Ranking alternatives means ordering them from the best to the worst alternative. The aim of the implemented method is to model a given ranking by means of a Choquet integral. The result of the function is an object of class Mobius.capacity. This function is an implementation of the TOMASO method (see Meyer and Roubens (2005)) in the particular ranking framework. The input data are given under the form of a set of alternatives and a partial weak order, each alternative being described according to a set of criteria. These well-known alternatives are called "prototypes". They represent alternatives for which the decision maker has an a priori knowledge and for which he/she is able to build a ranking. If the provided ranking (partial weak order) of the prototypes cannot be described by a Choquet integral, an approximative solution, which minimizes the "gap" between the given ranking and the one derived from the Choquet integral, is proposed. The problem is solved by quadratic programming.
ls.ranking.capa.ident(n, k, C, rk, d, A.Shapley.preorder = NULL,
A.Shapley.interval = NULL, A.interaction.preorder = NULL,
A.interaction.interval = NULL, A.inter.additive.partition = NULL,
sigf = 5, maxiter = 20, epsilon = 1e-6)The function returns a list structured as follows:
Object of class
Mobius.capacity containing the Moebius transform of the
k-additive solution.
The global evaluations satisfying the given ranking.
Information returned by ipop (cf. kernlab) on
the convergence of the solver.
The ranks of the prototypes
The Choquet integral of the prototypes
Object of class numeric containing the
number of elements of the set on which the object of class
Mobius.capacity is to be defined (in short, the number of criteria).
Object of class numeric imposing that the solution is at
most a k-additive capacity (the Moebius transform of subsets whose cardinal is
superior to k vanishes).
Object of class matrix containing the
n-column criteria matrix. Each line of this matrix
corresponds to a prototype.
Object of class matrix containing the
constraints relative to the preorder of the prototypes. Each line
of this 2-column matrix corresponds to one constraint of the type
"the alternative i is prefered to the alternative j". A line is structured as follows: the first element
encodes i, the second j.
Object of class numeric containing the threshold value, i.e. the minimal "distance" between two neighbor alternatives in the given ranking (e.g. the difference in terms of the Choquet integral of the
a prototype with rank 3 and a prototype with rank 4 should be at least d).
Object of class matrix containing the
constraints relative to the preorder of the criteria. Each line
of this 3-column matrix corresponds to one constraint of the type
"the Shapley importance index of criterion i is greater than
the Shapley importance index of criterion j with preference threshold
delta.S". A line is structured as follows: the first element
encodes i, the second j, and the third element contains
the preference threshold delta.S.
Object of class matrix containing the
constraints relative to the quantitative importance of the
criteria. Each line of this 3-column matrix corresponds to one
constraint of the type "the Shapley importance index of criterion
i lies in the interval [a,b]". The interval
[a,b] has to be included in [0,1]. A line of the
matrix is structured as follows: the first element encodes i,
the second a, and the third b.
Object of class matrix
containing the constraints relative to the preorder of the pairs of
criteria in terms of the Shapley interaction index. Each line of this 5-column matrix
corresponds to one constraint of the type "the Shapley interaction
index of the pair ij of criteria is greater than the Shapley interaction
index of the pair kl of criteria with preference threshold delta.I".
A line is structured as follows: the first two elements encode
ij, the second two kl, and the fifth element contains
the preference threshold delta.I.
Object of class matrix
containing the constraints relative to the type and the magnitude of
the Shapley interaction index for pairs of criteria. Each line of
this 4-column matrix corresponds to one constraint of the type
"the Shapley interaction index of the pair ij of criteria
lies in the interval [a,b]". The interval [a,b] has to
be included in [-1,1]. A line is structured as follows: the first two elements encode
ij, the third element encodes a, and the fourth element
encodes b.
Object of class numeric
encoding a partition of the set of criteria imposing that there be
no interactions among criteria belonging to different classes
of the partition. The partition is to be given under the form of a
vector of integers from {1,...,n} of length n such
that two criteria belonging to the same class are "marked" by the
same integer. For instance, the partition {{1,3},{2,4},{5}} can
be encoded as c(1,2,1,2,3). See Fujimoto and Murofushi (2000)
for more details on the concept of mu-inter-additive partition.
Precision (default: 5 significant figures). Parameter to
be passed to the ipop function (quadratic programming)
of the kernlab package.
Maximum number of iterations. Parameter to
be passed to the ipop function (quadratic programming)
of the kernlab package.
Object of class numeric containing the
threshold value for the monotonicity constraints, i.e. the
difference between the "weights" of two subsets whose cardinals
differ exactly by 1 must be greater than epsilon.
The quadratic program is solved using the ipop function of
the kernlab package.
K. Fujimoto and T. Murofushi (2000) Hierarchical decomposition of the Choquet integral, in: Fuzzy Measures and Integrals: Theory and Applications, M. Grabisch, T. Murofushi, and M. Sugeno Eds, Physica Verlag, pages 95-103.
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.
Mobius.capacity-class,
lin.prog.capa.ident,
mini.var.capa.ident,
mini.dist.capa.ident,
least.squares.capa.ident,
heuristic.ls.capa.ident,
entropy.capa.ident.
arthur <- c(1, 1, 0.75, 0.25)
lancelot <- c(0.75, 0.75, 0.75, 0.75)
yvain <- c(1, 0.625, 0.5, 1)
perceval <- c(0.25, 0.5, 0.75, 0.75)
erec <- c(0.375, 1, 0.5 , 0.75)
P <- rbind(arthur, lancelot, yvain, perceval, erec)
# lancelot > erec; yvain > erec, erec > perceval, erec > arthur
rk.proto <- rbind(c("lancelot","erec"), c("yvain","erec"), c("erec","perceval"), c("erec","arthur"))
n<-4
k<-2
d<-0.1
## search for a capacity which satisfies the constraints
lrc <- ls.ranking.capa.ident(n ,k, P, rk.proto, d)
lrc
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