MCDA (version 0.1.0)

LPDMRSort: MRSort that takes into account large performance differences.

Description

MRSort is a simplified ElectreTRI method that uses the pessimistic assignment rule, without indifference or preference thresholds attached to criteria. LPDMRSort considers both a binary discordance and a binary concordance conditions including several interactions between them.

Usage

LPDMRSort(
  performanceTable,
  categoriesLowerProfiles,
  categoriesRanks,
  criteriaWeights,
  criteriaMinMax,
  majorityThreshold,
  criteriaVetos = NULL,
  criteriaDictators = NULL,
  majorityRule = "M",
  alternativesIDs = NULL,
  criteriaIDs = NULL,
  categoriesIDs = NULL
)

Value

The function returns a vector containing the assignments of the alternatives to the categories.

Arguments

performanceTable

Matrix or data frame containing the performance table. Each row corresponds to an alternative, and each column to a criterion. Rows (resp. columns) must be named according to the IDs of the alternatives (resp. criteria).

categoriesLowerProfiles

Matrix containing, in each row, the lower profiles of the categories. The columns are named according to the criteria, and the rows are named according to the categories. The index of the row in the matrix corresponds to the rank of the category.

categoriesRanks

A vector containing the ranks of the categories (1 for the best, with higher values for increasingly less preferred categories). The vector needs to be named with the categories names, whereas the ranks need to be a range of values from 1 to the number of categories.

criteriaWeights

Vector containing the weights of the criteria. The elements are named according to the IDs of the criteria.

criteriaMinMax

Vector containing the preference direction on each of the criteria. "min" (resp. "max") indicates that the criterion has to be minimized (maximized). The elements are named according to the IDs of the criteria.

majorityThreshold

The cut threshold for the concordance condition. Should be at least half of the sum of the weights.

criteriaVetos

Matrix containing in each row a vector defining the veto values for the lower profile of the category. NA values mean that no veto is defined. A veto threshold for criterion i and category k represents the performance below which an alternative is forbidden to outrank the lower profile of category k, and thus is forbidden to be assigned to the category k. The rows are named according to the categories, whereas the columns are named according to the criteria.

criteriaDictators

Matrix containing in each row a vector defining the dictator values for the lower profile of the category. NA values mean that no veto is defined. A dictator threshold for criterion i and category k represents the performance above which an alternative is guaranteed to outrank the lower profile of category k, and thus may no be assigned below category k. The rows are named according to the categories, whereas the columns are named according to the criteria.

majorityRule

String denoting how the vetoes and dictators are combined in order to form the assignment rule. The values to choose from are "M", "V", "D", "v", "d", "dV", "Dv", "dv". "M" corresponds to using only the majority rule without vetoes or dictators, "V" considers only the vetoes, "D" only the dictators, "v" is like "V" only that a dictator may invalidate a veto, "d" is like "D" only that a veto may invalidate a dictator, "dV" is like "V" only that if there is no veto we may then consider the dictator, "Dv" is like "D" only that when there is no dictator we may consider the vetoes, while finally "dv" is identical to using both dictator and vetoes only that when both are active they invalidate each other, so the majority rule is considered in that case.

alternativesIDs

Vector containing IDs of alternatives, according to which the datashould be filtered.

criteriaIDs

Vector containing IDs of criteria, according to which the data should be filtered.

categoriesIDs

Vector containing IDs of categories, according to which the data should be filtered.

References

Bouyssou, D. and Marchant, T. An axiomatic approach to noncompensatory sorting methods in MCDM, II: more than two categories. European Journal of Operational Research, 178(1): 246--276, 2007.

Meyer, P. and Olteanu, A-L. Integrating large positive and negative performance differences in majority-rule sorting models. European Journal of Operational Research, submitted, 2015.

Examples

# the performance table

performanceTable <- rbind(c(10,10,9), c(10,9,10), c(9,10,10), c(9,9,10), 
                          c(9,10,9), c(10,9,9), c(10,10,7), c(10,7,10), 
                          c(7,10,10), c(9,9,17), c(9,17,9), c(17,9,9), 
                          c(7,10,17), c(10,17,7), c(17,7,10), c(7,17,10), 
                          c(17,10,7), c(10,7,17), c(7,9,17), c(9,17,7), 
                          c(17,7,9), c(7,17,9), c(17,9,7), c(9,7,17))

profilesPerformances <- rbind(c(10,10,10),c(0,0,0))

vetoPerformances <- rbind(c(7,7,7),c(0,0,0))

dictatorPerformances <- rbind(c(17,17,17),c(0,0,0))

rownames(performanceTable) <- c("a1", "a2", "a3", "a4", "a5", "a6", "a7", 
                                "a8", "a9", "a10", "a11", "a12",  "a13", 
                                "a14", "a15", "a16", "a17", "a18", "a19", 
                                "a20", "a21", "a22", "a23", "a24")

rownames(profilesPerformances) <- c("P","F")

rownames(vetoPerformances) <- c("P","F")

rownames(dictatorPerformances) <- c("P","F")

colnames(performanceTable) <- c("c1","c2","c3")

colnames(profilesPerformances) <- c("c1","c2","c3")

colnames(vetoPerformances) <- c("c1","c2","c3")

colnames(dictatorPerformances) <- c("c1","c2","c3")

lambda <- 0.5

weights <- c(1/3,1/3,1/3)

names(weights) <- c("c1","c2","c3")

categoriesRanks <-c(1,2)

names(categoriesRanks) <- c("P","F")

criteriaMinMax <- c("max","max","max")

names(criteriaMinMax) <- colnames(performanceTable)

assignments <-rbind(c("P","P","P","F","F","F","F","F","F","F","F","F",
                    "F","F","F","F","F","F","F","F","F","F","F","F"), 
                    c("P","P","P","F","F","F","P","P","P","P","P","P",
                    "P","P","P","P","P","P","P","P","P","P","P","P"), 
                    c("P","P","P","F","F","F","F","F","F","F","F","F",
                    "P","P","P","P","P","P","F","F","F","F","F","F"), 
                    c("P","P","P","F","F","F","P","P","P","P","P","P",
                    "P","P","P","P","P","P","F","F","F","F","F","F"), 
                    c("P","P","P","F","F","F","F","F","F","P","P","P",
                    "F","F","F","F","F","F","F","F","F","F","F","F"), 
                    c("P","P","P","F","F","F","F","F","F","P","P","P",
                    "P","P","P","P","P","P","P","P","P","P","P","P"), 
                    c("P","P","P","F","F","F","F","F","F","P","P","P",
                    "P","P","P","P","P","P","F","F","F","F","F","F"))

colnames(assignments) <- rownames(performanceTable)

majorityRules <- c("V","D","v","d","dV","Dv","dv")

for(i in 1:7)
{
  ElectreAssignments<-LPDMRSort(performanceTable, profilesPerformances, 
                                categoriesRanks,
                                weights, criteriaMinMax, lambda, 
                                criteriaVetos=vetoPerformances,
                                criteriaDictators=dictatorPerformances,
                                majorityRule = majorityRules[i])

  print(all(ElectreAssignments == assignments[i,]))
}