| Title: | Composition of Probabilistic Preferences (CPP) |
|---|---|
| Description: | CPP is a multiple criteria decision method to evaluate alternatives on complex decision making problems, by a probabilistic approach. The CPP was created and expanded by Sant'Anna, Annibal P. (2015) <doi:10.1007/978-3-319-11277-0>. |
| Authors: | Luiz O. Gavião, Annibal P. Sant'Anna, Gilson B.A. Lima and Pauli A.A. Garcia. |
| Maintainer: | Luiz O. Gavião <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 0.1.0 |
| Built: | 2025-03-05 04:02:37 UTC |
| Source: | https://github.com/cran/CPP |
This package supports the decision making in multicriteria problems. The CPP method uses a probabilistic approach to emulate the uncertainty that is inevitably present in preference evaluations. This uncertainty is modelled by probability distributions around the exact measurements of alternatives in all criteria considered in the problem. This package includes several functions described in the references, with a focus on ranking, sorting and analysing the evolution of alternatives.
Luiz Octavio Gaviao ([email protected]);
Annibal Parracho Sant'Anna ([email protected]);
Gilson Brito Alves Lima ([email protected]); and
Pauli Adriano de Almada Garcia ([email protected]).
Sant'Anna, Annibal P. (2015). Probabilistic Composition of Preferences: Theory and Applications, Springer.
Gaviao, Luiz O. & Lima, Gilson B.A. (2017) Support decision to player selection: an application of the CPP in soccer, Novas Edicoes Academicas [in Portuguese].
This function computes the aggregated value of different expert's estimatives, using Beta PERT distributions to randomize the decision matrix.
Agg.Sim(x, min, max, s, w, b)Agg.Sim(x, min, max, s, w, b)
x |
Decision matrix of expert estimatives (rows) and criteria (columns). Benefit criteria must be positive and cost criteria must be negative. |
min |
Vector of minimum values in each criterion scale. For common scales to all criteria, the vector must repeat the minimum value as many times as the number of criteria. |
max |
Vector of maximum values in each criterion scale. For common scales to all criteria, the vector must repeat the maximum value as many times as the number of criteria. |
s |
Shape of a Beta PERT distribution, as described in the package 'mc2d'. There is no default value, however the higher the shape the higher the kurtosis of the random variable. |
w |
Weights describing the expert experience in the subject matter. |
b |
Beta describes the balance between the expert weights and their opinions. Beta varies in the interval [0,1]. The higher the index, the higher the importance of weights. |
SM are the Similarity Matrices per criterion. CDC describes the Consensus Coefficient matrix. Agg.value gives the aggregated value of expert opinions per criterion.
## Expert's estimatives on four criteria Exp.1 = c(4,7,6,8) Exp.2 = c(4,3,6,5) Exp.3 = c(3,8,2,9) Exp.4 = c(6,8,9,7) Exp.5 = c(5,9,2,4) Exp.6 = c(7,6,5,5) x = rbind(Exp.1,Exp.2,Exp.3,Exp.4,Exp.5) # Decision matrix min = c(0,0,0,0) # Minimum scale values. max = c(10,10,10,10) # Maximum scale values. s = 4 # Shape w = c(0.4,0.3,0.2,0.06,0.04) # Expert relevance. b = 0.4 Agg.Sim(x,min,max,s,w,b)## Expert's estimatives on four criteria Exp.1 = c(4,7,6,8) Exp.2 = c(4,3,6,5) Exp.3 = c(3,8,2,9) Exp.4 = c(6,8,9,7) Exp.5 = c(5,9,2,4) Exp.6 = c(7,6,5,5) x = rbind(Exp.1,Exp.2,Exp.3,Exp.4,Exp.5) # Decision matrix min = c(0,0,0,0) # Minimum scale values. max = c(10,10,10,10) # Maximum scale values. s = 4 # Shape w = c(0.4,0.3,0.2,0.06,0.04) # Expert relevance. b = 0.4 Agg.Sim(x,min,max,s,w,b)
This function computes criteria weights, using AHP and randomic pair-wise evaluations by Beta PERT distributions.
AHP.Beta(n, s, list)AHP.Beta(n, s, list)
n |
Random numbers created from Beta PERT distributions, using the parameters 'min', 'mean' and 'max' of each pair-wise criteria comparison elicited from the experts. |
s |
Shape of a Beta PERT distribution, as described in package "mc2d". There is no default value, however the higher the shape the higher the kurtosis, which emulates the precision of data elicited from experts. |
list |
List of pair-wise comparison matrices of expert opinions. The function 'list' is embedded in R. |
Weights returned from a simulation of AHP with Beta PERT distributions. The weights are driven from the simulated matrix that gives the minimum AHP Consistent Ratio.
Saaty, Thomas L. (1980). The analytic hierarchy process: planning, priority setting, resource allocation, McGraw-Hill.
n=5000 # simulation s=6 # shape of Beta PERT distribution # Expert pair-wise evaluations Exp.1 = matrix(c(1,0.2,0.3,5,1,0.2,3,5,1),3,3) Exp.2 = matrix(c(1,2,8,0.5,1,6,0.12,0.16,1),3,3) Exp.3 = matrix(c(1,0.5,0.5,2,1,6,2,0.16,1),3,3) Exp.4 = matrix(c(1,3,4,0.3,1,0.5,0.25,0.3,1),3,3) Exp.5 = matrix(c(1,4,5,0.25,1,1,0.2,1,1),3,3) list = list(Exp.1,Exp.2,Exp.3,Exp.4,Exp.5) AHP.Beta(n,s,list)n=5000 # simulation s=6 # shape of Beta PERT distribution # Expert pair-wise evaluations Exp.1 = matrix(c(1,0.2,0.3,5,1,0.2,3,5,1),3,3) Exp.2 = matrix(c(1,2,8,0.5,1,6,0.12,0.16,1),3,3) Exp.3 = matrix(c(1,0.5,0.5,2,1,6,2,0.16,1),3,3) Exp.4 = matrix(c(1,3,4,0.3,1,0.5,0.25,0.3,1),3,3) Exp.5 = matrix(c(1,4,5,0.25,1,1,0.2,1,1),3,3) list = list(Exp.1,Exp.2,Exp.3,Exp.4,Exp.5) AHP.Beta(n,s,list)
This function computes criteria weights, using AHP and randomic pair-wise evaluations by Uniform distributions.
AHP.Unif(n, list)AHP.Unif(n, list)
n |
Random numbers created from Uniform distributions, using the parameters 'min' and 'max' of each pair-wise criteria comparison elicited from the experts. |
list |
Pair-wise comparison matrices of expert opinions. The function 'list' is embedded in R. |
Weights returned from a simulation of AHP with Uniform distributions. The weights are driven from the simulated matrix that gives the minimum AHP Consistent Index.
Saaty, Thomas L. (1980). The analytic hierarchy process: planning, priority setting, resource allocation, McGraw-Hill.
n=5000 # Simulation # Expert pair-wise evaluations Exp.1 = matrix(c(1,0.2,0.3,5,1,0.2,3,5,1),3,3) Exp.2 = matrix(c(1,2,8,0.5,1,6,0.12,0.16,1),3,3) Exp.3 = matrix(c(1,0.5,0.5,2,1,6,2,0.16,1),3,3) Exp.4 = matrix(c(1,3,4,0.3,1,0.5,0.25,0.3,1),3,3) Exp.5 = matrix(c(1,4,5,0.25,1,1,0.2,1,1),3,3) list = list(Exp.1,Exp.2,Exp.3,Exp.4,Exp.5) AHP.Unif(n,list)n=5000 # Simulation # Expert pair-wise evaluations Exp.1 = matrix(c(1,0.2,0.3,5,1,0.2,3,5,1),3,3) Exp.2 = matrix(c(1,2,8,0.5,1,6,0.12,0.16,1),3,3) Exp.3 = matrix(c(1,0.5,0.5,2,1,6,2,0.16,1),3,3) Exp.4 = matrix(c(1,3,4,0.3,1,0.5,0.25,0.3,1),3,3) Exp.5 = matrix(c(1,4,5,0.25,1,1,0.2,1,1),3,3) list = list(Exp.1,Exp.2,Exp.3,Exp.4,Exp.5) AHP.Unif(n,list)
This function computes CPP by additive weighting. Experts' estimatives are based on pair-wise comparisons of criteria and are joined in a list of matrices. The estimatives are used as parameters to probabilistic distributions. The minimum, mean, and maximum values of each pair of criteria are used to model Beta PERT distributions. Randomic values are generated and applied to the AHP method. The matrix that comprises de minimum AHP Consistent Index is used to return the criteria weights.
CPP.AHP.Beta(n, s, list, x)CPP.AHP.Beta(n, s, list, x)
n |
Random numbers created from Beta PERT distributions, using the parameters 'min', 'mean' and 'max' of each pair-wise criteria comparison elicited from the experts. |
s |
Shape of a Beta PERT distribution, as described in Package 'mc2d'. There is no default value, however the higher the shape the higher the kurtosis, which emulates the precision of data elicited from experts. |
list |
Pair-wise comparison matrices of expert opinions. The function 'list' is embedded in R. |
x |
Decision matrix of Alternatives (rows) and Criteria (columns). Benefit criteria must be positive and cost criteria must be negative. |
Weights returned from the AHP method. PMax are the joint probabilities of each alternative being higher than the others, per criterion. CPP gives the final scores and ranks of alternatives by weighted sum.
Sant'Anna, Annibal P. (2015). Probabilistic Composition of Preferences: Theory and Applications, Springer.
Saaty, Thomas L. (1980). The analytic hierarchy process: planning, priority setting, resource allocation, McGraw-Hill.
n=5000 # simulation s=6 # shape of Beta PERT distribution # Expert pair-wise evaluations Exp.1 = matrix(c(1,0.2,0.3,5,1,0.2,3,5,1),3,3) Exp.2 = matrix(c(1,2,8,0.5,1,6,0.12,0.16,1),3,3) Exp.3 = matrix(c(1,0.5,0.5,2,1,6,2,0.16,1),3,3) Exp.4 = matrix(c(1,3,4,0.3,1,0.5,0.25,0.3,1),3,3) Exp.5 = matrix(c(1,4,5,0.25,1,1,0.2,1,1),3,3) list = list(Exp.1,Exp.2,Exp.3,Exp.4,Exp.5) # Alternatives' original scores Alt.1 = c(30,86,-5) Alt.2 = c(26,77,-12) Alt.3 = c(22,93,-4) Alt.4 = c(34,65,-10) Alt.5 = c(31,80,-8) Alt.6 = c(29,79,-9) Alt.7 = c(37,55,-15) Alt.8 = c(21,69,-11) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) # Decision matrix CPP.AHP.Beta(n,s,list,x)n=5000 # simulation s=6 # shape of Beta PERT distribution # Expert pair-wise evaluations Exp.1 = matrix(c(1,0.2,0.3,5,1,0.2,3,5,1),3,3) Exp.2 = matrix(c(1,2,8,0.5,1,6,0.12,0.16,1),3,3) Exp.3 = matrix(c(1,0.5,0.5,2,1,6,2,0.16,1),3,3) Exp.4 = matrix(c(1,3,4,0.3,1,0.5,0.25,0.3,1),3,3) Exp.5 = matrix(c(1,4,5,0.25,1,1,0.2,1,1),3,3) list = list(Exp.1,Exp.2,Exp.3,Exp.4,Exp.5) # Alternatives' original scores Alt.1 = c(30,86,-5) Alt.2 = c(26,77,-12) Alt.3 = c(22,93,-4) Alt.4 = c(34,65,-10) Alt.5 = c(31,80,-8) Alt.6 = c(29,79,-9) Alt.7 = c(37,55,-15) Alt.8 = c(21,69,-11) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) # Decision matrix CPP.AHP.Beta(n,s,list,x)
This function computes CPP by additive weighting. Experts' estimatives are based on pair-wise comparisons of criteria and are joined in a list of matrices. The estimatives are used as parameters to probabilistic distributions. The minimum and maximum values of each pair of criteria are used to model Uniform distributions. Randomic values are generated and applied to the AHP method. The matrix that comprises de minimum AHP Consistent Index is used to return the criteria weights.
CPP.AHP.Unif(n, list, x)CPP.AHP.Unif(n, list, x)
n |
Random numbers based on Uniform distributions, using the parameters 'min' and 'max' of each pair-wise criteria comparison elicited from the experts. |
list |
Pair-wise comparison matrices of expert opinions. The function 'list' is embedded in R. |
x |
Decision matrix of Alternatives (rows) and Criteria (columns). Benefit criteria must be positive and cost criteria must be negative. |
Weights returned from the AHP method. PMax are the joint probabilities of each alternative being higher than the others, per criterion. CPP gives the final scores and ranks of alternatives by weighted sum.
Sant'Anna, Annibal P. (2015). Probabilistic Composition of Preferences: Theory and Applications, Springer
Saaty, Thomas L. (1980). The analytic hierarchy process: planning, priority setting, resource allocation, McGraw-Hill.
# Computing weights by the AHP method, with 'n' simulated matrices. n=5000 # simulation # Expert pair-wise evaluations Exp.1 = matrix(c(1,0.2,0.3,5,1,0.2,3,5,1),3,3) Exp.2 = matrix(c(1,2,8,0.5,1,6,0.12,0.16,1),3,3) Exp.3 = matrix(c(1,0.5,0.5,2,1,6,2,0.16,1),3,3) Exp.4 = matrix(c(1,3,4,0.3,1,0.5,0.25,0.3,1),3,3) Exp.5 = matrix(c(1,4,5,0.25,1,1,0.2,1,1),3,3) list = list(Exp.1,Exp.2,Exp.3,Exp.4,Exp.5) # Alternatives' original scores Alt.1 = c(30,86,-5) Alt.2 = c(26,77,-12) Alt.3 = c(22,93,-4) Alt.4 = c(34,65,-10) Alt.5 = c(31,80,-8) Alt.6 = c(29,79,-9) Alt.7 = c(37,55,-15) Alt.8 = c(21,69,-11) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) # Decision matrix CPP.AHP.Unif(n,list,x)# Computing weights by the AHP method, with 'n' simulated matrices. n=5000 # simulation # Expert pair-wise evaluations Exp.1 = matrix(c(1,0.2,0.3,5,1,0.2,3,5,1),3,3) Exp.2 = matrix(c(1,2,8,0.5,1,6,0.12,0.16,1),3,3) Exp.3 = matrix(c(1,0.5,0.5,2,1,6,2,0.16,1),3,3) Exp.4 = matrix(c(1,3,4,0.3,1,0.5,0.25,0.3,1),3,3) Exp.5 = matrix(c(1,4,5,0.25,1,1,0.2,1,1),3,3) list = list(Exp.1,Exp.2,Exp.3,Exp.4,Exp.5) # Alternatives' original scores Alt.1 = c(30,86,-5) Alt.2 = c(26,77,-12) Alt.3 = c(22,93,-4) Alt.4 = c(34,65,-10) Alt.5 = c(31,80,-8) Alt.6 = c(29,79,-9) Alt.7 = c(37,55,-15) Alt.8 = c(21,69,-11) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) # Decision matrix CPP.AHP.Unif(n,list,x)
This function computes the CPP by axes, using Beta PERT distributions to randomize the decision matrix. The CPP by axes is used to rank alternatives in multicriteria decision problems. The "Progressive-Conservative" and the "Optimist-Pessimist" axes emulate four decision maker's points of view.
CPP.Axes.Beta(x, s)CPP.Axes.Beta(x, s)
x |
Decision matrix of Alternatives (rows) and Criteria (columns). Benefit criteria must be positive and cost criteria must be negative. |
s |
Shape of a Beta PERT distribution, as described in the package 'mc2d'. There is no default value, however the higher the shape the higher the kurtosis, which emulates the precision of data. |
PMax are the joint probabilities of each alternative being higher than the others, per criterion. PMin are the joint probabilities of each alternative being lower than the others, also by criterion. Axes returns the alternatives' scores by axis and ranking for decisionmaking.
Sant'Anna, Annibal P. (2015). Probabilistic Composition of Preferences: Theory and Applications, Springer.
Garcia, Pauli A. A. & Sant'Anna, Annibal P. (2015). Vendor and logistics provider selection in the construction sector: A probabilistic preferences composition approach. Pesquisa Operacional 35.2: 363-375.
# Alternatives' original scores Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) # Decision matrix s = 4 # Shape CPP.Axes.Beta(x,s)# Alternatives' original scores Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) # Decision matrix s = 4 # Shape CPP.Axes.Beta(x,s)
This function computes the CPP by axes, using Normal distributions to randomize the decision matrix. The CPP by axes is used to rank alternatives in multicriteria decision problems. The "Progressive-Conservative" and the "Optimist-Pessimist" axes emulate four decision maker's points of view.
CPP.Axes.Normal(x)CPP.Axes.Normal(x)
x |
Decision matrix of Alternatives (rows) and Criteria (columns). Benefit criteria must be positive and cost criteria must be negative. |
PMax are the joint probabilities of each alternative being higher than the others, per criterion. PMin are the joint probabilities of each alternative being lower than the others, also by criterion. Axes returns the alternatives' scores by axis and ranking for decisionmaking.
Sant'Anna, Annibal P. (2015). Probabilistic Composition of Preferences: Theory and Applications, Springer.
Garcia, Pauli A. A. & Sant'Anna, Annibal P. (2015). Vendor and logistics provider selection in the construction sector: A probabilistic preferences composition approach. Pesquisa Operacional 35.2: 363-375.
# Alternatives' original scores Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) # Decision matrix CPP.Axes.Normal(x)# Alternatives' original scores Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) # Decision matrix CPP.Axes.Normal(x)
This function computes the CPP by Choquet integrals, using Beta PERT distributions to randomize the decision matrix. The CPP by Choquet integrals is used to rank alternatives in multicriteria decision problems.
CPP.Choquet.Beta(x, s)CPP.Choquet.Beta(x, s)
x |
Decision matrix of Alternatives (rows) and Criteria (columns). Benefit criteria must be positive and cost criteria must be negative. |
s |
Shape of a Beta PERT distribution, as described in the package 'mc2d'. There is no default value, however the higher the shape the higher the kurtosis, which emulates the precision of data elicited from experts. |
PMax are the joint probabilities of each alternative being higher than the others, per criterion. Capacities are the interactions of all combined criteria, computed by the Progressive-Optimistic (PO) point of view. Choq returns the alternatives' scores by Choquet integrals and their respetive rankings for decisionmaking. Shap returns the Shapley indices, which are associated with criteria weights.
Sant'Anna, Annibal P. (2015). Probabilistic Composition of Preferences: Theory and Applications, Springer.
# Alternatives' original scores Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) # Decision matrix s = 4 # Shape CPP.Choquet.Beta(x,s)# Alternatives' original scores Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) # Decision matrix s = 4 # Shape CPP.Choquet.Beta(x,s)
The CPP by the Gini Index is used to rank alternatives by evenness of evaluations, in multicriteria decision problems.
CPP.Gini(x, s)CPP.Gini(x, s)
x |
Decision matrix of Alternatives (rows) and Criteria (columns). Benefit criteria must be positive and cost criteria must be negative. |
s |
Shape of Beta PERT distribution, as described in the package 'mc2d'. There is no default value, however the higher the shape the higher the kurtosis. |
PMax are the joint probabilities of each alternative being higher than the others, per criterion. CPP.Gini returns the alternatives' scores by the Gini Index and their respective preference ranks for decisionmaking.
Sant'Anna, Annibal P. (2015). Probabilistic Composition of Preferences: Theory and Applications, Springer
Gaviao, Luiz O. & Lima, Gilson B.A. (2017) Support decision to player selection: an application of the CPP in soccer, Novas Edições Acadêmicas [in Portuguese].
# Alternatives' original scores Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) # Decision matrix s = 4 # Shape CPP.Gini(x,s)# Alternatives' original scores Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) # Decision matrix s = 4 # Shape CPP.Gini(x,s)
The CPP-Malmquist is used to dynamic evaluation of alternatives, in multicriteria problems, considering two different moments.
CPP.Malmquist.Beta(m1, m2, s)CPP.Malmquist.Beta(m1, m2, s)
m1 |
Decision matrix of Alternatives (rows) and Criteria (columns) in moment '1'. Benefit criteria must be positive and cost criteria must be negative. |
m2 |
Decision matrix of Alternatives (rows) and Criteria (columns) in the following moment '2'. Benefit criteria must be positive and cost criteria must be negative. |
s |
Shape of a Beta PERT distribution, as described in the package 'mc2d'. There is no default value, however the higher the shape the higher the kurtosis, which emulates the precision of data. |
MC gives the Malmquist Conservative index. MP gives the Malmquist Progressive index. Finally, Index gives the CPP-Malmquist of all alternatives and their rankings for decisionmaking. The indices greater than one represent a relative evolution of the alternative between the two periods, while the indices lower than one reveal the alternatives that decreased performance in relation to the others.
# Alternatives' original scores Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) m1 = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) # Decision matrix of the previous moment '1'. Alt.1 = c(3,29,82,-3) Alt.2 = c(6,28,70,-8) Alt.3 = c(2,20,99,-8) Alt.4 = c(5,31,62,-14) Alt.5 = c(9,27,73,-5) m2 = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) # Decision matrix of the following moment '2'. s = 4 # Shape CPP.Malmquist.Beta(m1,m2,s)# Alternatives' original scores Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) m1 = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) # Decision matrix of the previous moment '1'. Alt.1 = c(3,29,82,-3) Alt.2 = c(6,28,70,-8) Alt.3 = c(2,20,99,-8) Alt.4 = c(5,31,62,-14) Alt.5 = c(9,27,73,-5) m2 = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) # Decision matrix of the following moment '2'. s = 4 # Shape CPP.Malmquist.Beta(m1,m2,s)
The algorithm evaluates alternatives by integrating the CPP-Tri, the CPP-Malmquist, the CPP-Gini, the alternatives' market values and the CPP by axes. The CPP-mb was originally applied in sports science to evaluate players' performance.
CPP.mb(t1, t2, m, q, s)CPP.mb(t1, t2, m, q, s)
t1 |
Decision matrix of Alternatives (rows) and Criteria (columns) in the moment '1'. Benefit criteria must be positive and cost criteria negative. |
t2 |
Decision matrix of Alternatives (rows) and Criteria (columns) in the following moment '2'. Benefit criteria must be positive and cost criteria negative. |
m |
Vector of alternatives' market values. |
q |
Vector of quantiles, indicating the classes' profiles. |
s |
Shape of a Beta PERT distribution, as described in the package 'mc2d'. There is no default value, however the higher the shape the higher the kurtosis, which emulates the precision of data. |
Class assigns the alternatives to classes, defined by the indicated profiles. The list of classes also shows the decision matrices to be modeled by CPP-PP. CPP-mb indicates the final scores per class.
Sant'Anna, Annibal P. (2015). Probabilistic Composition of Preferences: Theory and Applications, Springer.
Lewis, Michael. (2004) Moneyball: The art of winning an unfair game. WW Norton & Company.
Gaviao, Luiz O. & Lima, Gilson B.A. (2017) Support decision to player selection: an application of the CPP in soccer, Novas Edições Acadêmicas [in Portuguese].
## Decision matrix of the previous moment '1'. Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) Alt.6 = c(6,29,79,-9) Alt.7 = c(8,37,55,-15) Alt.8 = c(10,21,69,-11) t1 = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) ## Decision matrix of the following moment '2'. Alt.1 = c(3,29,82,-3) Alt.2 = c(6,28,70,-8) Alt.3 = c(2,20,99,-8) Alt.4 = c(5,31,62,-14) Alt.5 = c(9,27,73,-5) Alt.6 = c(4,33,85,-13) Alt.7 = c(9,39,59,-10) Alt.8 = c(8,19,77,-9) t2 = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) m = c(100,120,150,140,90,70,110,130) # Market values q = c(0.65,0.35) # quantiles of class profiles s = 4 # Shape CPP.mb(t1,t2,m,q,s)## Decision matrix of the previous moment '1'. Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) Alt.6 = c(6,29,79,-9) Alt.7 = c(8,37,55,-15) Alt.8 = c(10,21,69,-11) t1 = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) ## Decision matrix of the following moment '2'. Alt.1 = c(3,29,82,-3) Alt.2 = c(6,28,70,-8) Alt.3 = c(2,20,99,-8) Alt.4 = c(5,31,62,-14) Alt.5 = c(9,27,73,-5) Alt.6 = c(4,33,85,-13) Alt.7 = c(9,39,59,-10) Alt.8 = c(8,19,77,-9) t2 = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) m = c(100,120,150,140,90,70,110,130) # Market values q = c(0.65,0.35) # quantiles of class profiles s = 4 # Shape CPP.mb(t1,t2,m,q,s)
This function computes the CPP-rh, using Beta PERT distributions to randomize the decision matrices. The CPP-rh is used to evaluate alternatives by integrating the CPP-Tri, the CPP-Malmquist, the CPP-Gini and the CPP by axes. The CPP-rh and the CPP-mb are very similar, but the CPP-rh does not include the alternatives's market value.
CPP.rh(t1, t2, q, s)CPP.rh(t1, t2, q, s)
t1 |
Decision matrix of Alternatives (rows) and Criteria (columns) in the previous moment '1'. Benefit criteria must be positive and cost criteria must be negative. |
t2 |
Decision matrix of Alternatives (rows) and Criteria (columns) in the following moment '2'. Benefit criteria must be positive and cost criteria must be negative. |
q |
Vector of quantiles, indicating the classes' profiles. |
s |
Shape of a Beta PERT distribution, as described in the package 'mc2d'. There is no default value, however the higher the shape the higher the kurtosis, which emulates the precision of data. |
Class identifies the alternatives' classes, according to the selected profiles. CPP-RH returns the alternatives' scores per class.
## Decision matrix of the previous moment '1'. Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) Alt.6 = c(6,29,79,-9) Alt.7 = c(8,37,55,-15) Alt.8 = c(10,21,69,-11) t1 = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) ## Decision matrix of the following moment '2'. Alt.1 = c(3,29,82,-3) Alt.2 = c(6,28,70,-8) Alt.3 = c(2,20,99,-8) Alt.4 = c(5,31,62,-14) Alt.5 = c(9,27,73,-5) Alt.6 = c(4,33,85,-13) Alt.7 = c(9,39,59,-10) Alt.8 = c(8,19,77,-9) t2 = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) q = c(0.65,0.35) # quantiles of class profiles s = 4 # Shape CPP.rh(t1,t2,q,s)## Decision matrix of the previous moment '1'. Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) Alt.6 = c(6,29,79,-9) Alt.7 = c(8,37,55,-15) Alt.8 = c(10,21,69,-11) t1 = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) ## Decision matrix of the following moment '2'. Alt.1 = c(3,29,82,-3) Alt.2 = c(6,28,70,-8) Alt.3 = c(2,20,99,-8) Alt.4 = c(5,31,62,-14) Alt.5 = c(9,27,73,-5) Alt.6 = c(4,33,85,-13) Alt.7 = c(9,39,59,-10) Alt.8 = c(8,19,77,-9) t2 = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) q = c(0.65,0.35) # quantiles of class profiles s = 4 # Shape CPP.rh(t1,t2,q,s)
This function computes the CPP-SAW, using Normal distributions and weights defined by the decision maker. The CPP-SAW is used to evaluate alternatives by weighted sum.
CPP.SAW(x, w)CPP.SAW(x, w)
x |
Decision matrix of Alternatives (rows) and Criteria (columns). Benefit criteria must be positive and cost criteria must be negative. |
w |
Vector of weights assigned by the decision maker. Weights are normalized, just in case their sum differs from the unity. |
Weights repeat the parameter 'w' if sum the unity, otherwise are normalized. PMax indicates the joint probabilities of each alternative being higher than the others, per criterion. CPP returns the alternatives' scores by weighted sum, indicating the preference ranks for decisionmaking.
Sant'Anna, Annibal P. (2015). Probabilistic Composition of Preferences: Theory and Applications, Springer.
# Decision matrix Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) Alt.6 = c(6,29,79,-9) Alt.7 = c(8,37,55,-15) Alt.8 = c(10,21,69,-11) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) w = c(0.2,0.3,0.4,0.1) CPP.SAW(x,w)# Decision matrix Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) Alt.6 = c(6,29,79,-9) Alt.7 = c(8,37,55,-15) Alt.8 = c(10,21,69,-11) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) w = c(0.2,0.3,0.4,0.1) CPP.SAW(x,w)
This function computes the CPP-SAW, using Normal distributions to randomize the decision matrix and weights defined by entropy. The CPP-SAW Entropy is used to evaluate alternatives by weighted sum.
CPP.SAW.Entropy(x)CPP.SAW.Entropy(x)
x |
Decision matrix of Alternatives (rows) and Criteria (columns). Benefit criteria must be positive and cost criteria must be negative. |
Weights by entropy.PMax are the joint probabilities of each alternative being higher than the others, per criterion. CPP returns the alternatives' scores by weighted sum, indicating the preference ranks for decisionmaking.
Sant'Anna, Annibal P. (2015). Probabilistic Composition of Preferences: Theory and Applications, Springer.
## Decision matrix. Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) Alt.6 = c(6,29,79,-9) Alt.7 = c(8,37,55,-15) Alt.8 = c(10,21,69,-11) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) CPP.SAW.Entropy(x)## Decision matrix. Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) Alt.6 = c(6,29,79,-9) Alt.7 = c(8,37,55,-15) Alt.8 = c(10,21,69,-11) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) CPP.SAW.Entropy(x)
This function computes the CPP-Tri, using Beta PERT distributions to randomize the decision matrix. The CPP-Tri is used to classify alternatives, indicating the order of classes, whose quantity is defined by the decision maker. The probabilities of each alternative being higher and lower than the classes' profiles are composed by the Progressive-Pessimist (PP) point of view.
CPP.Tri.Beta(x, q, s)CPP.Tri.Beta(x, q, s)
x |
Decision matrix of Alternatives (rows) and Criteria (columns). Benefit criteria must be positive and cost criteria must be negative. |
q |
Vector of quantiles, indicating the classes' profiles. |
s |
Shape of a Beta PERT distribution, as described in the package 'mc2d'. There is no default value, however the higher the shape the higher the kurtosis, which emulates the precision of data. |
Prob.Plus are the probabilities of each alternative being higher than the classes'profiles. Prob.Minus are the probabilities of each alternative being lower than the classes'profiles. CPP.Tri returns the alternatives' classes.
Sant'Anna, Annibal P. (2015). Probabilistic Composition of Preferences: Theory and Applications, Springer.
Sant'Anna, Annibal P.; Costa, Helder G.; Pereira, Valdecy (2015). CPP-TRI: a sorting method based on the probabilistic composition of preferences. International Journal of Information and Decision Sciences 7.3, 193-212.
Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) Alt.6 = c(6,29,79,-9) Alt.7 = c(8,37,55,-15) Alt.8 = c(10,21,69,-11) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) # Decision matrix. q = c(0.65,0.35) # quantiles of classes' profiles. s = 4 # Shape CPP.Tri.Beta(x,q,s)Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) Alt.6 = c(6,29,79,-9) Alt.7 = c(8,37,55,-15) Alt.8 = c(10,21,69,-11) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) # Decision matrix. q = c(0.65,0.35) # quantiles of classes' profiles. s = 4 # Shape CPP.Tri.Beta(x,q,s)
This function computes the CPP-Tri with Choquet integrals, using Beta PERT distributions to randomize the decision matrix. The CPP Tri is used to classify alternatives, indicating the order of a number of classes defined by the decision maker. The probabilities of each alternative being higher and lower than the classes' profiles are composed by Choquet integrals.
CPP.Tri.Choquet(x, q, s)CPP.Tri.Choquet(x, q, s)
x |
Decision matrix of Alternatives (rows) and Criteria (columns). Benefit criteria must be positive and cost criteria negative. |
q |
Vector of quantiles, indicating the classes' profiles. |
s |
Shape of a Beta PERT distribution, as described in the package 'mc2d'. There is no default value, however the higher the shape the higher the kurtosis, which emulates the precision of data. |
Choquet.O returns the probabilities of each alternative being higher than the classes' profiles, composed by Choquet integrals. Choquet.U indicates the probabilities of each alternative being lower than the classes' profiles. CPP.Tri.Chq returns the alternatives' classes.
Sant'Anna, Annibal P. (2015). Probabilistic Composition of Preferences: Theory and Applications, Springer.
Sant’Anna, Annibal P.; Lima, Gilson B. A.; Gavião, Luiz O. (2018) A Probabilistic approach to the inequality adjustment of the Human Development Index. Pesquisa Operacional 38.1: 99-116.
Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) Alt.6 = c(6,29,79,-9) Alt.7 = c(8,37,55,-15) Alt.8 = c(10,21,69,-11) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) # Decision matrix. q = c(0.65,0.35) # quantiles of classes' profiles. s = 4 # Shape CPP.Tri.Choquet(x,q,s)Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) Alt.6 = c(6,29,79,-9) Alt.7 = c(8,37,55,-15) Alt.8 = c(10,21,69,-11) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) # Decision matrix. q = c(0.65,0.35) # quantiles of classes' profiles. s = 4 # Shape CPP.Tri.Choquet(x,q,s)
This function computes weights by Shannon's entropy.
Entrop.weights(x)Entrop.weights(x)
x |
Decision matrix of Alternatives (rows) and Criteria (columns). Benefit criteria must be positive and cost criteria negative. |
Weights for each criterion.
Pomerol, Jean-Charles & Barba-Romero, Sergio. (2012) Multicriterion Decision in Management: Principles and Practice, Springer.
Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) Alt.6 = c(6,29,79,-9) Alt.7 = c(8,37,55,-15) Alt.8 = c(10,21,69,-11) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) # Decision matrix. Entrop.weights(x)Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) Alt.6 = c(6,29,79,-9) Alt.7 = c(8,37,55,-15) Alt.8 = c(10,21,69,-11) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5,Alt.6,Alt.7,Alt.8) # Decision matrix. Entrop.weights(x)
This function computes the probabilities of each alternative maximizing the preference per criterion, using Beta PERT distributions to randomize the decision matrix.
PMax.Beta(x, s)PMax.Beta(x, s)
x |
Decision matrix of Alternatives (rows) and Criteria (columns). Benefit criteria must be positive and cost criteria must be negative. |
s |
Shape of a Beta PERT distribution, as described in the package 'mc2d'. There is no default value, however the higher the shape the higher the kurtosis, which emulates the precision of data. |
PMax are the joint probabilities of each alternative being higher than the others, per criterion.
Sant'Anna, Annibal P. (2015). Probabilistic Composition of Preferences: Theory and Applications, Springer.
# Decision matrix Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) s = 4 # Shape PMax.Beta(x,s)# Decision matrix Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) s = 4 # Shape PMax.Beta(x,s)
This function computes the probabilities of each alternative maximizing the preference per criterion, using Normal distributions to randomize the decision matrix.
PMax.Normal(x)PMax.Normal(x)
x |
Decision matrix of Alternatives (rows) and Criteria (columns). Benefit criteria must be positive and cost criteria must be negative. |
PMax are the joint probabilities of each alternative being higher than the others, per criterion.
Sant'Anna, Annibal P. (2015). Probabilistic Composition of Preferences: Theory and Applications, Springer.
# Decision matrix Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) PMax.Normal(x)# Decision matrix Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) PMax.Normal(x)
This function computes the Probabilities of each alternative minimizing the preference per criterion, using Beta PERT distributions to randomize the decision matrix.
PMin.Beta(x, s)PMin.Beta(x, s)
x |
Decision matrix of Alternatives (rows) and Criteria (columns). Benefit criteria must be positive and cost criteria must be negative. |
s |
Shape of a Beta PERT distribution, as described in the package 'mc2d'. There is no default value, however the higher the shape the higher the kurtosis, which emulates the precision of data. |
PMin are the joint probabilities of each alternative being lower than the others, per criterion.
Sant'Anna, Annibal P. (2015). Probabilistic Composition of Preferences: Theory and Applications, Springer.
# Decision matrix Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) s = 4 # Shape PMin.Beta(x,s)# Decision matrix Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) s = 4 # Shape PMin.Beta(x,s)
This function computes the probabilities of each alternative minimizing the preference per criterion, using Normal distributions to randomize the decision matrix.
PMin.Normal(x)PMin.Normal(x)
x |
Decision matrix of Alternatives (rows) and Criteria (columns). Benefit criteria must be positive and cost criteria must be negative. |
PMin are the joint probabilities of each alternative being lower than the others, per criterion.
Sant'Anna, Annibal P. (2015). Probabilistic Composition of Preferences: Theory and Applications, Springer.
# Decision matrix Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) PMin.Normal(x)# Decision matrix Alt.1 = c(2,30,86,-5) Alt.2 = c(4,26,77,-12) Alt.3 = c(3,22,93,-4) Alt.4 = c(6,34,65,-10) Alt.5 = c(5,31,80,-8) x = rbind(Alt.1,Alt.2,Alt.3,Alt.4,Alt.5) PMin.Normal(x)