# Identification of profiles, weights, majority threshold and veto and dictator thresholds for the MRSort sorting approach extended to handle large performance differences. 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. The identification of the profiles, weights, majority threshold and veto and dictator thresholds are done by taking into account assignment examples. ```r LPDMRSortInferenceExact( performanceTable, assignments, categoriesRanks, criteriaMinMax, majorityRule = "M", readableWeights = FALSE, readableProfiles = FALSE, minmaxLPD = FALSE, alternativesIDs = NULL, criteriaIDs = NULL ) ``` ## 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). - `assignments`: Vector containing the assignments (IDs of the categories) of the alternatives to the categories. The elements are named according to the alternatives. - `categoriesRanks`: Vector containing the ranks of the categories. The elements are named according to the IDs of the categories. - `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. - `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. - `readableWeights`: Boolean parameter indicating whether the weights are to be spaced more evenly or not. - `readableProfiles`: Boolean parameter indicating whether the profiles are to be spaced more evenly or not. - `minmaxLPD`: Boolean parameter indicating whether the veto thresholds are to be minimized (or maximized if lower criteria values are preferred) while the dictator thresholds are to be maximized (or minimized if lower criteria values are preferred). - `alternativesIDs`: Vector containing IDs of alternatives, according to which the data should be filtered. - `criteriaIDs`: Vector containing IDs of criteria, according to which the data should be filtered. ## Returns The function returns a list structured as follows : - **lambda**: The majority threshold. - **weights**: A vector containing the weights of the criteria. The elements are named according to the criteria IDs. - **profilesPerformances**: A matrix containing the lower profiles of the categories. The columns are named according to the criteria, whereas the rows are named according to the categories. The lower profile of the lower category can be considered as a dummy profile. - **vetoPerformances**: A matrix containing the veto profiles of the categories. The columns are named according to the criteria, whereas the rows are named according to the categories. The veto profile of the lower category can be considered as a dummy profile. - **solverStatus**: The solver status as given by glpk. ## Examples ```r # 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)) 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") colnames(performanceTable) <- 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 1:1)# change to 7 in order to perform all tests { x<-LPDMRSortInferenceExact(performanceTable, assignments[i,], categoriesRanks, criteriaMinMax, majorityRule = majorityRules[i], readableWeights = TRUE, readableProfiles = TRUE, minmaxLPD = TRUE) ElectreAssignments<-LPDMRSort(performanceTable, x$profilesPerformances, categoriesRanks, x$weights, criteriaMinMax, x$lambda, criteriaVetos=x$vetoPerformances, criteriaDictators=x$dictatorPerformances, majorityRule = majorityRules[i]) print(x) print(all(ElectreAssignments == assignments[i,])) } ``` ## References Bouyssou, D. and Marchant, T. An axiomatic approach to noncompen- satory 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.

LPDMRSortInferenceExact function