consrank {ConsRank}

Branch-and-bound and heuristic algorithms to find consensus (median) ranking according to the Kemeny's axiomatic approach

Description

Branch-and-bound, Quick , FAST and DECOR algorithms to find consensus (median) ranking according to the Kemeny's axiomatic approach. The median ranking(s) can be restricted to be necessarily a full ranking, namely without ties

Usage

consrank(
  X,
  wk = NULL,
  ps = TRUE,
  algorithm = "BB",
  full = FALSE,
  itermax = 10,
  np = 15,
  gl = 100,
  ff = 0.4,
  cr = 0.9,
  proc = FALSE
)

Arguments

X

A n by m data matrix, in which there are n judges and m objects to be judged. Each row is a ranking of the objects which are represented by the columns. If X contains the rankings observed only once, the argument wk can be used

wk

Optional: the frequency of each ranking in the data

ps

If PS=TRUE, on the screen some information about how many branches are processed are displayed.

algorithm

Specifies the used algorithm. One among "BB", "quick", "fast" and "decor". algorithm="BB" is the default option.

full

Specifies if the median ranking must be searched in the universe of rankings including all the possible ties (full=FALSE) or in the restricted space of full rankings (permutations). full=FALSE is the default option.

itermax

maximum number of iterations for FAST and DECOR algorithms. itermax=10 is the default option.

np

For DECOR algorithm only: the number of population individuals. np=15 is the default option.

gl

For DECOR algorithm only: generations limit, maximum number of consecutive generations without improvement. gl=100 is the default option.

ff

For DECOR algorithm only: the scaling rate for mutation. Must be in [0,1]. ff=0.4 is the default option.

cr

For DECOR algorithm only: the crossover range. Must be in [0,1]. cr=0.9 is the default option.

proc

For BB algorithm only: proc=TRUE allows the branch and bound algorithm to work in difficult cases, i.e. when the number of objects is larger than 15 or 25. proc=FALSE is the default option

Details

The BB algorithm can take long time to find the solutions if the number objects to be ranked is large with some missing (>15-20 if full=FALSE, <25-30 if full=TRUE). quick algorithm works with a large number of items to be ranked. The solution is quite accurate. fast algorithm works with a large number of items to be ranked by repeating several times the quick algorithm with different random starting points. decor algorithm works with a very large number of items to be ranked. For decor algorithm, empirical evidence shows that the number of population individuals (the 'np' parameter) can be set equal to 10, 20 or 30 for problems till 20, 50 and 100 items. Both scaling rate and crossover ratio (parameters 'ff' and 'cr') must be set by the user. The default options (ff=0.4, cr=0.9) work well for a large variety of data sets All algorithms allow the user to set the option 'full=TRUE' if the median ranking(s) must be searched in the restricted space of permutations instead of in the unconstrained universe of rankings of n items including all possible ties

Value

a "list" containing the following components:

Consensus		the Consensus Ranking
Tau		averaged TauX rank correlation coefficient
Eltime		Elapsed time in seconds

#'

Author(s)

Antonio D'Ambrosio [e-mail]

References

Emond, E. J., and Mason, D. W. (2002). A new rank correlation coefficient with application to the consensus ranking problem. Journal of Multi-Criteria Decision Analysis, 11(1), 17-28. D'Ambrosio, A., Amodio, S., and Iorio, C. (2015). Two algorithms for finding optimal solutions of the Kemeny rank aggregation problem for full rankings. Electronic Journal of Applied Statistical Analysis, 8(2), 198-213. Amodio, S., D'Ambrosio, A. and Siciliano, R. (2016). Accurate algorithms for identifying the median ranking when dealing with weak and partial rankings under the Kemeny axiomatic approach. European Journal of Operational Research, 249(2), 667-676. D'Ambrosio, A., Mazzeo, G., Iorio, C., and Siciliano, R. (2017). A differential evolution algorithm for finding the median ranking under the Kemeny axiomatic approach. Computers and Operations Research, vol. 82, pp. 126-138.

See Also

iwquickcons

Examples

data(Idea)
RevIdea<-6-Idea 
# as 5 means "most associated", it is necessary compute the reverse ranking of 
# each rankings to have rank 1 = "most associated" and rank 5 = "least associated"
CR<-consrank(RevIdea)
CR<-consrank(RevIdea,algorithm="quick")
#CR<-consrank(RevIdea,algorithm="fast",itermax=10)
#not run
#data(EMD)
#CRemd<-consrank(EMD[,1:15],wk=EMD[,16],algorithm="decor",itermax=1)
#data(APAFULL)
#CRapa<-consrank(APAFULL,full=TRUE)