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 $$\displaystyle \left( \sigma ^-_i\left( \bm{\lambda}\right)\right) ^2 = \sum _{j,q=1}^n\lambda _j\lambda _q\mathrm{Cov}(\tilde{x}_{ij},\tilde{x}_{iq}) -2\sum _{j=1}^n\lambda _j\mathrm{Cov}(\tilde{x}_{ij},\tilde{x}_{io})$$ $$+\mathrm{Var}(\tilde{x}_{io}),\quad i=1,\ldots ,m,$$ $$\displaystyle \left( \sigma ^+_r\left( \bm{\lambda}\right)\right) ^2 = \sum _{j,q=1}^n\lambda _j\lambda _q\mathrm{Cov}(\tilde{y}_{rj},\tilde{y}_{rq}) -2\sum _{j=1}^n\lambda _j\mathrm{Cov}(\tilde{y}_{rj},\tilde{y}_{ro})$$ $$+\mathrm{Var}(\tilde{y}_{ro}),\quad r=1,\ldots ,s.$$
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,
n_attempts_max = 5,
returnqp = FALSE,
...)A list with the results for the evaluated DMUs and other parameters for reproducibility.
The data of class deadata_stoch, including n DMUs,
and the expected values of m inputs and s outputs.
A value for parameter alpha.
A numeric vector containing which DMUs have to be evaluated.
If NULL (default), all DMUs are considered.
A numeric vector containing which DMUs are the evaluation reference set.
If NULL (default), all DMUs are considered.
This parameter is either NULL (default) or a string, equal to
"io" (input-oriented) or "oo" (output-oriented). It is used to modify the weight slacks.
If input-oriented, weight_slack_o are taken 0.
If output-oriented, weight_slack_i are taken 0.
A value, vector of length m, or matrix m x
ne (where ne is the length of dmu_eval)
with the weights of the input slacks. If 0, output-oriented.
A value, vector of length s, or matrix s x
ne (where ne is the length of dmu_eval)
with the weights of the output slacks. If 0, input-oriented.
A string, determining the type of returns to scale, equal to "crs" (constant), "vrs" (variable), "nirs" (non-increasing), "ndrs" (non-decreasing) or "grs" (generalized).
Lower bound for the generalized returns to scale (grs).
Upper bound for the generalized returns to scale (grs).
Character string with the name of the solver used by function solvecop
from package optiSolve.
Logical. If it is TRUE, it uses an initial vector (given by
the evaluated DMU) for the solver, except for "cccp". If it is FALSE,
the initial vector is given internally by the solver and it is usually
randomly generated.
A value with the maximum number of attempts if the solver does not converge. Each attempt uses a different initial vector.
Logical. If it is TRUE, it returns the quadratic
problems (objective function and constraints).
Other parameters, like the initial vector X, to be passed
to the solver.
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)
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. tools:::Rd_expr_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. tools:::Rd_expr_doi("10.1023/A:1007701304281")