README.Rmd

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output: github_document
---

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```{r, echo = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.path = "README-"
)
```

## 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
```{r, eval=FALSE}
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:
```{r, eval=TRUE}
library( mau )
file <- system.file("extdata", "utilities.txt", package = "mau" )
lines <- readLines( file )
for ( i in 1:length( lines ) ) { 
  cat( lines[i], '\n' )
}
```

## 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.
```{r,eval=FALSE}
# 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 )
```