# Chance Constrained Directional Models with Deterministic Directions 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 [REMOVE_ME]$$ \max \limits_{\beta, \bm{\lambda}}\quad \beta [REMOVE_ME_2]$$ [REMOVE_ME]$$ \text{s.t.} \quad P\left\{ \left( \tilde{\mathbf{x}}_o-\beta \mathbf{g}^--\tilde{X}\bm{\lambda}\right) _i\geq 0\right\} \geq 1-\alpha,\quad i=1,\ldots ,m, [REMOVE_ME_2]$$ [REMOVE_ME]$$ P\left\{ \left( \tilde{\mathbf{y}}_{o}+\beta \mathbf{g}^+-\tilde{Y}\bm{\lambda}\right) _r\leq 0\right\} \geq 1-\alpha,\quad r=1,\ldots ,s, [REMOVE_ME_2]$$ [REMOVE_ME]$$ \bm{\lambda}\geq \mathbf{0}, [REMOVE_ME_2]$$ where $\bm{\lambda}=(\lambda_1,\ldots,\lambda_n)^\top$, c("$\\tilde{\\mathbf{x}}_o\n$", "$=(\\tilde{x}_{1o},\\ldots,\\tilde{x}_{mo})^\\top$") and c("$\\tilde{\\mathbf{y}}_o=\n$", "$(\\tilde{y}_{1o},\\ldots,\\tilde{y}_{so})^\\top$") 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), c("$0\\leq \\mathbf{e}\\bm{\\lambda}\n$", "$\\leq 1$") (NIRS), $\mathbf{e}\bm{\lambda}\geq 1$ (NDRS) or c("$L\\leq \\mathbf{e}\n$", "$\\bm{\\lambda}\\leq U$") (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 [REMOVE_ME]$$ \max \limits_{\bm{\lambda},\mathbf{s}^-,\mathbf{s}^+}\quad \mathbf{w}^-\mathbf{s}^-+\mathbf{w}^+\mathbf{s}^+ [REMOVE_ME_2]$$ [REMOVE_ME]$$ \text{s.t.} \quad P\left\{ \left( \tilde{\mathbf{x}}_o-\beta ^*\mathbf{g}^--\tilde{X}\bm{\lambda}-\mathbf{s}^-\right) _i\geq 0\right\} \geq 1-\alpha,\quad i=1,\ldots ,m, [REMOVE_ME_2]$$ [REMOVE_ME]$$ P\left\{ \left( \tilde{\mathbf{y}}_{o}+\beta ^*\mathbf{g}^+-\tilde{Y}\bm{\lambda}+\mathbf{s}^+\right) _r\leq 0\right\} \geq 1-\alpha,\quad r=1,\ldots ,s, [REMOVE_ME_2]$$ [REMOVE_ME]$$ \bm{\lambda}\geq \mathbf{0},\,\, \mathbf{s}^-\geq \mathbf{0},\,\,\mathbf{s}^+\geq \mathbf{0}, [REMOVE_ME_2]$$ 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 [REMOVE_ME]$$ \max \limits_{\beta, \bm{\lambda}} \quad \beta [REMOVE_ME_2]$$ [REMOVE_ME]$$ \text{s.t.}\quad \beta \mathbf{g}^-+X\bm{\lambda}-\Phi ^{-1}(\alpha)\bm{\sigma} ^-(\bm{\lambda}) \leq \mathbf{x}_o, [REMOVE_ME_2]$$ [REMOVE_ME]$$ -\beta \mathbf{g}^++Y\bm{\lambda}+\Phi ^{-1}(\alpha)\bm{\sigma} ^+(\bm{\lambda}) \geq \mathbf{y}_{o}, [REMOVE_ME_2]$$ [REMOVE_ME]$$ \bm{\lambda}\geq \mathbf{0}, [REMOVE_ME_2]$$ and for the second stage, [REMOVE_ME]$$ \max \limits_{\bm{\lambda},\mathbf{s}^-,\mathbf{s}^+} \quad \mathbf{w}^-\mathbf{s}^-+\mathbf{w}^+\mathbf{s}^+ [REMOVE_ME_2]$$ [REMOVE_ME]$$ \text{s.t.}\quad \beta ^*\mathbf{g}^-+X\bm{\lambda}+\mathbf{s}^--\Phi ^{-1}(\alpha)\bm{\sigma} ^-(\bm{\lambda}) = \mathbf{x}_o, [REMOVE_ME_2]$$ [REMOVE_ME]$$ -\beta ^*\mathbf{g}^++Y\bm{\lambda}-\mathbf{s}^++\Phi ^{-1}(\alpha)\bm{\sigma} ^+(\bm{\lambda}) = \mathbf{y}_{o}, [REMOVE_ME_2]$$ [REMOVE_ME]$$ \bm{\lambda}\geq \mathbf{0},\,\, \mathbf{s}^-\geq \mathbf{0},\,\,\mathbf{s}^+\geq \mathbf{0}, [REMOVE_ME_2]$$ where $\Phi $ is the standard normal distribution, and [REMOVE_ME]$$ \displaystyle \left( \sigma ^-_i\left( \bm{\lambda}\right)\right) ^2 =\sum _{j,q=1}^n\lambda _j\lambda _q\mathrm{Cov}(\tilde{x}_{ij},\tilde{x}_{iq})-2\sum _{j=1}^n\lambda _j\mathrm{Cov}(\tilde{x}_{ij},\tilde{x}_{io}) [REMOVE_ME_2]$$ [REMOVE_ME]$$ +\mathrm{Var}(\tilde{x}_{io}),\quad i=1,\ldots ,m, [REMOVE_ME_2]$$ [REMOVE_ME]$$ \displaystyle \left( \sigma ^+_r\left( \bm{\lambda}\right)\right) ^2 =\sum _{j,q=1}^n\lambda _j\lambda _q\mathrm{Cov}(\tilde{y}_{rj},\tilde{y}_{rq})-2\sum _{j=1}^n\lambda _j\mathrm{Cov}(\tilde{y}_{rj},\tilde{y}_{ro}) [REMOVE_ME_2]$$ [REMOVE_ME]$$ +\mathrm{Var}(\tilde{y}_{ro}),\quad r=1,\ldots ,s. [REMOVE_ME_2]$$ ```r 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, n_attempts_max = 5, maxslack = FALSE, weight_slack_i = 1, weight_slack_o = 1, compute_target = TRUE, returnqp = FALSE, silent_ud = FALSE, ...) ``` ## Arguments - `datadea`: The data of class `deadata_stoch`, including `n` DMUs, and the expected values of `m` inputs and `s` outputs. - `alpha`: A value for parameter alpha. - `dmu_eval`: A numeric vector containing which DMUs have to be evaluated. If `NULL` (default), all DMUs are considered. - `dmu_ref`: A numeric vector containing which DMUs are the evaluation reference set. If `NULL` (default), all DMUs are considered. - `dir_input`: A value, vector of length `m`, or matrix `m` x `ne` (where `ne` is the length of `dmu_eval`) with the input directions. If `dir_input` == input matrix (of DMUS in `dmu_eval`) and `dir_output` == 0, it is equivalent to input oriented (`beta` = 1 - `efficiency`). If `dir_input` is omitted, input matrix (of DMUS in `dmu_eval`) is assigned. - `dir_output`: A value, vector of length `s`, or matrix `s` x `ne` (where `ne` is the length of `dmu_eval`) with the output directions. If `dir_input` == 0 and `dir_output` == output matrix (of DMUS in `dmu_eval`), it is equivalent to output oriented (`beta` = `efficiency` - 1). If `dir_output` is omitted, output matrix (of DMUS in `dmu_eval`) is assigned. - `rts`: A string, determining the type of returns to scale, equal to "crs" (constant), "vrs" (variable), "nirs" (non-increasing), "ndrs" (non-decreasing) or "grs" (generalized). - `L`: Lower bound for the generalized returns to scale (grs). - `U`: Upper bound for the generalized returns to scale (grs). - `solver`: Character string with the name of the solver used by function `solvecop` from package `optiSolve`. - `give_X`: 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. - `n_attempts_max`: A value with the maximum number of attempts if the solver does not converge. Each attempt uses a different initial vector. - `maxslack`: Logical. If it is `TRUE`, it computes the max slack solution. - `weight_slack_i`: 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 for the max slack solution. - `weight_slack_o`: 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 for the max slack solution. - `compute_target`: Logical. If it is `TRUE`, it computes targets of the max slack solution. - `returnqp`: Logical. If it is `TRUE`, it returns the quadratic problems (objective function and constraints) of stage 1. - `silent_ud`: Logical, to avoid warnings related with undesirable variables. - `...`: Other parameters, like the initial vector `X`, to be passed to the solver. ## Returns A list with the results for the evaluated DMUs and other parameters for reproducibility. ## 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 $\bm{\lambda}=(\lambda_1,\ldots,\lambda_n)^\top$, c("$\\tilde{\\mathbf{x}}_o\n$", "$=(\\tilde{x}_{1o},\\ldots,\\tilde{x}_{mo})^\\top$") and c("$\\tilde{\\mathbf{y}}_o=\n$", "$(\\tilde{y}_{1o},\\ldots,\\tilde{y}_{so})^\\top$") 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), c("$0\\leq \\mathbf{e}\\bm{\\lambda}\n$", "$\\leq 1$") (NIRS), $\mathbf{e}\bm{\lambda}\geq 1$ (NDRS) or c("$L\\leq \\mathbf{e}\n$", "$\\bm{\\lambda}\\leq U$") (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 $$ \displaystyle \left( \sigma ^-_i\left( \bm{\lambda}\right)\right) ^2 =\sum _{j,q=1}^n\lambda _j\lambda _q\mathrm{Cov}(\tilde{x}_{ij},\tilde{x}_{iq})-2\sum _{j=1}^n\lambda _j\mathrm{Cov}(\tilde{x}_{ij},\tilde{x}_{io}) $$ $$ +\mathrm{Var}(\tilde{x}_{io}),\quad i=1,\ldots ,m, $$ $$ \displaystyle \left( \sigma ^+_r\left( \bm{\lambda}\right)\right) ^2 =\sum _{j,q=1}^n\lambda _j\lambda _q\mathrm{Cov}(\tilde{y}_{rj},\tilde{y}_{rq})-2\sum _{j=1}^n\lambda _j\mathrm{Cov}(\tilde{y}_{rj},\tilde{y}_{ro}) $$ $$ +\mathrm{Var}(\tilde{y}_{ro}),\quad r=1,\ldots ,s. $$ ## Examples ```r # 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) # Evaluate the sixth DMU Collstochdirdd <- modelstoch_dir_dd(data_stoch, dmu_eval = 6) efficiencies(Collstochdirdd) ``` ## References Bolós, V.J.; Benítez, R.; Coll-Serrano, V. (2024). “Chance constrained directional models in stochastic data envelopment analysis", Operations Research Perspectives, 12, 100307.. tools:::Rd_expr_doi("10.1016/j.orp.2024.100307") ## 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)

modelstoch_dir_dd function