Running a MAUT Model

Introduction

The decision models based in Multi Attribute Utility Theory (MAUT) 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 $$u(x_1,\ldots,x_n) = \sum_{i=1}^n\, w_i\, u_i\ ( x_i )$$ every utility u_i can be normalized to take values only between 0 and 1.

A model based in MAUT additionally has a decision tree that represents the relations between subsets of utilities, given an ordered and conceptual development of the decision model.

This mau package is designed to implement and test decision models based in (MAUT).

Utility definition

The utility functions for a MAUT model could be defined in a practical format when the utilities are piecewise defined like Constant Risk Averse Utilities (CRAU for short), satisfying the equation: $$ \frac{u''}{u'} = \lambda $$ being λ a constant. The previous equation only admits two kind of solutions, u(x) = a ⋅ x + b or u(x) = a ⋅ e^b ⋅ x + c. To completely determine an utility, it is only necessary to know the three parameters a, b, c. Additionally, the following convention is assumed, if c is 0 then the utility is linear otherwise is an exponential function.

For piecewise CRAU, it is only necessary to define the parameters a, b, c of the function for each part of the domain of definition. The mau package could load the utilities from an standardized text file with the following internal structure.

The capabilities of mau are not only restricted to work with CRAU class, you can define any utility by employing the correct R script.

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
…

Here an example of the structure of the standardized file for utility definitions

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 MAUT model, the package mau is employed to load utilities defined in the file utilities.txt, 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 regarding concentrated weights variation.

1. Loading necessary packages

library( mau )
library( data.table )
library( igraph )
library( ggplot2 )

2. Index definition

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 ) )

3. Loading file with 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' )

The functions data.table has the following structure with the piecewise definition of CRAU’s

3. Evaluation of utilities over every index

# 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 )

The utilities data.table has the following structure

4. Construction of the decision tree

file <- system.file("extdata", "tree.csv", package = "mau" )
tree.data <- read_tree( file, skip = 0, nrow = 8 )
tree <- make_decision_tree( tree.data )
utilities <- eval_utilities( index, columns, functions )

plot( tree, layout = layout_as_tree )

5. Computing the decision model

weights <- tree.data[ !is.na( weight ) ]$weight
model <- compute_model( tree, utilities, weights )

6. Bar plot for every utility

xlab <- 'Utility'
ylab <- 'Institutions'
title <- 'Criteria utilities'

colors <- c( 'dodgerblue4', 'orange', 'gold', 'red3' )
deep <- 2
bar <- bar_plot( model, deep, colors, title, xlab, ylab )
plot( bar )

7. Sensitivity analysis under weights change. The weights are simulated employing a Dirichlet distribution.

n <- 800
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 )