated considering additional weights w₁, …, w_n, the whole final utility is given by the sum

u(x₁, …, x_n)=∑_{1 ≤ i ≤ n}w_iu_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.

Installation

To install mau you can proceed in the following way making use of the devtools library

library( devtools )
install_github( "pedroguarderas/mau" )

Utility definition

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 ⋅ x + b or u(x)=a ⋅ e^b ⋅ x + c.

File format for the piecewise defintion 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  0.2824

Main example

In 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 utilies 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 standarized 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 utilies 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 sensitivy 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" )
script<-'utilities.R'
lines<-17
skip<-2
encoding<-'utf-8'
functions<-Read.Utilities( file, script, lines, skip, encoding )
source( 'utilities.R' )

# 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.Simulation.Weight( S$simulation, title = 'Simulations', 
                                xlab = 'ID', ylab = 'Utility' ) 
plot( plot.S )