# Chance Constrained Directional Models with Stochastic Directions 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 [REMOVE_ME]$$ \max \limits_{\beta, \bm{\lambda}}\quad \beta [REMOVE_ME_2]$$ [REMOVE_ME]$$ \text{s.t.}\quad P\left\{ \left( \Theta ^-(\beta )\tilde{\mathbf{x}}_o-\tilde{X}\bm{\lambda}\right) _i\geq 0\right\} \geq 1-\alpha,\quadi=1,\ldots ,m, [REMOVE_ME_2]$$ [REMOVE_ME]$$ P\left\{ \left( \tilde{Y}\bm{\lambda}-\Theta ^+(\beta )\tilde{\mathbf{y}}_{o}\right) _r\geq 0\right\} \geq 1-\alpha,\quadr=1,\ldots ,s, [REMOVE_ME_2]$$ [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, $\Theta ^-(\beta )=I_m-\beta D^-$, c("$\\Theta ^+(\\beta )=I_s+\\beta\n$", "$D^+$") (with $I_m,I_s$ identity matrices), and c("$D^-=\\mathrm{diag}(d^-_1,\n$", "$\\ldots ,d^-_m)$"), $D^+=\mathrm{diag}(d^+_1,\ldots ,d^+_s)$ 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), 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]$$ \text{s.t.}\quad P\left\{ \left( \Theta ^-(\beta ^*)\tilde{\mathbf{x}}_o-\tilde{X}\bm{\lambda}-\mathbf{s}^-\right) _i\geq 0\right\} = 1-\alpha,\quadi=1,\ldots ,m, [REMOVE_ME_2]$$ [REMOVE_ME]$$ P\left\{ \left( \tilde{Y}\bm{\lambda}-\Theta ^+(\beta ^*)\tilde{\mathbf{y}}_{o}-\mathbf{s}^+\right) _r\geq 0\right\} = 1-\alpha,\quadr=1,\ldots ,s, [REMOVE_ME_2]$$ [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]$$ \text{s.t.} \quad X\bm{\lambda}-\Phi ^{-1}(\alpha)\bm{\sigma} ^-(\beta ,\bm{\lambda}) \leq \Theta ^-(\beta )\mathbf{x}_o, [REMOVE_ME_2]$$ [REMOVE_ME]$$ Y\bm{\lambda}+\Phi ^{-1}(\alpha)\bm{\sigma} ^+(\beta ,\bm{\lambda})\geq \Theta ^+(\beta )\mathbf{y}_o, [REMOVE_ME_2]$$ [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]$$ \text{s.t.} \quad X\bm{\lambda}+\mathbf{s}^--\Phi ^{-1}(\alpha)\bm{\sigma} ^-(\beta ^*,\bm{\lambda}) =\Theta ^-(\beta ^*)\mathbf{x}_o, [REMOVE_ME_2]$$ [REMOVE_ME]$$ Y\bm{\lambda}-\mathbf{s}^++\Phi ^{-1}(\alpha)\bm{\sigma} ^+(\beta ^*,\bm{\lambda}) = \Theta ^+(\beta ^*)\mathbf{y}_{o}, [REMOVE_ME_2]$$ [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]$$ \displaystyle \left( \sigma ^-_i\left( \beta, \bm{\lambda}\right)\right) ^2 =\sum _{j,q=1}^n\lambda _j\lambda _q\mathrm{Cov}(\tilde{x}_{ij},\tilde{x}_{iq})-2(1-\beta d^-_i)\sum _{j=1}^n\lambda _j\mathrm{Cov}(\tilde{x}_{ij},\tilde{x}_{io}) [REMOVE_ME_2]$$ [REMOVE_ME]$$ +(1-\beta d^-_i)^2\mathrm{Var}(\tilde{x}_{io}),\quad i=1,\ldots ,m, [REMOVE_ME_2]$$ [REMOVE_ME]$$ \displaystyle \left( \sigma ^+_r\left( \beta, \bm{\lambda}\right)\right) ^2 =\sum _{j,q=1}^n\lambda _j\lambda _q\mathrm{Cov}(\tilde{y}_{rj},\tilde{y}_{rq})-2(1+\beta d^+_r)\sum _{j=1}^n\lambda _j\mathrm{Cov}(\tilde{y}_{rj},\tilde{y}_{ro}) [REMOVE_ME_2]$$ [REMOVE_ME]$$ +(1+\beta d^+_r)^2\mathrm{Var}(\tilde{y}_{ro}),\quad r=1,\ldots ,s. [REMOVE_ME_2]$$ ```r 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, 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. - `d_input`: A value, vector of length `m`, or matrix `m` x `ne` (where `ne` is the length of `dmu_eval`) with the input orientation parameters. If `d_input` == 1 (default) and `dir_output` == 0, it is equivalent to input oriented (`beta` = 1 - `efficiency`). - `d_output`: A value, vector of length `s`, or matrix `s` x `ne` (where `ne` is the length of `dmu_eval`) with the output orientation parameters. If `d_input` == 0 and `d_output` == 1 (default), it is equivalent to output oriented (`beta` = `efficiency` - 1). - `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 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 $$ \max \limits_{\beta, \bm{\lambda}}\quad \beta $$ $$ \text{s.t.}\quad P\left\{ \left( \Theta ^-(\beta )\tilde{\mathbf{x}}_o-\tilde{X}\bm{\lambda}\right) _i\geq 0\right\} \geq 1-\alpha,\quadi=1,\ldots ,m, $$ $$ P\left\{ \left( \tilde{Y}\bm{\lambda}-\Theta ^+(\beta )\tilde{\mathbf{y}}_{o}\right) _r\geq 0\right\} \geq 1-\alpha,\quadr=1,\ldots ,s, $$ $$ \bm{\lambda}\geq \mathbf{0}, $$ 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, $\Theta ^-(\beta )=I_m-\beta D^-$, c("$\\Theta ^+(\\beta )=I_s+\\beta\n$", "$D^+$") (with $I_m,I_s$ identity matrices), and c("$D^-=\\mathrm{diag}(d^-_1,\n$", "$\\ldots ,d^-_m)$"), $D^+=\mathrm{diag}(d^+_1,\ldots ,d^+_s)$ 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), 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 $$ \max \limits_{\bm{\lambda},\mathbf{s}^-,\mathbf{s}^+}\quad \mathbf{w}^-\mathbf{s}^-+\mathbf{w}^+\mathbf{s}^+ $$ $$ \text{s.t.}\quad P\left\{ \left( \Theta ^-(\beta ^*)\tilde{\mathbf{x}}_o-\tilde{X}\bm{\lambda}-\mathbf{s}^-\right) _i\geq 0\right\} = 1-\alpha,\quadi=1,\ldots ,m, $$ $$ P\left\{ \left( \tilde{Y}\bm{\lambda}-\Theta ^+(\beta ^*)\tilde{\mathbf{y}}_{o}-\mathbf{s}^+\right) _r\geq 0\right\} = 1-\alpha,\quadr=1,\ldots ,s, $$ $$ \bm{\lambda}\geq \mathbf{0},\,\, \mathbf{s}^-\geq \mathbf{0},\,\,\mathbf{s}^+\geq \mathbf{0}, $$ 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 $$ \displaystyle \left( \sigma ^-_i\left( \beta, \bm{\lambda}\right)\right) ^2 =\sum _{j,q=1}^n\lambda _j\lambda _q\mathrm{Cov}(\tilde{x}_{ij},\tilde{x}_{iq})-2(1-\beta d^-_i)\sum _{j=1}^n\lambda _j\mathrm{Cov}(\tilde{x}_{ij},\tilde{x}_{io}) $$ $$ +(1-\beta d^-_i)^2\mathrm{Var}(\tilde{x}_{io}),\quad i=1,\ldots ,m, $$ $$ \displaystyle \left( \sigma ^+_r\left( \beta, \bm{\lambda}\right)\right) ^2 =\sum _{j,q=1}^n\lambda _j\lambda _q\mathrm{Cov}(\tilde{y}_{rj},\tilde{y}_{rq})-2(1+\beta d^+_r)\sum _{j=1}^n\lambda _j\mathrm{Cov}(\tilde{y}_{rj},\tilde{y}_{ro}) $$ $$ +(1+\beta d^+_r)^2\mathrm{Var}(\tilde{y}_{ro}),\quad r=1,\ldots ,s. $$ ## 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 Collstochdir <- modelstoch_dir(data_stoch, dmu_eval = 6) efficiencies(Collstochdir) ``` ## 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 function