lin.prog.capa.ident: Capacity identification based on linear programming
In kappalab: Non-Additive Measure and Integral Manipulation Functions

lin.prog.capa.ident

R Documentation

Capacity identification based on linear programming

Description

Creates an object of class Mobius.capacity using the linear programming approach proposed by Marichal and Roubens (see reference hereafter). Roughly speaking, this function determines, if it exists, the capacity compatible with a set of linear constraints that "separates" the most the provided alternatives. The problem is solved using the lpSolve package.

Usage

lin.prog.capa.ident(n, k, A.Choquet.preorder = NULL,
A.Shapley.preorder = NULL, A.Shapley.interval = NULL,
A.interaction.preorder = NULL, A.interaction.interval = NULL,
A.inter.additive.partition = NULL, epsilon = 1e-6)

Arguments

`n`	Object of class `numeric` containing the number of elements of the set on which the object of class `Mobius.capacity` is to be defined.
`k`	Object of class `numeric` imposing that the solution is at most a k-additive capacity (the Möbius transform of subsets whose cardinal is superior to `k` vanishes).
`A.Choquet.preorder`	Object of class `matrix` containing the constraints relative to the preorder of the alternatives. Each line of the matrix corresponds to one constraint of the type "alternative `a` is preferred to alternative `b` with preference threshold `delta.C`". A line is structured as follows: the first `n` elements encode alternative `a`, the next `n` elements encode alternative `b`, and the last element contains the preference threshold `delta.C`.
`A.Shapley.preorder`	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`.
`A.Shapley.interval`	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`.
`A.interaction.preorder`	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`.
`A.interaction.interval`	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`.
`A.inter.additive.partition`	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.
`epsilon`	Object of class `numeric` containing the thresold 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`.

Details

The linear program is solved using the lp function of the lpSolve.

Value

The function returns a list structured as follows:

`solution`	Object of class `Mobius.capacity` containing the Möbius transform of the `k`-additive solution, if any.
`value`	Value of the objective function.
`lp.object`	Object of class `lp.object` returned by `lpSolve`.

References

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.

J-L. Marichal and M. Roubens (2000), Determination of weights of interacting criteria from a reference set, European Journal of Operational Research 124, pages 641-650.

Examples

## some alternatives
a <- c(18,11,18,11,11)
b <- c(18,18,11,11,11)
c <- c(11,11,18,18,11)
d <- c(18,11,11,11,18)
e <- c(11,11,18,11,18)
    
## preference threshold relative
## to the preorder of the alternatives
delta.C <- 1

## corresponding Choquet preorder constraint matrix 
Acp <- rbind(c(d,a,delta.C),
             c(a,e,delta.C),
             c(e,b,delta.C),
             c(b,c,delta.C)
            )

## a Shapley preorder constraint matrix
## Sh(1) - Sh(2) >= -delta.S
## Sh(2) - Sh(1) >= -delta.S
## Sh(3) - Sh(4) >= -delta.S
## Sh(4) - Sh(3) >= -delta.S
## i.e. criteria 1,2 and criteria 3,4
## should have the same global importances
delta.S <- 0.01    
Asp <- rbind(c(1,2,-delta.S),
             c(2,1,-delta.S),
             c(3,4,-delta.S),
             c(4,3,-delta.S)
            )

## a Shapley interval constraint matrix
## 0.3 <= Sh(1) <= 0.9 
Asi <- rbind(c(1,0.3,0.9))


## an interaction preorder constraint matrix
## such that I(12) = I(34)
delta.I <- 0.01
Aip <- rbind(c(1,2,3,4,-delta.I),
             c(3,4,1,2,-delta.I))

## an interaction interval constraint matrix
## i.e. -0.20 <= I(12) <= -0.15 
Aii <- rbind(c(1,2,-0.2,-0.15))


## Not run: 
## a LP 2-additive solution
lin.prog <- lin.prog.capa.ident(5,2,A.Choquet.preorder = Acp)              
m <- lin.prog$solution
m

## the resulting global evaluations
rbind(c(a,mean(a),Choquet.integral(m,a)),
      c(b,mean(b),Choquet.integral(m,b)),
      c(c,mean(c),Choquet.integral(m,c)),
      c(d,mean(d),Choquet.integral(m,d)),
      c(e,mean(e),Choquet.integral(m,e)))

## the Shapley value
Shapley.value(m)

## a LP 3-additive more constrained solution
lin.prog2 <- lin.prog.capa.ident(5,3,
                                   A.Choquet.preorder = Acp,
                                   A.Shapley.preorder = Asp)
m <- lin.prog2$solution
m
rbind(c(a,mean(a),Choquet.integral(m,a)),
      c(b,mean(b),Choquet.integral(m,b)),
      c(c,mean(c),Choquet.integral(m,c)),
      c(d,mean(d),Choquet.integral(m,d)),
      c(e,mean(e),Choquet.integral(m,e)))
Shapley.value(m)

## a LP 5-additive more constrained solution
lin.prog3 <- lin.prog.capa.ident(5,5,
                                   A.Choquet.preorder = Acp,
                                   A.Shapley.preorder = Asp,
                                   A.Shapley.interval = Asi,
                                   A.interaction.preorder = Aip,
                                   A.interaction.interval = Aii)

m <- lin.prog3$solution
m
rbind(c(a,mean(a),Choquet.integral(m,a)),
      c(b,mean(b),Choquet.integral(m,b)),
      c(c,mean(c),Choquet.integral(m,c)),
      c(d,mean(d),Choquet.integral(m,d)),
      c(e,mean(e),Choquet.integral(m,e)))
summary(m)

## End(Not run)

kappalab documentation built on Nov. 8, 2023, 1:07 a.m.