R Under development (unstable) (2023-11-06 r85483 ucrt) -- "Unsuffered Consequences"
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Platform: x86_64-w64-mingw32/x64

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> library(kappalab)
Loading required package: lpSolve
Loading required package: quadprog
Loading required package: kernlab
> 
> ## 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 
> delta.I <- 0.01
> Aii <- rbind(c(1,2,-0.2,-0.15))
> 
> ## the capacity that we want to approach
> x <- runif(31)
> for (i in 2:31)
+     x[i] <- x[i] + x[i-1]
> mu <- normalize(capacity(c(0,x)))
> ## and its Mobius transform
> a.mu <- Mobius(mu)
> 
> ## some basic checks
> mini.dist.capa.ident(a.mu,5)
Distance used: Choquet.coefficients 

$solution
		Mobius.capacity
{}		0.000000
{1}		0.055342
{2}		0.063938
{3}		0.079113
{4}		0.097020
{5}		0.145004
{1,2}		0.044110
{1,3}		0.063852
{1,4}		0.079085
{1,5}		0.072544
{2,3}		0.189393
{2,4}		0.183361
{2,5}		0.168287
{3,4}		0.201114
{3,5}		0.178744
{4,5}		0.184426
{1,2,3}		-0.011451
{1,2,4}		0.009589
{1,2,5}		0.042316
{1,3,4}		0.071664
{1,3,5}		0.062276
{1,4,5}		0.076514
{2,3,4}		-0.085626
{2,3,5}		-0.075071
{2,4,5}		-0.050022
{3,4,5}		-0.090779
{1,2,3,4}		-0.202157
{1,2,3,5}		-0.191898
{1,2,4,5}		-0.228444
{1,3,4,5}		-0.330815
{2,3,4,5}		-0.242649
{1,2,3,4,5}		0.441219

$value
[1] -0.137454

$iterations
[1] 1 0

$iact
[1] 0

> mini.dist.capa.ident(a.mu,5,"binary.alternatives")
Distance used: binary.alternatives 

$solution
		Mobius.capacity
{}		0.000000
{1}		0.055342
{2}		0.063938
{3}		0.079113
{4}		0.097020
{5}		0.145004
{1,2}		0.044110
{1,3}		0.063852
{1,4}		0.079085
{1,5}		0.072544
{2,3}		0.189393
{2,4}		0.183361
{2,5}		0.168287
{3,4}		0.201114
{3,5}		0.178744
{4,5}		0.184426
{1,2,3}		-0.011451
{1,2,4}		0.009589
{1,2,5}		0.042316
{1,3,4}		0.071664
{1,3,5}		0.062276
{1,4,5}		0.076514
{2,3,4}		-0.085626
{2,3,5}		-0.075071
{2,4,5}		-0.050022
{3,4,5}		-0.090779
{1,2,3,4}		-0.202157
{1,2,3,5}		-0.191898
{1,2,4,5}		-0.228444
{1,3,4,5}		-0.330815
{2,3,4,5}		-0.242649
{1,2,3,4,5}		0.441219

$value
[1] -5.417779

$iterations
[1] 1 0

$iact
[1] 0

> mini.dist.capa.ident(a.mu,5,"global.scores")
Distance used: global.scores 

$solution
		Mobius.capacity
{}		0.000000
{1}		0.055342
{2}		0.063938
{3}		0.079113
{4}		0.097020
{5}		0.145004
{1,2}		0.044110
{1,3}		0.063852
{1,4}		0.079085
{1,5}		0.072544
{2,3}		0.189393
{2,4}		0.183361
{2,5}		0.168287
{3,4}		0.201114
{3,5}		0.178744
{4,5}		0.184426
{1,2,3}		-0.011451
{1,2,4}		0.009589
{1,2,5}		0.042316
{1,3,4}		0.071664
{1,3,5}		0.062276
{1,4,5}		0.076514
{2,3,4}		-0.085626
{2,3,5}		-0.075071
{2,4,5}		-0.050022
{3,4,5}		-0.090779
{1,2,3,4}		-0.202157
{1,2,3,5}		-0.191898
{1,2,4,5}		-0.228444
{1,3,4,5}		-0.330815
{2,3,4,5}		-0.242649
{1,2,3,4,5}		0.441219

$value
[1] -0.1349528

$iterations
[1] 2 0

$iact
[1] 1

> mini.dist.capa.ident(a.mu,3)
Distance used: Choquet.coefficients 

$solution
		Mobius.capacity
{}		0.000000
{1}		0.045495
{2}		0.058500
{3}		0.068556
{4}		0.084636
{5}		0.133132
{1,2}		0.067760
{1,3}		0.107976
{1,4}		0.130519
{1,5}		0.121925
{2,3}		0.215884
{2,4}		0.217161
{2,5}		0.200034
{3,4}		0.255388
{3,5}		0.230966
{4,5}		0.243957
{1,2,3}		-0.082416
{1,2,4}		-0.079649
{1,2,5}		-0.041792
{1,3,4}		-0.068760
{1,3,5}		-0.073018
{1,4,5}		-0.077052
{2,3,4}		-0.181967
{2,3,5}		-0.166281
{2,4,5}		-0.159506
{3,4,5}		-0.251448

$value
[1] -0.1357366

$iterations
[1] 1 0

$iact
[1] 0

> mini.dist.capa.ident(a.mu,3,"binary.alternatives")
Distance used: binary.alternatives 

$solution
		Mobius.capacity
{}		0.000000
{1}		0.035998
{2}		0.055616
{3}		0.057994
{4}		0.071333
{5}		0.120599
{1,2}		0.076288
{1,3}		0.121623
{1,4}		0.145992
{1,5}		0.136886
{2,3}		0.225122
{2,4}		0.228227
{2,5}		0.210587
{3,4}		0.271572
{3,5}		0.246637
{4,5}		0.261456
{1,2,3}		-0.090820
{1,2,4}		-0.088053
{1,2,5}		-0.050196
{1,3,4}		-0.077164
{1,3,5}		-0.081422
{1,4,5}		-0.085456
{2,3,4}		-0.190371
{2,3,5}		-0.174686
{2,4,5}		-0.167910
{3,4,5}		-0.259852

$value
[1] -5.41362

$iterations
[1] 2 0

$iact
[1] 1

> mini.dist.capa.ident(a.mu,3,"global.scores")
Distance used: global.scores 

$solution
		Mobius.capacity
{}		0.000000
{1}		0.042930
{2}		0.057824
{3}		0.065686
{4}		0.080983
{5}		0.129699
{1,2}		0.070243
{1,3}		0.111921
{1,4}		0.134986
{1,5}		0.126245
{2,3}		0.218570
{2,4}		0.220369
{2,5}		0.203096
{3,4}		0.260058
{3,5}		0.235490
{4,5}		0.249003
{1,2,3}		-0.084937
{1,2,4}		-0.082170
{1,2,5}		-0.044313
{1,3,4}		-0.071281
{1,3,5}		-0.075539
{1,4,5}		-0.079573
{2,3,4}		-0.184488
{2,3,5}		-0.168803
{2,4,5}		-0.162027
{3,4,5}		-0.253969

$value
[1] -0.1349403

$iterations
[1] 2 0

$iact
[1] 1

> 
> ## a minimum distance 2-additive solution
> min.dist <- mini.dist.capa.ident(a.mu,2,"binary.alternatives",A.Choquet.preorder = Acp)              
Distance used: binary.alternatives 

> m <- min.dist$solution
> m
		Mobius.capacity
{}		0.000000
{1}		0.251923
{2}		0.247885
{3}		0.109066
{4}		0.000001
{5}		0.394780
{1,2}		-0.247884
{1,3}		0.176649
{1,4}		0.000000
{1,5}		0.033792
{2,3}		0.000000
{2,4}		0.133939
{2,5}		0.000000
{3,4}		0.000000
{3,5}		-0.109065
{4,5}		0.008913
> 
> ## a minimum distance 3-additive more constrained solution
> min.dist2 <- mini.dist.capa.ident(a.mu,3,"global.scores",
+                                    A.Choquet.preorder = Acp,
+                                    A.Shapley.preorder = Asp)
Distance used: global.scores 

> m <- min.dist2$solution
> m
		Mobius.capacity
{}		0.000000
{1}		0.086199
{2}		0.098042
{3}		0.020438
{4}		0.020438
{5}		0.168892
{1,2}		-0.020944
{1,3}		0.342373
{1,4}		0.046691
{1,5}		0.336777
{2,3}		0.282617
{2,4}		0.318979
{2,5}		0.052665
{3,4}		-0.020437
{3,5}		0.116823
{4,5}		0.320860
{1,2,3}		-0.287362
{1,2,4}		0.009913
{1,2,5}		0.007873
{1,3,4}		0.138990
{1,3,5}		-0.384513
{1,4,5}		-0.275996
{2,3,4}		-0.141218
{2,3,5}		-0.005600
{2,4,5}		-0.298466
{3,4,5}		0.065966
> 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)))
     [,1] [,2] [,3] [,4] [,5] [,6]     [,7]
[1,]   18   11   18   11   11 13.8 14.14308
[2,]   18   18   11   11   11 13.8 12.14308
[3,]   11   11   18   18   11 13.8 11.14308
[4,]   18   11   11   11   18 13.8 15.14308
[5,]   11   11   18   11   18 13.8 13.14308
> Shapley.value(m)
        1         2         3         4         5 
0.1749493 0.1764129 0.1765477 0.1865477 0.2855424 
> 
> ## a minimum distance 5-additive more constrained solution
> min.dist3 <- mini.dist.capa.ident(a.mu,5,
+                                    A.Choquet.preorder = Acp,
+                                    A.Shapley.preorder = Asp,
+                                    A.Shapley.interval = Asi,
+                                    A.interaction.preorder = Aip,
+                                    A.interaction.interval = Aii)
Distance used: Choquet.coefficients 

> 
> m <- min.dist3$solution
> m
		Mobius.capacity
{}		0.000000
{1}		0.175233
{2}		0.175233
{3}		0.000001
{4}		0.009941
{5}		0.045067
{1,2}		-0.175232
{1,3}		0.285714
{1,4}		0.152062
{1,5}		0.383506
{2,3}		0.234150
{2,4}		0.248535
{2,5}		0.181052
{3,4}		0.022436
{3,5}		0.273024
{4,5}		0.152487
{1,2,3}		0.036418
{1,2,4}		0.177214
{1,2,5}		-0.045745
{1,3,4}		-0.067413
{1,3,5}		-0.456008
{1,4,5}		-0.199818
{2,3,4}		-0.065360
{2,3,5}		-0.192016
{2,4,5}		-0.079045
{3,4,5}		-0.141797
{1,2,3,4}		-0.255352
{1,2,3,5}		0.079602
{1,2,4,5}		-0.200490
{1,3,4,5}		0.110230
{2,3,4,5}		-0.130436
{1,2,3,4,5}		0.266809
> 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)))
     [,1] [,2] [,3] [,4] [,5] [,6]     [,7]
[1,]   18   11   18   11   11 13.8 14.22664
[2,]   18   18   11   11   11 13.8 12.22664
[3,]   11   11   18   18   11 13.8 11.22664
[4,]   18   11   11   11   18 13.8 15.22664
[5,]   11   11   18   11   18 13.8 13.22664
> summary(m)
Shapley value :
        1         2         3         4         5 
0.3000000 0.2900000 0.1166434 0.1066434 0.1867131 

Shapley interaction indices :
            1            2           3          4            5
1          NA -0.150000000  0.08707493  0.0585512  0.095870141
2 -0.15000000           NA  0.08831090  0.1362151  0.005576357
3  0.08707493  0.088310899          NA -0.1400000 -0.035385829
4  0.05855120  0.136215076 -0.14000000         NA -0.064705805
5  0.09587014  0.005576357 -0.03538583 -0.0647058           NA

Orness :
0.493152

Veto indices :
        1         2         3         4         5 
0.5749499 0.5700200 0.4500915 0.4457237 0.4934547 

Favor indices :
        1         2         3         4         5 
0.5500501 0.5424800 0.4457128 0.4375806 0.4899367 

Normalized variance :
0.1573271

Normalized entropy :
0.7631456
> 
> proc.time()
   user  system elapsed 
   0.54    0.07    0.60