Package for calculation of classic and Fuzzy AHP

Description

FuzzyAHP is an open source (LGPL 3) package for R. The package is only suitable for AHP that uses categorical rating of criteria for alternatives instead of pairwise comparison of alternatives according to each criteria. This adaptation of AHP is common in situations when the number of alternatives is hight and the pairwise comparison is thus inadequate or impossible to construct. The weights for criteria are, however, still determined from the pairwise comparison matrix. This approach towards AHP is common in Geosciences as well as other fields.

The determination of criteria weights is done according to process described by Krejčí, Pavlačka, and Talašová (2016), which yelds significantly narrower fuzzy numbers than previously used approaches.

Details

Please see vignettes for more details about the package and examples of use.

Complete list of classes and methods call help(package="FuzzyAHP").

Author(s)

Jan Caha cahik@atlas.cz, with contributions from Aneta Drážná

References

Krejčí, Jana, Ondřej Pavlačka, and Jana Talašová. 2016. “A fuzzy extension of Analytic Hierarchy Process based on the constrained fuzzy arithmetic.” Fuzzy Optimization and Decision Making. doi:10.1007/s10700-016-9241-0.

Class "FuzzyData"

Description

An S4 class to represent fuzzy data.

Slots

fnMin

A numeric vector of minimal values of fuzzy data.

fnModal

A numeric vector of modal values of fuzzy data.

fnMax

A numeric vector of maximal values of fuzzy data.

Class "FuzzyPairwiseComparisonMatrix"

Description

An S4 class to represent a fuzzy pairwise comparison matrix.

Slots

fnMin

A matrix of minimal values of fuzzy preferences.

fnModal

A matrix of modal values of fuzzy preferences.

fnMax

A matrix of maximal values of fuzzy preferences.

variableNames

Names of variables in the pariwise comparison matrix obtained either as colnames or rownames.

Class "FuzzyWeights"

Description

An S4 class to represent fuzzy weights for fuzzy AHP calculation.

Slots

fnMin

Object of class numeric containing minimal values of fuzzy weights.

fnModal

Object of class numeric containing modal values of fuzzy weights.

fnMax

Object of class numeric containing maximal values of fuzzy weights.

Class "PairwiseComparisonMatrix"

Description

An S4 class to represent a pairwise comparison matrix.

Slots

valuesChar

A pairwise comparison matrix based on Saaty's method as characters.

values

A pairwise comparison matrix based on Saaty's method as numeric.

variableNames

Names of variables in the pariwise comparison matrix obtained either as colnames or rownames.

Class "Weights"

Description

An S4 class representing weights for AHP calculation. Each value in numeric vector represents one weight.

Slots

weights

Object of class numeric containing weights.

Function that binds two FuzzyData together into one FuzzyData

Description

This methods construct object FuzzyData based on two FuzzyData. The functions merges the sources into single output. This method should be used in situations when both weights and input data are fuzzy.

Usage

bindColums(data1, data2)

## S4 method for signature 'FuzzyData,FuzzyData'
bindColums(data1, data2)

Arguments

Value

An object of class FuzzyData

Function that builds Fuzzy Pairwise Comparison Matrix based on list of Pairwise Comparison Matrices

Description

This functions builds Fuzzy Pairwise Comparison Matrix based on list of Pairwise Comparison Matrices the resulting Fuzzy Pairwise Comparison Matrix is calculated as minimum, geometric mean and maximum of each cell of all Pairwise Comparison Matrices in listOfMatrices.

Usage

buildFuzzyPairwiseComparisonMatrix(listOfMatrices)

## S4 method for signature 'list'
buildFuzzyPairwiseComparisonMatrix(listOfMatrices)

Arguments

Value

An object of class FuzzyPairwiseComparisonMatrix

Function that builds Pairwise Comparison Matrix based on list of Pairwise Comparison Matrices

Description

This functions builds Pairwise Comparison Matrix based on list of Pairwise Comparison Matrices the resulting Pairwise Comparison Matrix is calculated as geometric mean of all Pairwise Comparison Matrices in listOfMatrices.

Usage

buildPairwiseComparisonMatrix(listOfMatrices)

## S4 method for signature 'list'
buildPairwiseComparisonMatrix(listOfMatrices)

Arguments

Value

An object of class PairwiseComparisonMatrix

Function to calculate result of AHP

Description

This function calculates output of AHP based on Weights or FuzzyWeights on data represented either by matrix or FuzzyData.

Usage

calculateAHP(weights, data)

## S4 method for signature 'Weights,matrix'
calculateAHP(weights, data)

## S4 method for signature 'FuzzyWeights,matrix'
calculateAHP(weights, data)

## S4 method for signature 'FuzzyWeights,FuzzyData'
calculateAHP(weights, data)

## S4 method for signature 'PairwiseComparisonMatrix,matrix'
calculateAHP(weights, data)

## S4 method for signature 'FuzzyPairwiseComparisonMatrix,matrix'
calculateAHP(weights, data)

## S4 method for signature 'FuzzyPairwiseComparisonMatrix,FuzzyData'
calculateAHP(weights, data)

Arguments

Value

Either a matrix (if Weights and matrix were used as inputs) or FuzzyData (if FuzzyWeights were used).

Function to calculate fuzzy weights based on comparison matrix

Description

This functions calculates Weights or FuzzyWeights based on input pairwise comparison matrix.

Usage

calculateWeights(comparisonMatrix)

## S4 method for signature 'PairwiseComparisonMatrix'
calculateWeights(comparisonMatrix)

## S4 method for signature 'FuzzyPairwiseComparisonMatrix'
calculateWeights(comparisonMatrix)

Arguments

References

KREJČÍ, Jana, PAVLAČKA, Ondřej and TALAŠOVÁ, Jana, 2016, A fuzzy extension of Analytic Hierarchy Process based on the constrained fuzzy arithmetic. Fuzzy Optimization and Decision Making. 2016. DOI 10.1007/s10700-016-9241-0.

See Also

PairwiseComparisonMatrix-class

Function to calculate fuzzy weights based on comparison matrix using older approaches

Description

This functions calculates FuzzyWeights based on input fuzzy pairwise comparison matrix.

Usage

calculateWeights_old_methods(comparisonMatrix, type = "Chang")

## S4 method for signature 'FuzzyPairwiseComparisonMatrix'
calculateWeights_old_methods(comparisonMatrix, type = "Chang")

Arguments

Value

FuzzyWeights. If fuzzy weighting vector is to be obtained please see calculate_weighting_vector function.

References

CHANG, Da-Yong, 1996, Applications of the extent analysis method on fuzzy AHP. European Journal of Operational Research. 1996. Vol. 95, no. 3, p. 649–655. DOI 10.1016/0377-2217(95)00300-2.

TESFAMARIAM, Solomon and SADIQ, Rehan, 2006, Risk-based environmental decision-making using fuzzy analytic hierarchy process (F-AHP). Stochastic Environmental Research and Risk Assessment. 2006. Vol. 21, no. 1, p. 35–50. DOI 10.1007/s00477-006-0042-9.

WANG, Tien-Chin and CHEN, Yueh-Hsiang, 2008, Applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy AHP. Information Sciences [online]. October 2008. Vol. 178, no. 19, p. 3755–3765. DOI 10.1016/j.ins.2008.05.028.

See Also

calculate_weighting_vector

Function to calculate fuzzy weighting vector

Description

This functions calculates fuzzy weighting vector from FuzzyWeights. The calculation was first described by Chang (1996).

Usage

calculate_weighting_vector(fuzzyWeights)

## S4 method for signature 'FuzzyWeights'
calculate_weighting_vector(fuzzyWeights)

Arguments

Value

weighting vector for defined FuzzyWeights.

References

CHANG, Da-Yong, 1996, Applications of the extent analysis method on fuzzy AHP. European Journal of Operational Research. 1996. Vol. 95, no. 3, p. 649–655. DOI 10.1016/0377-2217(95)00300-2.

Function to mutually compare fuzzy data

Description

This function compares fuzzy data FuzzyData to identify the optimal (best) cases. The FuzzyData can only contain one fuzzy number, otherwise the comparison is not possible. The calculation of type "possibilityTheory" can be time consuming as it is based on comparing each fuzzy number to a maximal fuzzy number obtained from the set of fuzzy numbers.

Usage

compareFuzzyNumbers(fuzzyData, type, progressBar = FALSE)

## S4 method for signature 'FuzzyData,character'
compareFuzzyNumbers(fuzzyData, type, progressBar = FALSE)

Arguments

Value

A "matrix" of numeric value in case of "Chen" method or a "matrix" with two values in case of "possibilityTheory".

Function to rank results

Description

This function ranks data from the highest value to the lowest. Essentially it does (nrow(data)+1) - rank(data, na.last ="keep", ties.method= "max").

Usage

compareResults(data)

## S4 method for signature 'matrix'
compareResults(data)

Arguments

Value

A "matrix" of numeric value that indicates ranking of each row, with the highest value ranked as 1.

Function to determine Consistency Index

Description

This methods calculates Consistency index for PairwiseComparisonMatrix.

Usage

consistencyIndex(comparisonMatrix)

## S4 method for signature 'PairwiseComparisonMatrix'
consistencyIndex(comparisonMatrix)

## S4 method for signature 'FuzzyPairwiseComparisonMatrix'
consistencyIndex(comparisonMatrix)

Arguments

Value

A numeric value of Consistency index.

Function to determine Consistency Ratio

Description

This methods calculates Consistency Ratio for PairwiseComparisonMatrix. The consistency ratio can only be provided for PairwiseComparisonMatrix with less than 10 rows. For bigger matrices the value is not known.

Usage

consistencyRatio(comparisonMatrix, print.report = TRUE)

## S4 method for signature 'PairwiseComparisonMatrix'
consistencyRatio(comparisonMatrix, print.report = TRUE)

## S4 method for signature 'FuzzyPairwiseComparisonMatrix'
consistencyRatio(comparisonMatrix, print.report = TRUE)

Arguments

Details

Generally pairwise comparison matrixes are considered to be consistent if the value of Consistency Ration is smaller than 0.1. For matrices comparing more then 10 elements then Consistency Ratio is unsuitable, because the values of random index, that is necessary to obtain Consistency Ratio, are only known for matrixes with size smaller than 10\times10.

Value

A numeric value of Consistency Ratio, for PairwiseComparisonMatrix with more than 10 an error is raised.

Function to defuzzify fuzzy data

Description

This function defuzzifies FuzzyData into single value. The FuzzyData can only contain one fuzzy number, otherwise the defuzzification is not possible.

Usage

defuzziffy(fuzzyData, type = "mean")

## S4 method for signature 'FuzzyData'
defuzziffy(fuzzyData, type = "mean")

Arguments

Details

The triangular fuzzy numbers \tilde{A} represented as triplet [a1,a2,a3] are defuzzified using these methods: "Yager":

def (\tilde{A}) = \frac{(a2-a1)(a1+2/3(a2-a1)+(a3-a2)(a2+1/3(a3-a2))}{(a2-a1)+(a3-a2))}

"modalvalue":

def (\tilde{A}) = a2

"modalValueDominancy":

def(\tilde{A}) = (a1 + 4*a2 + a3)/6

"mean":

def(\tilde{A}) = (a1 + a2 + a3)/3

.

Value

A numeric value of defuzzified value, based on deffuzification method.

Function that creates FuzzyData

Description

This methods construct object FuzzyData based on provided matrix. The matrix needs to be have rows represent individual fuzzy numbers and three colums that represent minimal, modal and maximal value of fuzzy number.

Usage

fuzzyData(data, single.value = TRUE)

## S4 method for signature 'matrix'
fuzzyData(data, single.value = TRUE)

Arguments

Value

An object of class FuzzyData

See Also

FuzzyData

Function that creates Fuzzy Pairwise Comparions Matrix

Description

This methods construct object FuzzyPairwiseComparisonMatrix based on provided PairwiseComparisonMatrix and an optional fuzzy scale.

Usage

fuzzyPairwiseComparisonMatrix(pairwiseComparisonMatrix, fuzzyScale,
comparsionNotInScale, width)

## S4 method for signature 'PairwiseComparisonMatrix'
fuzzyPairwiseComparisonMatrix(
  pairwiseComparisonMatrix,
  fuzzyScale = getFuzzyScale(type = "full"),
  comparsionNotInScale = FALSE,
  width = 1
)

## S4 method for signature 'matrix'
fuzzyPairwiseComparisonMatrix(pairwiseComparisonMatrix)

Arguments

Details

Parameter fuzzyScale is expected as a vector containing n*3 values that represent triangular fuzzy numbers used as fuzzy intensity of importance (only the values equal or higher than 1, inverse values are calculated automatically). The values need to be ordered by fuzzy values. Default value of this parameter is as.double(c(1/2,1,2,1,2,3,2,3,4,3,4,5,4,5,6,5,6,7,6,7,8,7,8,9,8,9,9)). Another possibility is eg. as.double(c(1/3,1,3,1,3,5,3,5,7,5,7,9,7,9,9))

If param pairwiseComparisonMatrix is matrix then it needs to be of character type. Each element in the matrix must be specified as triplet "x;y;z", where x<=y<=z. From this matrix a pairwise comparison is constructed from y values and x and z function as lower and upper limits of y respectively. In this case the optional parameter fuzzyScale is not taken into account at all.

Value

Object of class FuzzyPairwiseComparisonMatrix

Function that creates Fuzzy Pairwise Comparions Matrix

Description

This methods construct object FuzzyPairwiseComparisonMatrix based on provided PairwiseComparisonMatrix and two matrices that form lower an upper significant values of the PairwiseComparisonMatrix that form middle significant value.

Usage

fuzzyPairwiseComparisonMatrix1(lowerValues, pairwiseComparisonMatrix, upperValues)

## S4 method for signature 'matrix,PairwiseComparisonMatrix,matrix'
fuzzyPairwiseComparisonMatrix1(
  lowerValues,
  pairwiseComparisonMatrix,
  upperValues
)

Arguments

Details

This function allows user to specify fuzzy pairwise comparison matrix that is not based on fuzzy scale but rely more on user's specification. The middle significant values have to be definied by PairwiseComparisonMatrix to ensure some elementary properties. The significant values provided to this function have to be correctly ordered and fuzzy numbers have to be reciprocal otherwise the function fails.

Value

Object of class FuzzyPairwiseComparisonMatrix

Function to extract specific fuzzy numbers

Description

This methods helps with extracting fuzzy numbers from FuzzyData and FuzzyWeights.

Usage

getFuzzyNumber(object, index)

## S4 method for signature 'FuzzyData,integer'
getFuzzyNumber(object, index)

## S4 method for signature 'FuzzyWeights,integer'
getFuzzyNumber(object, index)

Arguments

Value

A matrix where rows are fuzzy numbers and columns are important values.

Function to create Fuzzy Scale

Description

This methods creates fuzzy scale that are used while fuzzifying Piecewise comparison matrix.

Full scale is:

Basic scale is:

Usage

getFuzzyScale(type)

## S4 method for signature 'character'
getFuzzyScale(type)

Arguments

Value

A matrix representing the fuzzy scale.

Function that inserts FuzzyData on specific column number in another FuzzyData

Description

This methods construct object FuzzyData based on two FuzzyData and index. The functions merges the sources into single output. It inserts data2 onto specific position (defined by index) in data1.

Usage

insertInto(data1, data2, index)

## S4 method for signature 'FuzzyData,FuzzyData'
insertInto(data1, data2, index)

Arguments

Value

An object of class FuzzyData

Examples

values = (c(1,2,3,4,5,6,7,8,9))
values = matrix(values, nrow = 3, ncol = 3, byrow = TRUE)
fData = fuzzyData(values, single.value = FALSE)

v = (c(15,16,17))
v = matrix(v, nrow = 3, ncol = 1, byrow = TRUE)
fd = fuzzyData(v, single.value = FALSE)

fData = insertInto(fData, fd, 2)

Function that creates Pairwise Comparions Matrix

Description

This methods construct object PairwiseComparisonMatrix based on provided matrix. The matrix needs to be square and reciprocal with the intensity of importance (comparisons). Since the version 0.6.9 the comparsions can be represented as either characters (e.g. "1", "9", "1/9") or numeric (e.g. 1, 9, 1/9) .

Usage

pairwiseComparisonMatrix(matrix)

## S4 method for signature 'matrix'
pairwiseComparisonMatrix(matrix)

Arguments

Value

An object of class PairwiseComparisonMatrix

See Also

PairwiseComparisonMatrix

Examples

comparisonMatrixValues = c("1","9","5","1/9","1","1/3","1/5","3","1")
comparisonMatrix = matrix(comparisonMatrixValues, nrow = 3, ncol = 3, byrow = TRUE)
matrix = pairwiseComparisonMatrix(comparisonMatrix)

comparisonMatrixValues = c(1,9,5,1/9,1,1/3,1/5,3,1)
comparisonMatrix = matrix(comparisonMatrixValues, nrow = 3, ncol = 3, byrow = TRUE)
matrix = pairwiseComparisonMatrix(comparisonMatrix)

Function to assess strict consistency of Comparison Matrix

Description

Check if

a_{ik} = a_{ij} \times a_{jk}

applies for all i,j,k = 1,2,\dots,n, where n is size of a.

Usage

strictConsistency(PairwiseComparisonMatrix, print.report = TRUE)

## S4 method for signature 'FuzzyPairwiseComparisonMatrix'
strictConsistency(PairwiseComparisonMatrix, print.report = TRUE)

## S4 method for signature 'PairwiseComparisonMatrix'
strictConsistency(PairwiseComparisonMatrix, print.report = TRUE)

Arguments

Value

Boolean value indicating if Comparison Matrix passed the weak consistency test and a warning message listing the problematic triplets if the matrix is not consisten.

Function to create text representation of (Fuzzy) Pairwise comparison matrix

Description

This methods creates text representation of Pairwise comparison matrices.

Usage

textRepresentation(x, whole = TRUE)

## S4 method for signature 'FuzzyPairwiseComparisonMatrix'
textRepresentation(x, whole = TRUE)

## S4 method for signature 'PairwiseComparisonMatrix'
textRepresentation(x, whole = TRUE)

Arguments

Value

A dataframe of character type.

Function to assess Weak Consistency of Comparison Matrix

Description

Check if for a_{ij}>1,a_{jk}>1 applies that

a_{ik}>=\max(a_{ij},a_{jk})

for all i,j,k = 1,2,\dots,n, where n is size of a.

Usage

weakConsistency(PairwiseComparisonMatrix, print.report = TRUE)

## S4 method for signature 'FuzzyPairwiseComparisonMatrix'
weakConsistency(PairwiseComparisonMatrix, print.report = TRUE)

## S4 method for signature 'PairwiseComparisonMatrix'
weakConsistency(PairwiseComparisonMatrix, print.report = TRUE)

Arguments

Value

Boolean value indicating if Comparison Matrix passed the weak consistency test and a warning message listing the problematic triplets if the matrix is not consisten.