ls.sorting.treatment: Least squares capacity identification in the framework of a sorting procedure: evaluation of the determined capacity

Description

This function assigns alternatives to classes and optionally compares the obtained classification to a given one. The classes are described by a set of prototypes (well-known alternatives for the decision maker) and a capacity (which can, for instance, be determined by ls.sorting.capa.ident). This function (in combination with ls.sorting.capa.ident) is an implementation of the TOMASO method; see Meyer and Roubens (2005).

Usage

ls.sorting.treatment(P, cl.proto, a, A, cl.orig.A = NULL)

Value

The function returns a list structured as follows:

correct.A: Object of class matrix which contains the different types of assignments of the elements of A. In columns the alternatives. First line: correct assignment to a single class (assignment of degree 0). Second line: correct ambiguous assignment to two classes (assignment of degree 1), etc. Last line: bad assignment. In case no orig.class.A is given, correct.A is NULL.
class.A: Object of class matrix which contains the assignments of the elements of A. In columns, the alternatives. First line, lower class. Second line, upper class.
eval.correct: Object of class numeric which contains the ratio assignments over number of elements of A(for each type of assignment; the first element is the ratio of correct assignments). In case no orig.class.A is given, eval.correct is NULL.
minmax.P: Object of class matrix which contains the min and the max of each class, according to the prototypes. In columns, the classes, first line: the minimum, second line: the maximum.
Choquet.A: Object of class numeric which contains the Choquet integral of the evaluations of the alternatives of A.

Arguments

P: Object of class matrix containing the n-column criteria matrix. Each line of this matrix corresponds to a prototype. A prototype is an alternative for which the class is known beforehand.
cl.proto: Object of class numeric containing the indexes of the classes the prototypes P are belonging to (the greater the class index, the better the prototype is considered by the decision maker).
a: Object of class Mobius.capacity used to model the classes determined by P by a Choquet integral.
A: Object of class matrix containing the n-column criteria matrix representing the alternatives which have to be classified.
cl.orig.A: Object of class numeric containing the "true" classes of the alternatives of A. This can be used for the evaluation of the quality of the model.

References

P. Meyer, M. Roubens (2005), Choice, Ranking and Sorting in Fuzzy Multiple Criteria Decision Aid, in: J. Figueira, S. Greco, and M. Ehrgott, Eds, Multiple Criteria Decision Analysis: State of the Art Surveys, volume 78 of International Series in Operations Research and Management Science, chapter 12, pages 471-506. Springer Science + Business Media, Inc., New York.

Examples

Run this code


## generate a random problem with 10 prototypes and 4 criteria
n.proto <- 10 ## prototypes
n <- 4  ## criteria
P <- matrix(runif(n.proto*n,0,1),n.proto,n)

## the corresponding global scores, based on a randomly generated
## capacity a
glob.eval <- numeric(n.proto)
a <- capacity(c(0:(2^n-3),(2^n-3),(2^n-3))/(2^n-3))
for (i in 1:n.proto)
  glob.eval[i] <- Choquet.integral(a,P[i,])

## based on these global scores, let us create a classification (3 classes)
cl.proto<-numeric(n.proto)
cl.proto[glob.eval <= 0.33] <- 1
cl.proto[glob.eval > 0.33 & glob.eval<=0.66] <-2
cl.proto[glob.eval > 0.66] <- 3

## search for a capacity which satisfies the constraints
lsc <- ls.sorting.capa.ident(n ,4, P, cl.proto, 0.1)

## output of the QP
lsc$how

## analyse the quality of the model (classify the prototypes by the
## model and compare both assignments)
lst <- ls.sorting.treatment(P,cl.proto,lsc$solution,P,cl.proto)

## assignments of the prototypes
lst$class.A
## assignment types
lst$correct.A
## evaluation
lst$eval.correct

## generate a second set of random alternatives (A)
## their "correct" class is determined as beforehand with the
## randomly generated capacity a
## the goal is to see if we can reproduce this classification
## by the capacity learnt from the prototypes

## a randomly generated criteria matrix of 10 alternatives
A <- matrix(runif(10*n,0,1),10,n)
cl.orig.A <-numeric(10)
	
## the corresponding global scores
glob.eval.A <- numeric(10)
for (i in 1:10)
  glob.eval.A[i] <- Choquet.integral(a,A[i,])

## based on these global scores, let us determine a classification
cl.orig.A[glob.eval.A <= 0.33] <- 1
cl.orig.A[glob.eval.A>0.33 & glob.eval.A<=0.66] <-2
cl.orig.A[glob.eval.A > 0.66] <- 3

## let us now classify the alternatives of A according to the model
## built on P

lst <- ls.sorting.treatment(P,cl.proto,lsc$solution,A,cl.orig.A)

## assignment of the alternatives of A
lst$class.A
## type of assignments
lst$correct.A
## evaluation
lst$eval.correct

## show the learnt capacity
## x11()
## barplot(Shapley.value(lsc$solution), main="Learnt capacity", sub="Shapley")
## summary of the learnt capacity
lsc$solution
summary(lsc$solution)