Decision Models with Multi Attribute Utility Theory

Provides functions for the creation, evaluation and test of decision models based in Multi Attribute Utility Theory (MAUT). Can process and evaluate local risk aversion utilities for a set of indexes, compute utilities and weights for the whole decision tree defining the decision model and simulate weights employing Dirichlet distributions under addition constraints in weights. Also includes other rating analysis methods as for example the Colley, Offensive - Defensive ratings and the ranking aggregation with Borda count.

Introduction

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.

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 $u(x) = a \cdot x + b$ 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  0.2824

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 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 )