R/cpp_axes_normal.R

In CPP: Composition of Probabilistic Preferences (CPP)

#' CPP by axes using Normal distributions
#' @description This function computes the CPP by axes, using Normal distributions to randomize the decision matrix. The CPP by axes is used to rank alternatives in multicriteria decision problems. The "Progressive-Conservative" and the "Optimist-Pessimist" axes emulate four decision maker's points of view.
#' @param x Decision matrix of Alternatives (rows) and Criteria (columns). Benefit criteria must be positive and cost criteria must be negative.
#' @return PMax are the joint probabilities of each alternative being higher than the others, per criterion. PMin are the joint probabilities of each alternative being lower than the others, also by criterion. Axes returns the alternatives' scores by axis and ranking for decisionmaking.
#' @references Sant'Anna, Annibal P. (2015). Probabilistic Composition of Preferences: Theory and Applications, Springer.
#' @references Garcia, Pauli A. A. & Sant'Anna, Annibal P. (2015). Vendor and logistics provider selection in the construction sector: A probabilistic preferences composition approach. Pesquisa Operacional 35.2: 363-375.
#' @examples
#' # Alternatives' original scores
#' Alt.1 = c(2,30,86,-5)
#' Alt.2 = c(4,26,77,-12)
#' Alt.3 = c(3,22,93,-4)
#' Alt.4 = c(6,34,65,-10)
#' Alt.5 = c(5,31,80,-8)
#' x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) # Decision matrix
#' CPP.Axes.Normal(x)
#' @export
CPP.Axes.Normal = function (x) {

  ### Normalization of the decision matrix

  y = t(as.matrix(apply(x,2,sum)))
  dadosn=x
  for (j in 1:ncol(x))
  {
    for (i in 1:nrow(x))
    {
      dadosn[i,j] = x[i,j]/y[j]
    }}
  dadosn = replace(dadosn, dadosn == 0, 0.0000000001)
  apply(dadosn,2,sum)

  x = dadosn
  PMax = x
  mat = x

  sd = apply(x,2,sd)

  for (j in 1:ncol(x))
  {
    for (i in 1:nrow(x))
    {
      PMax[i,j] = (integrate(Vectorize(function(x) {prod(pnorm(x,mat[,j][-i],sd[j]))*dnorm(x,mat[,j][[i]],sd[j])}),-2,2)) $value
    }}
  PMax = PMax[,]

  ####

  PMin = x
  mat = x

  for (j in 1:ncol(x))
  {
    for (i in 1:nrow(x))
    {
      PMin[i,j] = (integrate(Vectorize(function(x) {prod(1-pnorm(x,mat[,j][-i],sd[j]))*dnorm(x,mat[,j][[i]],sd[j])}),-2,2)) $value
    }}
  PMin = PMin[,]

  ### Composition by axes

  # PP point of view

  PP = apply(PMax,1,prod)
  PP.rank = rank(-PP)

  # PO point of view

  Probs.m = 1-PMax
  PO = 1-(apply(Probs.m,1,prod))
  PO.rank = rank(-PO)

  # CP point of view

  Probs.mm = 1-PMin
  CP = apply(Probs.mm,1,prod)
  CP.rank = rank(-CP)

  # CO point of view

  CO = 1-(apply(PMin,1,prod))
  CO.rank = rank(-CO)

  Result = cbind(PP,PP.rank,PO,PO.rank,CP,CP.rank,CO,CO.rank)
  colnames(Result) = c("PP","Rank","PO","Rank","CP","Rank","CO","Rank")

  Result = list(PMax=PMax, PMin=PMin, Axes=Result)
  Result
}