Data: Cooper et al. (2001).

Description

Data of automobile industries collected from the Statistical Year Book of China of the Chinese Bureau of Statistics. Each year is treated as a DMU.

Usage

data("Automobile")

Format

Data frame with 17 rows and 4 columns. Definition of inputs (X) and output (Y):

X1 = Labor

Expressed in units of 1000 persons.

X2 = Capital

Stated in units of 1 million Ren Min Bi (Chinese monetary unit) adjusted to 1991 prices.

Y = Output

Stated in units of 1 million Ren Min Bi (Chinese monetary unit) adjusted to 1991 prices.

Author(s)

Vicente Bolos (vicente.bolos@uv.es). Department of Business Mathematics

Rafael Benitez (rafael.suarez@uv.es). Department of Business Mathematics

Vicente Coll-Serrano (vicente.coll@uv.es). Quantitative Methods for Measuring Culture (MC2). Applied Economics.

University of Valencia (Spain)

Source

Cooper, W.W.; Denga, H.; Gua, B.; Lib, S.; Thrall, R.M. (2001). "Using DEA to improve the management of congestion in Chinese industries (1981-1997)", Socio-Economic Planning Sciences, 35(4), 227-242.

See Also

make_deadata_stoch, modelstoch_radial

Data: Cooper et al. (2001).

Description

Data of textile industries collected from the Statistical Year Book of China of the Chinese Bureau of Statistics. Each year is treated as a DMU.

Usage

data("Textile")

Format

Data frame with 17 rows and 4 columns. Definition of inputs (X) and output (Y):

X1 = Labor

Expressed in units of 1000 persons.

X2 = Capital

Stated in units of 1 million Ren Min Bi (Chinese monetary unit) adjusted to 1991 prices.

Y = Output

Stated in units of 1 million Ren Min Bi (Chinese monetary unit) adjusted to 1991 prices.

Author(s)

Vicente Bolos (vicente.bolos@uv.es). Department of Business Mathematics

Rafael Benitez (rafael.suarez@uv.es). Department of Business Mathematics

Vicente Coll-Serrano (vicente.coll@uv.es). Quantitative Methods for Measuring Culture (MC2). Applied Economics.

University of Valencia (Spain)

Source

Cooper, W.W.; Denga, H.; Gua, B.; Lib, S.; Thrall, R.M. (2001). "Using DEA to improve the management of congestion in Chinese industries (1981-1997)", Socio-Economic Planning Sciences, 35(4), 227-242.

See Also

make_deadata_stoch, modelstoch_radial

Objective scores

Description

Extract the scores (optimal objective values) of the evaluated DMUs from a stochastic DEA solution. Note that these scores may not always be interpreted as efficiencies.

Usage

## S3 method for class 'dea_stoch'
efficiencies(x, ...)

Arguments

Value

A matrix with the scores (optimal objective values).

Author(s)

Vicente Bolós (vicente.bolos@uv.es). Department of Business Mathematics

Rafael Benítez (rafael.suarez@uv.es). Department of Business Mathematics

Vicente Coll-Serrano (vicente.coll@uv.es). Quantitative Methods for Measuring Culture (MC2). Applied Economics.

University of Valencia (Spain)

Examples

# Example 1.
library(deaR)
data("Coll_Blasco_2006")
ni <- 2 # number of inputs
no <- 2 # number of outputs
data_example <- make_deadata(datadea = Coll_Blasco_2006,
                             ni = ni,
                             no = no)
nd <- length(data_example$dmunames) # number of DMUs
var_input <- matrix(1, nrow = ni, ncol = nd)
var_output <- matrix(1, nrow = no, ncol = nd)
data_stoch <- make_deadata_stoch(datadea = data_example,
                                 var_input = var_input,
                                 var_output = var_output)
Collstoch <- modelstoch_radial(data_stoch)
efficiencies(Collstoch)

make_deadata_stoch

Description

This function creates, from a deadata object, a deadata_stoch object by adding the corresponding covariance matrices. These objects are prepared to be passed to a modelstoch_* function.

We consider \mathcal{D}=\left\{ \textrm{DMU}_1, \ldots ,\textrm{DMU}_n \right\} a set of n DMUs with m stochastic inputs and s stochastic outputs. Matrices \tilde{X}=(\tilde{x}_{ij}) and \tilde{Y}=(\tilde{y}_{rj}) are the input and output data matrices, respectively, where \tilde{x}_{ij} and \tilde{y}_{rj} represent the i-th input and r-th output of the j-th DMU. Moreover, we denote by X=(x_{ij}) and Y=(y_{rj}) their expected values. We suppose that \tilde{x}_{ij} and \tilde{y}_{rj} follow a multivariate probability distribution with means E(\tilde{x}_{ij})=x_{ij}, E(\tilde{y}_{rj})=y_{rj} and covariance matrix

\Delta = \begin{pmatrix} \Delta ^{II}_{11} & \cdots & \Delta ^{II}_{1m} & \Delta ^{IO}_{11} & \cdots & \Delta ^{IO}_{1s} \\ \vdots & & \vdots & \vdots & & \vdots \\ \Delta ^{II}_{m1} & \cdots & \Delta ^{II}_{mm} & \Delta ^{IO}_{m1} & \cdots & \Delta ^{IO}_{ms} \\ \\ \Delta ^{OI}_{11} & \cdots & \Delta ^{OI}_{1m} & \Delta ^{OO}_{11} & \cdots & \Delta ^{OO}_{1s} \\ \vdots & & \vdots & \vdots & & \vdots \\ \Delta ^{OI}_{s1} & \cdots & \Delta ^{OI}_{sm} & \Delta ^{OO}_{s1} & \cdots & \Delta ^{OO}_{ss} \\ \end{pmatrix}_{n(m+s)\times n(m+s)}

where \Delta ^{II}_{ik}, \Delta ^{OO}_{rp}, \Delta ^{IO}_{ir} and \Delta ^{OI}_{ri} are n\times n matrices, for 1\leq i,k\leq m and 1\leq r,p\leq s, such that

\left( \Delta ^{II}_{ik}\right) _{jq}=\mathrm{Cov}(\tilde{x}_{ij}, \tilde{x}_{kq}),

\left( \Delta ^{OO}_{rp}\right) _{jq}=\mathrm{Cov}(\tilde{y}_{rj}, \tilde{y}_{pq}),

\left( \Delta ^{IO}_{ir}\right) _{jq}=\left( \Delta ^{OI}_{ri}\right) _{qj} =\mathrm{Cov}(\tilde{x}_{ij}, \tilde{y}_{rq}),

with 1\leq j,q\leq n.

- If we have the covariances matrix in the general form above, it can be introduced directly by parameter cov_matrix. Since this matrix is supposed to be symmetric, only values above the diagonal are read, ignoring values below the diagonal.

- Alternatively, we can introduce the covariances matrix using parameters cov_II, cov_OO and cov_IO, that are 4-dimensional arrays of size m\times m\times n\times n, s\times s\times n\times n and m\times s\times n\times n, respectively, such that

\texttt{cov\_II[i, k, , ]}=\Delta ^{II}_{ik},

\texttt{cov\_OO[r, p, , ]}=\Delta ^{OO}_{rp},

\texttt{cov\_IO[i, r, , ]}=\Delta ^{IO}_{ir},

for 1\leq i,k\leq m and 1\leq r,p\leq s. Since matrices \Delta ^{II}_{ii} and \Delta ^{OO}_{rr} are supposed to be symmetric, only values above the diagonal are read, ignoring values below the diagonal. Moreover, since \Delta ^{II}_{ki} is the transpose of \Delta ^{II}_{ik}, and \Delta ^{OO}_{pr} is the transpose of \Delta ^{OO}_{rp}, only matrices \Delta ^{II}_{ik} and \Delta ^{OO}_{rp} with i\leq k and r\leq p are necessary, ignoring those with i>k and r>p.

- If covariances between different inputs/outputs are zero, we can make use of parameters cov_input and cov_output, that are arrays of size m\times n\times n and s\times n\times n, respectively, such that

\texttt{cov\_input[i, , ]}=\Delta ^{II}_{ii},

\texttt{cov\_output[r, , ]}=\Delta ^{OO}_{rr},

for 1\leq i\leq m and 1\leq r\leq s. By symmetry of \Delta ^{II}_{ii} and \Delta ^{OO}_{rr}, only values above the diagonal are read, ignoring values below the diagonal.

- Finally, if all the variables are independent then the covariances matrix is diagonal. Hence, we might use parameters var_input and var_output, that are matrices of size m\times n and s\times n, respectively, such that

\texttt{var\_input[i, j]}=\mathrm{Var}\left( \tilde{x}_{ij}\right) ,

\texttt{var\_output[r, j]}=\mathrm{Var}\left( \tilde{y}_{rj}\right) ,

for 1\leq i\leq m, 1\leq r\leq s and 1\leq j\leq n.

Usage

make_deadata_stoch(datadea = NULL,
          var_input = NULL,
          var_output = NULL,
          cov_input = NULL,
          cov_output = NULL,
          cov_II = NULL,
          cov_OO = NULL,
          cov_IO = NULL,
          cov_matrix = NULL)

Arguments

Value

An object of class deadata_stoch.

Author(s)

Vicente Bolós (vicente.bolos@uv.es). Department of Business Mathematics

Rafael Benítez (rafael.suarez@uv.es). Department of Business Mathematics

Vicente Coll-Serrano (vicente.coll@uv.es). Quantitative Methods for Measuring Culture (MC2). Applied Economics.

University of Valencia (Spain)

References

Cooper, W.W.; Huang, Z.; Lelas, V.; Li, S.X.; Olesen, O.B. (1998). “Chance Constrained Programming Formulations for Stochastic Characterizations of Efficiency and Dominance in DEA", Journal of Productivity Analysis, 9, 53-79.

El-Demerdash, B.E.; El-Khodary, I.A.; Tharwat, A.A. (2013). "Developing a Stochastic Input Oriented Data Envelopment Analysis (SIODEA) Model", International Journal of Advanced Computer Science and Applications, Vol.4, No. 4, 40-44.

Examples

# Example 1
library(deaR)
data("Coll_Blasco_2006")
ni <- 2 # number of inputs
no <- 2 # number of outputs
data_example <- make_deadata(datadea = Coll_Blasco_2006,
                             ni = ni,
                             no = no)
nd <- length(data_example$dmunames) # number of DMUs
# All variances are 1.
var_input <- matrix(1, nrow = ni, ncol = nd)
var_output <- matrix(1, nrow = no, ncol = nd)
data_stoch <- make_deadata_stoch(datadea = data_example,
                                 var_input = var_input,
                                 var_output = var_output)
# All covariances are 1.
cov_input <- array(1, dim = c(ni, nd, nd))
cov_output <- array(1, dim = c(no, nd, nd))
data_stoch2 <- make_deadata_stoch(datadea = data_example,
                                  cov_input = cov_input,
                                  cov_output = cov_output)

# Example 2. Deterministic data with one stochastic input,
# from El-Demerdash et al. (2013).
library(deaR)
dmunames <- c("A", "B", "C")
nd <- length(dmunames) # Number of DMUs
inputnames <- c("Professors", "Budget")
ni <- length(inputnames) # Number of Inputs
outputnames <- c("Diplomas", "Bachelors", "Masters")
no <- length(outputnames) # Number of Outputs
X <- matrix(c(5, 14, 8, 15, 7, 12),
            nrow = ni, ncol = nd, dimnames = list(inputnames, dmunames))
Y <- matrix(c(9, 4, 16, 5, 7, 10, 4, 9, 13),
            nrow = no, ncol = nd, dimnames = list(outputnames, dmunames))
datadea <- make_deadata(inputs = X,
                        outputs = Y)
covX <- array(0, dim = c(2, 3, 3))
# The 2nd input is stochastic.
# Since the corresponding 3x3 covariances matrix is symmetric, only values
# above the diagonal are necessary.
covX[2, 1, ] <- c(1.4, 0.9, 0.6)
covX[2, 2, 2:3] <- c(1.5, 0.7)
covX[2, 3, 3] <- 1.2
# Alternatively (note that values below the diagonal are ignored).
covX[2, , ] <- matrix(c(1.4, 0.9, 0.6, 0, 1.5, 0.7, 0, 0, 1.2),
                      byrow = TRUE)
datadea_stoch <- make_deadata_stoch(datadea,
                                    cov_input = covX)

# Example 3. 5 random observations of 3 DMUs with 2 inputs and 2 outputs.
library(deaR)
# Generate random observations.
input1 <- data.frame(I1D1 = rnorm(5, mean = sample(5:10, 1)),
                     I1D2 = rnorm(5, mean = sample(5:10, 1)),
                     I1D3 = rnorm(5, mean = sample(5:10, 1)))
input2 <- data.frame(I2D1 = rnorm(5, mean = sample(5:10, 1)),
                     I2D2 = rnorm(5, mean = sample(5:10, 1)),
                     I2D3 = rnorm(5, mean = sample(5:10, 1)))
output1 <- data.frame(O1D1 = rnorm(5, mean = sample(5:10, 1)),
                      O1D2 = rnorm(5, mean = sample(5:10, 1)),
                      O1D3 = rnorm(5, mean = sample(5:10, 1)))
output2 <- data.frame(O2D1 = rnorm(5, mean = sample(5:10, 1)),
                      O2D2 = rnorm(5, mean = sample(5:10, 1)),
                      O2D3 = rnorm(5, mean = sample(5:10, 1)))
# Generate deadata with means of observations.
inputs <- matrix(mapply(mean, cbind(input1, input2)),
                 nrow = 2,
                 ncol = 3,
                 byrow = TRUE,
                 dimnames = list(c("I1", "I2"), c("D1", "D2", "D3")))
outputs <- matrix(mapply(mean, cbind(output1, output2)),
                  nrow = 2,
                  ncol = 3,
                  byrow = TRUE,
                  dimnames = list(c("O1", "O2"), c("D1", "D2", "D3")))
datadea <- make_deadata(inputs = inputs,
                        outputs = outputs)
# Generate covariances matrix cov_matrix.
cov_matrix <- cov(cbind(input1, input2, output1, output2))
# Generate deadata_stoch
datadea_stoch <- make_deadata_stoch(datadea,
                                    cov_matrix = cov_matrix)

Chance Constrained Additive E-model.

Description

It solves the chance constrained additive E-model based on the deterministic additive model from Charnes et al. (1985), under constant and non-constant returns to scale.

Besides, the user can set weights for the input and/or output slacks. So, it is also possible to solve chance constrained versions of weighted additive models like Measure of Inefficiency Proportions (MIP) or Range Adjusted Measure (RAM), see Cooper et al. (1999).

We consider \mathcal{D}=\left\{ \textrm{DMU}_1, \ldots ,\textrm{DMU}_n \right\} a set of n DMUs with m stochastic inputs and s stochastic outputs. Matrices \tilde{X}=(\tilde{x}_{ij}) and \tilde{Y}=(\tilde{y}_{rj}) are the input and output data matrices, respectively, where \tilde{x}_{ij} and \tilde{y}_{rj} represent the i-th input and r-th output of the j-th DMU. Moreover, we denote by X=(x_{ij}) and Y=(y_{rj}) their expected values. In general, we denote vectors by bold-face letters and they are considered as column vectors unless otherwise stated. The 0-vector is denoted by \bm{0} and the context determines its dimension.

Given 0<\alpha <1, the program for \text{DMU}_o with constant returns to scale is given by

\max \limits_{\bm{\lambda},\mathbf{s}^-,\mathbf{s}^+}\quad \mathbf{w}^-\mathbf{s}^-+\mathbf{w}^+\mathbf{s}^+

\text{s.t.}\quad P\left\{ \left( \tilde{\mathbf{x}}_o-\tilde{X} \bm{\lambda}-\mathbf{s}^-\right) _i\geq 0\right\}\geq 1-\alpha ,\qquad i=1,\ldots ,m,

P\left\{ \left( \tilde{Y}\bm{\lambda}-\tilde{\mathbf{y}}_o-\mathbf{s}^+\right) _r \geq 0\right\}\geq 1-\alpha ,\qquad r=1,\ldots ,s,

\bm{\lambda}\geq \mathbf{0},\,\, \mathbf{s}^-\geq \mathbf{0},\,\, \mathbf{s}^+\geq \mathbf{0},

where \bm{\lambda}=(\lambda_1,\ldots,\lambda_n)^\top, \tilde{\mathbf{x}}_o =(\tilde{x}_{1o},\ldots,\tilde{x}_{mo})^\top and \tilde{\mathbf{y}}_o= (\tilde{y}_{1o},\ldots,\tilde{y}_{so})^\top are column vectors. Moreover, \mathbf{s}^-,\mathbf{s}^+ are column vectors with the slacks, and \mathbf{w}^-,\mathbf{w}^+ are positive row vectors with the weights for the slacks. Different returns to scale can be easily considered by adding the corresponding constraints: \mathbf{e}\bm{\lambda}=1 (VRS), 0\leq \mathbf{e}\bm{\lambda} \leq 1 (NIRS), \mathbf{e}\bm{\lambda}\geq 1 (NDRS) or L\leq \mathbf{e} \bm{\lambda}\leq U (GRS), with 0\leq L\leq 1 and U\geq 1, where \mathbf{e}=(1,\ldots ,1) is a row vector.

The deterministic equivalent for a multivariate normal distribution of inputs/outputs is given by

\max \limits_{\bm{\lambda},\mathbf{s}^-,\mathbf{s}^+}\quad \mathbf{w}^- \mathbf{s}^-+\mathbf{w}^+\mathbf{s}^+

\text{s.t.}\quad \mathbf{x}_o-X\bm{\lambda}-\mathbf{s}^-+\Phi ^{-1}(\alpha) \bm{\sigma} ^-(\bm{\lambda})\geq \mathbf{0},

Y\bm{\lambda}-\mathbf{y}_o-\mathbf{s}^++\Phi ^{-1}(\alpha)\bm{\sigma}^+ (\bm{\lambda})\geq \mathbf{0},

\bm{\lambda}\geq \mathbf{0},\,\, \mathbf{s}^-\geq \mathbf{0},\,\, \mathbf{s}^+\geq \mathbf{0},

where \Phi is the standard normal distribution, and

\displaystyle \left( \sigma ^-_i\left( \bm{\lambda}\right)\right) ^2 = \sum _{j,q=1}^n\lambda _j\lambda _q\mathrm{Cov}(\tilde{x}_{ij},\tilde{x}_{iq}) -2\sum _{j=1}^n\lambda _j\mathrm{Cov}(\tilde{x}_{ij},\tilde{x}_{io})

+\mathrm{Var}(\tilde{x}_{io}),\quad i=1,\ldots ,m,

\displaystyle \left( \sigma ^+_r\left( \bm{\lambda}\right)\right) ^2 = \sum _{j,q=1}^n\lambda _j\lambda _q\mathrm{Cov}(\tilde{y}_{rj},\tilde{y}_{rq}) -2\sum _{j=1}^n\lambda _j\mathrm{Cov}(\tilde{y}_{rj},\tilde{y}_{ro})

+\mathrm{Var}(\tilde{y}_{ro}),\quad r=1,\ldots ,s.

Usage

modelstoch_additive(datadea,
               alpha = 0.05,
               dmu_eval = NULL,
               dmu_ref = NULL,
               orientation = NULL,
               weight_slack_i = 1,
               weight_slack_o = 1,
               rts = c("crs", "vrs", "nirs", "ndrs", "grs"),
               L = 1,
               U = 1,
               solver = c("alabama", "cccp", "cccp2", "slsqp"),
               give_X = TRUE,
               returnqp = FALSE,
               ...)

Arguments

Value

A list with the results for the evaluated DMUs and other parameters for reproducibility.

Note

A DMU is \alpha-stochastically efficient if and only if the optimal objective value of the problem, (objval), is zero (or less than zero).

Author(s)

Vicente Bolós (vicente.bolos@uv.es). Department of Business Mathematics

Rafael Benítez (rafael.suarez@uv.es). Department of Business Mathematics

Vicente Coll-Serrano (vicente.coll@uv.es). Quantitative Methods for Measuring Culture (MC2). Applied Economics.

University of Valencia (Spain)

References

Charnes, A.; Cooper, W.W.; Golany, B.; Seiford, L.; Stuz, J. (1985) "Foundations of Data Envelopment Analysis for Pareto-Koopmans Efficient Empirical Production Functions", Journal of Econometrics, 30(1-2), 91-107. doi:10.1016/0304-4076(85)90133-2

Cooper, W.W.; Park, K.S.; Pastor, J.T. (1999). "RAM: A Range Adjusted Measure of Inefficiencies for Use with Additive Models, and Relations to Other Models and Measures in DEA". Journal of Productivity Analysis, 11, p. 5-42. doi:10.1023/A:1007701304281

Chance Constrained Additive P-model.

Description

It solves the "max" version of the "almost 100 chance constrained problem from Cooper et al. (1998), under constant and non-constant returns to scale.

Besides, the user can set weights for the input and/or output slacks. So, it is also possible to solve chance constrained versions of weighted additive models like Measure of Inefficiency Proportions (MIP) or Range Adjusted Measure (RAM), see Cooper et al. (1999).

We consider \mathcal{D}=\left\{ \textrm{DMU}_1, \ldots ,\textrm{DMU}_n \right\} a set of n DMUs with m stochastic inputs and s stochastic outputs. Matrices \tilde{X}=(\tilde{x}_{ij}) and \tilde{Y}=(\tilde{y}_{rj}) are the input and output data matrices, respectively, where \tilde{x}_{ij} and \tilde{y}_{rj} represent the i-th input and r-th output of the j-th DMU. Moreover, we denote by X=(x_{ij}) and Y=(y_{rj}) their expected values. In general, we denote vectors by bold-face letters and they are considered as column vectors unless otherwise stated. The 0-vector is denoted by \bm{0} and the context determines its dimension.

Given 0<\alpha <1 and \varepsilon a positive non-Archimedean infinitesimal, the program for \text{DMU}_o with constant returns to scale is given by

\max \limits_{\bm{\lambda}}\quad P\left\{ \mathbf{w}^-\cdot \left( \tilde{\mathbf{x}}_o-\tilde{X}\bm{\lambda}\right) +\mathbf{w}^+\cdot \left( \tilde{Y}\bm{\lambda}-\tilde{\mathbf{y}}_o\right) > 0\right\}

\text{s.t.}\quad P\left\{ \left( \tilde{\mathbf{x}}_o-\tilde{X} \bm{\lambda}\right) _i> 0\right\} \geq 1-\varepsilon,\quad i=1,\ldots ,m,

P\left\{ \left( \tilde{Y}\bm{\lambda}-\tilde{\mathbf{y}}_o\right) _r> 0\right\} \geq 1-\varepsilon,\quad r=1,\ldots ,s,

\bm{\lambda}\geq \mathbf{0},

where \bm{\lambda}=(\lambda_1,\ldots,\lambda_n)^\top, \tilde{\mathbf{x}}_o =(\tilde{x}_{1o},\ldots,\tilde{x}_{mo})^\top and \tilde{\mathbf{y}}_o= (\tilde{y}_{1o},\ldots,\tilde{y}_{so})^\top are column vectors. Moreover, \mathbf{w}^-,\mathbf{w}^+ are positive row vectors with the weights for the slacks. Different returns to scale can be easily considered by adding the corresponding constraints: \mathbf{e}\bm{\lambda}=1 (VRS), 0\leq \mathbf{e}\bm{\lambda} \leq 1 (NIRS), \mathbf{e}\bm{\lambda}\geq 1 (NDRS) or L\leq \mathbf{e} \bm{\lambda}\leq U (GRS), with 0\leq L\leq 1 and U\geq 1, where \mathbf{e}=(1,\ldots ,1) is a row vector.

The deterministic equivalent for a multivariate normal distribution of inputs/outputs is given by

\max \limits_{\bm{\lambda}}\quad \mathbf{w}^-\cdot \left( \mathbf{x}_o- X\bm{\lambda}\right)+\mathbf{w}^+\cdot \left( Y\bm{\lambda}-\mathbf{y}_o\right) -\Phi ^{-1}(\alpha)\sigma(\bm{\lambda})

\text{s.t.}\quad \mathbf{x}_o-X\bm{\lambda}+\Phi ^{-1}(\varepsilon) \bm{\sigma} ^-(\bm{\lambda})\geq \mathbf{0},

Y\bm{\lambda}-\mathbf{y}_o+\Phi ^{-1}(\varepsilon)\bm{\sigma}^+ (\bm{\lambda})\geq \mathbf{0},

\bm{\lambda}\geq \mathbf{0},

where \Phi is the standard normal distribution, and

\left( \sigma \left( \bm{\lambda}\right)\right) ^2=\bm{\lambda}^\top \left[ \displaystyle \sum _{i,k=1}^m\Delta ^{II}_{ik}+\sum _{r,p=1}^s \Delta ^{OO}_{rp}-2\sum _{i=1}^m\sum _{r=1}^s \Delta ^{IO}_{ir}\right] \bm{\lambda}

\displaystyle +2\sum _{j=1}^n\lambda _j\left[ \sum _{i=1}^m\sum _{r=1}^s\left( \mathrm{Cov}(\tilde{x}_{io},\tilde{y}_{rj})+\mathrm{Cov}(\tilde{x}_{ij}, \tilde{y}_{ro})\right) \right.

\displaystyle \left.-\sum _{i,k=1}^m\mathrm{Cov}(\tilde{x}_{io},\tilde{x}_{kj}) -\sum _{r,p=1}^s\mathrm{Cov}(\tilde{y}_{ro},\tilde{y}_{pj})\right]

\displaystyle +\sum _{i,k=1}^m\mathrm{Cov}(\tilde{x}_{io},\tilde{x}_{ko}) +\sum _{r,p=1}^s \mathrm{Cov}(\tilde{y}_{ro},\tilde{y}_{po})-2\sum _{i=1}^m \sum _{r=1}^s \mathrm{Cov}(\tilde{x}_{io}, \tilde{y}_{ro}),

\displaystyle \left( \sigma ^-_i\left( \bm{\lambda}\right)\right) ^2=\bm{\lambda}^\top \Delta ^{II}_{ii}\bm{\lambda}-2\sum _{j=1}^n\lambda _j\mathrm{Cov}(\tilde{x}_{ij}, \tilde{x}_{io})+\mathrm{Var}(\tilde{x}_{io}),\quad i=1,\ldots ,m,

\displaystyle \left( \sigma ^+_r\left( \bm{\lambda}\right)\right) ^2=\bm{\lambda}^\top \Delta ^{OO}_{rr}\bm{\lambda}-2\sum _{j=1}^n\lambda _j\mathrm{Cov}(\tilde{y}_{rj}, \tilde{y}_{ro})+\mathrm{Var}(\tilde{y}_{ro}),\quad r=1,\ldots ,s.

such that

\left( \Delta ^{II}_{ik}\right) _{jq}=\mathrm{Cov}(\tilde{x}_{ij}, \tilde{x}_{kq}),

\left( \Delta ^{OO}_{rp}\right) _{jq}=\mathrm{Cov}(\tilde{y}_{rj}, \tilde{y}_{pq}),

\left( \Delta ^{IO}_{ir}\right) _{jq}=\left( \Delta ^{OI}_{ri}\right) _{qj} =\mathrm{Cov}(\tilde{x}_{ij}, \tilde{y}_{rq}),

with 1\leq j,q\leq n.

Usage

modelstoch_additive_p(datadea,
               alpha = 0.05,
               epsilon = 0.05,
               dmu_eval = NULL,
               dmu_ref = NULL,
               orientation = NULL,
               weight_slack_i = 1,
               weight_slack_o = 1,
               rts = c("crs", "vrs", "nirs", "ndrs", "grs"),
               L = 1,
               U = 1,
               solver = c("alabama", "cccp", "cccp2", "slsqp"),
               give_X = TRUE,
               returnqp = FALSE,
               ...)

Arguments

Value

A list with the results for the evaluated DMUs and other parameters for reproducibility.

Note

The model in this function is the "max" version of the model in Cooper et al. (1998), in the sense that maximizes the sum of positive slacks like the conventional additive model in Cooper et al. (1985). Hence, a DMU is \alpha-stochastically efficient if and only if the optimal objective value of the problem, (objval), is zero (or less than zero).

Author(s)

Vicente Bolós (vicente.bolos@uv.es). Department of Business Mathematics

Rafael Benítez (rafael.suarez@uv.es). Department of Business Mathematics

Vicente Coll-Serrano (vicente.coll@uv.es). Quantitative Methods for Measuring Culture (MC2). Applied Economics.

University of Valencia (Spain)

References

Charnes, A.; Cooper, W.W.; Golany, B.; Seiford, L.; Stuz, J. (1985) "Foundations of Data Envelopment Analysis for Pareto-Koopmans Efficient Empirical Production Functions", Journal of Econometrics, 30(1-2), 91-107. doi:10.1016/0304-4076(85)90133-2

Cooper, W.W.; Huang, Z.; Lelas, V.; Li, S.X.; Olesen, O.B. (1998) "Chance Constrained Programming Formulations for Stochastic Characterizations of Efficiency and Dominance in DEA", Journal of Productivity Analysis, 9, 53–79. doi:10.1023/A:1018320430249

Cooper, W.W.; Park, K.S.; Pastor, J.T. (1999). "RAM: A Range Adjusted Measure of Inefficiencies for Use with Additive Models, and Relations to Other Models and Measures in DEA". Journal of Productivity Analysis, 11, p. 5-42. doi:10.1023/A:1007701304281

Chance Constrained Super-efficiency Additive E-model.

Description

It solves the chance constrained super-efficiency additive E-model based on the deterministic super-efficiency additive model from Du et al. (2010), under constant and non-constant returns to scale. Besides, the user can set weights for the input and/or output slacks.

We consider \mathcal{D}=\left\{ \textrm{DMU}_1, \ldots ,\textrm{DMU}_n \right\} a set of n DMUs with m stochastic inputs and s stochastic outputs. Matrices \tilde{X}=(\tilde{x}_{ij}) and \tilde{Y}=(\tilde{y}_{rj}) are the input and output data matrices, respectively, where \tilde{x}_{ij} and \tilde{y}_{rj} represent the i-th input and r-th output of the j-th DMU. Moreover, we denote by X=(x_{ij}) and Y=(y_{rj}) their expected values. In general, we denote vectors by bold-face letters and they are considered as column vectors unless otherwise stated. The 0-vector is denoted by \bm{0} and the context determines its dimension.

Given 0<\alpha <1, the program for \text{DMU}_o with constant returns to scale is given by

\min \limits_{\bm{\lambda},\mathbf{t}^-,\mathbf{t}^+}\quad \mathbf{w}^-\mathbf{t}^-+\mathbf{w}^+\mathbf{t}^+

\text{s.t.}\quad P\left\{ \left( \tilde{\mathbf{x}}_o-\tilde{X}_{-o} \bm{\lambda}+\mathbf{t}^-\right) _i\geq 0\right\}\geq 1-\alpha ,\qquad i=1,\ldots ,m,

P\left\{ \left( \tilde{Y}_{-o}\bm{\lambda}-\tilde{\mathbf{y}}_o+ \mathbf{t}^+\right) _r\geq 0\right\}\geq 1-\alpha ,\qquad r=1,\ldots ,s,

\bm{\lambda}\geq \mathbf{0},\,\, \mathbf{t}^-\geq \mathbf{0},\,\, \mathbf{t}^+\geq \mathbf{0},

where \tilde{X}_{-o},\tilde{Y}_{-o} are the input and output data matrices, respectively, defined by \mathcal{D}-\left\{ \textrm{DMU}_o\right\}, \bm{\lambda} =(\lambda_1,\ldots ,\lambda_{o-1},\lambda_{o+1},\ldots ,\lambda_n)^{\top}, \tilde{\mathbf{x}}_o=(\tilde{x}_{1o},\ldots,\tilde{x}_{mo})^\top and \tilde{\mathbf{y}}_o=(\tilde{y}_{1o},\ldots,\tilde{y}_{so})^\top are column vectors. Moreover, \mathbf{s}^-,\mathbf{s}^+ are column vectors with the slacks, and \mathbf{w}^-,\mathbf{w}^+ are positive row vectors with the weights for the slacks. Different returns to scale can be easily considered by adding the corresponding constraints: \mathbf{e}\bm{\lambda}=1 (VRS), 0\leq \mathbf{e}\bm{\lambda} \leq 1 (NIRS), \mathbf{e}\bm{\lambda}\geq 1 (NDRS) or L\leq \mathbf{e} \bm{\lambda}\leq U (GRS), with 0\leq L\leq 1 and U\geq 1, where \mathbf{e}=(1,\ldots ,1) is a row vector.

The deterministic equivalent for a multivariate normal distribution of inputs/outputs is given by

\min \limits_{\bm{\lambda},\mathbf{t}^-,\mathbf{t}^+}\quad \mathbf{w}^-\mathbf{t}^-+\mathbf{w}^+\mathbf{t}^+

\text{s.t.}\quad \mathbf{x}_o-X_{-o}\bm{\lambda}+\mathbf{t}^-+\Phi ^{-1} (\alpha)\bm{\sigma} ^-(\bm{\lambda})\geq \mathbf{0},

Y_{-o}\bm{\lambda}-\mathbf{y}_o+\mathbf{t}^++\Phi ^{-1}(\alpha) \bm{\sigma}^+(\bm{\lambda})\geq \mathbf{0},

\bm{\lambda}\geq \mathbf{0},\,\, \mathbf{t}^-\geq \mathbf{0},\,\, \mathbf{t}^+\geq \mathbf{0},

where \Phi is the standard normal distribution, and

\displaystyle \left( \sigma ^-_i\left( \bm{\lambda}\right)\right) ^2=\sum _ {\substack{j,q=1\\j,q\neq o}}^n\lambda _j\lambda _q\mathrm{Cov}(\tilde{x}_{ij}, \tilde{x}_{iq})-2\sum _{\substack{j=1\\j\neq o}}^n\lambda _j\mathrm{Cov} (\tilde{x}_{ij},\tilde{x}_{io})

+\mathrm{Var}(\tilde{x}_{io}),\quad i=1,\ldots ,m,

\displaystyle \left( \sigma ^+_r\left( \bm{\lambda}\right)\right) ^2=\sum _ {\substack{j,q=1\\j,q\neq o}}^n\lambda _j\lambda _q\mathrm{Cov} (\tilde{y}_{rj},\tilde{y}_{rq})-2\sum _{\substack{j=1\\j\neq o}}^n \lambda _j\mathrm{Cov}(\tilde{y}_{rj},\tilde{y}_{ro})

+\mathrm{Var}(\tilde{y}_{ro}),\quad r=1,\ldots ,s.

Usage

modelstoch_addsupereff(datadea,
               alpha = 0.05,
               dmu_eval = NULL,
               dmu_ref = NULL,
               orientation = NULL,
               weight_slack_i = NULL,
               weight_slack_o = NULL,
               rts = c("crs", "vrs", "nirs", "ndrs", "grs"),
               L = 1,
               U = 1,
               solver = c("alabama", "cccp", "cccp2", "slsqp"),
               compute_target = TRUE,
               returnqp = FALSE,
               ...)

Arguments

Value

A list with the results for the evaluated DMUs and other parameters for reproducibility.

Note

A DMU is \alpha-stochastically efficient if and only if the optimal objective value of the problem, (objval), is zero (or less than zero).

Author(s)

Vicente Bolós (vicente.bolos@uv.es). Department of Business Mathematics

Rafael Benítez (rafael.suarez@uv.es). Department of Business Mathematics

Vicente Coll-Serrano (vicente.coll@uv.es). Quantitative Methods for Measuring Culture (MC2). Applied Economics.

University of Valencia (Spain)

References

Du, J.; Liang, L.; Zhu, J. (2010). "A Slacks-based Measure of Super-efficiency in Data Envelopment Analysis. A Comment", European Journal of Operational Research, 204, 694-697. doi:10.1016/j.ejor.2009.12.007

Chance Constrained Directional Models with Stochastic Directions

Description

It solves chance constrained directional models with stochastic directions, under constant, variable, non-increasing, non-decreasing or generalized returns to scale. Inputs and outputs must follow a multivariate normal distribution. By default, models are solved in a two-stage process (slacks are maximized).

We consider \mathcal{D}=\left\{ \textrm{DMU}_1, \ldots ,\textrm{DMU}_n \right\} a set of n DMUs with m stochastic inputs and s stochastic outputs. Matrices \tilde{X}=(\tilde{x}_{ij}) and \tilde{Y}=(\tilde{y}_{rj}) are the input and output data matrices, respectively, where \tilde{x}_{ij} and \tilde{y}_{rj} represent the i-th input and r-th output of the j-th DMU. Moreover, we denote by X=(x_{ij}) and Y=(y_{rj}) their expected values. In general, we denote vectors by bold-face letters and they are considered as column vectors unless otherwise stated. The 0-vector is denoted by \bm{0} and the context determines its dimension.

Given 0<\alpha <1, the first stage program for \text{DMU}_o with constant returns to scale is given by

\max \limits_{\beta, \bm{\lambda}}\quad \beta

\text{s.t.}\quad P\left\{ \left( \Theta ^-(\beta )\tilde{\mathbf{x}}_o- \tilde{X}\bm{\lambda}\right) _i\geq 0\right\} \geq 1-\alpha,\quad i=1,\ldots ,m,

P\left\{ \left( \tilde{Y}\bm{\lambda}-\Theta ^+(\beta ) \tilde{\mathbf{y}}_{o}\right) _r\geq 0\right\} \geq 1-\alpha,\quad r=1,\ldots ,s,

\bm{\lambda}\geq \mathbf{0},

where \bm{\lambda}=(\lambda_1,\ldots,\lambda_n)^\top, \tilde{\mathbf{x}}_o =(\tilde{x}_{1o},\ldots,\tilde{x}_{mo})^\top and \tilde{\mathbf{y}}_o= (\tilde{y}_{1o},\ldots,\tilde{y}_{so})^\top are column vectors, \Theta ^-(\beta )=I_m-\beta D^-, \Theta ^+(\beta )=I_s+\beta D^+ (with I_m,I_s identity matrices), and D^-=\mathrm{diag}(d^-_1, \ldots ,d^-_m), D^+=\mathrm{diag}(d^+_1,\ldots ,d^+_s) are diagonal matrices with orientation parameters d^-_1,\ldots ,d^-_m, d^+_1,\ldots ,d^+_s \geq 0. Different returns to scale can be easily considered by adding the corresponding constraints: \mathbf{e}\bm{\lambda}=1 (VRS), 0\leq \mathbf{e}\bm{\lambda} \leq 1 (NIRS), \mathbf{e}\bm{\lambda}\geq 1 (NDRS) or L\leq \mathbf{e} \bm{\lambda}\leq U (GRS), with 0\leq L\leq 1 and U\geq 1, where \mathbf{e}=(1,\ldots ,1) is a row vector.

The corresponding second stage program is given by

\max \limits_{\bm{\lambda},\mathbf{s}^-,\mathbf{s}^+}\quad \mathbf{w}^- \mathbf{s}^-+\mathbf{w}^+\mathbf{s}^+

\text{s.t.}\quad P\left\{ \left( \Theta ^-(\beta ^*)\tilde{\mathbf{x}}_o- \tilde{X}\bm{\lambda}-\mathbf{s}^-\right) _i\geq 0\right\} = 1-\alpha,\quad i=1,\ldots ,m,

P\left\{ \left( \tilde{Y}\bm{\lambda}-\Theta ^+(\beta ^*) \tilde{\mathbf{y}}_{o}-\mathbf{s}^+\right) _r\geq 0\right\} = 1-\alpha,\quad r=1,\ldots ,s,

\bm{\lambda}\geq \mathbf{0},\,\, \mathbf{s}^-\geq \mathbf{0},\,\, \mathbf{s}^+\geq \mathbf{0},

where \beta ^* is the optimal objective function of the first stage program, \mathbf{s}^-,\mathbf{s}^+ are column vectors with the slacks, and \mathbf{w}^-,\mathbf{w}^+ are positive row vectors with the weights for the slacks.

The deterministic equivalents for a multivariate normal distribution of inputs/outputs are given by

\max \limits_{\beta, \bm{\lambda}} \quad \beta

\text{s.t.} \quad X\bm{\lambda}-\Phi ^{-1}(\alpha)\bm{\sigma} ^-(\beta , \bm{\lambda}) \leq \Theta ^-(\beta )\mathbf{x}_o,

Y\bm{\lambda}+\Phi ^{-1}(\alpha)\bm{\sigma} ^+(\beta ,\bm{\lambda}) \geq \Theta ^+(\beta )\mathbf{y}_o,

\bm{\lambda}\geq \mathbf{0},

and for the second stage,

\max \limits_{\bm{\lambda},\mathbf{s}^-,\mathbf{s}^+} \quad \mathbf{w}^- \mathbf{s}^-+\mathbf{w}^+\mathbf{s}^+

\text{s.t.} \quad X\bm{\lambda}+\mathbf{s}^--\Phi ^{-1}(\alpha) \bm{\sigma} ^-(\beta ^*,\bm{\lambda}) =\Theta ^-(\beta ^*)\mathbf{x}_o,

Y\bm{\lambda}-\mathbf{s}^++\Phi ^{-1}(\alpha)\bm{\sigma} ^+(\beta ^*, \bm{\lambda}) = \Theta ^+(\beta ^*)\mathbf{y}_{o},

\bm{\lambda}\geq \mathbf{0},\,\, \mathbf{s}^-\geq \mathbf{0},\,\, \mathbf{s}^+\geq \mathbf{0},

where \Phi is the standard normal distribution, and

\displaystyle \left( \sigma ^-_i\left( \beta, \bm{\lambda}\right)\right) ^2 = \sum _{j,q=1}^n\lambda _j\lambda _q\mathrm{Cov}(\tilde{x}_{ij},\tilde{x}_{iq})- 2(1-\beta d^-_i)\sum _{j=1}^n\lambda _j\mathrm{Cov}(\tilde{x}_{ij},\tilde{x}_{io})

+(1-\beta d^-_i)^2\mathrm{Var}(\tilde{x}_{io}),\quad i=1,\ldots ,m,

\displaystyle \left( \sigma ^+_r\left( \beta, \bm{\lambda}\right)\right) ^2 = \sum _{j,q=1}^n\lambda _j\lambda _q\mathrm{Cov}(\tilde{y}_{rj},\tilde{y}_{rq})- 2(1+\beta d^+_r)\sum _{j=1}^n\lambda _j\mathrm{Cov}(\tilde{y}_{rj},\tilde{y}_{ro})

+(1+\beta d^+_r)^2\mathrm{Var}(\tilde{y}_{ro}),\quad r=1,\ldots ,s.

Usage

modelstoch_dir(datadea,
            alpha = 0.05,
            dmu_eval = NULL,
            dmu_ref = NULL,
            d_input = 1,
            d_output = 1,
            rts = c("crs", "vrs", "nirs", "ndrs", "grs"),
            L = 1,
            U = 1,
            solver = c("alabama", "cccp", "cccp2", "slsqp"),
            give_X = TRUE,
            maxslack = TRUE,
            weight_slack_i = 1,
            weight_slack_o = 1,
            compute_target = TRUE,
            returnqp = FALSE,
            silent_ud = FALSE,
            ...)

Arguments

Value

A list with the results for the evaluated DMUs and other parameters for reproducibility.

Author(s)

Vicente Bolós (vicente.bolos@uv.es). Department of Business Mathematics

Rafael Benítez (rafael.suarez@uv.es). Department of Business Mathematics

Vicente Coll-Serrano (vicente.coll@uv.es). Quantitative Methods for Measuring Culture (MC2). Applied Economics.

University of Valencia (Spain)

References

Bolós, V.J.; Benítez, R.; Coll-Serrano, V. (2024). “Chance constrained directional models in stochastic data envelopment analysis", Operations Research Perspectives, 12, 100307.. doi:10.1016/j.orp.2024.100307

Examples

# Example 1.
library(deaR)
data("Coll_Blasco_2006")
ni <- 2 # number of inputs
no <- 2 # number of outputs
data_example <- make_deadata(datadea = Coll_Blasco_2006,
                             ni = ni,
                             no = no)
nd <- length(data_example$dmunames) # number of DMUs
var_input <- matrix(1, nrow = ni, ncol = nd)
var_output <- matrix(1, nrow = no, ncol = nd)
data_stoch <- make_deadata_stoch(datadea = data_example,
                                 var_input = var_input,
                                 var_output = var_output)
Collstochdir <- modelstoch_dir(data_stoch)

Chance Constrained Directional Models with Deterministic Directions

Description

It solves chance constrained directional models with deterministic directions, under constant, variable, non-increasing, non-decreasing or generalized returns to scale. Inputs and outputs must follow a multivariate normal distribution. By default, models are solved in a two-stage process (slacks are maximized).

We consider \mathcal{D}=\left\{ \textrm{DMU}_1, \ldots ,\textrm{DMU}_n \right\} a set of n DMUs with m stochastic inputs and s stochastic outputs. Matrices \tilde{X}=(\tilde{x}_{ij}) and \tilde{Y}=(\tilde{y}_{rj}) are the input and output data matrices, respectively, where \tilde{x}_{ij} and \tilde{y}_{rj} represent the i-th input and r-th output of the j-th DMU. Moreover, we denote by X=(x_{ij}) and Y=(y_{rj}) their expected values. In general, we denote vectors by bold-face letters and they are considered as column vectors unless otherwise stated. The 0-vector is denoted by \bm{0} and the context determines its dimension.

Given 0<\alpha <1, the first stage program for \text{DMU}_o with constant returns to scale is given by

\max \limits_{\beta, \bm{\lambda}}\quad \beta

\text{s.t.} \quad P\left\{ \left( \tilde{\mathbf{x}}_o-\beta \mathbf{g}^-- \tilde{X}\bm{\lambda}\right) _i\geq 0\right\} \geq 1-\alpha,\quad i=1,\ldots ,m,

P\left\{ \left( \tilde{\mathbf{y}}_{o}+\beta \mathbf{g}^+-\tilde{Y}\bm{\lambda} \right) _r\leq 0\right\} \geq 1-\alpha,\quad r=1,\ldots ,s,

\bm{\lambda}\geq \mathbf{0},

where \bm{\lambda}=(\lambda_1,\ldots,\lambda_n)^\top, \tilde{\mathbf{x}}_o =(\tilde{x}_{1o},\ldots,\tilde{x}_{mo})^\top and \tilde{\mathbf{y}}_o= (\tilde{y}_{1o},\ldots,\tilde{y}_{so})^\top are column vectors, and \mathbf{g}=(-\mathbf{g}^-,\mathbf{g}^+)\neq \mathbf{0} is a preassigned direction (with \mathbf{g}^-\in \mathbb{R}^m and \mathbf{g}^+\in\mathbb{R}^s non-negative column vectors). Different returns to scale can be easily considered by adding the corresponding constraints: \mathbf{e}\bm{\lambda}=1 (VRS), 0\leq \mathbf{e}\bm{\lambda} \leq 1 (NIRS), \mathbf{e}\bm{\lambda}\geq 1 (NDRS) or L\leq \mathbf{e} \bm{\lambda}\leq U (GRS), with 0\leq L\leq 1 and U\geq 1, where \mathbf{e}=(1,\ldots ,1) is a row vector.

The corresponding second stage program is given by

\max \limits_{\bm{\lambda},\mathbf{s}^-,\mathbf{s}^+}\quad \mathbf{w}^- \mathbf{s}^-+\mathbf{w}^+\mathbf{s}^+

\text{s.t.} \quad P\left\{ \left( \tilde{\mathbf{x}}_o-\beta ^*\mathbf{g}^-- \tilde{X}\bm{\lambda}-\mathbf{s}^-\right) _i\geq 0\right\} \geq 1-\alpha,\quad i=1,\ldots ,m,

P\left\{ \left( \tilde{\mathbf{y}}_{o}+\beta ^*\mathbf{g}^+-\tilde{Y}\bm{\lambda} +\mathbf{s}^+\right) _r\leq 0\right\} \geq 1-\alpha,\quad r=1,\ldots ,s,

\bm{\lambda}\geq \mathbf{0},\,\, \mathbf{s}^-\geq \mathbf{0},\,\, \mathbf{s}^+\geq \mathbf{0},

where \beta ^* is the optimal objective function of the first stage program, \mathbf{s}^-,\mathbf{s}^+ are column vectors with the slacks, and \mathbf{w}^-,\mathbf{w}^+ are positive row vectors with the weights for the slacks.

The deterministic equivalents for a multivariate normal distribution of inputs/outputs are given by

\max \limits_{\beta, \bm{\lambda}} \quad \beta

\text{s.t.}\quad \beta \mathbf{g}^-+X\bm{\lambda}-\Phi ^{-1} (\alpha)\bm{\sigma} ^-(\bm{\lambda}) \leq \mathbf{x}_o,

-\beta \mathbf{g}^++Y\bm{\lambda}+\Phi ^{-1}(\alpha) \bm{\sigma} ^+(\bm{\lambda}) \geq \mathbf{y}_{o},

\bm{\lambda}\geq \mathbf{0},

and for the second stage,

\max \limits_{\bm{\lambda},\mathbf{s}^-,\mathbf{s}^+} \quad \mathbf{w}^- \mathbf{s}^-+\mathbf{w}^+\mathbf{s}^+

\text{s.t.}\quad \beta ^*\mathbf{g}^-+X\bm{\lambda}+\mathbf{s}^--\Phi ^{-1} (\alpha)\bm{\sigma} ^-(\bm{\lambda}) = \mathbf{x}_o,

-\beta ^*\mathbf{g}^++Y\bm{\lambda}-\mathbf{s}^++\Phi ^{-1}(\alpha) \bm{\sigma} ^+(\bm{\lambda}) = \mathbf{y}_{o},

\bm{\lambda}\geq \mathbf{0},\,\, \mathbf{s}^-\geq \mathbf{0},\,\, \mathbf{s}^+\geq \mathbf{0},

where \Phi is the standard normal distribution, and

\displaystyle \left( \sigma ^-_i\left( \bm{\lambda}\right)\right) ^2 = \sum _{j,q=1}^n\lambda _j\lambda _q\mathrm{Cov}(\tilde{x}_{ij},\tilde{x}_{iq})- 2\sum _{j=1}^n\lambda _j\mathrm{Cov}(\tilde{x}_{ij},\tilde{x}_{io})

+\mathrm{Var}(\tilde{x}_{io}),\quad i=1,\ldots ,m,

\displaystyle \left( \sigma ^+_r\left( \bm{\lambda}\right)\right) ^2 = \sum _{j,q=1}^n\lambda _j\lambda _q\mathrm{Cov}(\tilde{y}_{rj},\tilde{y}_{rq})- 2\sum _{j=1}^n\lambda _j\mathrm{Cov}(\tilde{y}_{rj},\tilde{y}_{ro})

+\mathrm{Var}(\tilde{y}_{ro}),\quad r=1,\ldots ,s.

Usage

modelstoch_dir_dd(datadea,
            alpha = 0.05,
            dmu_eval = NULL,
            dmu_ref = NULL,
            dir_input = NULL,
            dir_output = NULL,
            rts = c("crs", "vrs", "nirs", "ndrs", "grs"),
            L = 1,
            U = 1,
            solver = c("alabama", "cccp", "cccp2", "slsqp"),
            give_X = TRUE,
            maxslack = TRUE,
            weight_slack_i = 1,
            weight_slack_o = 1,
            compute_target = TRUE,
            returnqp = FALSE,
            silent_ud = FALSE,
            ...)

Arguments

Value

A list with the results for the evaluated DMUs and other parameters for reproducibility.

Author(s)

Vicente Bolós (vicente.bolos@uv.es). Department of Business Mathematics

Rafael Benítez (rafael.suarez@uv.es). Department of Business Mathematics

Vicente Coll-Serrano (vicente.coll@uv.es). Quantitative Methods for Measuring Culture (MC2). Applied Economics.

University of Valencia (Spain)

References

Bolós, V.J.; Benítez, R.; Coll-Serrano, V. (2024). “Chance constrained directional models in stochastic data envelopment analysis", Operations Research Perspectives, 12, 100307.. doi:10.1016/j.orp.2024.100307

Examples

# Example 1.
library(deaR)
data("Coll_Blasco_2006")
ni <- 2 # number of inputs
no <- 2 # number of outputs
data_example <- make_deadata(datadea = Coll_Blasco_2006,
                             ni = ni,
                             no = no)
nd <- length(data_example$dmunames) # number of DMUs
var_input <- matrix(1, nrow = ni, ncol = nd)
var_output <- matrix(1, nrow = no, ncol = nd)
data_stoch <- make_deadata_stoch(datadea = data_example,
                                 var_input = var_input,
                                 var_output = var_output)
Collstochdirdd <- modelstoch_dir_dd(data_stoch)

Chance Constrained Radial Models

Description

It solves input and output oriented chance constrained radial DEA models under constant (CCR model), variable (BCC model), non-increasing, non-decreasing or generalized returns to scale, based on the model in Cooper et al. (2002). By default, models are solved in a two-stage process (slacks are maximized).

We consider \mathcal{D}=\left\{ \textrm{DMU}_1, \ldots ,\textrm{DMU}_n \right\} a set of n DMUs with m stochastic inputs and s stochastic outputs. Matrices \tilde{X}=(\tilde{x}_{ij}) and \tilde{Y}=(\tilde{y}_{rj}) are the input and output data matrices, respectively, where \tilde{x}_{ij} and \tilde{y}_{rj} represent the i-th input and r-th output of the j-th DMU. Moreover, we denote by X=(x_{ij}) and Y=(y_{rj}) their expected values. In general, we denote vectors by bold-face letters and they are considered as column vectors unless otherwise stated. The 0-vector is denoted by \bm{0} and the context determines its dimension.

Given 0<\alpha <1, the first stage program for \text{DMU}_o with constant returns to scale is given by

\min \limits_{\theta, \bm{\lambda}}\quad \theta

\text{s.t.}\quad P\left\{ \left( \theta \tilde{\mathbf{x}}_o-\tilde{X} \bm{\lambda}\right) _i\geq 0\right\} \geq 1-\alpha,\quad i=1,\ldots ,m,

P\left\{ \left( \tilde{Y}\bm{\lambda}-\tilde{\mathbf{y}}_{o}\right) _r \geq 0\right\} \geq 1-\alpha,\quad r=1,\ldots ,s,

\bm{\lambda}\geq \mathbf{0},

where \bm{\lambda}=(\lambda_1,\ldots,\lambda_n)^\top, \tilde{\mathbf{x}}_o =(\tilde{x}_{1o},\ldots,\tilde{x}_{mo})^\top and \tilde{\mathbf{y}}_o= (\tilde{y}_{1o},\ldots,\tilde{y}_{so})^\top are column vectors. Different returns to scale can be easily considered by adding the corresponding constraints: \mathbf{e}\bm{\lambda}=1 (VRS), 0\leq \mathbf{e}\bm{\lambda} \leq 1 (NIRS), \mathbf{e}\bm{\lambda}\geq 1 (NDRS) or L\leq \mathbf{e} \bm{\lambda}\leq U (GRS), with 0\leq L\leq 1 and U\geq 1, where \mathbf{e}=(1,\ldots ,1) is a row vector.

The corresponding second stage program is given by

\max \limits_{\bm{\lambda},\mathbf{s}^-,\mathbf{s}^+}\quad \mathbf{w}^-\mathbf{s}^-+\mathbf{w}^+\mathbf{s}^+

\text{s.t.}\quad P\left\{ \left( \theta ^*\tilde{\mathbf{x}}_o-\tilde{X} \bm{\lambda}-\mathbf{s}^-\right) _i\geq 0\right\} = 1-\alpha,\quad i=1,\ldots ,m,

P\left\{ \left( \tilde{Y}\bm{\lambda}-\tilde{\mathbf{y}}_{o}-\mathbf{s}^+ \right) _r\geq 0\right\} = 1-\alpha,\quad r=1,\ldots ,s,

\bm{\lambda}\geq \mathbf{0},\,\, \mathbf{s}^-\geq \mathbf{0},\,\, \mathbf{s}^+\geq \mathbf{0},

where \theta ^* is the optimal objective function of the first stage program, \mathbf{s}^-,\mathbf{s}^+ are column vectors with the slacks, and \mathbf{w}^-,\mathbf{w}^+ are positive row vectors with the weights for the slacks.

The deterministic equivalents for a multivariate normal distribution of inputs/outputs are given by

\min \limits_{\theta, \bm{\lambda}}\quad \theta

\text{s.t.}\quad X\bm{\lambda}-\Phi ^{-1}(\alpha)\bm{\sigma} ^-(\theta, \bm{\lambda}) \leq \theta \mathbf{x}_o,

Y\bm{\lambda}+\Phi ^{-1}(\alpha)\bm{\sigma} ^+(\bm{\lambda}) \geq \mathbf{y}_o,

\bm{\lambda}\geq \mathbf{0},

and for the second stage,

\max \limits_{\bm{\lambda},\mathbf{s}^-,\mathbf{s}^+}\quad \mathbf{w}^-\mathbf{s}^-+\mathbf{w}^+\mathbf{s}^+

\text{s.t.}\quad X\bm{\lambda}+\mathbf{s}^--\Phi ^{-1}(\alpha) \bm{\sigma} ^-(\theta ^*,\bm{\lambda}) =\theta ^* \mathbf{x}_o,

Y\bm{\lambda}-\mathbf{s}^++\Phi ^{-1}(\alpha)\bm{\sigma} ^+ (\bm{\lambda}) = \mathbf{y}_{o},

\bm{\lambda}\geq \mathbf{0},\,\, \mathbf{s}^-\geq \mathbf{0},\,\, \mathbf{s}^+\geq \mathbf{0},

where \Phi is the standard normal distribution, and

\displaystyle \left( \sigma ^-_i\left( \theta, \bm{\lambda}\right)\right) ^2 = \sum _{j,q=1}^n\lambda _j\lambda _q\mathrm{Cov}(\tilde{x}_{ij},\tilde{x}_{iq}) -2\theta\sum _{j=1}^n\lambda _j\mathrm{Cov}(\tilde{x}_{ij},\tilde{x}_{io})

+\theta ^2\mathrm{Var}(\tilde{x}_{io}),\quad i=1,\ldots ,m,

\displaystyle \left( \sigma ^+_r\left( \bm{\lambda}\right)\right) ^2 = \sum _{j,q=1}^n\lambda _j\lambda _q\mathrm{Cov}(\tilde{y}_{rj},\tilde{y}_{rq}) -2\sum _{j=1}^n\lambda _j\mathrm{Cov}(\tilde{y}_{rj},\tilde{y}_{ro})

+\mathrm{Var}(\tilde{y}_{ro}),\quad r=1,\ldots ,s.

Usage

modelstoch_radial(datadea,
            alpha = 0.05,
            dmu_eval = NULL,
            dmu_ref = NULL,
            orientation = c("io", "oo"),
            rts = c("crs", "vrs", "nirs", "ndrs", "grs"),
            L = 1,
            U = 1,
            solver = c("alabama", "cccp", "cccp2", "slsqp"),
            give_X = TRUE,
            maxslack = TRUE,
            weight_slack_i = 1,
            weight_slack_o = 1,
            vtrans_i = NULL,
            vtrans_o = NULL,
            compute_target = TRUE,
            returnqp = FALSE,
            silent_ud = FALSE,
            ...)

Arguments

Value

A list with the results for the evaluated DMUs and other parameters for reproducibility.

Author(s)

Vicente Bolós (vicente.bolos@uv.es). Department of Business Mathematics

Rafael Benítez (rafael.suarez@uv.es). Department of Business Mathematics

Vicente Coll-Serrano (vicente.coll@uv.es). Quantitative Methods for Measuring Culture (MC2). Applied Economics.

University of Valencia (Spain)

References

Cooper, W.W.; Deng, H.; Huang, Z.; Li, S.X. (2002). “Chance constrained programming approaches to technical efficiencies and inefficiencies in stochastic data envelopment analysis", Journal of the Operational Research Society, 53:12, 1347-1356. doi:10.1057/palgrave.jors.2601433

El-Demerdash, B.E.; El-Khodary, I.A.; Tharwat, A.A. (2013). "Developing a Stochastic Input Oriented Data Envelopment Analysis (SIODEA) Model", International Journal of Advanced Computer Science and Applications, Vol.4, No. 4, 40-44.

Examples

# Example 1.
library(deaR)
data("Coll_Blasco_2006")
ni <- 2 # number of inputs
no <- 2 # number of outputs
data_example <- make_deadata(datadea = Coll_Blasco_2006,
                             ni = ni,
                             no = no)
nd <- length(data_example$dmunames) # number of DMUs
var_input <- matrix(1, nrow = ni, ncol = nd)
var_output <- matrix(1, nrow = no, ncol = nd)
data_stoch <- make_deadata_stoch(datadea = data_example,
                                 var_input = var_input,
                                 var_output = var_output)
Collstoch <- modelstoch_radial(data_stoch)

# Example 2. Deterministic data with one stochastic input.
# Replication of results in El-Demerdash et al. (2013).
library(deaR)
dmunames <- c("A", "B", "C")
nd <- length(dmunames) # Number of DMUs
inputnames <- c("Professors", "Budget")
ni <- length(inputnames) # Number of Inputs
outputnames <- c("Diplomas", "Bachelors", "Masters")
no <- length(outputnames) # Number of Outputs
X <- matrix(c(5, 14, 8, 15, 7, 12),
            nrow = ni, ncol = nd, dimnames = list(inputnames, dmunames))
Y <- matrix(c(9, 4, 16, 5, 7, 10, 4, 9, 13),
            nrow = no, ncol = nd, dimnames = list(outputnames, dmunames))
datadea <- make_deadata(inputs = X,
                        outputs = Y)
covX <- array(0, dim = c(2, 3, 3))
# The 2nd input is stochastic.
# Since the corresponding 3x3 covariances matrix is symmetric, only values
# above the diagonal are necessary.
covX[2, 1, ] <- c(1.4, 0.9, 0.6)
covX[2, 2, 2:3] <- c(1.5, 0.7)
covX[2, 3, 3] <- 1.2
# Alternatively (note that values below the diagonal are ignored).
covX[2, , ] <- matrix(c(1.4, 0.9, 0.6, 0, 1.5, 0.7, 0, 0, 1.2),
                      byrow = TRUE)
datadea_stoch <- make_deadata_stoch(datadea,
                                    cov_input = covX)
res <- modelstoch_radial(datadea_stoch, rts = "vrs")

Chance Constrained Radial Super-efficiency Models

Description

Solve chance constrained radial super-efficiency DEA models, based on the Cooper et al. (2002) chance constrained radial efficiency models. Analogously to the deterministic case, it removes the evaluated DMU from the set of reference DMUs dmu_ref with respect to which it is evaluated.

Usage

modelstoch_radial_supereff(datadea,
                                  dmu_eval = NULL,
                                  dmu_ref = NULL,
                                  ...)

Arguments

Value

A list with the results for the evaluated DMUs and other parameters for reproducibility.

Note

Radial super-efficiency chance constrained model under non constant (vrs, nirs, ndrs, grs) returns to scale can be unfeasible for certain DMUs.

Author(s)

Vicente Bolós (vicente.bolos@uv.es). Department of Business Mathematics

Rafael Benítez (rafael.suarez@uv.es). Department of Business Mathematics

Vicente Coll-Serrano (vicente.coll@uv.es). Quantitative Methods for Measuring Culture (MC2). Applied Economics.

University of Valencia (Spain)

References

Cooper, W.W.; Deng, H.; Huang, Z.; Li, S.X. (2002). “Chance constrained programming approaches to technical efficiencies and inefficiencies in stochastic data envelopment analysis", Journal of the Operational Research Society, 53:12, 1347-1356.

See Also

modelstoch_radial

Examples

# Example 1.
library(deaR)
ni = 2
no = 1
datatext <- make_deadata(Textile, ni = 2, no = 1)
nd <- length(datatext$dmunames)
# Compute variances
mean_i <- apply(datatext$input, MARGIN = 1, FUN = mean)
mean_o <- mean(datatext$output)
var_i1 <- sum((datatext$input[1, ] - mean_i[1]) ^ 2) / (nd - 1)
var_i2 <- sum((datatext$input[2, ] - mean_i[2]) ^ 2) / (nd - 1)
var_o <- sum((datatext$output - mean_o) ^ 2) / (nd - 1)
var_input <- matrix(rep(c(var_i1, var_i2), nd),
                    nrow = ni,
                    ncol = nd)
var_output <- matrix(var_o,
                     nrow = no,
                     ncol = nd)
datatext_stoch <- make_deadata_stoch(datatext,
                                     var_input = var_input,
                                     var_output = var_output)
res <- modelstoch_radial_supereff(datatext_stoch, orientation = "oo")