Package {mcdabench}

Benchmarking for Multi-Criteria Decision Analysis Methods

Description

Comparative benchmarking of Multi-Criteria Decision Analysis (MCDA) methods. These methods are designed to evaluate and rank alternatives based on multiple criteria, applying normalization, weighting, and aggregation techniques. The package includes popular decision-making methods such as MEGAN, TOPSIS, VIKOR, WASPAS, WSM, WPM, and others, facilitating flexible and efficient analyses for multi-criteria problems.

Details

The mcdabench package aims to simplify the application of MCDA methods for researchers, analysts, and decision-makers. The package includes functionality for:

• Data normalization (e.g., "maxmin", "sum", "linear")

• Weight calculation using various techniques ("equal", "entropy", "critic", etc.)

• Methods for ranking alternatives based on multiple criteria:

  • edas - Evaluation based on Distance from Average Solution

  • mabac - Multi-Attributive Border Approximation Area Comparison

  • marcos - Measurement of Alternatives and Ranking According to Compromise Solution

  • ram - Root Assessment Method

  • topsis - Technique for Order of Preference by Similarity to Ideal Solution

  • vikor - Compromise ranking based on utility and regret measures

  • waspas - Combines Weighted Sum Model (WSM) and Weighted Product Model (WPM)

  • wsm - Weighted Sum Method

  • wpm - Weighted Product Method

• Flexible support for benefit-cost criteria, customizable weight schemes, and normalization methods.

This package is suited for applications in various decision-making contexts, such as supply chain management, project evaluation, and resource allocation.

For detailed examples and method-specific information, refer to the help pages of individual functions.

Value

No return value. This documentation describes the mcdabench package and its functionality.

Author(s)

Cagatay Cebeci

Examples

data(egrids)
dm <- as.matrix(egrids$dmat) # Decision matrix
bc <- egrids$bcvec # Benefit and cost vector
wei <- egrids$weights # Weigths of the criteria

# Perform MEGAN analysis
resmegan <- megan(dmatrix=dm, bcvec=bc, weights=wei, thr=0)
print(resmegan$superiority)  # MEGAN superiority scores
print(resmegan$rank) # Ranks of alternatives by MEGAN

# Perform VIKOR analysis
resvikor <- vikor(dmatrix=dm, bcvec=bc, weights=wei, v=0.5)
print(resvikor$Q)  # VIKOR compromise scores
print(resvikor$rank)  # Ranks of alternatives by VIKOR

# Perform TOPSIS analysis
restopsis <- topsis(dmatrix=dm, bcvec=bc, weights=wei)
print(restopsis$p)  # TOPSIS closeness coefficient
print(restopsis$rank) # Ranks of alternatives by TOPSIS

AHP: Analytic Hierarchy Process

Description

Computes criteria weights using the Analytic Hierarchy Process (AHP). Supports multiple weighting methods and consistency ratio calculation.

Usage

ahp(pmatrix, method = "maxeig", weights = NULL, tiesmethod = "average")

Arguments

Details

The AHP method calculates weights based on a comparison matrix. If weights are provided, direct ranking is performed without AHP computation. Otherwise, weights are computed using method, with "maxeig" using the principal eigenvector. The consistency ratio (CR) is also calculated to assess decision matrix reliability.

Value

A list with the following components:

Author(s)

Cagatay Cebeci

References

Saaty, T. L. (1990). How to make a decision: the analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26. <doi:10.1016/0377-2217(90)90057-I>

Examples

dataset <- matrix(c(
  1, 6, 2, 5, 7,   
  1/6, 1, 4, 8, 3,  
  1/2, 1/4, 1, 9, 6,  
  1/5, 1/8, 1/9, 1, 2,  
  1/7, 1/3, 1/6, 1/2, 1  
), nrow=5, byrow=TRUE)

resahp_1 <- ahp(dataset, method="maxeig")
print(resahp_1)

resahp_2 <- ahp(dataset, method="mean")
print(resahp_2)

ARAS: Additive Ratio Assessment Method

Description

The ARAS method, developed by Zavadskas and Turskis in 2010, aims to evaluate and rank alternatives based on given criteria using normalization, weighting, and utility degree computation. Due to its simple calculation procedures and clear logic, ARAS is widely used in multi-criteria decision-making (MCDA) problems.

Usage

aras(dmatrix, bcvec, weights, tiesmethod="average")

Arguments

Details

The ARAS method extends the decision matrix with ESP values, calcnormals the scores, applies weights, calculates utility values (S) and relative utilities (K) for ranking alternatives.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Zavadskas, E. K., & Turskis, Z. (2010). A new additive ratio assessment (ARAS) method in multicriteria decision-making. Technological and Economic Development of Economy, 16(2), 159-172. <doi:10.3846/tede.2010.10>.

Examples

# Sample decision matrix
dmat <- matrix(c(
  84, 10, 5, 20,  
  66, 18, 8, 15,
  74, 12, 6, 25,
  90, 22, 9, 18, 
  68, 18, 7, 15  
), nrow = 5, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4", "A5")
colnames(dmat) <- c("C1", "C2", "C3", "C4")

# Benefit-cost vector (1: benefit, -1: cost)
bc <- c(1, -1, 1, -1)

# User-defined weights
uw <- c(0.1, 0.4, 0.3, 0.2)

# Apply ARAS with user-defined weights
resaras_1 <- aras(dmatrix=dmat, bcvec=bc, weights=uw)
print(resaras_1)

# Apply ARAS with CRITIC weights
criwei <- calcweights(dmatrix=dmat, bcvec=bc, type="critic", normethod="ratio")

resaras_2 <- aras(dmatrix=dmat, bcvec=bc, weights=criwei)
print(resaras_2$Ki)    # Relative utilities
print(resaras_2$rank)  # Ranks of relative utilities

AROMAN: Alternative Ranking Order Method Accounting for Two Step Normalization

Description

AROMAN applies a two-step normalization process to rank alternatives based on multiple criteria, combining max-min and vector normalization methods.

Usage

aroman(dmatrix, bcvec, weights, beta = 0.5, lambda = 0.5, tiesmethod="average")

Arguments

Details

The decision matrix dmatrix is normalized using both max-min and vector normalization techniques. A weighted normalized matrix is created using the given criteria weights. The aggregated average normalization values are calculated.The sum of weighted normalized values is computed separately for benefit and cost criteria. A final ranking score (R_i) is calculated using lambda weighting. Alternatives are ranked based on decreasing R_i values.

Value

A list containing:

Author(s)

Cagatay Cebeci

References

Bošković, S., Švadlenka, L., Jovčić, S., Dobrodolac, M., Simić, V., & Bacanin, N. (2023). An alternative ranking order method accounting for two-step normalization (AROMAN) - A case study of the electric vehicle selection problem. IEEE Access, 11, 39496-39507. <doi:10.1109/ACCESS.2023.3265818>

Examples

# Define the decision matrix
dmat <- matrix(c(
  10, 5, 8,
  6, 9, 7,
  8, 7, 9,
  5, 8, 6
), nrow = 4, byrow = TRUE)

# Profit-cost vector
bc <- c(1, -1, 1)

# Weights vector
uw <- c(0.4, 0.3, 0.3)

# Apply AROMAN
resaroman <- aroman(dmatrix = dmat, bcvec = bc, weights = uw, 
    beta = 0.6, lambda = 0.5)
print(resaroman)

MCDA Boxplot

Description

A function to create boxplots for a Multi-criteria Decision Analysis (MCDA) data matrices with customizable labels, colors, and additional arguments.

Usage

boxplotmcda(datamatrix, labels = NULL, colors = NULL, mt = NULL, ...)

Arguments

Details

This function generates boxplots for the columns of the given data matrix. It allows customization through labels, colors, and additional boxplot arguments. The labels are displayed on the x-axis, and a grid is added for better visual clarity.

Value

No return value. This function is called for its side effect of producing a boxplot visualization of the MCDA data matrix.

Author(s)

Cagatay Cebeci

Examples

set.seed(123)
datamatrix <- matrix(runif(50, 0, 1), nrow = 10, ncol = 5)
bcvec <- c(-1, 1, 1, -1)

# Data boxplot with default labels and colors
boxplotmcda(datamatrix)

# Data boxplot with custom labels and colors
labels <- c("Criteria 1", "Criteria 2", "Criteria 3", "Criteria 4", "Criteria 5")
colors <- c("red", "blue", "green", "purple", "orange")
boxplotmcda(datamatrix, labels = labels, colors = colors)

# Normalized data boxplot with with default labels and colors
nmatrix <- calcnormal(X=datamatrix, bcvec=bcvec, type="maxmin")
boxplotmcda(nmatrix, mt="MaxMin Normalized Matrix")

Normalization of Decision Matrices

Description

This function provides various normalization techniques to preprocess decision matrices for multi-criteria decision-making methods.

Usage

calcnormal(X, bcvec, type = "maxmin")

Arguments

Details

The supported normalization methods are:

"null"

No normalization.

"maxmin"

Max-Min normalization. Based on Ersoy (2021); Baydaş et al. (2024).

"vector"

Vector normalization. Referenced in Nijkamp & van Delft (1977); Milani et al. (2005).

"sum"

Sum normalization. Jee & Kang (2000); Milani et al. (2005); ÖWang & Luo (2010); Stanujkic et al. (2013); Ersoy (2021).

"zavadskas"

Zavadskas-Turskis normalization. Zavadskas & Turskis (2011).

"enhanced"

Enhanced accuracy normalization. Proposed by Zeng et al. (2013).

"linear"

Linear normalization.

"nonlinear"

Non-linear normalization.

"logarithmic"

Logarithmic normalization.

"ratio"

Linear ratio normalization.

Value

A normalized decision matrix of the same dimension as X, the original decision matrix.

Author(s)

Cagatay Cebeci

References

Baydas, M., Yilmaz, M., Jovic, Z., Stevic, Z., Ozuyar, S. E. G., & Ozcil, A. (2024). A comprehensive MCDM assessment for economic data: success analysis of maximum normalization, CODAS, and fuzzy approaches. Financial Innovation, 10(1), 105.

Ersoy, N. (2021). Selecting the best normalization technique for ROV method: Towards a real life application. Gazi University Journal of Science, 34(2), 592-609.

Jee, D. H., & Kang, K. J. (2000). A method for optimal material selection aided with decision making theory. Materials & Design, 21(3), 199-206.

Milani, A. S., Shanian, A., Madoliat, R., & Nemes, J. A. (2005). The effect of normalization norms in multiple attribute decision making models: a case study in gear material selection. Structural and Multidisciplinary Optimization, 29, 312-318.

Nijkamp, P., & van Delft, A. (1977). Multi-criteria analysis and regional decision-making (Vol. 8). Springer Science & Business Media.

Stanujkic, D., Djordjevic, B., & Djordjevic, M. (2013). Comparative analysis of some prominent MCDM methods: A case of ranking Serbian banks. Serbian Journal of Management, 8(2), 213-241.

Wang, Y. M., & Luo, Y. (2010). Integration of correlations with standard deviations for determining attribute weights in multiple attribute decision making. Mathematical and Computer Modelling, 51(1-2), 1-12.

Zavadskas, E. K., & Turskis, Z. (2011). Multiple criteria decision making (MCDM) methods in economics: an overview. Technological and Economic Development of Economy, 17(2), 397-427.

Zeng, Q. L., Li, D. D., & Yang, Y. B. (2013). VIKOR method with enhanced accuracy for multiple criteria decision making in healthcare management. Journal of Medical Systems, 37(2), 9908.

See Also

calcweights

Examples

data(egrids)
dmatrix <- egrids$dmat
bcvec <- egrids$bcvec

# Max-min normalization
normmaxmin <- calcnormal(X=dmatrix, bcvec=bcvec, type="maxmin")
print(normmaxmin)

# Vector normalization
normvector <- calcnormal(X=dmatrix, bcvec=bcvec, type="vector")
print(normvector)

# Zavadskas normalization
normzavad <- calcnormal(X=dmatrix, bcvec=bcvec, type="zavadskas")
print(normzavad)

Calculate Consensus Ranks using Various Methods

Description

This function calculates consensus ranks for a set of alternatives, given their rankings from multiple MCDM methods. It supports the Score method, Neustadt's method, and Buchholz's method for aggregating ranks.

Usage

calcranks(rankmatrix, rankmethod = "score")

Arguments

Value

A list containing the results of the ranking calculation:

• rankmethod: A character string indicating the ranking method used.

• rating_scores: A named numeric vector of the calculated rating scores for each alternative based on the chosen method.

• ordered_alternatives: A character vector of alternative names, ordered from the best-performing to the worst-performing according to the calculated rating_scores.

• score_ranks: A numeric vector of the final consensus ranks using the "minimum" ties method.

• average_score_ranks: A numeric vector of the final consensus ranks for each alternative the "average" tie method.

Author(s)

Cagatay Cebeci

References

Gogodze, J. (2019). Ranking‐Theory Methods for Solving Multicriteria Decision‐Making Problems. Advances in Operations Research, 2019(1), 3217949. <doi:10.1155/2019/3217949>

Examples

# Sample ranking matrix (rows are methods, columns are alternatives R1-R10)
rmat <- matrix(c(
  5, 10, 3, 9, 7, 8, 1, 2, 6, 4,
  10, 9, 6, 3, 7, 1, 5, 2, 4, 8,
  10, 9, 8, 4, 6, 1, 7, 5, 2, 3,
  8, 10, 4, 5, 9, 7, 1, 3, 2, 6,
  9, 7, 1, 8, 3, 4, 2, 6, 10, 5,
  8, 10, 5, 9, 6, 7, 1, 2, 3, 4,
  3, 1, 4, 6, 2, 7, 9, 10, 8, 5,
  2, 1, 5, 3, 4, 6, 8, 10, 9, 7,
  4, 1, 8, 2, 5, 3, 10, 9, 6, 7,
  2, 1, 2, 6, 4, 5, 3, 7, 9, 8,
  2, 1, 7, 4, 3, 5, 10, 9, 8, 6,
  7, 10, 3, 9, 5, 8, 1, 2, 6, 4,
  1, 2, 3, 7, 5, 10, 4, 6, 9, 8,
  4, 1, 6, 3, 2, 5, 10, 9, 8, 7
), nrow = 14, byrow = TRUE)

rownames(rmat) <- c("TOPSIS", "MAUT", "MACBETH", "ORESTE", "VIKOR", "SIR", "EVAMIX", 
   "ARAS", "MOORA", "COPRAS", "WASPAS", "TODIM", "MABAC", "MARE")
colnames(rmat) <- paste0("R", 1:10)

# Calculate ranks using the Score method
results_score <- calcranks(rmat, rankmethod = "score")
print(results_score)

# Calculate ranks using Neustadt's method
results_neustadt <- calcranks(rmat, rankmethod = "neustadt")
print(results_neustadt)

# Calculate ranks using Buchholz's method
results_buchholz <- calcranks(rmat, rankmethod = "buchholz")
print(results_buchholz)

Calculation of Criteria Weights

Description

This function calculates criteria weights using different methods such as Equal weighting, and weighting based on Angle, CRITIC, CILOS, Entropy, Geometric mean, GINI coefficient, MEREC, MPSI, ROC, RS, Standard deviation, and Variance.

Usage

calcweights(dmatrix, bcvec = NULL, type = "equal", normethod = NULL)

Arguments

Details

In multi-criteria decision-making methods, normalization is often necessary when calculating weights, but it depends on the specific approach used. Normalization ensures that criteria with different scales become comparable, which is crucial for accurate decision analysis. For instance, the entropy method applies normalization to make data scale-independent. Similarly, the CRITIC method typically requires normalization since it considers the variability and correlation between criteria. Additionally, various normalization techniques, such as Min-Max normalization, Z-score normalization, and vector normalization, can be employed depending on the method's requirements. Min-Max normalization scales data within a fixed range, making comparisons straightforward. Z-score normalization adjusts data based on mean and standard deviation, which helps in standardizing different distributions. Vector normalization converts values into unit vectors, ensuring proportional representation across criteria. Selecting the most appropriate normalization technique is key to ensuring accurate and meaningful results in MCDA. See link{normalize} for available methods of normalization.

The current version of function includes the following weighting methods:

angle

Angular weighting method

critic

CRITIC: CRiteria Importance Through Intercriteria Correlation method

cilos

CLIOS: Criterion Impact LOSs method

entropy

Entropy: Shannon entropy method

entropy2

Entropy method II

equal

Equal weighting

geom

Geometric mean based weighting

gini

GINI coefficient based weighting

merec

MEREC: MEthod based on the Removal Effects of Criteria)

mpsi

MPSI: Modified Preference Selection Index

roc

ROC: Rank Order Centroid weighting

rs

RS: Rank Sum weighting

sdev

Standard deviation-based weighting

var

Variance-based weighting

Value

A vector of weights corresponding to each column of the decision matrix.

Author(s)

Cagatay Cebeci

References

Diakoulaki, D., Mavrotas, G., & Koumoutsos, K. (1995). Determining objective weights in multiple criteria problems using the CRITIC method. Computers & Operations Research, 22(7), 763-770. <doi:10.1016/0305-0548(94)00059-H>

Ghorabaee, M. K., Zavadskas, E. K., Olfat, L., & Turskis, Z. (2017). Multi-criteria inventory classification using a novel method of evaluation based on ratios and comparisons (MEREC). Informatica, 28(3), 599-616.

Gligoric, M., Gligoric, Z., Lutovac, S., Negovanovic, M., & Langovic, Z. (2022). Novel hybrid MPSI-MARA decision-making model for support system selection in an underground mine. Systems, 10(6), 248. <doi:10.3390/systems10060248>

Keshavarz-Ghorabaee, M., Amiri, M., Zavadskas, E. K., Turskis, Z., & Antucheviciene, J. (2021). Determination of objective weights using a new method based on the removal effects of criteria (MEREC). Symmetry, 13(4), 525. <doi:10.3390/sym13040525>

Shuai, D., Zongzhun, Z., Yongji, W., & Lei, L. (2012, May). A new angular method to determine the objective weights. In 2012 24th Chinese Control and Decision Conference (CCDC) (pp. 3889-3892). IEEE.

Zavadskas, E. K., & Podvezko, V. (2016). Integrated determination of objective criteria weights in MCDM. International Journal of Information Technology & Decision Making, 15(2), 267-283. <doi:10.1142/S0219622016500036>

Zeleny, M. (1982). Multiple criteria decision making. McGraw-Hill.

Jia, J., Fischer, G. W., & Dyer, J. S. (1998). Attribute weighting methods and decision quality in the presence of response error: a simulation study. Journal of Behavioral Decision Making, 11(2), 85-105. <doi:10.1002/(SICI)1099-0771(199806)11:2<85::AID-BDM282>3.0.CO;2-K>

Examples

# Load the dataset
data(egrids)
dmat <- egrids$dmat  # Decision matrix
bc <- egrids$bcvec   # Benefit-cost vector

# Calculate weights using the equal weighting method
calcweights(dmatrix=dmat, type="equal")

# Calculate weights using the entropy method
calcweights(dmatrix=dmat, type="entropy")

# Calculate weights using the entropy method with MaxMin normalization
calcweights(dmatrix=dmat, bcvec=bc, type="entropy", normethod="maxmin")

# Calculate weights using the std. deviation method
calcweights(dmatrix=dmat, type="sdev")

# Calculate weights using the MEREC method
calcweights(dmatrix=dmat, bcvec=bc, type="merec")

# Calculate weights using the MPSI method
calcweights(dmatrix=dmat, bcvec=bc, type="mpsi")

# Calculate weights using the angular method
calcweights(dmatrix=dmat, bcvec=bc, type="angle")

COCOSO: Combined Compromise Solution Method

Description

COCOSO proposed by Yazdani et al (2019) is an MCDA method that normalizes the decision matrix and ranks alternatives based on specified criterion weights.

Usage

  cocoso(dmatrix, bcvec, weights, lambda = 0.5, tiesmethod="average")

Arguments

Value

Returns a list containing:

Author(s)

Cagatay Cebeci

References

Yazdani, M., Zarate, P., Kazimieras Zavadskas, E., & Turskis, Z. (2019). A combined compromise solution (CoCoSo) method for multi-criteria decision-making problems. Management Decision, 57(9), 2501-2519. <doi:10.1108/MD-05-2017-0458>

See Also

calcweights, calcnormal

Examples

# Sample decision matrix
dmat <- matrix(c(
  84, 10, 5, 20,  
  66, 18, 8, 15,
  74, 12, 6, 25,
  90, 22, 9, 18, 
  68, 18, 7, 15  
), nrow = 5, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4", "A5")
colnames(dmat) <- c("C1", "C2", "C3", "C4")

# Benefit-cost vector (1: benefit, -1: cost)
bc <- c(1, -1, 1, -1)

# User-defined weights
uw <- c(0.1, 0.4, 0.3, 0.2)

# Apply COCOSO with user-defined weights
rescocoso_1 <- cocoso(dmatrix=dmat, bcvec = bc, weights = uw, lambda=0.1)
print(rescocoso_1)

# Apply COCOSO with CRITIC weights
cw <- calcweights(dmatrix=dmat, bcvec=bc, type="critic", normethod="linear")
rescocoso_2 <- cocoso(dmatrix=dmat, bcvec = bc, weights = cw, lambda=0.8, tiesmethod="average")
print(rescocoso_2)

CODAS: COmbinative Distance-based ASsessment

Description

Implements the COmbinative Distance-based ASsessment (CODAS) method for multi-criteria decision analysis (Keshavarz et al, 2016). This method ranks alternatives based on a combination of Euclidean and Taxicab distances from the negative ideal solution.

Usage

codas(dmatrix, bcvec, weights, thr = 0.1, tiesmethod="average")

Arguments

Details

The CODAS method evaluates alternatives by calculating their distances from the negative ideal solution using both Euclidean and Taxicab metrics. A combinative appraisal score is then computed, giving more preference to the Euclidean distance. The alternatives are ranked based on these appraisal scores in descending order.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Keshavarz Ghorabaee, M., Zavadskas, E. K., Turskis, Z., & Antuchevičienė, J. (2016). A new combinative distance-based assessment (CODAS) method for multi-criteria decision-making. <https://etalpykla.vilniustech.lt/handle/123456789/116529>.

See Also

calcweights, calcnormal

Examples

# Decision matrix (5 alternatives, 4 criteria)
dmat <- matrix(c(
  84, 10, 5, 20,  
  66, 18, 8, 15,
  74, 12, 6, 25,
  90, 22, 9, 18, 
  68, 18, 7, 15  
), nrow = 5, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4", "A5")
colnames(dmat) <- c("C1", "C2", "C3", "C4")

# Benefit-cost vector (1: benefit, -1: cost)
bc <- c(1, -1, 1, -1)

# User-defined weights
uw <- c(0.1, 0.4, 0.3, 0.2)

# Apply CODAS with user-defined weights, maxmin normalization, and thr = 0.1
rescodas_1 <- codas(dmatrix=dmat, bcvec = bc, weights = uw)
print(rescodas_1)

# Apply CODAS with entropy weights
entwei <- calcweights(dmatrix=dmat, bcvec = bc, type="entropy", normethod="sum")
rescodas_2 <- codas(dmatrix=dmat, bcvec = bc, weights = entwei, thr = 0.5)
print(rescodas_2)

COPRAS: Complex Proportional Assessment

Description

Applies the COPRAS method for decision makings.

Usage

copras(dmatrix, bcvec, weights, normethod = "sum", tiesmethod = "average")

Arguments

Value

A list with the following components:

Author(s)

Cagatay Cebeci

References

Razavi Hajiagha, S. H., Hashemi, S. S., & Zavadskas, E. K. (2013). A complex proportional assessment method for group decision making in an interval-valued intuitionistic fuzzy environment. Technological and Economic Development of Economy, 19(1), 22-37. <doi:10.3846/20294913.2012.762953>

Examples

dmatrix <- matrix(c(
  10, 50, 30, 80,
  20, 60, 20, 70,
  30, 40, 40, 60,
  40, 30, 50, 50
), nrow=4, byrow=TRUE)

bcvec <- c(1, 1, -1, 1)
weights <- c(0.3, 0.4, 0.1, 0.2)

rescopras <- copras(dmatrix, bcvec, weights)
print(rescopras)

Correlation Matrix Visualization

Description

Visualizes a correlation or similarity matrix using heatmap-style plots, allowing customization of colors, shapes, axis labels, and text annotations.

Usage

corplot(cormat, labsize = 10, fsize = 3, fcolor = "black", 
   colpal = c("red", "white", "dodgerblue"), title = "Correlation Matrix", 
   xlab = "Coef", ylab = "Coef", shape = "square", displabels = TRUE)

Arguments

Value

No return value.

Examples

# Sample correlation matrix
set.seed(123)
df <- data.frame(
  X1 = rnorm(20), X2 = rnorm(20), X3 = rnorm(20), X4 = rnorm(20),
  X5 = rnorm(20), X6 = rnorm(20), X7 = rnorm(20), X8 = rnorm(20),
  X9 = rnorm(20), X10 = rnorm(20)
)
cormat <- cor(df, method = "spearman")

corplot(cormat, labsize = 12, fsize = 4, fcolor = "black", 
        title = "Spearman Correlation Matrix", xlab = "Methods", ylab = "Methods",
        shape = "square", displabels = TRUE)

EDAS: Evaluation based on Distance from Average Solution

Description

This function implements the method Evaluation based on Distance from Average Solution (EDAS) (Keshavarz et al, 2015) for multi-criteria decision-making by calculating distances from the average solution for each criterion, and evaluating alternatives based on positive and negative distances.

Usage

edas(dmatrix, bcvec, weights, tiesmethod="average")

Arguments

Details

The EDAS method evaluates alternatives based on their positive and negative distances from the average solution for each criterion. Positive distances (pda) measure how much better an alternative is compared to the average, while negative distances (nda) measure how much worse it is. These values are combined into scores for ranking alternatives.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Keshavarz Ghorabaee, M., Zavadskas, E. K., Olfat, L., & Turskis, Z. (2015). Multi-criteria inventory classification using a new method of evaluation based on distance from average solution (EDAS). Informatica, 26(3), 435-451.<doi: 10.3233/INF-2015-1070>.

See Also

calcweights

Examples

# Decision matrix (5 alternatives, 4 criteria)
dmat <- matrix(c(
  84, 10, 5, 20,  
  66, 18, 8, 15,
  74, 12, 6, 25,
  90, 22, 9, 18, 
  68, 18, 7, 15  
), nrow = 5, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4", "A5")
colnames(dmat) <- c("C1", "C2", "C3", "C4")

# Benefit-cost vector (1: benefit, -1: cost)
bc <- c(1, -1, 1, -1)

# User-defined weights
userwei <- c(0.1, 0.4, 0.3, 0.2)

# Apply EDAS with user-defined weights
resedas_1 <- edas(dmatrix=dmat, bcvec = bc, weights = userwei)
print(resedas_1)

# Apply EDAS with GINI weights
giniwei <- calcweights(dmatrix=dmat, bcvec = bc, type="gini")
resedas_2 <- edas(dmatrix=dmat, bcvec = bc, weights = giniwei)
print(resedas_2) 

System Configuration for Optimizing Smart Grids

Description

The egrids dataset contains a decision matrix (egrids), a benefit-cost vector (bcvec), and a weights vector (weights) designed for analyzing and ranking grid alternatives (Gs) using multi-criteria decision-making methods.

Usage

data(egrids)

Format

A list with the following components:

dmat

A numeric matrix where rows represent grid alternatives (G) and columns represent criteria. Criteria include:

• C1: Energy Efficiency (

• C2: System Reliability (

• C3: Renewable Integration (

• C4: Grid Stability Score. Higher is better

• C5: Environmental Impact (Score). Lower is better

• C6: Operational Cost ($M). Lower is better

• C7: Emissions (CO_2 tons/year). Lower is better

• C8: Response Time (seconds). Lower is better

• C9: Infrastructure Investment ($M). Lower is better

• C10: Maintenance Cost ($M). Lower is better

bcvec

A numeric benefit and cost vector specifying the type of each criterion:

• 1: Benefit criterion meaning higher value is better.

• -1: Cost criterion meaning lower value is better.

weights

A numeric vector containing the weights assigned to each criterion.

Details

This synthetic dataset provides basic criteria for evaluating and comparing smart grids using various multi-criteria decision analysis methods such as TOPSIS, VIKOR, WASPAS, MEGAN, and others. Each row in the decision matrix (dmat) corresponds to a different grid, while each column represents a specific criterion.

The bcvec vector identifies whether each criterion is a benefit (1) or a cost (-1), enabling proper handling in decision-making methods.

The weights vector contains the weights given to criteria.

Author(s)

Cagatay Cebeci

Examples

data(egrids)
dmat <- egrids$dmat # Decision matrix
bcvec <- egrids$bcvec # Benefit and cost vector
weights <- egrids$weights # Criteria weights

print(dmat) 
print(bcvec) 
print(weights) 

ELECTRE I (ELimination Et Choix Traduisant la REalité)

Description

Performs outranking analysis using the ELECTRE I method, identifying dominant alternatives based on concordance and discordance indices.

Usage

electre1(dmatrix, bcvec, weights, ct = 0.65, dt = 0.35, tiesmethod = "average")

Arguments

Details

ELECTRE I is the original outranking-based multi-criteria decision method. It constructs binary pairwise relations based on:

• Concordance index: weighted agreement over criteria supporting the preference of one alternative over another.

• Discordance index: the strongest opposition to that preference.

An alternative \(a_i\) outranks \(a_j\) only if:

• The concordance index is at least ct, and

• The discordance index is at most dt.

The final result includes the outranking relations and the kernel (alternatives that are not strongly dominated by others). Additionally, a basic ranking is computed based on how many alternatives each option outranks, using tiesmethod.

Value

A list with the following elements:

Author(s)

Cagatay Cebeci

See Also

electre2, electre3, electre4, calcnormal

Examples

dmatrix <- matrix(c(70, 3, 500,
                    90, 4, 400,
                    60, 2, 550), 
                  nrow = 3, byrow = TRUE)
rownames(dmatrix) <- c("A1", "A2", "A3")
bcvec <- c(1, 1, -1)
weights <- c(0.4, 0.3, 0.3)
result <- electre1(dmatrix, bcvec, weights)
result$Kernel
result$rank

ELECTRE II (ELimination Et Choix Traduisant la REalité)

Description

Implements the ELECTRE II outranking method to rank a set of alternatives based on concordance, discordance, and credibility analysis.

Usage

electre2(dmatrix, bcvec, weights, p = NULL, q = NULL, v = NULL, 
         st = 0.65, wt = 0.5, tiesmethod = "average")

Arguments

Details

ELECTRE II uses pairwise credibility analysis with two credibility thresholds to distinguish strong and weak outranking relations among alternatives.

The st and wt arguments define the strong and weak credibility cutoffs. While there is no fixed default specified in the original ELECTRE II methodology, values such as st = 0.65 and wt = 0.5 are commonly used in the literature due to their balance between discrimination and robustness.

However, users are encouraged to tune these thresholds based on the decision context, the number of alternatives, and how sensitive the comparison process should be. In highly uncertain decision environments, lower thresholds may be appropriate.

The final ranking is computed as the average rank between upward and downward outranking scores, providing a balanced assessment of dominance and vulnerability.

Value

A list with the following elements:

Author(s)

Cagatay Cebeci

See Also

electre3, electre4, calcnormal, rank

Examples

dmatrix <- matrix(c(70, 3, 500,
                    90, 4, 400,
                    60, 2, 550),
                  nrow = 3, byrow = TRUE)
rownames(dmatrix) <- c("A1", "A2", "A3")
bcvec <- c(1, 1, -1)
weights <- c(0.4, 0.3, 0.3)
result <- electre2(dmatrix, bcvec, weights)
result$rank

ELECTRE III (ELimination Et Choix Traduisant la REalité)

Description

The ELECTRE III method applies fuzzy outranking by evaluating concordance and discordance between alternatives using user-defined or default pseudo-criteria. It generates a credibility matrix and provides net scores and a partial ranking.

Usage

electre3(dmatrix, bcvec, weights, p = NULL, q = NULL, v = NULL, 
    thr = 0.5, tiesmethod = "average")

Arguments

Details

ELECTRE III uses the pseudo-criteria and fuzzy thresholds.

Value

A list with the following components:

Author(s)

Cagatay Cebeci

See Also

electre1, electre2, electre4, calcnormal

Examples

dmatrix <- matrix(c(70, 3, 500,
                    90, 4, 400,
                    60, 2, 550), 
                  nrow = 3, byrow = TRUE)
rownames(dmatrix) <- c("A1", "A2", "A3")
bcvec <- c(1, 1, -1)
weights <- c(0.4, 0.3, 0.3)
result <- electre3(dmatrix, bcvec, weights)
result$rank

ELECTRE IV (ELimination Et Choix Traduisant la REalité)

Description

The ELECTRE IV is a multi-criteria decision-making technique that ranks alternatives using concordance and discordance principles with progressive elimination.

Usage

electre4 (dmatrix, bcvec, weights, p = NULL, q = NULL, v = NULL, 
    thr = 0.5, tiesmethod = "average")

Arguments

Details

The ELECTRE IV method constructs concordance and discordance matrices based on pseudo-criteria thresholds. A credibility matrix is then derived, followed by a threshold-based filtering process to eliminate weaker alternatives.

Value

A list containing:

Author(s)

Cagatay Cebeci

See Also

calcweights, calcnormal

Examples

dmat <- matrix(c(5, 150, 3, 200, 4, 180), nrow = 3, byrow = TRUE)
bc <- c(-1, 1)
uw <- c(0.6, 0.4)

# Pseudo-criterion threshold values
pt <- list(list(p = 2), list(p = 30))
qt <- list(list(q = 1), list(q = 15))
vt <- list(list(v = 3), list(v = 45))
thr <- 0.6

reselectre4_1 <- electre4(dmatrix = dmat, bcvec = bc, weights = uw,
   p = pt, q = qt, v = vt, thr=thr)
print(reselectre4_1)

criwei <- calcweights(dmat, bcvec = bc, type = "critic")
reselectre4_2 <- electre4(dmatrix = dmat, bcvec = bc, weights = criwei)
print(reselectre4_2)

Visualize Dominance Flow of Ranked Alternatives

Description

Creates a directional plot illustrating the dominance flow among alternatives based on their ranks. Equal ranks are marked with "=" and dominance with ">".

Usage

flowplot(rankvec, colpal, txtcol = "black", orientation = "horizontal")

Arguments

Details

The function draws a graph in which each node represents an alternative. Arrows indicate flow of preference: a ">" means dominance, while "=" indicates tie. The layout is determined by the specified orientation. Edge labels are automatically positioned slightly above the edge direction based on flow geometry.

Value

No return value.

Author(s)

Cagatay Cebeci

Examples

rankvec <- c(1, 2, 3, 4, 5.5, 5.5, 7)
colpal <- rainbow(length(unique(rankvec)))

# Horizontal flow
flowplot(rankvec, colpal, orientation = "horizontal")

# Vertical flow
flowplot(rankvec, colpal, orientation = "vertical")

Fuzzy TOPSIS Method

Description

Computes rankings for decision alternatives using the Fuzzy TOPSIS method, which evaluates alternatives based on triangular fuzzy numbers (l, m, u).

Usage

ftopsis(fuzzydmatrix, bcvec, fuzzyweights, tiesmethod = "average")

Arguments

Value

A list containing:

Author(s)

Cagatay Cebeci

References

Nadaban, S., Dzitac, S., & Dzitac, I. (2016). Fuzzy TOPSIS: a general view. Procedia Computer Science, 91, 823-831. <doi:10.1016/j.procs.2016.07.088>

Examples

fuzzydmatrix <- matrix(c(
  8, 10, 12,    # A1-C1
  18, 20, 22,   # A2-C1
  28, 30, 32,   # A3-C1
  38, 40, 42,   # A4-C1
  45, 50, 55,   # A1-C2
  55, 60, 65,   # A2-C2
  35, 40, 45,   # A3-C2
  25, 30, 35,   # A4-C2
  28, 30, 32,   # A1-C3
  18, 20, 22,   # A2-C3
  38, 40, 42,   # A3-C3
  48, 50, 52,   # A4-C3
  75, 80, 85,   # A1-C4
  65, 70, 75,   # A2-C4
  55, 60, 65,   # A3-C4
  45, 50, 55    # A4-C4
), nrow = 4, byrow = TRUE)

bcvec <- c(1, 1, -1, 1) 

fuzzyweights <- matrix(c(
  0.25, 0.30, 0.35, 
  0.35, 0.40, 0.45, 
  0.05, 0.10, 0.15, 
  0.15, 0.20, 0.25  
), nrow = 4, byrow = TRUE)

resftopsis <- ftopsis(fuzzydmatrix, bcvec, fuzzyweights)
print(resftopsis)

FUCA: Faire Un Choix Adéquat (Make an Adequate Choice)

Description

Implements the Faire Un Choix Adéquat (FUCA) method that ranks alternatives based on the weighted sum of their ranks within each criterion for Multi-Criteria Decision Analysis (MCDA).

Usage

fuca(dmatrix, bcvec, weights, tiesmethod="average")

Arguments

Details

The Faire Un Choix Adéquat (FUCA), which translates from French to "Make an Adequate Choice," is a straightforward and intuitive MCDA method that avoids the need for data normalization (Fernando et al, 2010). The algorithm operates by ranking alternatives within each criterion and then aggregating these ranks using criterion weights.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Fernando, M. M. L., Escobedo, J. L. P., Azzaro-Pantel, C., Pibouleau, L., Domenech, S., & Aguilar-Lasserre, A. (2011). Selecting the best portfolio alternative from a hybrid multiobjective GA-MCDM approach for New Product Development in the pharmaceutical industry. In 2011 IEEE Symposium on Computational Intelligence in Multicriteria Decision-Making (MDCM) (pp. 159-166). IEEE.

See Also

calcweights

Examples

# Decision matrix (5 alternatives, 4 criteria)
dmat <- matrix(c(
  84, 10, 5, 20,  
  66, 18, 8, 15,
  74, 12, 6, 25,
  90, 22, 9, 18, 
  68, 18, 7, 15  
), nrow = 5, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4", "A5")
colnames(dmat) <- c("C1", "C2", "C3", "C4")

# Benefit-cost vector (1: benefit, -1: cost)
bc <- c(1, -1, 1, -1)

# User-defined weights
userwei <- c(0.1, 0.4, 0.3, 0.2)

# Run FUCA with user-defined weights
resfuca_1 <- fuca(dmat, bcvec=bc, weights=userwei)
print(resfuca_1)

# Run FUCA with the calculated entropy weights
entwei <- calcweights(dmat, bcvec=bc, type="entropy")

resfuca_2 <- fuca(dmat, bcvec=bc, weights=entwei)
print(resfuca_2)

Gradual Modification of Initial Weights

Description

Generates a matrix of weights by gradually modifying one criterion at a time while rescaling others accordingly.

Usage

gengradwei(initwei, rp = seq(0.1, 0.5, 0.1))

Arguments

Details

This function iteratively modifies the initial weights, reducing one criterion at a time while proportionally adjusting others to maintain a valid weight distribution. Negative weights are prevented by setting a lower bound. The resulting matrix provides different weighting scenarios, which can be useful for sensitivity analysis in multi-criteria decision-making models.

Value

A numeric matrix where each row represents a different weight scenario. The first row contains the initial weights, while subsequent rows show modified weights based on the reduction percentages.

Author(s)

Cagatay Cebeci

References

Stevic, Z., Pamucar, D., Puška, A., & Chatterjee, P. (2020). Sustainable supplier selection in healthcare industries using a new MCDM method: Measurement of alternatives and ranking according to COmpromise solution (MARCOS). Computers & Industrial Engineering, 140, 106231. doi:10.1016/j.cie.2019.106231

Demir, G., Chatterjee, P., & Pamucar, D. (2024). Sensitivity analysis in multi-criteria decision making: A state-of-the-art research perspective using bibliometric analysis. Expert Systems with Applications, 237, 121660. doi:10.1016/j.eswa.2023.121660

Rachman, A. P., Ichwania, C., Mangkuto, R. A., Pradipta, J., Koerniawan, M. D., & Sarwono, J. (2024). Comparison of multi-criteria decision-making methods for selection of optimum passive design strategy. Energy and Buildings, 314, 114285. doi:10.1016/j.enbuild.2024.114285

See Also

genrandwei

Examples

# Define initial weights
init_weights <- c(C1 = 0.5, C2 = 0.3, C3 = 0.2)

# Generate gradual weight modifications
scenarios <- gengradwei(init_weights)
print(scenarios)

Random Modification of Initial Weights

Description

Generates a set of weights by random modifying the initial weights over multiple iterations using a defined step size.

Usage

genrandwei(initwei, ss = NULL, niters = 5)

Arguments

Details

This function iteratively adjusts the initial weight values by adding random perturbations within the defined step size. The modified weights are normalized to ensure their sum remains equal to 1. Additionally, negative weights are prevented.

Value

A numeric matrix where each row represents a different weight scenario. The first row contains the initial weights, while subsequent rows show random modifications.

Author(s)

Cagatay Cebeci

See Also

gengradwei

Examples

# Define initial weights
init_weights <- c(C1 = 0.4, C2 = 0.3, C3 = 0.3)

# Generate random weight modifications
scenarios <- genrandwei(init_weights, ss = 0.05, niters = 10)

# View resulting scenarios
print(scenarios)

GRA: Grey Relational Analysis

Description

Implements the Grey Relational Analysis (GRA) method for multi-criteria decision analysis by ranking alternatives based on their grey relation degrees with the ideal solution.

Usage

gra(dmatrix, bcvec, weights, idesol = NULL, grdmethod = "sum", 
    rho = 0.5, tiesmethod="average")

Arguments

Details

The Grey Relational Analysis (GRA) method normalizes the decision matrix and then calculates the grey relational coefficients between each alternative and the ideal solution. The grey relation degree for each alternative is the average of its grey relational coefficients across all criteria. Alternatives are ranked based on their grey relation degrees in descending order (higher degree implies closer to the ideal solution). The grey relational coefficients for each criterion are multiplied by their respective weights before calculating the grey relation degree. To calculate Grey Relational Degrees, "sum" applies a weighted sum. If grdmethod is set to "mean", a weighted mean is used instead. This choice affects the relative influence of criteria in the final ranking.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Julong, D. (1989). Introduction to grey system theory. The Journal of Grey System, 1(1), 1-24. Deng, J.L. (1982), Control problems of grey systems, Systems and Control Letters, 5, 288‐294.

See Also

calcweights, calcnormal

Examples

# Sample decision matrix
dmat <- matrix(c(
  84, 10, 5, 20,  
  66, 18, 8, 15,
  74, 12, 6, 25,
  90, 22, 9, 18, 
  68, 18, 7, 15  
), nrow = 5, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4", "A5")
colnames(dmat) <- c("C1", "C2", "C3", "C4")

# Benefit-cost vector (1: benefit, -1: cost)
bc <- c(1, -1, 1, -1)

# User-defined weights
uw <- c(0.1, 0.4, 0.3, 0.2)

# Apply GRA with default settings
resgra_1 <- gra(dmatrix=dmat, bcvec=bc, weights=uw)
print(resgra_1)

# Apply GRA with a low rho value
resgra_2 <- gra(dmat, bc, weights = uw, rho = 0.3)
print(resgra_2)

# Apply GRA with a higher rho value
resgra_3 <- gra(dmatrix=dmat, bcvec=bc, weights=uw, grdmethod="sum", rho = 0.7)
print(resgra_3)

# Apply GRA with ideal solutions
ideal_solutions <- c(80, 15, 6, 16)
resgra_4 <- gra(dmatrix=dmat, bcvec=bc, weights=uw, idesol=ideal_solutions, rho=0.5)
print(resgra_4)

MABAC: Multi-Attributive Border Approximation Area Comparison

Description

This function is an implementation of the Multi-Attributive Border Approximation Area Comparison (MABAC) method to evaluate and rank alternatives based on given criteria through normalization, weighting, and border approximation area calculations.

Usage

mabac(dmatrix, bcvec, weights, tiesmethod="average")

Arguments

Details

The MABAC method involves normalizing the decision matrix, applying weights to criteria, and calculating the border approximation area. The border approximation area represents the average value of the alternatives for each criterion. A distance matrix is computed by subtracting the border approximation area from the weighted normalized matrix, and the alternatives are ranked based on their scores.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Pamučar, D., & Ćirović, G. (2015). The selection of transport and handling resources in logistics centers using Multi-Attributive Border Approximation area Comparison (MABAC). Expert Systems with Applications, 42(6), 3016-3028. <doi:10.1016/j.eswa.2014.11.057>

See Also

calcweights, calcnormal

Examples

# Sample decision matrix
dmat <- matrix(c(
  84, 10, 5, 20,  
  66, 18, 8, 15,
  74, 12, 6, 25,
  90, 22, 9, 18, 
  68, 18, 7, 15  
), nrow = 5, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4", "A5")
colnames(dmat) <- c("C1", "C2", "C3", "C4")

# Benefit-cost vector (1: benefit, -1: cost)
bc <- c(1, -1, 1, -1)

# User-defined weights
uw <- c(0.1, 0.4, 0.3, 0.2)

# Apply MABAC with user-defined weights
resmabac_1 <- mabac(dmatrix=dmat, bcvec=bc, weights=uw)
print(resmabac_1) 

# Apply MABAC with MEREC weighting method
mw <- calcweights(dmat, bcvec=bc, type="merec", normethod="maxmin")
resmabac_2 <- mabac(dmatrix=dmat, bcvec=bc, weights=mw)
print(resmabac_2$score) # Relative utilities
print(resmabac_2$rank)  # Ranks of relative utilities

MACONT T6: Multi-Criteria Decision-Making with Comprehensive Normalization (6 steps)

Description

MACONT 6 applies a multi-step normalization process to rank alternatives based on multiple criteria using a combination of arithmetic and geometric weighted averaging.

Usage

macont6(dmatrix, bcvec, weights, p=0.5, q=0.5, delta=0.5, theta=0.5, tiesmethod="average")

Arguments

Details

1. The decision matrix dmatrix is normalized using three different techniques: ratio-based, max-min, and vector normalization.

2. A balanced normalized matrix is computed using weighted averages of the three normalization methods.

3. The S1 and S2 ranking values are calculated using rho and zeta components.

4. A final ranking score (S) is derived, adjusting for preference weighting.

5. Alternatives are ranked in descending order based on comprehensive scores.

Value

A list containing:

Author(s)

Cagatay Cebeci

References

Wen, Z., Liao, H., & Zavadskas, E. K. (2020). MACONT: Mixed aggregation by comprehensive normalization technique for multi-criteria analysis. Informatica, 31(4), 857-880. <doi:10.15388/20-INFOR417>

Nguyen, A. T. (2023). Expanding the data normalization strategy to the MACONT method for multi-criteria decision making. Engineering, Technology & Applied Science Research, 13(2), 10489-10495. <doi:10.48084/etasr.5672>

See Also

topsis, vikor, electre1

Examples

# Decision matrix
dmat <- matrix(c(
  5, 3, 4,
  4, 5, 2,
  3, 4, 5,
  5, 2, 3
), nrow = 4, byrow = TRUE)

bcvec <- c(1, -1, 1)
weights <- c(0.3, 0.4, 0.3)

# Run MACONT
res_macont <- macont6(dmatrix = dmat, bcvec = bcvec, weights = weights,
     p = 0.4, q = 0.3, delta = 0.6, theta = 0.7)

print(res_macont$rank)

MAIRCA: Multi-Attribute Ideal-Real Comparative Analysis

Description

The MAIRCA is a multi-criteria decision-making method that normalizes the decision matrix and ranks alternatives based on specified criterion weights.

Usage

  mairca(dmatrix, bcvec, weights, tiesmethod="average")

Arguments

Value

Returns a list containing:

Author(s)

Cagatay Cebeci

References

Gigović, L., Pamučar, D., Bajić, Z., & Milićević, M. (2016). The combination of expert judgment and GIS-MAIRCA analysis for the selection of sites for ammunition depots. Sustainability, 8(4), 372. <doi:10.1080/1331677X.2018.1506706>

See Also

calcweights

Examples

# Sample decision matrix
dmat <- matrix(c(
  84, 10, 5, 20,  
  66, 18, 8, 15,
  74, 12, 6, 25,
  90, 22, 9, 18, 
  68, 18, 7, 15  
), nrow = 5, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4", "A5")
colnames(dmat) <- c("C1", "C2", "C3", "C4")

# Benefit-cost vector (1: benefit, -1: cost)
bc <- c(1, -1, 1, -1)

# User-defined weights
uw <- c(0.1, 0.4, 0.3, 0.2)
  
# Apply MAIRCA with user-defined weights
resmairca_1 <- mairca(dmatrix=dmat, bcvec=bc, weights = uw)
print(resmairca_1)

# Apply MAIRCA with Critic weights
cw <- calcweights(dmat, bcvec=bc, type="critic")
resmairca_2 <- mairca(dmatrix=dmat, bcvec=bc, weights = cw)
print(resmairca_2)

MARCOS: Measurement of Alternatives and Ranking According to Compromise Solution

Description

Implements the MARCOS method for MCDM by evaluating alternatives based on their relationships to ideal and anti-ideal solutions.

Usage

marcos(dmatrix, bcvec, weights, normethod = "ratio", tiesmethod="average")

Arguments

Details

The MARCOS method (Stević et al, 2020) constructs an extended decision matrix by adding ideal and anti-ideal solutions. The matrix is normalized, and weights are applied to each criterion. Scores are calculated based on the relative distance of each alternative to the ideal and anti-ideal solutions. Final rankings are determined using the compromise solution, which integrates the positive and negative distances.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Stević, Ž., Pamučar, D., Puška, A., & Chatterjee, P. (2020). Sustainable supplier selection in healthcare industries using a new MCDM method: Measurement of alternatives and ranking according to COmpromise solution (MARCOS). Computers & Industrial Engineering, 140, 106231. <doş:10.1016/j.cie.2019.106231>

See Also

calcweights, calcnormal

Examples

# Sample decision matrix
dmat <- matrix(c(
  84, 10, 5, 20,  
  66, 18, 8, 15,
  74, 12, 6, 25,
  90, 22, 9, 18, 
  68, 18, 7, 15  
), nrow = 5, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4", "A5")
colnames(dmat) <- c("C1", "C2", "C3", "C4")

# Benefit-cost vector (1: benefit, -1: cost)
bc <- c(1, -1, 1, -1)

# User-defined weights
uw <- c(0.1, 0.4, 0.3, 0.2)

# Apply MARCOS with user-defined weights
resmarcos_1 <- marcos(dmatrix=dmat, bcvec=bc, weights=uw)

print(resmarcos_1$f_k)   # Final compromise scores for alternatives
print(resmarcos_1$rank)  # Ranks of the final compromise scores for alternatives

MAUT: Multi-Attribute Utility Theory

Description

The MAUT evaluates alternatives based on weighted aggregation of their utilities across multiple criteria.

Usage

maut(dmatrix, bcvec, weights, utilfuncs = NULL, normutil = TRUE,  
    ss = 1, tiesmethod = "average")

Arguments

Details

The MAUT method involves defining a utility function for each criterion that reflects the decision maker's preferences. These utility functions transform the raw performance values into utility scores. The overall utility of each alternative is then calculated as the weighted sum of its utility scores across all criteria. Alternatives are ranked based on their overall utility scores in descending order.

In MAUT based on utility theory, the decision-maker calculates the utility of each criterion and strives to best reflect their preferences in the decision-making process. Utility is shaped by subjective perceptions and risk factors. On the other hand, MAVT based on value theory, the decision-maker determines the value of each criterion and optimizes these values to achieve specific objectives. MAVT generally focuses on the direct value of each criterion and adopts a more objective approach. Conceptual differences are related to the decision-maker’s preferences and risk perceptions. While working with the same matrix, different decision-makers may use either MAUT or MAVT, but the chosen approach affects how criteria should be evaluated. For example, if a decision-maker is risk-averse, they may prefer to make decisions using MAUT, as it flexibly models utility functions by taking into account risk aversion or risk-taking. On the other hand, if a decision-maker prefers a more objective approach and wants to directly optimize the value of each criterion, MAVT may be more suitable. In this case, the obtained value for each criterion takes precedence.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Keeney, R. L., & Raiffa, H. (1993). Decisions with multiple objectives: preferences and value trade-offs. Cambridge University Press. Ishizaka, A., & Nemery, P. (2013). Multi-criteria decision analysis: methods and software. John Wiley & Sons.

See Also

calcweights

Examples

# Sample decision matrix
dmat <- matrix(c(
  84, 10, 5, 20,  
  66, 18, 8, 15,
  74, 12, 6, 25,
  90, 22, 9, 18, 
  68, 18, 7, 15  
), nrow = 5, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4", "A5")
colnames(dmat) <- c("C1", "C2", "C3", "C4")

# Benefit-cost vector (1: benefit, -1: cost)
bc <- c(1, -1, 1, -1)

# User-defined weights
uw <- c(0.1, 0.4, 0.3, 0.2)

# Utility functions
uf <- c("linear", "linear","quad","step")

# Apply MAUT with default linear utility functions
resmaut_linear <- maut(dmatrix=dmat, bcvec=bc, weights=uw)
print(resmaut_linear)

# Apply MAUT without normalizing the utility matrix
resmaut_nonorm <- maut(dmatrix=dmat, bcvec=bc, weights=uw, normutil = FALSE)
print(resmaut_nonorm)

# Apply MAUT without normalizing the utility matrix with utility functions
resmaut_utils <- maut(dmatrix=dmat, bcvec=bc, weights=uw, utilfuncs=uf, normutil = FALSE)
print(resmaut_utils)

MAVT: Multi-Attribute Value Theory

Description

Implements the Multi-Attribute Value Theory (MAVT) method for multi-criteria decision analysis. This method evaluates alternatives based on weighted aggregation of their values across multiple criteria.

Usage

mavt(dmatrix, bcvec, weights, valfuncs = NULL, normvals = TRUE, 
    ss = 1, tiesmethod = "average")

Arguments

Details

The MAVT method involves defining a value function for each criterion that reflects the decision maker's preferences or judgments about the desirability of different levels of performance. These value functions transform the raw performance values into value scores. The overall value of each alternative is then calculated as the weighted sum of its value scores across all criteria. Alternatives are ranked based on their overall value scores in descending order.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Keeney, R. L., & Raiffa, H. (1993). Decisions with multiple objectives: preferences and value trade-offs. Cambridge University Press.

See Also

calcweights, calcnormal

Examples

# Sample decision matrix
dmat <- matrix(c(
  84, 10, 5, 20,  
  66, 18, 8, 15,
  74, 12, 6, 25,
  90, 22, 9, 18, 
  68, 18, 7, 15  
), nrow = 5, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4", "A5")
colnames(dmat) <- c("C1", "C2", "C3", "C4")

# Benefit-cost vector (1: benefit, -1: cost)
bc <- c(1, -1, 1, -1)

# User-defined weights
uw <- c(0.1, 0.4, 0.3, 0.2)

# Apply MAVT with default linear value functions
resmavt_linear <- mavt(dmatrix=dmat, bcvec=bc, weights=uw)
print(resmavt_linear)

# Apply MAVT without normalizing the value matrix
resmavt_nonorm <- mavt(dmatrix=dmat, bcvec=bc, weights=uw, normvals = FALSE)
print(resmavt_nonorm)

MEGAN: Multi-criteria Evaluation via Gini-weighted Attributes in Numerical Decision Matrices

Description

  A novel method to make multi-criteria decision analysis (MCDA) using various criteria, weights, and thresholds on numerical decision matrices.

Usage

megan(dmatrix, bcvec, weights="gini", normethod = NULL, thr = NULL, 
      tht="p25", tiesmethod="average")

Arguments

Details

The MEGAN method involves the following steps:

1. Normalize the decision matrix.    

2. Calculate criteria weights (if not provided).    

3. Calculate the weighted normalized decision matrix.    

4. Convert cost criteria to benefit criteria.    

5. Calculate distances to the minimum value for each criterion.    

6. Normalize the distance matrix.    

7. Calculate row sums of the normalized distances.    

8. Calculate superiority scores.    

9. Rank the alternatives based on their superiority scores and the given threshold. If threshold value is zero (thr=0, the rank of scores is returned using rank function with the ties method "average".)

Value

A list containing:

Author(s)

  Cagatay Cebeci

See Also

  calcweights, calcnormal

Examples

data(egrids)
dmat <- as.matrix(egrids$dmat) # Decision matrix
bc <- egrids$bcvec # Benefit and cost vector
wei <- egrids$weights # Criteria weights

# Apply MEGAN without threshold
resmegan_1 <- megan(dmatrix=dmat, bcvec=bc, weights=wei, thr=0)
print(resmegan_1)

# Apply MEGAN with a fixed threshold of 10%
resmegan_2 <- megan(dmatrix=dmat, bcvec=bc, weights=wei, thr=10)
print(resmegan_2$superiority)
print(resmegan_2$rank)

# Apply MEGAN with the 'p5' threshold
resmegan_3 <- megan(dmatrix=dmat, bcvec=bc, weights=wei, thr=NULL, tht="p5")
print(resmegan_3$superiority)
print(resmegan_3$rank)

# Apply MEGAN with the standard deviation threshold
resmegan_4 <- megan(dmatrix=dmat, bcvec=bc, weights=wei, thr=NULL, tht="sdev")
print(resmegan_4$superiority)
print(resmegan_4$rank)

Multi-Criteria Evaluation with Gradually-weighted Aggregation

Description

Performs multi-criteria evaluation with gradually modified weights, aggregating numerical decision matrices for ranking.

Usage

megan2 (dmatrix, bcvec, weights, weimethod="random", normethod=NULL, thr=0, tht="p25", 
   tiesmethod="average", ss=0.05, niters=10, rp=c(0.1, 0.5, 0.1))

Arguments

Details

This function applies gradual modifications to the initial weights, redistributing the importance of criteria step by step. Rankings are aggregated based on preference evaluations and adjusted to handle ties.

Value

A list containing:

Author(s)

Cagatay Cebeci

See Also

gengradwei, rankaggregate, megan

Examples

# Decision matrix
dmat <- matrix(c(5, 150, 3, 200, 4, 180), nrow = 3, byrow = TRUE)
rownames(dmat) <- paste0("A", 1:3)
colnames(dmat) <- c("C1", "C2")

# Benefit-Cost vector
bc <- c(-1, 1)

# Initial weights
init_weights <- c(C1 = 0.6, C2 = 0.4)

# Run megan2 function with gradual weight modifications
result_grad <- megan2(dmatrix = dmat, bcvec = bc, weights = init_weights, rp = c(0.1, 0.3, 0.2))

# Generated weight scenarios
print(result_grad$weimat)

print(result_grad$rank) # Aggregated rankings

# Run megan2 function with random weight modifications
result_rand <- megan2(dmatrix = dmat, bcvec = bc, weights = init_weights, ss=0.02, niters=10)

# Generated weight scenarios
print(result_rand$weimat)

print(result_rand$rank) # Aggregated rankings

Benchmark Multiple MCDM Methods

Description

Runs and compares multiple Multi-Criteria Decision-Making (MCDM) methods on a given decision matrix, using consistent criteria weights and types. Supports both predefined and user-supplied methods with optional parameter customization.

Usage

methodbench(dmatrix,bcvec, weights, normethod = "maxmin",
  mcdm = "all", params = NULL, tiesmethod = "average")

Arguments

Details

This function is designed to support benchmarking of multiple MCDM methods under consistent problem setups. It supports built-in methods "aras","aroman","cocoso","codas","copras","edas","electre4", "fuca","gra","mabac", "macont","marcos","mairca","maut","mavt", "megan","megan2","moora","ocra","oretes", "promethee1","promethee2", "promethee3","promethee4","promethee5","promethee6", "ram","rov","smart","topsis","vikor","waspas","wpm","wsm", and also allows integration of external functions if they return a named list including a rank vector.

For each method, base arguments (dmatrix, bcvec, weights) are passed, and if any params are specified, they are merged in before calling the method via do.call().

If a method fails or does not return a rank element, a warning is displayed and the result is skipped.

Value

A list containing:

Author(s)

Cagatay Cebeci

See Also

calcweights, rankaggregate, rankcompare, rankheatmap

Examples

# Sample decision matrix
dm <- matrix(c(
  10, 20, 30, 1.5, 102, 55,
  15, 25, 35, 1.6, 90, 60,
  12, 22, 32, 1.7, 100, 58,
  13, 24, 33, 1.8, 95, 57,
  14, 26, 37, 1.9, 98, 59,
  11, 23, 31, 1.65, 101, 56,
  16, 27, 36, 1.55, 97, 61,
  17, 28, 38, 1.7, 99, 63,
  18, 29, 39, 1.8, 94, 62,
  19, 30, 40, 1.75, 96, 64
), nrow = 10, byrow = TRUE)
colnames(dm) <- paste0("C", 1:ncol(dm))
rownames(dm) <- paste0("ALT", 1:nrow(dm))

# Benefit-Cost vector
bc <- c(1, -1, 1, -1, 1, 1)

# User-defined weights
userwei <- c(0.3, 0.1, 0.2, 0.1, 0.2, 0.1)

prmlist <- list(
      aras      = list(),
      aroman    = list(lambda = 0.5, beta = 0.5),
      cocoso    = list(lambda = 0.5),
      codas     = list(thr = 0.1),
      copras    = list(),
      electre1  = list(),
      electre2  = list(),
      electre3  = list(),
      electre4  = list(),
      fuca      = list(),
      gra       = list(idesol = NULL, grdmethod = "sum", rho = 0.5),
      mabac     = list(),
      macont6   = list(p = 0.5, q = 0.5, delta = 0.5, theta = 0.5),
      marcos    = list(),
      mairca    = list(),
      maut      = list(utilfuncs = NULL, normutil = TRUE, ss = 1),
      mavt      = list(valfuncs = NULL, normvals = TRUE, ss = 1),
      megan     = list(normethod = "ratio", thr = 0, tht = "sdev"),
      megan2    = list(normethod = "ratio", thr = 0, tht = "sdev"),
      moora     = list(),
      ocra      = list(),
      oretes    = list(domplot = FALSE),
      promethee1 = list(),
      promethee2 = list(),
      promethee3 = list(strict = FALSE),
      promethee4 = list(alpha = 0.2),
      promethee5 = list(g = 0, l = 100),
      promethee6 = list(varmethod = "abs_sum"),
      ram       = list(normethod = "sum"),
      rov       = list(normethod = "maxmin"),
      smart     = list(),
      topsis    = list(normethod = "maxmin"),
      vikor     = list(normethod = "maxmin", v = 0.8),
      waspas    = list(normethod = "linear", v = 0.5),
      wpm       = list(normethod = "vector"),
      wsm       = list()
)

# prmlist$electre4 <- list(
#      p = c(0.6, 0.7, 0.5, 0.6, 0.6, 0.8),
#      q = c(0.3, 0.35, 0.25, 0.3, 0.3, 0.4),
#      v = c(0.9, 1.0, 0.8, 0.9, 0.9, 1.1))

# Compare all methods with default settings using user-defined weights
rescomp_1 <- methodbench(dmatrix=dm, bcvec=bc, weights=userwei, params=prmlist, mcdm="all")

print(rescomp_1$rankmat)

rankheatmap(rescomp_1$rankmat, colpal=1, cellnotes=TRUE, tcol="black")

# Overall ranks and flow of dominance
resoverall <- rankaggregate(rescomp_1$rankmat, tiesmethod="average", 
   topk = 3, damp = 0.5, niters = 200, tol = 1e-4)
str(resoverall)

# Compare selected methods with 'equal' weights
methodlist <- c("aras", "cocoso", "copras", "edas", "electre4", "mairca", 
   "mabac", "marcos", "megan", "ocra", "oretes", "promet1", "promet6", 
   "topsis", "vikor", "waspas")

equwei <- calcweights(dm, bcvec=bc, type="equal")
rescomp_2 <- methodbench(dmatrix = dm, bcvec = bc, weights = equwei,
   mcdm = methodlist, params=prmlist)

print(rescomp_2$rankmat)
rankheatmap(rescomp_2$rankmat, colpal=4, cellnotes=TRUE, tcol="white")

# Overall ranks and outranking
resoverall <- rankaggregate(rescomp_2$rankmat, tiesmethod="average", 
   topk = 3, damp = 0.5, niters = 200, tol = 1e-4)
print(resoverall$preference_ranking)
print(resoverall$preference_table)

# Compare selected methods with 'gini' weights
giniwei <- calcweights(dm, bcvec=bc, type="gini")
rescomp_3 <- methodbench(dmatrix = dm, bcvec = bc, weights = giniwei,
   mcdm = methodlist, params=prmlist)
print(rescomp_3$rankmat)
rankheatmap(rescomp_3$rankmat, colpal=1, cellnotes=TRUE, tcol="white")

# Overall ranks and outranking
resoverall <- rankaggregate(rescomp_3$rankmat, tiesmethod="average", topk = 3,
   damp = 0.5, niters = 200, tol = 1e-4)
print(resoverall)

# Custom function example
# A Dummy user-defined method
johndoe <- function(dmatrix, bcvec, weights, janefactor = 1) {
  score <- rowSums(scale(dmatrix) * weights) * janefactor
  ranking <- rank(-score, ties.method="average")
  results <- list(score = score, rank=ranking)
  return(results)
}

prmlist <- list(
   aras = list(),
   vikor = list(v = 0.5),
   waspas = list(v = 0.5),
   johndoe = list(jane = 0.25)
)

# Compare user-defined 'johndoe' with popular MCDM methods
rescomp_user <- methodbench(
  dmatrix = dm,
  bcvec = bc,
  weights = userwei,
  mcdm = c("edas", "topsis", "vikor", "aras", "waspas", "oreste", "johndoe"),
  params = prmlist
)

print(rescomp_user$rankmat)

rankheatmap(rescomp_user$rankmat, colpal = 3, cellnotes = TRUE, tcol = "black")

# Aggregate ranking
resoverall_user <- rankaggregate(rescomp_user$rankmat, tiesmethod = "average", topk = 3)
print(resoverall_user$preference_ranking)
print(resoverall_user$preference_table)

MOORA: Multi-Objective Optimization by Ratio Analysis

Description

The Multi-Objective Optimization by Ratio Analysis (MOORA) method for multi-criteria decision analysis ranks alternatives based on the balance between their performance on benefit and cost criteria.

Usage

moora(dmatrix, bcvec, weights, tiesmethod="average")

Arguments

Details

The Multi-Objective Optimization by Ratio Analysis (MOORA) method first normalizes the decision matrix using vector normalization. Then, for each alternative, it calculates an optimization value by summing the weighted normalized values of the benefit criteria and subtracting the weighted normalized values of the cost criteria. Alternatives are ranked based on these optimization values in descending order (higher value implies better performance).

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Brauers, W. K., & Zavadskas, E. K. (2006). The MOORA method and its application to privatization in a transition economy. Control and Cybernetics, 35(2), 445-469.

See Also

calcweights

Examples

# Sample decision matrix
dmat <- matrix(c(
  84, 10, 5, 20,  
  66, 18, 8, 15,
  74, 12, 6, 25,
  90, 22, 9, 18, 
  68, 18, 7, 15  
), nrow = 5, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4", "A5")
colnames(dmat) <- c("C1", "C2", "C3", "C4")

# Benefit-cost vector (1: benefit, -1: cost)
bc <- c(1, -1, 1, -1)

# User-defined weights
uw <- c(0.1, 0.4, 0.3, 0.2)

# MOORA with user-defined weights
resmoora_1 <- moora(dmatrix = dmat, bcvec = bc, weights = uw)
print(resmoora_1)

# MOORA with equal weights
equw <- calcweights(dmat, bcvec=bc, type="equal")
resmoora_2 <- moora(dmatrix =  dmat, bcvec = bc, weights = equw)
print(resmoora_2)

# MOORA with entropy weights
entw <- calcweights(dmat, bcvec=bc, type="entropy")
resmoora_3 <- moora(dmatrix = dmat, bcvec = bc, weights = entw)
print(resmoora_3)

OCRA: Operational Competitiveness Rating

Description

Computes rankings for decision alternatives using the OCRA method, which evaluates benefit and cost criteria relative to reference values.

Usage

ocra(dmatrix, bcvec, weights, tiesmethod = "average")

Arguments

Value

A list with the following components:

Author(s)

Cagatay Cebeci

References

Madic, M., Petkovic, D., & Radovanovic, M. (2015). Selection of non-conventional machining processes using the OCRA method. Serbian Journal of Management, 10(1), 61-73.

Examples

dmatrix <- matrix(c(
  10, 50, 30,  
  20, 60, 20,  
  30, 40, 40,  
  40, 30, 50   
), nrow = 4, byrow = TRUE)

bcvec <- c(1, -1, 1)  # Benefit (1), Cost (-1), Benefit (1)
weights <- c(0.3, 0.4, 0.3)  # Criterion weights

resocra <- ocra(dmatrix, bcvec, weights)
print(resocra)

ORETES: Organísation, Rangemént et Estímatíon En Synthèse

Description

The "Organísation, Rangemént et Estímatíon En Synthèse (ORETES)" is an MCDM technique based on outranking relations and graph theory.The function calculates lambda values, psi ranks, a normalized dominance matrix, outranking flows, and provides a final ranking of alternatives. It also offers an optional visualization of the dominance graph.

Usage

oretes(dmatrix, bcvec, weights, tiesmethod = "average", domplot = FALSE)

Arguments

Details

The ORETES method proceeds through several steps:

1. Lambda Values Calculation: Normalizes criterion values for each alternative to a scale of 0 to 1, where 1 represents the ideal performance and 0 the worst.

2. Psi Matrix Calculation: Ranks the lambda values for each alternative across its criteria. This provides an internal ranking of criteria importance for each alternative.

3. Dominance Matrix Calculation: Computes the strength of preference (dominance) of one alternative over another. For each pair of alternatives (A, B), it sums the weights of the criteria where A performs better than B. The matrix is then normalized.

4. Outranking Flow Calculation: Determines the net outranking flow for each alternative by subtracting its incoming dominance strength from its outgoing dominance strength. A higher net flow indicates a stronger overall preference.

5. Final Ranking: Alternatives are ranked based on their net outranking flow, from highest to lowest.

The dominance graph, if plotted, visually represents the normalized dominance matrix, with arrows indicating the direction and strength of preference between alternatives. Thicker arrows indicate stronger dominance.

Value

A list containing the following components:

Note

This implementation of ORETES is an adaptation focused on the core principles of lambda values, dominance relations, and outranking flow. It simplifies some of the more intricate aspects of the original method, such as specific indifference/preference thresholds and detailed preference functions, for broader applicability in MCDM.

Author(s)

Cagatay Cebeci

References

Roubens, M. (1982). Preference relations on actions and criteria in multicriteria decision making. European Journal of Operational Research, 10(1), 51-55. <doi:10.1016/0377-2217(82)90131-X>

See Also

graph_from_data_frame, plot.igraph

Examples

# Sample decision matrix
dmatrix <- matrix(c(
  10, 50, 30,  
  20, 60, 20, 
  30, 40, 40, 
  40, 30, 50), 
  nrow = 4, byrow = TRUE, 
  dimnames = list(paste0("A", 1:4), paste0("C", 1:3)
))

# Define benefit (1) or cost (-1) for each criterion
bcvec <- c(1, -1, 1) 

# Define weights for each criterion
weights <- c(0.3, 0.4, 0.3) # Sum does not necessarily need to be 1

# ORETES without plotting the dominance graph
resoreste_1 <- oretes(dmatrix, bcvec, weights, domplot = FALSE)
print(resoreste_1)

# ORETES with the dominance graph
resoreste_2 <- oretes(dmatrix, bcvec, weights, domplot = TRUE)
print(resoreste_2)

# Zero weights (no dominance relationships)
  zerowei <- c(0, 0, 0)
  resoreste_3 <- oretes(dmatrix, bcvec, zerowei, domplot = TRUE)
  print(resoreste_3)

Parallel Coordinate Plot

Description

This function creates a parallel coordinate plot using the provided data frame or matrix. Each row represents a different method, and each line in the plot corresponds to the values for that method.

Usage

parcorplot(x, xl = NULL, yl = NULL, lt = NULL, colpal = NULL)

Arguments

Value

No return value, but a plot displaying the parallel coordinates for each method.

Author(s)

Cagatay Cebeci

Examples

# Sample rank matrix
rankmat <- data.frame(
  ALT1 = c(3.0, 3.0, 2.5, 3.0, 3.0, 3.0, 2.0, 1.0),
  ALT2 = c(1, 1, 1, 1, 1, 1, 1, 2),
  ALT3 = c(2.0, 2.0, 2.5, 2.0, 2.0, 2.0, 3.0, 3.0)
)
rownames(rankmat) <- c(
  "ARAS", "EDAS", "ELECTRE4", "MABAC", "MARCOS", "MEGAN", "TOPSIS", "VIKOR"
)

parcorplot(rankmat, xl = "Alternatives", yl = "Ranks", lt = "MCDA Methods")

PROMETHEE I Method for Partial Ranking in Multi-Criteria Decision Analysis

Description

Implements the PROMETHEE I method to compute positive and negative preference flows, generate pairwise outranking comparisons, and determine a partial ranking of alternatives.

Usage

promethee1(dmatrix, bcvec, weights, normethod = NULL, 
    prefuncs = NULL, thr = NULL, tiesmethod = "average")

Arguments

Details

PROMETHEE I allows pairwise comparison of alternatives under multiple criteria using weighted preference functions. The method computes how much an alternative is preferred to others (Phi+) and how much it is outranked (Phi-), without producing a complete ordering in all cases. Indifference and incomparability are explicitly represented.

You can customize the behavior via different preference functions and thresholds for each criterion. When no thresholds are provided, default ones based on column differences are used.

Value

A list containing:

Author(s)

Cagatay Cebeci

Examples

# Sample decision matrix
dm <- matrix(c(10, 5, 8,
               6, 9, 7,
               8, 7, 9), nrow = 3, byrow = TRUE)
rownames(dm) <- paste0("A", 1:3)
colnames(dm) <- paste0("C", 1:3)

# All criteria are benefits
bc <- c(1, 1, 1)

# Weights sum to 1
userwei <- c(0.3, 0.4, 0.3)

# Preference functions and thresholds
prefuncs <- c("linear", "v-shape", "level")
thr <- list(
  list(p = 5),         # Linear
  list(p = 4),         # V-shape
  list(q = 1, p = 3)   # Level
)

# Apply PROMETHEE I
respromethee1 <- promethee1(dmatrix=dm, bcvec=bc, weights=userwei, 
   prefuncs = prefuncs, thr = thr)
print(respromethee1)

PROMETHEE II: Preference Ranking Organization METHod for Enrichment Evaluations (Method II)

Description

Performs the PROMETHEE II method for complete ranking of alternatives in a Multi-Criteria Decision Analysis (MCDA) problem.

Usage

promethee2(dmatrix, bcvec, weights, normethod = NULL, prefuncs = NULL, 
    thr = NULL, tiesmethod="average")

Arguments

Details

The PROMETHEE II method calculates a complete preorder of alternatives. It extends PROMETHEE I by calculating the net outranking flow (Phi_net) for each alternative, which is the difference between positive (Phi_plus) and negative (Phi_minus) outranking flows. Then, alternatives can be ranked based on their Phi_net values.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Brans, J. P., & Mareschal, B. (1988). Geometrical representations for MCDA. European Journal of Operational Research, 34(1), 69-77. <doi:10.1016/0377-2217(88)90456-0>

Brans, J. P., Mareschal, B. (2005). Promethee Methods. In: Multiple Criteria Decision Analysis: State of the Art Surveys. International Series in Operations Research & Management Science, vol 78. Springer, New York, NY. <doi:10.1007/0-387-23081-5_5>

Behzadian, M., Kazemzadeh, R. B., Albadvi, A., & Aghdasi, M. (2010). PROMETHEE: A comprehensive literature review on methodologies and applications. European Journal of Operational Research, 200(1), 198-215. <doi:10.1016/j.ejor.2009.01.021>

Brans, J. P., De Smet, Y. (2016). PROMETHEE Methods. In: Greco, S., Ehrgott, M., Figueira, J. (eds) Multiple Criteria Decision Analysis. International Series in Operations Research & Management Science, 233. Springer, New York, NY. <doi:10.1007/978-1-4939-3094-4_6>

See Also

promethee1, promethee3, promethee4, promethee5, promethee6

Examples

# Decision matrix
dmat <- matrix(c(
  7, 12, 8, 6, 14,   
  12, 18, 11, 8, 22, 
  8, 14, 11, 7, 10,  
  1.2, 1.9, 1.2, 0.8, 1.1, 
  0, 0, 1, 0, 1,  
  0, 1, 0, 1, 2  
), 
nrow = 5, byrow = FALSE)

colnames(dmat) <- c("C1", "C2", "C3", "C4", "C5", "C6")
rownames(dmat) <- c("A", "B", "C", "D", "E")

# Benefit-Cost vector
bc <- c(1, 1, 1, 1, -1, 1)

# Weights
uw <- c(0.13, 0.08, 0.20, 0.17, 0.12, 0.3)

# Threshold values
repeat_list <- list(list(p = 50), list(p = 30))
tvals <- rep(repeat_list, length.out = ncol(dmat))

# Preference function
pf <- rep("linear", ncol(dmat))

# Run PROMETHEE II function
respromethee2 <- promethee2(dmatrix=dmat, bcvec=bc, weights=uw,
    thr = tvals, prefuncs = pf)

# Phi+ values
print(respromethee2$Phi_plus)
 
# Phi- values
print(respromethee2$Phi_minus)

# Phi_net values
print(respromethee2$Phi_net)

# Complete ranking (PROMETHEE II)
print(respromethee2$rank)

# Run with a single threshold value applied to all criteria (linear)
tvals_single <- list(p = 40)
respromethee2_single <- promethee2(dmatrix=dmat, bcvec=bc, weights=uw,
    thr = tvals_single, prefuncs = "linear")

# Phi_net values for single threshold
print(respromethee2_single$Phi_net)

# Complete ranking for single threshold
print(respromethee2_single$rank)

# Run with different preference functions and mixed thresholds
tvals_mixed <- list(list(q = 10, p = 30), list(p = 20), list(q = 10, p = 30), 
   list(q = 10, p = 30), list(q = 10, p = 30), list(q = 10, p = 30))
pf_mixed <- c("linear", "linear", "linear", "linear", "level", "level")
respromethee2_mixed <- promethee2(dmatrix=dmat, bcvec=bc, weights=uw,
   thr = tvals_mixed, prefuncs = pf_mixed)

# Phi_net values with mixed thresholds
print(respromethee2_mixed$Phi_net)

# Complete ranking with mixed thresholds
print(respromethee2_mixed$rank)

PROMETHEE III: Preference Ranking Organization METHod for Enrichment Evaluations (Method III)

Description

Performs the PROMETHEE III method for complete ranking of alternatives in a Multi-Criteria Decision Analysis (MCDM) problem, allowing for interval-based ranking.

Usage

promethee3(dmatrix, bcvec, weights, normethod = NULL, 
   prefuncs = NULL, thr = NULL, strict = FALSE, tiesmethod="average")

Arguments

Details

The PROMETHEE III method extends PROMETHEE II to allow for interval-based ranking of alternatives, providing a more nuanced result that can account for uncertainty. It calculates positive (Phi_plus), negative (Phi_minus), and net (Phi_net) outranking flows. The final ranking depends on the strict parameter. If strict = FALSE, alternatives are ranked based on intervals (which, in this basic implementation, are point estimates equal to Phi_net). If strict = TRUE, the ranking is based on Phi_net, as in PROMETHEE II.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Brans, J. P., & Mareschal, B. (1988). Geometrical representations for MCDA. European Journal of Operational Research, 34(1), 69-77. <doi:10.1016/0377-2217(88)90456-0>

Brans, J. P., Mareschal, B. (2005). Promethee Methods. In: Multiple Criteria Decision Analysis: State of the Art Surveys. International Series in Operations Research & Management Science, vol 78. Springer, New York, NY. <doi:10.1007/0-387-23081-5_5>

Behzadian, M., Kazemzadeh, R. B., Albadvi, A., & Aghdasi, M. (2010). PROMETHEE: A comprehensive literature review on methodologies and applications. European Journal of Operational Research, 200(1), 198-215. <doi:10.1016/j.ejor.2009.01.021>

Brans, J. P., De Smet, Y. (2016). PROMETHEE Methods. In: Greco, S., Ehrgott, M., Figueira, J. (eds) Multiple Criteria Decision Analysis. International Series in Operations Research & Management Science, 233. Springer, New York, NY. <doi:10.1007/978-1-4939-3094-4_6>

See Also

promethee1, promethee2, promethee4, promethee5, promethee6

Examples

# Decision matrix
dmat <- matrix(c(5, 150, 3, 200, 4, 180), nrow = 3, byrow = TRUE)
rownames(dmat) <- paste0("A", 1:3)
colnames(dmat) <- c("C1", "C2")

# Benefit-Cost vector
bc <- c(-1, 1)

# User-defined weights
uw <- c(0.6, 0.4)

# Threshold values
tvals <- list(list(p = 50), list(p = 30))

# Preference functions
pf <- c("linear", "linear")

# Run PROMETHEE III function (strict = FALSE for interval ranking)
respromethee3_interval <- promethee3(dmatrix=dmat, bcvec=bc, weights=uw, 
   prefuncs = pf, thr = tvals, strict = FALSE)
print(respromethee3_interval)

# Run PROMETHEE III function (strict = TRUE for strict ranking)
respromethee3_strict <- promethee3(dmatrix=dmat, bcvec=bc, weights=uw, 
   prefuncs = pf, thr = tvals, strict = TRUE)
print(respromethee3_strict)

PROMETHEE IV: Preference Ranking Organization METHod for Enrichment Evaluations (Method IV)

Description

Performs the PROMETHEE IV method for complete ranking of alternatives in a Multi-Criteria Decision Analysis (MCDM) problem, considering the average outranking flow.

Usage

promethee4(dmatrix, bcvec, weights, normethod = NULL, 
   prefuncs = NULL, thr = NULL, alpha = 0.2, tiesmethod="average")

Arguments

Details

PROMETHEE IV ranks alternatives by considering their deviation from the average outranking flow. It calculates positive (Phi_plus), negative (Phi_minus), and net (Phi_net) outranking flows, and then computes the average net outranking flow (Phi_avg). The final ranking is based on the adjusted net outranking flow: Phi_net - alpha * Phi_avg.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Brans, J. P., & Mareschal, B. (1988). Geometrical representations for MCDA. European Journal of Operational Research, 34(1), 69-77. <doi:10.1016/0377-2217(88)90456-0>

Brans, J. P., Mareschal, B. (2005). Promethee Methods. In: Multiple Criteria Decision Analysis: State of the Art Surveys. International Series in Operations Research & Management Science, vol 78. Springer, New York, NY. <doi:10.1007/0-387-23081-5_5>

Behzadian, M., Kazemzadeh, R. B., Albadvi, A., & Aghdasi, M. (2010). PROMETHEE: A comprehensive literature review on methodologies and applications. European Journal of Operational Research, 200(1), 198-215. <doi:10.1016/j.ejor.2009.01.021>

Brans, J. P., De Smet, Y. (2016). PROMETHEE Methods. In: Greco, S., Ehrgott, M., Figueira, J. (eds) Multiple Criteria Decision Analysis. International Series in Operations Research & Management Science, 233. Springer, New York, NY. <doi:10.1007/978-1-4939-3094-4_6>

See Also

promethee1, promethee2, promethee3, promethee5, promethee6

Examples

# Decision matrix
dmat <- matrix(c(5, 150, 3, 200, 4, 180), nrow = 3, byrow = TRUE)
rownames(dmat) <- paste0("A", 1:3)
colnames(dmat) <- c("C1", "C2")

# Benefit-Cost vector
bc <- c(-1, 1)

# Weights
uw <- c(0.6, 0.4)

# Threshold values
tvals <- list(list(p = 50), list(p = 30))

# Preference function
pf <- c("linear", "linear")

# Run PROMETHEE IV function with alpha = 0.2
respromethee4_1 <- promethee4(dmatrix=dmat, bcvec=bc, weights=uw,
  thr= tvals, prefuncs = pf, alpha = 0.2)
print(respromethee4_1)

# Run PROMETHEE IV function with alpha = 0.8
respromethee4_2 <- promethee4(dmatrix=dmat, bcvec=bc, weights=uw,
  thr = tvals, prefuncs = pf, alpha = 0.8)
print(respromethee4_2)

PROMETHEE V: Preference Ranking Organization METHod for Enrichment Evaluations (Method V)

Description

PROMETHEE V method applies multi-criteria decision analysis, which ranks alternatives considering both their net outranking flow and feasibility constraints.

Usage

promethee5(dmatrix, bcvec, weights, normethod = NULL,
    prefuncs = NULL, thr = NULL, g = 0, l = 100, tiesmethod="average")

Arguments

Details

PROMETHEE V method, an extension of the PROMETHEE II method is used for ranking alternatives in multi-criteria decision problems. PROMETHEE V incorporates feasibility constraints, defined by lower (g) and upper (l) limits on the net outranking flow.

The prefuncs argument allows for different preference functions to be used for each criterion. The preference function transforms the difference between the evaluations of two alternatives on a given criterion into a preference degree.

The thr argument specifies the threshold parameters for the preference functions. The required parameters vary depending on the chosen preference function.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Brans, J. P., & Mareschal, B. (1988). Geometrical representations for MCDA. European Journal of Operational Research, 34(1), 69-77. <doi:10.1016/0377-2217(88)90456-0>

Brans, J. P., Mareschal, B. (2005). Promethee Methods. In: Multiple Criteria Decision Analysis: State of the Art Surveys. International Series in Operations Research & Management Science, vol 78. Springer, New York, NY. <doi:10.1007/0-387-23081-5_5>

Behzadian, M., Kazemzadeh, R. B., Albadvi, A., & Aghdasi, M. (2010). PROMETHEE: A comprehensive literature review on methodologies and applications. European Journal of Operational Research, 200(1), 198-215. <doi:10.1016/j.ejor.2009.01.021>

Brans, J. P., De Smet, Y. (2016). PROMETHEE Methods. In: Greco, S., Ehrgott, M., Figueira, J. (eds) Multiple Criteria Decision Analysis. International Series in Operations Research & Management Science, 233. Springer, New York, NY. <doi:10.1007/978-1-4939-3094-4_6>

See Also

promethee1, promethee2, promethee3, promethee4, promethee6

Examples

# Decision matrix
dmat <- matrix(c(5, 150, 3, 200, 4, 180, 6, 170), nrow = 4, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4")
colnames(dmat) <- c("C1", "C2")

# Benefit-Cost vector
bc <- c(-1, 1)

# Weights
uw <- c(0.6, 0.4)

# Threshold values
tvals <- list(list(p = 50), list(p = 30))

# Parameters for PROMETHEE V
gval <- 0
lval <- 100

# Preference function
pf <- c("linear", "linear")

# Run PROMETHEE V function
respromethee5 <- promethee5(dmatrix=dmat, bcvec=bc, weights=uw, 
     prefunc = pf, thr = tvals, g = gval, l = lval)
print(respromethee5)

PROMETHEE VI: Preference Ranking Organization METHod for Enrichment Evaluations (Method VI)

Description

PROMETHEE VI method for multi-criteria decision analysis, ranks alternatives based on a balance between their central tendency and variability.

Usage

promethee6 (dmatrix, bcvec, weights, normethod = NULL, prefuncs = NULL, thr = NULL,
   varmethod = "abs_sum", p = NULL, q = NULL, tiesmethod="average")

Arguments

Details

The promethee6 function implements the PROMETHEE VI method, an extension of the PROMETHEE II method. PROMETHEE VI considers both the central tendency and the variability of the net outranking flows to rank alternatives.

The prefuncs argument allows for different preference functions to be used for each criterion. The preference function transforms the difference between the evaluations of two alternatives on a given criterion into a preference degree.

The thr argument specifies the parameters for the preference functions. The required parameters vary depending on the chosen preference function.

The arguments p and q are specific to PROMETHEE VI. They represent the preference and indifference parameters, respectively, used to calculate the Lambda values, which adjust the final ranking based on the variability of the net outranking flows.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Brans, J. P., & Mareschal, B. (1988). Geometrical representations for MCDA. European Journal of Operational Research, 34(1), 69-77. <doi:10.1016/0377-2217(88)90456-0> Brans, J. P., Mareschal, B. (2005). Promethee Methods. In: Multiple Criteria Decision Analysis: State of the Art Surveys. International Series in Operations Research & Management Science, vol 78. Springer, New York, NY. <doi:10.1007/0-387-23081-5_5> Behzadian, M., Kazemzadeh, R. B., Albadvi, A., & Aghdasi, M. (2010). PROMETHEE: A comprehensive literature review on methodologies and applications. European Journal of Operational Research, 200(1), 198-215. <doi:10.1016/j.ejor.2009.01.021> Brans, J. P., De Smet, Y. (2016). PROMETHEE Methods. In: Greco, S., Ehrgott, M., Figueira, J. (eds) Multiple Criteria Decision Analysis. International Series in Operations Research & Management Science, 233. Springer, New York, NY. <doi:10.1007/978-1-4939-3094-4_6>

See Also

promethee1, promethee2, promethee3, promethee4, promethee5

Examples

# Decision matrix
dmat <- matrix(c(5, 150, 3, 200, 4, 180), nrow = 3, byrow = TRUE)
rownames(dmat) <- paste0("A", 1:3)
colnames(dmat) <- paste0("C", 1:2)

# Benefit-Cost vector
bc <- c(-1, 1)

# Predefined weights
uw <- c(0.5, 0.5)

# Threshold values
tvals <- list(list(p = 50), list(p = 30))

# Preference function
pf <- c("linear", "linear")

# PROMETHEE VI p and q parameters
pval <- 120
qval <- 20

# Run PROMETHEE VI function
respromethee6_1 <- promethee6(dmatrix=dmat, bcvec=bc, weights=uw,
     prefuncs = pf, thr = tvals, p = pval, q = qval)
print(respromethee6_1)

# Using different preference functions
pf2 <- c("v-shape", "level")
thr2 <- list(list(p = 60), list(q = 15, p = 40))
respromethee6_2 <- promethee6(dmatrix=dmat, bcvec=bc, weights=uw, 
      prefuncs = pf2, thr = thr2, p = pval, q = qval)
print(respromethee6_2)

# Different weights and normalization
uw <- c(0.7, 0.3)
respromethee6_3 <- promethee6(dmatrix=dmat, bcvec=bc, weights=uw, 
  normethod = "ratio", p = pval, q = qval)
print(respromethee6_3)

RAM: Root Assessment Method

Description

Implements the Root Assessment Method (RAM) to evaluate and rank alternatives based on normalized scores, weighted contributions, and root-based calculations.

Usage

ram(dmatrix, bcvec, weights, normethod = "sum", tiesmethod="average")

Arguments

Details

The RAM method evaluates alternatives by normalizing the decision matrix, applying weights to criteria, and calculating scores based on the root formula that integrates positive and negative contributions. The final ranking is derived from the computed scores (ri), which reflect the balance between benefit and cost criteria.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Sotoudeh-Anvari, A. (2023). Root Assessment Method (RAM): A novel multi-criteria decision making method and its applications in sustainability challenges. Journal of Cleaner Production, 423, 138695.

See Also

calcweights, calcnormal

Examples

# Sample decision matrix
dmat <- matrix(c(
  84, 10, 5, 20,  
  66, 18, 8, 15,
  74, 12, 6, 25,
  90, 22, 9, 18, 
  68, 18, 7, 15  
), nrow = 5, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4", "A5")
colnames(dmat) <- c("C1", "C2", "C3", "C4")

# Benefit-cost vector (1: benefit, -1: cost)
bc <- c(1, -1, 1, -1)

# User-defined weights
uw <- c(0.1, 0.4, 0.3, 0.2)
  
# Apply RAM with user-defined weights
resram_1 <- ram(dmatrix=dmat, bcvec=bc, weights=uw)
print(resram_1)

# Apply RAM with equal weights
equw <- calcweights(dmat, bcvec=bc, type="equal")
resram_2 <- ram(dmatrix=dmat, bcvec=bc, weights=equw)
print(resram_2)

Aggregat Ranks of Alternatives for Decision Making

Description

Aggregates and summarizes the ranking information of alternatives obtained from multiple methods. Given a rank matrix where rows represent evaluation methods (e.g., MCDA methods) and columns represent alternatives, the function computes aggregated overall ranks, produces hierarchical outranking flow, and counts the frequency of each alternative appearing in the top tk ranks.

Usage

rankaggregate(rankmat, topk=3, damp=0.5, niters=100, tol=1e-4, tiesmethod="average")

Arguments

Details

The prefsummary function aggregates ranking information for a set of alternatives evaluated by multiple methods. In particular, it performs the following tasks:

1. Computes alternative rankings based on the Top-tk counts method.

2. Computes the overall ranks based on the sum of the ranks across methods using the Rank Sum approach.

3. Produces aggregated ranks using the Median of ranks method.

4. Computes pairwise comparisons between alternatives based on the Copeland method (assigning 1 point for a win and 0.5 points for a tie in pairwise comparisons).

5. Calculates overall scores using the Kemeny & Young method based on Kendall Tau distances.

6. Computes a stationary score and ranking using a Markov chain (PageRank-like) approach.

7. Combines the rankings from the above methods into a matrix and constructs a hierarchical (outranking) string (preference hierarchy) for each method.

Value

A list with the following components:

Author(s)

Cagatay Cebeci

References

Saari, D. G., & Merlin, V. R. (1996). The Copeland method: I.: Relationships and the dictionary. Economic Theory, 8, 51-76. <doi:10.1007/BF01212012>

Examples

# Example rank matrix (rows: methods, columns: alternatives)
ranks <- c(
  5.0, 7.0, 6.0, 10.0, 8.0, 9.0, 3.0, 1.0, 4.0, 2.0,
  7.0, 8.0, 3.5, 3.5, 9.0, 3.5, 3.5, 3.5, 3.5, 10.0,
  5.0, 7.0, 6.0, 10.0, 9.0, 8.0, 3.0, 2.0, 4.0, 1.0,
  6.0, 5.0, 3.0, 9.0, 8.0, 10.0, 1.0, 2.0, 7.0, 4.0,
  5.5, 7.0, 4.0, 10.0, 9.0, 8.0, 3.0, 1.0, 5.5, 2.0,
  9.0, 7.0, 6.0, 8.0, 2.0, 10.0, 4.0, 5.0, 1.0, 3.0,
  10.0, 9.0, 6.0, 7.0, 4.0, 8.0, 5.0, 3.0, 2.0, 1.0,
  5.0, 7.0, 6.0, 10.0, 8.0, 9.0, 3.0, 1.0, 4.0, 2.0,
  5.0, 7.0, 6.0, 10.0, 8.0, 9.0, 3.0, 1.0, 4.0, 2.0,
  5.0, 7.0, 6.0, 10.0, 8.0, 9.0, 3.0, 1.0, 4.0, 2.0,
  2.0, 8.0, 3.0, 9.0, 5.0, 10.0, 1.0, 6.0, 4.0, 7.0,
  2.0, 8.0, 3.0, 9.0, 5.0, 10.0, 1.0, 6.0, 4.0, 7.0,
  9.0, 6.0, 7.0, 5.0, 3.0, 8.0, 4.0, 2.0, 2.0, 1.0,
  10.0, 6.0, 8.0, 7.0, 5.0, 9.0, 4.0, 3.0, 2.0, 1.0,
  10.0, 6.0, 7.0, 9.0, 5.0, 8.0, 4.0, 3.0, 2.0, 1.0,
  7.0, 10.0, 5.0, 9.0, 6.0, 8.0, 3.0, 1.0, 4.0, 2.0,
  6.0, 9.0, 5.0, 10.0, 7.0, 8.0, 3.0, 1.0, 4.0, 2.0,
  7.0, 10.0, 5.0, 9.0, 6.0, 8.0, 3.0, 1.0, 4.0, 2.0,
  3.0, 3.0, 10.0, 3.0, 3.0, 3.0, 8.0, 6.0, 9.0, 7.0,
  3.0, 9.0, 5.0, 10.0, 8.0, 7.0, 6.0, 2.0, 4.0, 1.0,
  10.0, 6.0, 8.0, 7.0, 5.0, 9.0, 4.0, 3.0, 2.0, 1.0,
  2.0, 8.0, 3.0, 9.0, 5.0, 10.0, 1.0, 6.0, 4.0, 7.0,
  2.0, 8.0, 3.0, 9.0, 5.0, 10.0, 1.0, 6.0, 4.0, 7.0,
  8.0, 6.0, 5.0, 10.0, 7.0, 9.0, 3.0, 1.0, 4.0, 2.0,
  9.0, 3.0, 7.0, 5.0, 10.0, 1.0, 8.0, 2.0, 6.0, 4.0,
  8.0, 9.0, 3.0, 4.0, 10.0, 6.0, 2.0, 1.0, 5.0, 7.0,
  8.5, 8.5, 3.0, 5.0, 8.5, 6.0, 2.0, 1.0, 4.0, 8.5,
  5.0, 7.0, 6.0, 10.0, 8.0, 9.0, 3.0, 1.0, 4.0, 2.0
)

rankmat <- matrix(ranks, nrow = 28, byrow = TRUE)
rownames(rankmat) <- c(
  "ARAS", "COCOSO", "CODAS", "EDAS", "ELECTRE4", "FUCA", "GRA", "MABAC",
  "MAIRCA", "MARCOS", "MAUT", "MAVT", "MEGAN", "MOORA", "PROMET1",
  "PROMET2", "PROMET3", "PROMET4", "PROMET5", "PROMET6", "RAM", "ROV",
  "SMART", "TOPSIS", "VIKOR", "WASPAS", "WPM", "WSM"
)
colnames(rankmat) <- paste0("ALT_", 1:ncol(rankmat))

# Aggregate the ranks
res_summary <- rankaggregate(rankmat, tiesmethod = "average", 
   topk = 1, tol = 1e-4, niters = 10, damp = 0.4)
print(res_summary)

Comparison of Ranking Results

Description

Computes and returns multiple ranking comparison measures, including Spearman correlation, Bootstrap tests based-on Jensen-Shannon divergences, Permutation test based-on Shannon-entropy differences, Wojciech Salabun similarity coefficient, and Wilcoxon rank sum test statistics.

Usage

rankcompare(rankmat, nperms = 1000, nboot = 1000, 
    entropyopt = "jsd", j = 1, k = 3, 
    alpha = 0.05, padjmethod = "fdr",
    biplot = TRUE)

Arguments

Value

A list with five elements:

Author(s)

Cagatay Cebeci

See Also

sensana, rankentboot, rankentperm, rankmia, rankwilcox, rankwssim, rankspearman, rankrangesim

Examples

rankmat <- matrix(sample(1:10, 50, replace=TRUE), ncol=5)
rownames(rankmat) <- paste0("A", 1:10)
colnames(rankmat) <- paste0("M", 1:5)

result <- rankcompare(rankmat, entropyopt="jsd", nboot=100, nperms=100, 
   padjmethod="bonferroni", k=3, j=1)
print(result)

Bootstrap-Based Rank Entropy Comparison

Description

Computes pairwise entropy differences for ranking matrices using Shannon entropy or Jensen-Shannon divergence. The function applies a bootstrap-based statistical test and adjusts p-values using various correction methods.

Usage

rankentboot(rankmat, nboot = 1000, entropyopt = "shannon", padjmethod = "none")

Arguments

Details

This function computes entropy-based differences between ranking methods and uses bootstrap sampling to generate p-values. P-values can be adjusted using multiple correction methods. The output matrix contains:

• Lower triangle: Similarity scores (inverse entropy differences).

• Upper triangle: Bootstrap p-values (adjusted if selected).

• Diagonal: Empty values.

Value

A list containing:

Author(s)

Cagatay Cebeci

See Also

p.adjust, rankcompare, ranksrd, rankentperm, rankmia, rankwilcox, rankwssim, rankspearman, rankrangesim

Examples

# Rank matrix
rankmat <- as.matrix(read.table(text = "
                     G1   G2 G3 G4   G5 G6 G7   G8   G9 G10  G11  G12
ARAS              3 12.0  1  8  4.0  7  6 10.0  2.0   5 11.0  9.0
EDAS              3 12.0  2  8  4.0  9  6 10.0  1.0   5 11.0  7.0
ELECTRE4          3 11.5  4  8  1.5  7  5  9.5  1.5   6 11.5  9.5
FUCA              4  3.0 10  7  2.0  6  1  9.0  5.0  11  8.0 12.0
GRA               6  3.0 10  5  2.0  7  1  9.0  4.0  11  8.0 12.0
MABAC             3 12.0  4  8  1.0  7  6  9.0  2.0   5 10.0 11.0
MAIRCA           10  1.0  9  5 12.0  6  7  4.0 11.0   8  3.0  2.0
MARCOS            3 12.0  1  8  4.0  7  6 10.0  2.0   5 11.0  9.0
MEGAN             5  3.0 11  7  1.0  6  2  9.0  4.0  10  8.0 12.0
MOORA             7  1.0 11  9  2.0  4  3  8.0  6.0  10  5.0 12.0
PROMET2           3  9.0  6  8  1.0  5  4 10.0  2.0   7 11.0 12.0
RAM               7  1.0 11  9  2.0  4  3  8.0  6.0  10  5.0 12.0
ROV               3 12.0  4  8  1.0  7  6  9.0  2.0   5 10.0 11.0
SMART             5  3.0 11  7  2.0  6  1  9.0  4.0  10  8.0 12.0
TOPSIS            2 12.0  4  8  1.0  7  6  9.0  3.0   5 11.0 10.0
VIKOR             1 12.0  4  8  2.0  5  7 10.0  3.0   6  9.0 11.0
WASPAS            2 12.0  1  8  4.0  7  6 10.0  3.0   5 11.0  9.0
", header = TRUE, row.names = 1))

print(rankmat)

res_shannon_adj_none <- rankentboot(rankmat, nboot = 100, 
   entropyopt = "shannon", padjmethod="none")
print(res_shannon_adj_none)

res_shannon_adj_bonf <- rankentboot(rankmat, nboot = 100, 
   entropyopt = "shannon", padjmethod="bonferroni")
print(res_shannon_adj_bonf)

res_jsd_adj_fdr <- rankentboot(rankmat, nboot = 100, 
   entropyopt = "jsd", padjmethod="fdr")
print(res_jsd_adj_fdr)

res_jsd_adj_holm <- rankentboot(rankmat, nboot = 100, 
   entropyopt = "jsd", padjmethod="holm")
print(res_jsd_adj_holm)

Entropy-Based Pairwise Rank Comparison

Description

Compares pairwise rankings using either Shannon Entropy or Jensen-Shannon Divergence (JSD), with permutation testing for statistical significance.

Usage

rankentperm(rankmat, nperms = 1000, entropyopt = "shannon", padjmethod="none")

Arguments

Details

This function analyzes ranking stability by computing entropy-based differences between rank distributions across weighting scenarios or MCDA methods. The function allows the choice of Shannon Entropy or Jensen-Shannon Divergence, offering different sensitivity levels. Shannon Entropy was introduced the concept of entropy in Shannon's paper, 'A Mathematical Theory of Communication' in 1949, laying the foundation for information theory. The Jensen-Shannon Divergence (JSD) is based on the work of Johan Jensen and Claude Shannon. It is a symmetric version of the Kullback-Leibler Divergence: The KL Divergence is an asymmetric metric D(P||Q) \neq D(Q||P) The JSD was designed to address this asymmetry issue, making it a symmetric measure. A permutation test evaluates the statistical significance of obtained entropy based ranking differences.

Value

A list containing:

Note

Please note that JSD can provide a slightly more sensitive difference assessment compared to Shannon Entropy.

Author(s)

Cagatay Cebeci

References

Shannon, C.E., Weaver, W. (1949). The Mathematical Theory of Communication, Univ of Illinois Press. <ISBN:0-252-72548-4>

Maasoumi, E., & Racine, J. (2002). Entropy and predictability of stock market returns. Journal of Econometrics, 107(1-2), 291-312.

See Also

rankcompare, rankentboot, rankmia, rankwilcox, rankwssim, rankspearman, rankrangesim

Examples

# Example rank matrix
rankmat <- matrix(c(
  1, 1, 1, 1, 2, 2, 1, 1, 2, 2,
  2, 2, 2, 3, 3, 1, 2, 2, 3, 3,
  3, 3, 3, 2, 1, 3, 3, 3, 1, 1,
  4, 4, 4, 3, 4, 4, 4, 4, 4, 4,
  5, 5, 5, 5, 5, 5, 5, 5, 5, 5
), nrow = 5, byrow = TRUE)

rownames(rankmat) <- c("OptA", "OptB", "OptC", "OptD", "OptE")
colnames(rankmat) <- paste0("SW_", 1:10)

# Shannon Entropy method
res_shannon <- rankentperm(rankmat, nperms=100, entropyopt = "shannon", padjmethod="fdr")
print(res_shannon$displaymat)
print(res_shannon$pval_matrix)

# Jensen-Shannon Divergence method
res_jsd <- rankentperm(rankmat, nperms=100, entropyopt = "jsd", padjmethod="bonferroni")
print(res_jsd$displaymat)
print(res_jsd$pval_matrix)

Compute Goodman & Kruskal's Gamma Statistic

Description

Computes the Goodman & Kruskal's Gamma statistic between all pairs of rankings in a given matrix.

Usage

rankgamma(rankmat)

Arguments

Value

A list with the following components:

Author(s)

Cagatay Cebeci

Examples

# Rank matrix based on MCDA methods
rankmat <- matrix(c(
    3, 12.0, 1, 8, 4.0, 7, 6, 10.0, 2.0, 5, 11.0, 9.0,
    3, 12.0, 2, 8, 4.0, 9, 6, 10.0, 1.0, 5, 11.0, 7.0,
    3, 11.5, 4, 8, 1.5, 7, 5, 9.5, 1.5, 6, 11.5, 9.5,
    4, 3.0, 10, 7, 2.0, 6, 1, 9.0, 5.0, 11, 8.0, 12.0,
    6, 3.0, 10, 5, 2.0, 7, 1, 9.0, 4.0, 11, 8.0, 12.0,
    3, 12.0, 4, 8, 1.0, 7, 6, 9.0, 2.0, 5, 10.0, 11.0,
    10, 1.0, 9, 5, 12.0, 6, 7, 4.0, 11.0, 8, 3.0, 2.0,
    3, 12.0, 1, 8, 4.0, 7, 6, 10.0, 2.0, 5, 11.0, 9.0,
    3, 12.0, 4, 8, 1.0, 7, 6, 9.0, 2.0, 5, 10.0, 11.0,
    5, 3.0, 11, 7, 1.0, 6, 2, 9.0, 4.0, 10, 8.0, 12.0,
    7, 1.0, 11, 9, 2.0, 4, 3, 8.0, 6.0, 10, 5.0, 12.0,
    4, 10.0, 6, 8, 1.0, 5, 3, 9.0, 2.0, 7, 11.0, 12.0,
    3, 9.0, 6, 8, 1.0, 5, 4, 10.0, 2.0, 7, 11.0, 12.0,
    4, 10.0, 6, 8, 1.0, 5, 3, 9.0, 2.0, 7, 11.0, 12.0,
    7, 1.0, 11, 9, 2.0, 4, 3, 8.0, 6.0, 10, 5.0, 12.0,
    3, 12.0, 4, 8, 1.0, 7, 6, 9.0, 2.0, 5, 10.0, 11.0,
    5, 3.0, 11, 7, 2.0, 6, 1, 9.0, 4.0, 10, 8.0, 12.0,
    2, 12.0, 4, 8, 1.0, 7, 6, 9.0, 3.0, 5, 11.0, 10.0,
    1, 12.0, 4, 8, 2.0, 5, 7, 10.0, 3.0, 6, 9.0, 11.0,
    2, 12.0, 1, 8, 4.0, 7, 6, 10.0, 3.0, 5, 11.0, 9.0,
    2, 12.0, 1, 7, 4.0, 8, 6, 10.0, 3.0, 5, 11.0, 9.0,
    4, 7.0, 9, 5, 1.0, 6, 2, 8.0, 3.0, 11, 10.0, 12.0
), nrow=22, byrow=TRUE) # Changed nrow to 22 as there are 22 rows of data

rownames(rankmat) <- c("ARAS", "EDAS", "ELECTRE4", "FUCA", "GRA", "MABAC", "MAIRCA", 
    "MARCOS", "MAUT", "MEGAN", "MOORA", "PROMET1", "PROMET2", "PROMET6", "RAM", "ROV",
    "SMART", "TOPSIS", "VIKOR", "WASPAS", "WPM", "WSM")
colnames(rankmat) <- c("G1", "G2", "G3", "G4", "G5", 
    "G6", "G7", "G8", "G9", "G10", "G11", "G12")

print(rankmat)

result <- rankgamma(rankmat)
print(round(result$gammamat, 2))
gammamat <- round(result$gammamat, 2)

corplot(gammamat, labsize = 10, fsize = 3, fcolor = "black", 
   colpal = c("red", "white", "dodgerblue"), title = "Goodman- Kruskall's Gamma Statistics", 
   xlab = "Methods", ylab = "Methods", shape = "square", displabels = TRUE)

Rank Heatmap Plot

Description

Draws a heatmap of the overall ranks of n number of alternatives across p metrics.

Usage

  rankheatmap(rankmat, colpal = NULL, yl = NULL, xl = NULL, htitle = NULL, 
    tcol = "white", dendro = "row", cellnotes = FALSE)

Arguments

Details

A heatmap is a type of graphic where the values of elements in a matrix are represented by colors. It visualizes a data matrix with two dimensions, applying gradual colors to indicate different values of the items. Typically, darker colors signal higher values while lighter colors denote lower values. Heatmaps are popular in exploratory data analysis and data science applications. It's essential to decide on proper color encoding and meaningful reordering of rows and columns to yield informative heatmaps (Gehlenborg & Wong, 2012).

Heatmaps are often used to identify patterns, correlations, and outliers in datasets at a glance. They find applications in fields like finance, geography, data analytics, and genomics. Additionally, they can be extremely useful for model evaluation, letting users quickly gauge the performance of alternative models based on multiple criteria.

Value

No value returned.

Author(s)

Cagatay Cebeci

References

Gehlenborg, N., & Wong, B. (2012). Heat maps. Nature Methods, 9(3), 213.

See Also

boxplotmcda, rankcompare

Examples

# Example rank matrix
alternatives <- paste0("Alt", LETTERS[1:8])
criteria <- paste0("C", 1:15)

set.seed(24)
rankmat <- data.frame(Method = alternatives)
for (criterion in criteria) {
  rankmat[[criterion]] <- sample(1:8, 8, replace = FALSE)
}
rownames(rankmat) <- rankmat[,1]
rankmat <- as.matrix(rankmat[,-1])
print(rankmat)
rankheatmap(rankmat)

# Heatmap with color palette number 1 and row dendrogram only
rankheatmap(rankmat, colpal = 1, dendro = "row", cellnotes = TRUE)

# Heatmap with user-defined color palette, column dendrogram, and cell notes 
colpalcustom <- colorRampPalette(c("orange", "green", "blue"))
rankheatmap(rankmat, colpal = colpalcustom, dendro = "row", cellnotes = TRUE)

Modified Index of Agreement

Description

Calculates the Modified Index of Agreement (MIA) between multiple rankings by MCDA methods.

Usage

rankmia(rankmat, j = 1, na.rm = TRUE)

Arguments

Details

The Modified Index of Agreement (MOIA) is an improved version of the Index of Agreement (IA), originally developed by Willmott (1981). It was designed as a standardized measure to evaluate the accuracy of model predictions compared to observed data. However, limitations led to modifications that enhance its sensitivity and reliability.

This function extends the application of MOIA to the comparison of rank orderings, useful in Multi-Criteria Decision Analysis (MCDA) problems. The Index of Agreement and Modified Index of Agreement values range between 0 and 1, where 1 indicates perfect agreement and 0 indicates complete disagreement.

MIA is calculated using the following formula:

MOIA = 1 - \frac{\sum_{i=1}^{N} (|S_i - \bar{O}|^j + |O_i - \bar{O}|^j)}{\sum_{i=1}^{N} |O_i - S_i|^j}

where:

MIA

The Modified Index of Agreement value (ranging from 0 to 1).

N

The total number of items being ranked.

S_i

The rank assigned by one MCDA method.

O_i

The rank assigned by another MCDA method.

\bar{O}

The mean rank value, calculated as (N+1)/2.

j

Exponent used for absolute differences (typically j=1).

|\cdot|

The absolute value function.

\sum

Summation over all ranked items.

Higher values indicate greater agreement between rankings. This formula allows quantification of ranking similarity across MCDA methods.

Value

A data frame where each row and column represents a ranking from the input rankmat. The values in the data frame are the Modified Index of Agreement scores between the corresponding pairs of rankings, formatted to two decimal places.

Author(s)

Cagatay Cebeci

See Also

rankcompare, rankentboot,rankentperm, rankwilcox, rankwssim, rankspearman, rankrangesim

Examples

# Rank matrix based on MCDA methods
rankmat <- matrix(c(
    3, 12.0, 1, 8, 4.0, 7, 6, 10.0, 2.0, 5, 11.0, 9.0,
    3, 12.0, 2, 8, 4.0, 9, 6, 10.0, 1.0, 5, 11.0, 7.0,
    3, 11.5, 4, 8, 1.5, 7, 5, 9.5, 1.5, 6, 11.5, 9.5,
    4, 3.0, 10, 7, 2.0, 6, 1, 9.0, 5.0, 11, 8.0, 12.0,
    6, 3.0, 10, 5, 2.0, 7, 1, 9.0, 4.0, 11, 8.0, 12.0,
    3, 12.0, 4, 8, 1.0, 7, 6, 9.0, 2.0, 5, 10.0, 11.0,
    10, 1.0, 9, 5, 12.0, 6, 7, 4.0, 11.0, 8, 3.0, 2.0,
    3, 12.0, 1, 8, 4.0, 7, 6, 10.0, 2.0, 5, 11.0, 9.0,
    3, 12.0, 4, 8, 1.0, 7, 6, 9.0, 2.0, 5, 10.0, 11.0,
    5, 3.0, 11, 7, 1.0, 6, 2, 9.0, 4.0, 10, 8.0, 12.0,
    7, 1.0, 11, 9, 2.0, 4, 3, 8.0, 6.0, 10, 5.0, 12.0,
    4, 10.0, 6, 8, 1.0, 5, 3, 9.0, 2.0, 7, 11.0, 12.0,
    3, 9.0, 6, 8, 1.0, 5, 4, 10.0, 2.0, 7, 11.0, 12.0,
    4, 10.0, 6, 8, 1.0, 5, 3, 9.0, 2.0, 7, 11.0, 12.0,
    7, 1.0, 11, 9, 2.0, 4, 3, 8.0, 6.0, 10, 5.0, 12.0,
    3, 12.0, 4, 8, 1.0, 7, 6, 9.0, 2.0, 5, 10.0, 11.0,
    5, 3.0, 11, 7, 2.0, 6, 1, 9.0, 4.0, 10, 8.0, 12.0,
    2, 12.0, 4, 8, 1.0, 7, 6, 9.0, 3.0, 5, 11.0, 10.0,
    1, 12.0, 4, 8, 2.0, 5, 7, 10.0, 3.0, 6, 9.0, 11.0,
    2, 12.0, 1, 8, 4.0, 7, 6, 10.0, 3.0, 5, 11.0, 9.0,
    2, 12.0, 1, 7, 4.0, 8, 6, 10.0, 3.0, 5, 11.0, 9.0,
    4, 7.0, 9, 5, 1.0, 6, 2, 8.0, 3.0, 11, 10.0, 12.0
), nrow=22, byrow=TRUE) # Changed nrow to 22 as there are 22 rows of data

rownames(rankmat) <- c("ARAS", "EDAS", "ELECTRE4", "FUCA", "GRA", "MABAC", "MAIRCA", 
    "MARCOS", "MAUT", "MEGAN", "MOORA", "PROMET1", "PROMET2", "PROMET6", "RAM", "ROV",
    "SMART", "TOPSIS", "VIKOR", "WASPAS", "WPM", "WSM")
colnames(rankmat) <- c("G1", "G2", "G3", "G4", "G5", 
    "G6", "G7", "G8", "G9", "G10", "G11", "G12")

print(rankmat)

# Calculate the Modified Index of Agreement matrix
resmia <- rankmia(rankmat)
print(resmia)

Principal Component Analysis for Ranking Data with Geladi's Approach

Description

Performs Principal Component Analysis (PCA) on a matrix of ranks, where rows represent alternatives and columns represent criteria. This function adapts Geladi's approach to interpreting PCA results in the context of multi-criteria performance, focusing on overall performance (PC1) and uniformity of behavior (PC2) of alternatives.

Usage

rankpca(rankmat, biplot = TRUE, reverse = FALSE)

Arguments

Details

This function applies PCA to ranking data, offering insights into the underlying patterns and relationships among alternatives based on their performance across multiple criteria. It extends Geladi's concept of using PCA to analyze "regression vectors" to ranking vectors.

The interpretation of the first two principal components (PC1 and PC2) follows Geladi's framework:

• PC1 (Overall Performance): This component typically captures the largest amount of variance and differentiates alternatives based on their overall performance. Alternatives clustering at one end of PC1 generally perform better or worse across most criteria. The exact interpretation (e.g., positive PC1 score = better/worse) depends on the signs of the criteria loadings.

• PC2 (Uniformity of Behavior): This component represents the consistency of an alternative's performance across different criteria. Alternatives with PC2 scores close to zero exhibit uniform behavior, meaning their ranking profile is consistent. Alternatives with PC2 scores significantly deviating from zero show non-uniform behavior, indicating their performance varies considerably depending on the criterion.

Note: PCA on ranking data is primarily a discovery and understanding tool, not a direct ranking method. It helps identify clusters, outliers, and key driving criteria, complementing explicit ranking methods like SRD.

Value

A list containing the following components:

References

Geladi, P. (1995). The regression model comparison plot (REMOCOP), Proc. Spectroscopy Across the Spectrum IV, Norwich, UK, 11–14 July 1994, in: D. Andrews, A. Davies (Editors), Frontiers inAnalytical Spectroscopy, The Royal Society of Chemistry, Cambridge, UK, 1995, pp. 225–236.

Héberger, K. (2010). Sum of Ranking Differences (SRD) for method comparison and its relationship to the principle of parsimony. Chemometrics and Intelligent Laboratory Systems, 101(1), 1-8.

See Also

prcomp, fviz_pca_biplot, ranksrd

Examples

# Sample rank matrix (8 alternatives, 15 criteria)
rankmat <- data.frame(
  C1 = c(7, 3, 2, 8, 1, 6, 5, 4), C2 = c(2, 1, 4, 7, 6, 3, 8, 5),
  C3 = c(5, 8, 1, 7, 4, 6, 2, 3), C4 = c(8, 2, 1, 4, 3, 5, 6, 7),
  C5 = c(1, 5, 4, 6, 2, 3, 7, 8), C6 = c(6, 8, 2, 3, 1, 4, 5, 7),
  C7 = c(6, 7, 1, 8, 2, 3, 5, 4), C8 = c(6, 3, 4, 5, 2, 1, 7, 8),
  C9 = c(1, 8, 3, 4, 6, 5, 7, 2), C10 = c(7, 8, 5, 2, 4, 6, 3, 1),
  C11 = c(6, 2, 3, 8, 5, 7, 4, 1), C12 = c(6, 8, 3, 7, 4, 1, 5, 2),
  C13 = c(2, 3, 7, 5, 6, 1, 8, 4), C14 = c(2, 1, 4, 5, 3, 8, 6, 7),
  C15 = c(1, 6, 4, 5, 8, 7, 3, 2),
  row.names = c("AltA", "AltB", "AltC", "AltD", "AltE", "AltF", "AltG", "AltH")
)

# Perform Geladi PCA and show the biplot
respca <- rankpca(rankmat, biplot = TRUE, reverse=TRUE)

# Geladi PCA Analysis Results
print(paste("Proportion of Explained Variance:", paste(round(respca$expvar, 3), collapse = ", ")))

# Alternative Scores (PC1 and PC2)
print(round(respca$ascores[, 1:2], 2))

# Criteria Loadings (PC1 and PC2)
print(round(respca$critloads[, 1:2], 2))

# PC1 Interpretation
print(respca$intprtpc1)

# PC2 Interpretation
print(respca$intprtpc2)

Calculate Similarity of Ranked Alternatives in a Given Range

Description

Calculates the similarity between ranking methods by selecting alternatives from a given range and comparing them across different methods. Additionally, it provides an option to visualize the similarity matrix as a heatmap.

Usage

rankrangesim(mat, k = 3, plotsim = FALSE, 
   low_color = "blue", high_color = "red", text_color = "white")

Arguments

Details

This function takes a ranking matrix and extracts the top or bottom alternatives for each ranking method within the specified range. It then calculates how often methods select the same alternatives, forming a similarity matrix.

If plotsim = TRUE, the function generates a heatmap using ggplot2 to visually display the similarity percentages.

Value

A list containing:

See Also

rankcompare, rankentboot,rankentperm, rankmia, rankwilcox, rankwssim, rankspearman

Examples

# Example ranking matrix
rankmat <- matrix(c(
   3, 12, 1, 8, 4, 7, 6, 10, 2, 5, 11, 9,
   3, 12, 2, 8, 4, 9, 6, 10, 1, 5, 11, 7,
   3, 11.5, 4, 8, 1.5, 7, 5, 9.5, 1.5, 6, 11.5, 9.5,
   4, 3, 10, 7, 2, 6, 1, 9, 5, 11, 8, 12,
   6, 3, 10, 5, 2, 7, 1, 9, 4, 11, 8, 12,
   3, 12, 4, 8, 1, 7, 6, 9, 2, 5, 10, 11,
   3, 12, 1, 8, 4, 7, 6, 10, 2, 5, 11, 9,
   5, 3, 11, 7, 1, 6, 2, 9, 4, 10, 8, 12,
   7, 1, 11, 9, 2, 4, 3, 8, 6, 10, 5, 12,
   3, 9, 6, 8, 1, 5, 4, 10, 2, 7, 11, 12
), nrow = 10, byrow = TRUE)

rownames(rankmat) <- c("ARAS", "EDAS", "ELECT4", "FUCA", "GRA", 
   "MABAC", "MARCOS", "MEGAN", "MOORA", "PROMT2")
colnames(rankmat) <- paste0("G", 1:12)

# Similarity for the best and worst k
k1 <- 3 # Best 3 alternatives 
k2 <- -3 # Worst 3 alternatives 

res_k1 <- rankrangesim(rankmat, k=k1, plotsim = TRUE)  # Best 3 
res_k2 <- rankrangesim(rankmat, k=k2, plotsim = FALSE) # Worst 3 

print(res_k1$selected_k) 
print(res_k1$similarity) 
print(res_k1$percentage) 

print(res_k2$selected_k) 
print(res_k2$similarity) 
print(res_k2$percentage) 

# Similarity for a specific range 

kr <- c(5, 8) 
res_kr <- rankrangesim(rankmat, k=kr, plotsim = TRUE, 
   low_color = "dodgerblue", high_color = "orange", text_color = "gray")

print(res_kr$selected_k) 
print(res_kr$similarity) 
print(res_kr$percentage)

Spearman's Rank Correlation Matrix

Description

Calculates Spearman's rank correlation matrix for a ranking matrix.

Usage

rankspearman(rankmat)

Arguments

Details

In the resulting matrix, statistical significance stars appear above the diagonal, while correlation coefficients appear below the diagonal. Elements on the diagonal show the correlation coefficient (usually 1.00). The function calculates Spearman's rank correlation and associated p-values for each pair of variables using cor.test(method = "spearman"). Significance stars are assigned based on p-values:

***

$p < 0.001$

**

$p < 0.01$

*

$p < 0.05$

Blank

$p >= 0.05$ (not significant)

The stars appear in cells above the diagonal show the signifiance of correlations. Cells below the diagonal contain rounded correlation coefficients. Diagonal cells show a variable's correlation with itself (1.00).

Value

Returns a list:

Note

If ties exist in the ranking matrix, the cor.test function may be unable to compute exact p-values and will instead use approximate values. This usually does not affect the validity of the analysis but is important to note.

Author(s)

Cagatay Cebeci

See Also

rankcompare, rankentboot,rankentperm, rankmia, rankwilcox, rankwssim, rankrangesim

Examples

# Sample ranking matrix
rankmat <- as.matrix(read.table(text = "
         G1   G2 G3 G4   G5 G6 G7   G8   G9 G10  G11  G12
ARAS       3 12.0  1  8  4.0  7  6 10.0  2.0   5 11.0  9.0
EDAS       3 12.0  2  8  4.0  9  6 10.0  1.0   5 11.0  7.0
ELECTRE4   3 11.5  4  8  1.5  7  5  9.5  1.5   6 11.5  9.5
FUCA       4  3.0 10  7  2.0  6  1  9.0  5.0  11  8.0 12.0
GRA        6  3.0 10  5  2.0  7  1  9.0  4.0  11  8.0 12.0
MABAC      3 12.0  4  8  1.0  7  6  9.0  2.0   5 10.0 11.0
MAIRCA    10  1.0  9  5 12.0  6  7  4.0 11.0   8  3.0  2.0
MARCOS     3 12.0  1  8  4.0  7  6 10.0  2.0   5 11.0  9.0
MAUT       3 12.0  4  8  1.0  7  6  9.0  2.0   5 10.0 11.0
MEGAN      5  3.0 11  7  1.0  6  2  9.0  4.0  10  8.0 12.0
MOORA      7  1.0 11  9  2.0  4  3  8.0  6.0  10  5.0 12.0
PROMET1    4 10.0  6  8  1.0  5  3  9.0  2.0   7 11.0 12.0
PROMET2    3  9.0  6  8  1.0  5  4 10.0  2.0   7 11.0 12.0
PROMET6    4 10.0  6  8  1.0  5  3  9.0  2.0   7 11.0 12.0
RAM        7  1.0 11  9  2.0  4  3  8.0  6.0  10  5.0 12.0
ROV        3 12.0  4  8  1.0  7  6  9.0  2.0   5 10.0 11.0
SMART      5  3.0 11  7  2.0  6  1  9.0  4.0  10  8.0 12.0
TOPSIS     2 12.0  4  8  1.0  7  6  9.0  3.0   5 11.0 10.0
VIKOR      1 12.0  4  8  2.0  5  7 10.0  3.0   6  9.0 11.0
WASPAS     2 12.0  1  8  4.0  7  6 10.0  3.0   5 11.0  9.0
WPM        2 12.0  1  7  4.0  8  6 10.0  3.0   5 11.0  9.0
WSM        4  7.0  9  5  1.0  6  2  8.0  3.0  11 10.0 12.0",
header = TRUE, row.names = 1))

result <- rankspearman(rankmat)

print(result$displaymat)
print(result$p_values)

SRD: Sum of Ranking Differences

Description

Calculates the Sum of Ranking Differences (SRD) for a set of alternatives across multiple criteria. The SRD method assesses the performance of alternatives by comparing their actual rankings to a defined reference ranking. A lower SRD value indicates better agreement with the reference and thus, better performance.

Usage

ranksrd(rankmat, refopt = "mean", defrank = NULL, tiesmethod="average")

Arguments

Details

The SRD method, as proposed by Károly Héberger (Héberger, 2010), offers a fair and unambiguous way to compare methods, models, or alternatives. It quantifies the "goodness" of an alternative by summing the absolute differences between its actual rank in each criterion and its corresponding reference rank. The procedure is repeated for each alternative, and the resulting SRD values are used to rank them.

The formula for SRD for a single alternative is:

SRD_i = \sum_{j=1}^{m} |R_{ij} - R_{ref,i}|

where R_{ij} is the rank of alternative i in criterion j, R_{ref,i} is the reference rank for alternative i, and m is the total number of criteria. A smaller SRD value indicates that the alternative's ranking is closer to the reference ranking, implying better performance.

Value

A list containing the following components:

Note

Ensure that the input rankmat contains actual rank values (e.g., 1, 2, 3...) and not raw performance scores. The ranks should be consistent (e.g., 1 is always best, higher numbers are worse).

Author(s)

Cagatay Cebeci

References

Héberger, K. (2010). Sum of Ranking Differences (SRD) for method comparison and its relationship to the principle of parsimony. Chemometrics and Intelligent Laboratory Systems, 101(1), 1-8.

See Also

rank

Examples

# Sample rank matrix (8 alternatives, 15 criteria)
rankmat <- data.frame(
  C1 = c(7, 3, 2, 8, 1, 6, 5, 4), C2 = c(2, 1, 4, 7, 6, 3, 8, 5),
  C3 = c(5, 8, 1, 7, 4, 6, 2, 3), C4 = c(8, 2, 1, 4, 3, 5, 6, 7),
  C5 = c(1, 5, 4, 6, 2, 3, 7, 8), C6 = c(6, 8, 2, 3, 1, 4, 5, 7),
  C7 = c(6, 7, 1, 8, 2, 3, 5, 4), C8 = c(6, 3, 4, 5, 2, 1, 7, 8),
  C9 = c(1, 8, 3, 4, 6, 5, 7, 2), C10 = c(7, 8, 5, 2, 4, 6, 3, 1),
  C11 = c(6, 2, 3, 8, 5, 7, 4, 1), C12 = c(6, 8, 3, 7, 4, 1, 5, 2),
  C13 = c(2, 3, 7, 5, 6, 1, 8, 4), C14 = c(2, 1, 4, 5, 3, 8, 6, 7),
  C15 = c(1, 6, 4, 5, 8, 7, 3, 2),
  row.names = c("AltA", "AltB", "AltC", "AltD", "AltE", "AltF", "AltG", "AltH")
)

# Using mean rank as reference (default)
ressrd_mean <- ranksrd(rankmat, refopt = "mean")
print(ressrd_mean)

# Using the best possible rank (rank 1) as reference
ressrd_best <- ranksrd(rankmat, refopt = "best")
print(ressrd_best$srdvals)
print(ressrd_best$srdrank)
print(ressrd_best$rank)

# Using the worst possible rank (max rank in data) as reference
ressrd_worst <- ranksrd(rankmat, refopt = "worst")
print(ressrd_worst$srdvals)
print(ressrd_worst$srdrank)

# Using user defined reference
usrrank <- c(4, 6, 1, 8, 2, 3, 7, 5)
names(usrrank) <- rownames(rankmat) 

ressrd_user <- ranksrd(rankmat, refopt = "custom", defrank = usrrank)
print(ressrd_user$srdvals)
print(ressrd_user$srdrank)

Pairwise Wilcoxon Rank Sum Tests for Comparing Rows of a Matrix

Description

Performs pairwise Wilcoxon Rank Sum tests /WSRT) to compare the rows of a given matrix. P-values are adjusted for multiple comparisons using the specified method.

Usage

rankwilcox(rankmat, alpha = 0.05, padjmethod = "none")

Arguments

Details

This function iterates through all unique pairs of rows in the input matrix and performs a Wilcoxon signed-rank test for each pair. The Wilcoxon signed-rank test is used to assess the significance of the difference between two related samples. The p-values obtained from these pairwise tests are then adjusted for multiple comparisons to control the family-wise error rate. The results are presented in a matrix format, with adjusted p-values in the upper triangle and test statistics in the lower triangle.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

See Also

rankcompare, rankentboot,rankentperm, rankmia, rankwssim, rankspearman, rankrangesim

Examples

# Rank matrix
rankmat <- matrix(c(
    3, 2, 3, 5, 6, 3, 3, 5, 5, 2, 5, 5, 2, 1, 3,
    7, 6, 7, 8, 8, 7, 7, 7, 7, 4, 7, 8, 7, 7, 7,
    1, 1, 1, 1, 2, 1, 1, 1, 1, 5, 1, 2, 1, 2, 1,
    8, 8, 8, 3, 5, 6, 8, 4, 3, 7, 3, 3, 8, 3, 3,
    2, 5, 2, 2, 1, 2, 2, 2, 2, 8, 2, 1, 3, 4, 2,
    4, 7, 5, 7, 7, 8, 4, 8, 8, 1, 8, 7, 6, 8, 8,
    6, 4, 6, 4, 3, 5, 6, 3, 4, 6, 4, 4, 5, 6, 6,
    5, 3, 4, 6, 4, 4, 5, 6, 6, 3, 6, 6, 4, 5, 5),
    nrow = 5, byrow = TRUE)
rownames(rankmat) <- paste0("MET", 1:nrow(rankmat))
colnames(rankmat) <- paste0("ALT", 1:ncol(rankmat))

res_wilcox <- rankwilcox(rankmat)
print(res_wilcox$displaymat)

Wojciech Sałabun Coefficient of Rankings Similarity (WS)

Description

This function calculates the Wojciech Sałabun Coefficient of Rankings Similarity (WS) between different rankings of examined MCDA methods. The WS coefficient quantifies the similarity between two rankings of MCDA methods, giving higher priority to the similarity of positions at the top of the ranking.

Usage

rankwssim(rankmat)

Arguments

Details

The wranksim function takes a rank matrix as input and computes the WS coefficient for each pair of rankings within the matrix. The WS coefficient, introduced by Sałabun and Urbaniak, is calculated using the following formula: WS = 1 - \sum_{i=1}^{n} \left( 2^{-R_{xi}} \cdot \frac{|R_{xi} - R_{yi}|}{\max(|1 - R_{xi}|, |N - R_{xi}|)} \right) where:

WS

represents the value of the similarity coefficient.

N

denotes the length of the ranking (number of alternatives).

R_{xi}

represents the rank of the i-th element in ranking x.

R_{yi}

represents the rank of the i-th element in ranking y.

The WS coefficient ranges from 0 to 1, where 1 indicates perfect similarity and 0 indicates complete dissimilarity between the two rankings. This metric prioritizes the agreement in the top ranks, which is often crucial in Multiple Criteria Decision Making (MCDM) for identifying the most preferred alternatives.

Due to the dependence of the WS coefficient value on the reference dataset (x), the resulting matrix of WS coefficient values is asymmetrical, unlike Spearman's rank correlation coefficient. Therefore, for accurate and fair comparisons, each MCDM method's ranking set should be used as the reference (x) when determining the WS coefficient value.

Based on Sałabun and Urbaniak (2020), Rachman et al (2024) states that two rankings are considered to have high ranking similarity if WS > 0.689, while WS < 0.352 indicates low similarity. MCDA methods are assumed to have a medium level of likeness if the WS value falls within the range from 0.352 to 0.689. Two MCDA methods are defined having totally similar if they reach absolute consensus by achieving Spearman Rank Correlation = +1 with WS = 1.

Value

The function returns a data frame containing the WS coefficients between all pairs of input rankings. The row and column names correspond to the names of the rankings from the input matrix. Each cell \((i, j)\) in the data frame represents the WS coefficient calculated with the \(i\)-th ranking as the reference (x) and the \(j\)-th ranking as the compared ranking (y). The coefficients are rounded to two decimal places and formatted as character strings.

Author(s)

Cagatay Cebeci

References

Rachman, A. P., Ichwania, C., Mangkuto, R. A., Pradipta, J., Koerniawan, M. D., & Sarwono, J. (2024). Comparison of multi-criteria decision-making methods for selection of optimum passive design strategy. Energy and Buildings, 314, 114285. <doi:10.1016/j.enbuild.2024.114285>

Sałabun, W., Urbaniak, K. (2020). A New Coefficient of Rankings Similarity in Decision-Making Problems. In: Krzhizhanovskaya, V., et al. Computational Science – ICCS 2020. ICCS 2020. Lecture Notes in Computer Science, vol 12138. Springer, Cham. <doi:10.1007/978-3-030-50417-5_47>

See Also

rankcompare, rankentboot,rankentperm, rankmia, rankwilcox, rankspearman, rankrangesim

Examples

# Rank matrix
rankmat <- as.matrix(read.table(text = "
                     G1   G2 G3 G4   G5 G6 G7   G8   G9 G10  G11  G12
ARAS              3 12.0  1  8  4.0  7  6 10.0  2.0   5 11.0  9.0
EDAS              3 12.0  2  8  4.0  9  6 10.0  1.0   5 11.0  7.0
ELECTRE4          3 11.5  4  8  1.5  7  5  9.5  1.5   6 11.5  9.5
FUCA              4  3.0 10  7  2.0  6  1  9.0  5.0  11  8.0 12.0
GRA               6  3.0 10  5  2.0  7  1  9.0  4.0  11  8.0 12.0
MABAC             3 12.0  4  8  1.0  7  6  9.0  2.0   5 10.0 11.0
MAIRCA           10  1.0  9  5 12.0  6  7  4.0 11.0   8  3.0  2.0
MARCOS            3 12.0  1  8  4.0  7  6 10.0  2.0   5 11.0  9.0
MAUT              3 12.0  4  8  1.0  7  6  9.0  2.0   5 10.0 11.0
MEGAN             5  3.0 11  7  1.0  6  2  9.0  4.0  10  8.0 12.0
MOORA             7  1.0 11  9  2.0  4  3  8.0  6.0  10  5.0 12.0
PROMET1           4 10.0  6  8  1.0  5  3  9.0  2.0   7 11.0 12.0
PROMET2           3  9.0  6  8  1.0  5  4 10.0  2.0   7 11.0 12.0
PROMET6           4 10.0  6  8  1.0  5  3  9.0  2.0   7 11.0 12.0
RAM               7  1.0 11  9  2.0  4  3  8.0  6.0  10  5.0 12.0
ROV               3 12.0  4  8  1.0  7  6  9.0  2.0   5 10.0 11.0
SMART             5  3.0 11  7  2.0  6  1  9.0  4.0  10  8.0 12.0
TOPSIS            2 12.0  4  8  1.0  7  6  9.0  3.0   5 11.0 10.0
VIKOR             1 12.0  4  8  2.0  5  7 10.0  3.0   6  9.0 11.0
WASPAS            2 12.0  1  8  4.0  7  6 10.0  3.0   5 11.0  9.0
WPM               2 12.0  1  7  4.0  8  6 10.0  3.0   5 11.0  9.0
WSM               4  7.0  9  5  1.0  6  2  8.0  3.0  11 10.0 12.0
", header = TRUE, row.names = 1))

print(rankmat)

# Calculate the WS similarity matrix using rank matrix
reswsm <- rankwssim(rankmat)
print(reswsm$displaymat)

Renewable Energy Grids

Description

The renewable dataset provides a decision matrix (dmat), a benefit-cost vector (bcvec), and a weights vector (weights) for evaluating and ranking renewable energy grid alternatives using multi-criteria decision-making methods.

Usage

data(egrids)

Format

A list with the following components:

dmat

A numeric matrix with 10 rows and 14 columns, where each row represents a grid alternative (G1 to G10) and each column represents a criterion. The criteria based on Prior et al (2025), are defined as follows:

• C1 - Local Community (%): Measures the impact of the energy system on improving local living standards and economic opportunities by providing reliable and clean energy services. Higher percentages indicate greater benefits for the community.

• C2 - Social Equity (%): Evaluates the fairness of energy access and benefit distribution among community members. Higher percentages reflect more equitable outcomes across different social groups.

• C3 - Education Awareness (%): Assesses the public’s knowledge and awareness about renewable energy and sustainable practices. An elevated percentage implies a better-informed community ready to embrace clean energy solutions.

• C4 - Emission (Score): Quantifies the harmful emissions (e.g., CO_2 or other pollutants) generated by the energy system. This is a cost-type criterion; lower scores are more favourable as they signify reduced environmental harm.

• C5 - Biodiversity Loss (Score): Captures the negative impact of the energy project on local ecosystems and species diversity. Being a cost indicator, lower scores are preferable since they represent less biodiversity degradation.

• C6 - Sustainable Resources (%): Represents the efficient and sustainable use of natural resources within the energy system, including aspects like recycling and conservation. Higher percentages demonstrate improved resource sustainability.

• C7 - Technological Innovations (Score): Reflects the level of new technology adoption and innovations integrated into the energy system. Higher scores indicate more advanced technological implementations.

• C8 - Energy Efficiency (%): Measures how effectively the energy system converts input into useful output with minimal losses. A higher percentage shows a more efficient operation of the grid.

• C9 - Support Infrastructure (Score): Assesses the quality and modernity of the supporting infrastructure, such as smart grids, communication systems, and storage facilities. Higher scores denote a stronger, more modern infrastructure.

• C10 - Implementation Cost ($M): Quantifies the initial capital investment required for deploying the energy system, measured in millions of dollars. As a cost-type criterion, lower amounts are typically more desirable. In our simulation, advanced systems may involve higher costs indicative of superior technology.

• C11 - Economic Efficiency (

• C12 - Governmental Incentives (Score): Gauges the level and strength of governmental support, such as subsidies and incentives, available for the energy project. Higher scores suggest a more supportive policy environment.

• C13 - Macroeconomics (Score): Represents the broader economic benefits of the energy system, including impacts on investment, job creation, and infrastructure development. Higher scores are associated with positive macroeconomic contributions.

• C14 - Microeconomics (Score): Focuses on project-specific financial indicators (e.g., ROI, NPV) that reflect the viability and profitability of the energy system. Higher scores indicate more attractive economic prospects at the micro level.

For example, the first row (G1) of dmat contains the values: 78, 74, 66, 18, 11, 75, 65, 78, 78, 171, 69, 44, 59, and 50.

bcvec

A numeric vector of length 14 indicating the type of each criterion:

• 1: Benefit criterion (higher is better).

• -1: Cost criterion (lower is better).

In this dataset, the benefit-cost vector is: 1, 1, 1, -1, -1, 1, 1, 1, 1, -1, 1, 1, 1, 1.

weights

A numeric vector of length 14 containing the weights assigned to each criterion. The weights are as follows: 0.10, 0.05, 0.08, 0.08, 0.09, 0.13, 0.05, 0.06, 0.08, 0.09, 0.08, 0.04, 0.03, and 0.04, which sum to 1.

Details

The dataset is a synthetic dataset designed for evaluating and comparing renewable grid alternatives using multi-criteria decision-making methods. Each row in the decision matrix (dmat) represents a grid alternative (labeled G1 through G10) and each column corresponds to a specific criterion as described above. The benefit-cost vector (bcvec) indicates whether a criterion is to be maximized (benefit, 1) or minimized (cost, -1), while the weights vector (weights) reflects the relative importance of each criterion.

Author(s)

Cagatay Cebeci

References

Prior, J. F. P., Siluk, J. C. M. , Rigo, P. D. & Numes-Ramos, V. (2025). Factor for Decision-Making on Renewable Energy Modes in Rural Properties. Power System Techonlogy, 49(2), 354-393.

Examples

data(egrids)
# Display the decision matrix.
print(egrids$dmat)
# Display the benefit-cost vector.
print(egrids$bcvec)
# Display the criteria weights.
print(egrids$weights)

ROV: Range of Value

Description

Applies the Rank of Value (ROV) method on a decision matrix using specified weights and a benefit-cost vector.

Usage

rov(dmatrix, bcvec, weights, normethod = "maxmin", tiesmethod="average")

Arguments

Details

The rov function implements the Rank Order Centroid (ROV) method, a multi-criteria decision-making (MCDM) technique used to rank alternatives. The function takes a decision matrix, a benefit-cost vector, and criterion weights as input. The decision matrix is first normalized using the specified normalization method. Then, the criterion weights are either provided directly or calculated using a specified method. Finally, a utility score is calculated for each alternative, and the alternatives are ranked based on these scores.

Value

A list includes the following results:

Author(s)

Cagatay Cebeci

References

Madic, M., Radovanovic, M., & Manic, M. (2016). Application of the ROV method for the selection of cutting fluids. Decision Science Letters, 5(2), 245-254.

See Also

calcweights, calcnormal

Examples

# Sample decision matrix
dmat <- matrix(c(
  84, 10, 5, 20,  
  66, 18, 8, 15,
  74, 12, 6, 25,
  90, 22, 9, 18, 
  68, 18, 7, 15  
), nrow = 5, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4", "A5")
colnames(dmat) <- c("C1", "C2", "C3", "C4")

# Benefit-cost vector (1: benefit, -1: cost)
bc <- c(1, -1, 1, -1)

# User-defined weights
uw <- c(0.1, 0.4, 0.3, 0.2)
  
# Apply ROV using user-defined weights
resrov_1 <- rov(dmatrix=dmat, bcvec=bc, weights=uw)
print(resrov_1)

# Apply ROV using equal weights
equw <- calcweights(dmatrix=dmat, bcvec=bc, type="equal", normethod="maxmin")
resrov_2 <- rov(dmatrix=dmat, bcvec=bc, weights = equw)
print(resrov_2)

Sankey Diagram

Description

Generates a Sankey diagram to visualize the relationship between ranking methods and alternatives.

Usage

sankeyplot(rankmat, fs = 12, nw = 30)

Arguments

Details

Ideal for MCDM contexts where ranking comparison is crucial. The Sankey diagram shows the mapping between ranking methods (sources) and alternatives (targets), where link thickness reflects the strength of preference. Each link's width is computed as k - Rank, meaning that lower ranks (i.e., better positions) result in thicker flows.

This visualization allows quick identification of which alternatives are consistently favored across multiple methods, and highlights divergence in method behavior. For example, an alternative consistently ranked highly will attract wider links from several methods, creating a visually dominant path toward that target.

Value

An interactive Sankey diagram rendered using the networkD3 package.

See Also

sankeyNetwork

Examples

rankmat <- matrix(c(
  5,9,7,8,4,2,10,11,3,1,12,6,
  3,12,8,4,9,6,7,10,2,5,11,1,
  2,12,4,8,6,7,9,10,5,1,11,3,
  2,11,6,8,3,5,9,10,1,4,12,7,
  7,9,6,8,5,2,11,10,3,1,12,4,
  5,9,7,8,4,2,10,11,3,1,12,6,
  5,8,6,9,4,2,10,11,3,1,12,7,
  4,6,9,8,1,2,7,10,3,5,11,12,
  5,8,6,9,1,3,7,12,2,4,11,10,
  5,7,6,9,1,2,8,12,3,4,11,10
), nrow = 10, byrow = TRUE)
rownames(rankmat) <- c("Equal","Critic","StdDev","Gini","Geometric",
  "Merec","Mpsi","Entropy","Roc","Rs")
colnames(rankmat) <- paste0("G",1:12)
sankeyplot(rankmat)

Selection of Multi-Criteria Decision-Making Methods

Description

This functions selects the most suitable MCDM method for a given problem by evaluating the ranking results of various MCDM methods (rankmat) against the decision matrix of alternatives (dmatrix). The function proposes the MCDM method that best ranks the alternatives with the highest composite criterion value among the "top m" alternatives. The number of "top" alternatives (mval) can be determined manually or dynamically using various methods (percentage-based, entropy-based, rule-based, or top-k frequency).

Usage

selectmcdm(rankmat, dmatrix, bcvec, weights, mval = NULL,
  mvmet = "percent", mvpars = list(perc = 0.25, rule = c(2, 4, 5), k=5, r=3),
  verbose = TRUE)

Arguments

Details

The function evaluates multiple MCDM methods and selects the most suitable one based on cumulative performance values. As the extensions to the proposed method by Bandyopadhyay (2021), if mval is not provided, it is determined dynamically using one of the following methods:

• "percent": Selects a percentage of alternatives based on mvpars$perc.

• "entropy": Uses Shannon entropy to determine mval dynamically.

• "rule": Applies predefined thresholds to determine mval.

• "topk": Selects alternatives appearing frequently in the top k positions.

Value

Returns a list with the following components:

Author(s)

Cagatay Cebeci

References

Bandyopadhyay, S. (2021). Comparison among multi-criteria decision analysis techniques: A novel method. Progress in Artificial Intelligence, 10(2), 195-216. <doi:10.1007/s13748-021-00235-5>

Examples

# Decison matrix
datadf <- data.frame(
  P = c(4800, 10000, 3890, 7800, 4450, 5000, 1990, 3000, 4400, 4000),
  MSS = c(2700, 3000, 4000, 2000, 3000, 2700, 3000, 2200, 2400, 2800),
  MSL = c(12.7, 20, 13, 8, 14.5, 12.8, 40, 12.7, 14.3, 15.2),
  LC = c(50, 70, 10, 8, 6.8, 10, 6, 6.5, 14.3, 8),
  MC = c(3000, 4000, 500, 1500, 1500, 2000, 350, 1500, 500, 1000),
  MR = c(980, 1000, 800, 900, 1040, 500, 510, 1200, 850, 900),
  W = c(150, 200, 210, 100, 500, 350, 200, 230, 200, 250),
  WRT = c(6, 1, 3, 1, 5, 1, 2, 1, 3, 2)
)
rownames(datadf) <- paste0("Rob.", 1:10)
dmat <- as.matrix(datadf)

# Benefit-Cost vector
bc <- c(
  P = -1,  # Price
  MSS = 1, # Maximum sewing speed (rpm)
  MSL = 1, # Maximum stitch length (mm)
  LC = 1,  # Load capacity (kg)
  MC = 1,  # Memory capacity
  MR = 1,  # Manipulator reach
  W = -1,  # Weight (kg)
  WRT = 1  # Warranty (years)
)

# Weights
uwei <- c(
  P = 0.172414,
  MSS = 0.172414,
  MSL = 0.137931,
  LC = 0.12069,
  MC = 0.013448,
  MR = 0.137931,
  W = 0.068966,
  WRT = 0.086207
)

rankdf <- data.frame(
  R1 = c(5, 10, 10, 8, 9, 8, 3, 2, 4, 2, 2, 7, 1, 4),
  R2 = c(10, 9, 9, 10, 7, 10, 1, 1, 1, 1, 1, 10, 2, 1),
  R3 = c(3, 6, 8, 4, 1, 5, 4, 5, 8, 2, 7, 3, 3, 6),
  R4 = c(9, 3, 4, 5, 8, 9, 6, 3, 2, 6, 4, 9, 7, 3),
  R5 = c(7, 7, 6, 9, 3, 6, 2, 4, 5, 4, 3, 5, 5, 2),
  R6 = c(8, 1, 1, 7, 4, 7, 7, 6, 3, 5, 5, 8, 10, 5),
  R7 = c(1, 5, 7, 1, 2, 1, 9, 8, 10, 3, 10, 1, 4, 10),
  R8 = c(2, 2, 5, 3, 6, 2, 10, 10, 9, 7, 9, 2, 6, 9),
  R9 = c(6, 4, 2, 2, 10, 3, 8, 9, 6, 9, 8, 6, 9, 8),
  R10 = c(4, 8, 3, 6, 5, 4, 5, 7, 7, 8, 6, 4, 8, 7)
)
rownames(rankdf) <- c("TOPSIS", "MAUT", "MACBETH", 
   "ORESTE", "VIKOR", "SIR", "EVAMIX", "ARAS", "MOORA",
   "COPRAS", "WASPAS", "TODIM", "MABAC", "MARE")
rankmat <- as.matrix(rankdf)

# With a user-defined specified mval
results_user <- selectmcdm(rankmat, dmat, bc, uwei, mval = 4, verbose = FALSE)
print(results_user)

# With percentage-based mval
results_percent <- selectmcdm(rankmat, dmat, bc, uwei, 
   mvmet = "percent", mvpars = list(perc = 0.25), verbose = FALSE)
print(results_percent)

# With entropy-based mval
results_entropy <- selectmcdm(rankmat, dmat, bc, uwei, 
   mvmet = "entropy", verbose = FALSE)
print(results_entropy)

# With rule-based mval
results_rule <- selectmcdm(rankmat, dmat, bc, uwei, mvmet = "rule",
   mvpars = list(rule = c(2, 6, 8)), verbose = FALSE)
print(results_rule)

# With top-k based mval
results_topk_m <- selectmcdm(rankmat, dmat, bc, uwei, mvmet = "topk",
   mvpars = list(k = 3, r = nrow(dmat)/2), verbose = FALSE)
print(results_topk_m)

Sensitivity and Stability Analyses for MCDA

Description

Performs various sensitivity and stability analyses on a matrix of rank results from Multi-Criteria Decision Analysis (MCDA).

Usage

sensana(rankmat)

Arguments

Details

This function calculates several measures to assess the sensitivity and stability of alternative rankings across different scenarios. The computed measures include:

• SD: The standard deviation of the ranks for each alternative across all scenarios, indicating the dispersion of its ranks.

• RSI: Rank Stability Index (Stability scores) for each alternative, calculated as 1 minus the proportion of rank changes across scenarios. A higher score indicates greater rank stability.

• RVOL: Rank Volatility scores for each alternative, representing the proportion of rank changes between consecutive scenarios. A lower score indicates less rank volatility.

• sensscores: Weight Sensitivity scores for each scenario, representing the average absolute rank change of all alternatives when compared to the first scenario. Lower scores suggest higher stability of the overall ranking relative to the first scenario.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

Examples

# Example rank matrix
rankmat <- matrix(
  c(1, 2, 1, 3,
    2, 1, 3, 1,
    3, 3, 2, 2),
  nrow = 3,
  byrow = TRUE,
  dimnames = list(c("ALT1", "ALT2", "ALT3"), paste0("S_", 1:4))
)

# Perform sensitivity and stability analysis
results <- sensana(rankmat)

print(results)

Plot Weight Sensitivity Analysis Results

Description

Creates a bar or pie chart to visualize the frequency of ranking patterns resulting from a sensitivity analysis performed by weisana.

Usage

sensplot(senstable, topn = 10, colpal = NULL, type = "bar", mtitle = NULL)

Arguments

Details

This function is a companion to weisana() and is designed to offer a quick graphical summary of how frequently different ranking outcomes appear across weight scenarios. The bar plot also includes percentage labels above the bars for easy interpretation.

Value

No return value.

Author(s)

Cagatay Cebeci

See Also

weisana for generating the sensitivity table input.

Examples

# Simulated example
set.seed(123)
results <- data.frame(
  Pattern = c("2,1,2,4", "2,1,3,4", "1,3,2,4", "1,3,4,2"),
  Count = c(50, 30, 15, 5),
  Percent = c(50, 15, 20, 5),
  stringsAsFactors = FALSE
)

sensplot(results, topn = 5, type = "bar")
sensplot(results, type = "pie", mtitle = "Ranking Pattern Distribution")

SMART: Simple Multi-Attribute Rating Technique

Description

This function applies the Simple Multi-Attribute Rating Technique (SMART) for multi-criteria decision analysis. The SMART evaluates alternatives based on weighted aggregation of their values determined by value functions.

Usage

smart(dmatrix, bcvec=NULL, weights, normethod=NULL, valfuncs = NULL, tiesmethod="average")

Arguments

Details

The SMART method involves defining value functions for each criterion to scale the performance values and assigning weights to the criteria to reflect their importance. The overall score for each alternative is then calculated as the weighted sum of its values across all criteria. Alternatives are ranked based on their overall scores in descending order.

If weights is a numeric vector, these weights are used directly after normalization.

If valfuncs is not provided, linear value functions (min-max scaling) are used to normalize the performance values of each criterion to a 0-1 range.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Olson, D.L. (1996). Smart. In: Decision Aids for Selection Problems. Springer Series in Operations Research. Springer, New York, NY. <doi:10.1007/978-1-4612-3982-6_4>

See Also

calcweights, calcnormal

Examples

# Sample decision matrix
dmat <- matrix(c(
  84, 10, 5, 20,  
  66, 18, 8, 15,
  74, 12, 6, 25,
  90, 22, 9, 18, 
  68, 18, 7, 15  
), nrow = 5, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4", "A5")
colnames(dmat) <- c("C1", "C2", "C3", "C4")

# Benefit-cost vector (1: benefit, -1: cost)
bc <- c(1, -1, 1, -1)

# User-defined weights
uw <- c(0.1, 0.4, 0.3, 0.2)
ressmart_1 <- smart(dmatrix=dmat, bcvec=bc, weights=uw)
print(ressmart_1)

# Criteria weights with std. dev. method
sdw <- calcweights(dmat, bcvec=bc, type="sdev")
ressmart_2 <- smart(dmatrix=dmat, bcvec=bc, weights = sdw)
print(ressmart_2)

# Custom value functions
vf <- list(
  function(x) 1 - (x - min(dmat[, 1])) / (max(dmat[, 1]) - min(dmat[, 1])),
  function(x) (x - min(dmat[, 2])) / (max(dmat[, 2]) - min(dmat[, 2])),
  function(x) 1 - (x - min(dmat[, 1])) / (max(dmat[, 1]) - min(dmat[, 1])),
  function(x) (x - min(dmat[, 2])) / (max(dmat[, 2]) - min(dmat[, 2]))
)
ressmart_3 <- smart(dmatrix=dmat, bcvec=bc, weights=uw, valfuncs = vf)
print(ressmart_3)

TOPSIS: Technique for Order of Preference by Similarity to Ideal Solution

Description

This function implements the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method for multi-criteria decision-making by evaluating alternatives based on their distances from the ideal and anti-ideal solutions.

Usage

topsis(dmatrix, bcvec, weights, normethod = "maxmin", tiesmethod = "average")

Arguments

Details

The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method for Multi-Criteria Decision Making (MCDM) was originally developed by Ching-Lai Hwang and Kwangsun Yoon in 1981. Their seminal work, "Multiple Attribute Decision Making: Methods and Applications", published by Springer-Verlag in 1981, introduced the fundamental concepts and methodology of TOPSIS. Later, there were further developments and refinements of the method by Yoon in 1987, and by Hwang, Lai, and Liu in 1993. The TOPSIS method evaluates alternatives by normalizing the decision matrix, applying weights, and calculating distances to the positive ideal solution (pis) and the negative ideal solution (nis). The closeness coefficient (p) is computed as the ratio of the distance to the negative ideal solution (Dm) to the sum of the distances to both ideal solutions (Dm + Dp).

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Hwang, C.L. & Yoon, K. (1981). Multiple Attribute Decision Making: Methods and Applications. New York: Springer-Verlag.

Yoon, K. (1987). A reconciliation among discrete compromise solutions. Journal of the Operational Research Society, 38(3), 277-286. <doi:10.1057/jors.1987.44>

Hwang, C. L., Lai, Y. J., & Liu, T. Y. (1993). A new approach for multiple objective decision making. Computers & Operations Research, 20(8), 889-899.

See Also

calcweights, calcnormal

Examples

# Sample decision matrix
dmat <- matrix(c(
  84, 10, 5, 20,  
  66, 18, 8, 15,
  74, 12, 6, 25,
  90, 22, 9, 18, 
  68, 18, 7, 15  
), nrow = 5, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4", "A5")
colnames(dmat) <- c("C1", "C2", "C3", "C4")

# Benefit-cost vector (1: benefit, -1: cost)
bc <- c(1, -1, 1, -1)

# User-defined weights
uw <- c(0.1, 0.4, 0.3, 0.2)

# Apply TOPSIS with user-defined weights
restopsis_1 <- topsis(dmatrix = dmat, bcvec = bc, weights = uw)
print(restopsis_1) 

# Apply TOPSIS with entropy weighting method
entw <- calcweights(dmat, bcvec=bc, type="entropy")
restopsis_2 <- topsis(dmatrix = dmat, bcvec = bc, weights = entw)
print(restopsis_2$p) # Closeness coefficients for ranking alternatives
print(restopsis_2$rank) # Rank of alternatives

VIKOR: VIsekriterijumska optimizacija i KOmpromisno Resenje

Description

Implements the VIsekriterijumska optimizacija i KOmpromisno Resenje (VIKOR) (Multi-criteria optimization and Compromise Solution in English) method to rank alternatives by evaluating their compromise solutions based on the closeness to ideal solutions.

Usage

vikor(dmatrix, bcvec, weights, normethod = "maxmin", v = 0.5, tiesmethod = "average")

Arguments

Details

The VIKOR method normalizes the decision matrix, applies weights, and calculates the utility measure (S), the regret measure (R), and the compromise measure (Q). These measures are used to rank the alternatives, with the best compromise solution identified as the one with the smallest Q value.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Opricović, S. (1979). Multicriteria Optimization in Civil Engineering. (Ph.D. Dissertation, in Serbian), Faculty of Civil Engineering, Belgrade.

Duckstein, L., & Opricovic, S. (1980). Multiobjective optimization in river basin development. Water Resources Research, 16(1), 14-20.

Opricović, S. (1998). Multicriteria optimization of civil engineering systems. Faculty of civil engineering, Belgrade, 2(1), 5-21.

Opricovic, S., & Tzeng, G. H. (2004). Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS. European journal of Operational Research, 156(2), 445-455. <doi:10.1016/S0377-2217(03)00020-1>

See Also

calcweights, calcnormal

Examples

# Sample decision matrix
dmat <- matrix(c(
  84, 10, 5, 20,  
  66, 18, 8, 15,
  74, 12, 6, 25,
  90, 22, 9, 18, 
  68, 18, 7, 15  
), nrow = 5, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4", "A5")
colnames(dmat) <- c("C1", "C2", "C3", "C4")

# Benefit-cost vector (1: benefit, -1: cost)
bc <- c(1, -1, 1, -1)

# User-defined weights
uw <- c(0.1, 0.4, 0.3, 0.2)

resvikor_1 <- vikor(dmatrix=dmat, bcvec=bc, weights=uw, v = 0.5)
print(resvikor_1)

resvikor_2 <- vikor(dmatrix=dmat, bcvec=bc, weights=uw, v = 0.1)
print(resvikor_2)

criw <- calcweights(dmat, bcvec=bc, type="critic")
resvikor_3 <- vikor(dmatrix=dmat, bcvec=bc, weights=criw, 
  normethod="vector", v = 0.5)
print(resvikor_3)

WASPAS: Weighted Aggregated Sum Product Assessment

Description

An implementation of the Weighted Aggregated Sum Product Assessment (WASPAS) method, which combines the Weighted Sum Model (WSM) and the Weighted Product Model (WPM), to evaluate and rank alternatives based on normalized scores and assigned weights.

Usage

waspas(dmatrix, bcvec, weights, normethod = "linear", v = 0.5, tiesmethod = "average")

Arguments

Details

The WASPAS method integrates the advantages of the Weighted Sum Model (WSM) and the Weighted Product Model (WPM) through a parameterized combination. Alternatives are evaluated based on a sum-based score (qsum) and a product-based score (qprod), with the final score (p) calculated as a weighted average of the two.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Chakraborty, S., Zavadskas, E. K., & Antuchevičienė, J. (2015). Applications of WASPAS method as a multi-criteria decision-making tool. URI <https://etalpykla.vilniustech.lt/handle/123456789/151097>.

Zavadskas, E. K., Vilutienė, T., Turskis, Z., & Šaparauskas, J. (2014). Multi-criteria analysis of Projects’ performance in construction. Archives of Civil and Mechanical Engineering, 14, 114-121. <doi:10.1016/j.acme.2013.07.006>

See Also

calcweights, calcnormal

Examples

# Sample decision matrix
dmat <- matrix(c(
  84, 10, 5, 20,  
  66, 18, 8, 15,
  74, 12, 6, 25,
  90, 22, 9, 18, 
  68, 18, 7, 15  
), nrow = 5, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4", "A5")
colnames(dmat) <- c("C1", "C2", "C3", "C4")

# Benefit-cost vector (1: benefit, -1: cost)
bc <- c(1, -1, 1, -1)

# User-defined weights
uw <- c(0.1, 0.4, 0.3, 0.2)

# Apply WASPAS with user-defined weights
reswaspas_1 <- waspas(dmatrix = dmat, bcvec=bc, 
   weights=uw, normethod = "linear", v = 0.5)
print(reswaspas_1)

# Apply WASPAS with GINI weights
giniw <- calcweights(dmat, bcvec=bc, type="gini")
reswaspas_2 <- waspas(dmatrix = dmat, bcvec=bc, 
   weights=giniw, normethod = "linear", v = 0.5)
print(reswaspas_2)

Weight Sensitivity Analysis for Multi-Criteria Decision-Making

Description

Performs a weight sensitivity analysis by generating modified weight scenarios and applying a user-defined MCDA ranking method (e.g., TOPSIS, VIKOR) to each. Summarizes unique ranking patterns in rankings.

Usage

weisana(dmatrix, bcvec, weights, weimethod = "gradual", weipars = NULL,
        mcdamethod = topsis, methodpars, sensplot = TRUE)

Arguments

Details

This function allows for analyzing the robustness of MCDA results under different weight variations. It supports both gradual and random perturbation schemes and is agnostic to the MCDA method used.

Value

A named list with the following components:

Author(s)

Cagatay Cebeci

Examples

# Sample decision matrix and parameters
dmat <- matrix(runif(20), nrow=5)
rownames(dmat) <- paste0("A", 1:nrow(dmat))
colnames(dmat) <- paste0("C", 1:ncol(dmat))
print(dmat)
# Benefit-cost vector
bc <- c(1, 1, -1, 1)

# A very simple custom MCDA function
customfunc <- function(dmatrix, bcvec, weights, v=0.5, tiesmethod="average") {
  scores <- (rowSums(scale(dmatrix) * weights))^v
  ranks <- rank(-scores, ties.method=tiesmethod)
  list(rank = ranks)
}

# Sensitivity of the custom MCDM with gradual weight modification
mp <- list(v = 0.6, tiesmethod="average")
wp <- list(rp = c(0.1, 0.5, 0.7))
resgrawei <- weisana(dmatrix = dmat, bcvec = bc,
     weimethod = "gradual", weipars = wp,
     mcdamethod = customfunc, methodpars = mp)
print(resgrawei)

# Sensitivity of the custom MCDM with random weight modification
mp <- list(v = 0.6)
wp <- list(ss = 0.1, niters=20)
resrandwei <- weisana(dmatrix = dmat, bcvec = bc,
     weimethod = "random", weipars = wp,
     mcdamethod = customfunc, methodpars = mp)
print(resrandwei)

# Test TOPSIS
mp <- list()
wp <- list(rp = seq(0.01, 0.5, 0.05))

topsisgrawei <- weisana(dmatrix = dmat, bcvec = bc,
     weimethod = "gradual", weipars = wp,
     mcdamethod = topsis, methodpars = mp)
print(topsisgrawei)

# Test VIKOR
mp <- list(v=0.5)
wp <- list(rp = seq(0.01, 0.5, 0.05))

vikorgrawei <- weisana(dmatrix = dmat, bcvec = bc,
     weimethod = "gradual", weipars = wp,
     mcdamethod = vikor, methodpars = mp, sensplot=FALSE)
print(vikorgrawei)
sensplot(vikorgrawei$sensitivity_table, 
   mtitle="Weight Sensivity Analysis for VIKOR", colpal=terrain.colors(10))

# Test WASPAS
mp <- list(v=0.5, normethod="linear", tiesmethod="average")
wp <- list(rp = seq(0.01, 0.6, 0.01))

waspasgrawei <- weisana(dmatrix = dmat, bcvec = bc,
     weimethod = "gradual", weipars = wp,
     mcdamethod = waspas, methodpars = mp)
print(waspasgrawei)
sensplot(waspasgrawei$sensitivity_table, 
   mtitle="Weight Sensivity Analysis for WASPAS", colpal=terrain.colors(10))
   
waspasrandwei <- weisana(dmatrix = dmat, bcvec = bc,
     weimethod = "random", weipars = list(ss=0.05, niters=50),
     mcdamethod = waspas, methodpars = mp)
print(waspasrandwei)
sensplot(waspasrandwei$sensitivity_table, 
   mtitle="Weight Sensivity Analysis (random) for WASPAS", colpal=terrain.colors(10))

WPM: Weighted Product Method

Description

Implements the Weighted Product Method (WPM) to evaluate and rank alternatives using multiplicative aggregation of weighted and normalized criteria.

Usage

wpm(dmatrix, bcvec, weights, normethod = "vector", tiesmethod = "average")

Arguments

Details

The WPM evaluates alternatives by normalizing the decision matrix, raising each criterion to the power of its respective weight, and aggregating the values multiplicatively. This method calculates the relative performance (p) of each alternative for ranking purposes.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Miller, D. W., & Starr, M. K. (1960). Executive decisions and operations research (Vol. 40). Englewood Cliffs, NJ: Prentice-Hall.

Zavadskas, E. K., Vilutiene, T., Turskis, Z., & Šaparauskas, J. (2014). Multi-criteria analysis of Projects' performance in construction. Archives of Civil and Mechanical Engineering, 14, 114-121. <doi:10.1016/j.acme.2013.07.006>

See Also

calcweights, calcnormal

Examples

# Sample decision matrix
dmat <- matrix(c(
  84, 10, 5, 20,  
  66, 18, 8, 15,
  74, 12, 6, 25,
  90, 22, 9, 18, 
  68, 18, 7, 15  
), nrow = 5, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4", "A5")
colnames(dmat) <- c("C1", "C2", "C3", "C4")

# Benefit-cost vector (1: benefit, -1: cost)
bc <- c(1, -1, 1, -1)

# User-defined weights
uw <- c(0.1, 0.4, 0.3, 0.2)

# Apply WPM with user-defined weights
reswpm_1 <- wpm(dmatrix=dmat, bcvec=bc, weights=uw)
print(reswpm_1)

# Apply WPM with MEREC weights
merecw <- calcweights(dmat, bcvec=bc, type="merec")
reswpm_2 <- wpm(dmatrix=dmat, bcvec=bc, weights=merecw, normethod="vector")
print(reswpm_2)

WSM: Weighted Sum Method / SAW: Simple Additive Weighting

Description

Implementation of the Weighted Sum Method (WSM) or Simple Addiitive Weighting (SAW) to evaluate and rank alternatives by aggregating weighted normalized scores for each criterion.

Usage

wsm(dmatrix, bcvec, weights, normethod = NULL, tiesmethod = "average")

Arguments

Details

Simple Additive Weighting method emerged from the early developments in multi-attribute utility theory and decision analysis in the 1960s, with significant contributions from researchers like Ward Edwards and David E. Bell. The method was then further popularized and applied in various contexts within the MCDM field. The WSM evaluates alternatives by normalizing the decision matrix, applying weights to criteria using multiplication, and aggregating the weighted scores (p) for each alternative. This straightforward method provides a clear ranking of alternatives based on their overall performance.

Value

A list containing the following components:

Author(s)

Cagatay Cebeci

References

Zavadskas, E. K., Vilutiene, T., Turskis, Z., & Šaparauskas, J. (2014). Multi-criteria analysis of Projects' performance in construction. Archives of Civil and Mechanical Engineering, 14, 114-121. <doi:10.1016/j.acme.2013.07.006>

Panjaitan, M. I. (2019). Simple Additive Weighting (SAW) method in determining beneficiaries of foundation benefits. Login, 13(1), 19-25. <doi:10.24224/login.v13i1.22>

See Also

calcweights, calcnormal

Examples

# Sample decision matrix
dmat <- matrix(c(
  84, 10, 5, 20,  
  66, 18, 8, 15,
  74, 12, 6, 25,
  90, 22, 9, 18, 
  68, 18, 7, 15  
), nrow = 5, byrow = TRUE)
rownames(dmat) <- c("A1", "A2", "A3", "A4", "A5")
colnames(dmat) <- c("C1", "C2", "C3", "C4")

# Benefit-cost vector (1: benefit, -1: cost)
bc <- c(1, -1, 1, -1)

# User-defined weights
uw <- c(0.1, 0.4, 0.3, 0.2)

reswsm_1 <- wsm(dmatrix=dmat, bcvec=bc, weights=uw)
print(reswsm_1)

reswsm_2 <- wsm(dmatrix=dmat, bcvec=bc, weights=uw, normethod = "sum")
print(reswsm_2)

reswsm_3 <- wsm(dmatrix=dmat, bcvec=bc, weights=uw, normethod = "maxmin")
print(reswsm_3)