# Slacks-Based Measure (SBM) of Efficiency Calculate additive Data Envelopment Analysis (DEA) efficiency with the Slacks-Based Measure (SBM) of efficiency (Tone, 2001) ```r dea.sbm(base, noutput, fixed = NULL, rts = 2, bound = NULL, whichDMUs = NULL, print.status = FALSE) ``` ## Arguments - `base`: A data frame with N rows and S+M columns, where N is the number of Decision-Making Units (DMUs), S is the number of outputs and M is the number of inputs. - `noutput`: The number of outputs produced by the DMUs. All DMUs must produce the same number of outputs. - `fixed`: A numeric vector containing column indices for fixed (non-controllable) outputs and/or inputs (if any) in the data. Defaults to NULL. - `rts`: Returns to scale specification. 1 for constant returns to scale and 2 (default) for variable returns to scale. - `bound`: A data frame with N rows and S+M columns containing user-defined bounds on the slacks of each DMU. If bounds are supplied by the user in cases where some outputs and/or inputs are fixed, values should be 0 for these fixed variables. Same for slacks that do not require bounds. Defaults to NULL. - `whichDMUs`: Numeric vector specifying the line numbers of the DMUs for which efficiency should be calculated. Defaults to NULL, i.e. by default efficiency is calculated for all DMUs in the dataset. - `print.status`: Defaults to FALSE. If the solution of the linear program is NA for one or more DMUs, print.status can be set to TRUE to find out why. ## Details The Slacks-Based Measure (SBM) of efficiency (Tone, 2001) is an additive DEA model that maximizes the sum of input and output slacks for each DMU. Unlike other additive DEA models, SBM's objective function has a ratio form, with input slacks summed in the numerator and output slacks summed in the denominator. These sums in the ratio result in an aggregate measure of all inefficiencies that a DMU may exhibit in its inputs and outputs and can thus be seen as a holistic measure of (Pareto-Koopmans) efficiency. SBM weighs input and output slacks in the objective function with the respective inputs used and outputs produced by each DMU. SBM is units invariant, that is, the value of the aggregate inefficiency score of a DMU is independent of the units in which the inputs and outputs are measured, as long as these units are the same for every DMU. For each DMU, SBM returns an efficiency score that is bounded by 0 and 1. ## Returns If print.status=FALSE, returns a data frame with N rows and 1+N+S+M columns. Column 1 reports the inefficiency score of each DMU. Columns 2 to N+1 contain the lambda values (intensity variables) indicating the DMU(s) that serve as reference for each DMU. Columns 1+N+1 to 1+N+S report the optimal output slacks for each DMU. Columns 1+N+S+1 to 1+N+S+M report the optimal input slacks for each DMU. When the linear program of a DMU is infeasible, the row of this DMU in the data frame will have NA values. If print.status=TRUE, returns a list with two elements. The first element is the aforementioned data frame. The second element is a numeric vector indicating the solution status of the linear program (e.g. 0: solution found; 2: problem is infeasible; 5: problem needs scaling; etc. See [https://CRAN.R-project.org/package=lpSolveAPI](https://CRAN.R-project.org/package=lpSolveAPI)). ## Author(s) Andreas Diomedes Soteriades, andreassot10@yahoo.com ## References Tone K. (2001) A slacks-based measure of efficiency in data envelopment analysis. **European Journal of Operational Research**, 130 , 498--509 ## See Also `dea.gem`, `dea.fast` ## Examples ```r # Twelve DMUs, 2 inputs, 2 outputs # (see Table 1.5 in Cooper et al., 2007): base <- data.frame( y1= c(100,150,160,180,94,230,220,152,190,250,260,250), y2= c(90,50,55,72,66,90,88,80,100,100,147,120), x1= c(20,19,25,27,22,55,33,31,30,50,53,38), x2= c(151,131,160,168,158,255,235,206,244,268,306,284)) # Example 1: Get inefficiency scores, # lambdas and slacks for all DMUs: dea.sbm(base, noutput= 2, rts= 1) dea.sbm(base, noutput= 2, rts= 2) # Example 2: Same as above, but consider output y2 and input x1 as fixed: dea.sbm(base, noutput= 2, rts= 1, fixed= c(2,3)) dea.sbm(base, noutput= 2, rts= 2, fixed= c(2,3)) # Example 3: Impose an upper bound to all slacks: # results with bounds dea.sbm(base, noutput= 2, rts= 1, bound= base/12)$eff # removing bounds allows for larger inefficiencies: dea.sbm(base, noutput= 2, rts= 1, bound= NULL)$eff # check solution status of linear programs when y2 and x1 are fixed # and y1, x2 slacks are bounded: bound <- base fixed <- c(2,3) bound[fixed] <- 0 dea.sbm(base, noutput= 2, bound= bound, fixed= c(2, 3), rts= 1, print.status= TRUE)[[2]] dea.sbm(base, noutput= 2, bound= bound, fixed= c(2, 3), rts= 2, print.status= TRUE)[[2]] # Example 4: Get inefficiency scores for DMUs 11 and 12: bound <- base fixed <- c(2,3) bound[fixed] <- 0 dea.sbm(base, noutput= 2, bound= bound, fixed= c(2, 3), rts= 1, whichDMUs= c(11, 12))$eff dea.sbm(base, noutput= 2, bound= bound, fixed= c(2, 3), rts= 2, whichDMUs= c(11, 12))$eff ## The function is currently defined as function (base, noutput, fixed = NULL, rts = 2, bound = NULL, whichDMUs = NULL, print.status = FALSE) { s <- noutput m <- ncol(base) - s n <- nrow(base) ifelse(!is.null(whichDMUs), nn <- length(whichDMUs), nn <- n) if (is.null(whichDMUs)) { whichDMUs <- 1:n } re <- data.frame(matrix(0, nrow = nn, ncol = 1 + n + s + m)) names(re) <- c("eff", paste("lambda", 1:n, sep = ""), paste("slack.y", 1:s, sep = ""), paste("slack.x", 1:m, sep = "")) slacks <- diag(s + m) slacks[1:s, ] <- -slacks[1:s, ] type <- rep("=", s + m) if (!is.null(fixed)) { slacks[, fixed] <- 0 type[fixed[fixed <= s]] <- ">=" type[fixed[fixed > s]] <- "<=" } k <- 0 A.aux <- cbind(t(base), slacks) index.fixed.y <- fixed[which(fixed %in% 1:s)] index.fixed.x <- fixed[which(fixed %in% (s + 1):(s + m))] S <- s - length(index.fixed.y) M <- m - length(index.fixed.x) if (print.status == TRUE) { ifelse(!is.null(whichDMUs), solution.status <- rep(0, nn), solution.status <- rep(0, n)) } if (rts == 2 & is.null(bound)) { for (i in whichDMUs) { k <- k + 1 lpmodel <- make.lp(nrow = 0, ncol = 1 + n + s + m) A <- cbind(-t(base)[, i], A.aux) xt <- as.numeric((1/S) * (1/base[i, 1:s])) if (!is.null(fixed)) { xt[index.fixed.y] <- 0 } xt <- c(1, rep(0, n), xt, rep(0, m)) add.constraint(lpmodel, xt = xt, type = "=", rhs = 0) for (j in 1:(s + m)) { add.constraint(lpmodel, xt = A[j, ], type = type[j], rhs = 0) } add.constraint(lpmodel, xt = c(-1, rep(1, n), rep(0, s + m)), type = "=", rhs = 0) set.rhs(lpmodel, b = c(1, rep(0, s + m + 1))) obj <- as.numeric((-1/M) * (1/base[i, (s + 1):(s + m)])) if (!is.null(fixed)) { obj[index.fixed.x - s] <- 0 } obj <- c(1, rep(0, n + s), obj) set.objfn(lpmodel, obj = obj) x <- solve(lpmodel) if (print.status == TRUE) { solution.status[k] <- x } re[k, ] <- c(get.objective(lpmodel), tail(get.primal.solution(lpmodel), n + s + m)/get.primal.solution(lpmodel)[1 + 1 + s + m + 1 + 1]) if (x != 0) { re[k, ] <- rep(NA, ncol(re)) } } } if (rts == 2 & !is.null(bound)) { index <- which(colSums(bound) != 0) nrows <- length(index) A.bound <- matrix(0, nrow = nrows, ncol = n + s + m) kk <- 0 for (i in index) { kk <- kk + 1 A.bound[kk, n + index[kk]] <- 1 } A.bound <- cbind(0, A.bound) for (i in whichDMUs) { A.bound <- matrix(0, nrow = nrows, ncol = n + s + m) kk <- 0 for (j in index) { kk <- kk + 1 A.bound[kk, n + index[kk]] <- 1 } A.bound <- cbind(0, A.bound) k <- k + 1 lpmodel <- make.lp(nrow = 0, ncol = 1 + n + s + m) A <- cbind(-t(base)[, i], A.aux) A.bound[, 1] <- as.numeric(-bound[i, index]) A.bound[A.bound[, 1] == 0, ] <- 0 A <- rbind(A, A.bound) xt <- as.numeric((1/S) * (1/base[i, 1:s])) if (!is.null(fixed)) { xt[index.fixed.y] <- 0 } xt <- c(1, rep(0, n), xt, rep(0, m)) add.constraint(lpmodel, xt = xt, type = "=", rhs = 0) for (j in 1:(s + m)) { add.constraint(lpmodel, xt = A[j, ], type = type[j], rhs = 0) } add.constraint(lpmodel, xt = c(-1, rep(1, n), rep(0, s + m)), type = "=", rhs = 0) for (l in 1:kk) { add.constraint(lpmodel, xt = A[s + m + l, ], type = "<=", rhs = 0) } set.rhs(lpmodel, b = c(1, rep(0, s + m + 1 + l))) obj <- as.numeric((-1/M) * (1/base[i, (s + 1):(s + m)])) if (!is.null(fixed)) { obj[index.fixed.x - s] <- 0 } obj <- c(1, rep(0, n + s), obj) set.objfn(lpmodel, obj = obj) x <- solve(lpmodel) if (print.status == TRUE) { solution.status[k] <- x } re[k, ] <- c(get.objective(lpmodel), tail(get.primal.solution(lpmodel), n + s + m)/get.primal.solution(lpmodel)[1 + 1 + s + m + 1 + sum(colSums(bound) != 0) + 1]) if (x != 0) { re[k, ] <- rep(NA, ncol(re)) } } } if (rts == 1 & is.null(bound)) { for (i in whichDMUs) { k <- k + 1 lpmodel <- make.lp(nrow = 0, ncol = 1 + n + s + m) A <- cbind(-t(base)[, i], A.aux) xt <- as.numeric((1/S) * (1/base[i, 1:s])) if (!is.null(fixed)) { xt[index.fixed.y] <- 0 } xt <- c(1, rep(0, n), xt, rep(0, m)) add.constraint(lpmodel, xt = xt, type = "=", rhs = 0) for (j in 1:(s + m)) { add.constraint(lpmodel, xt = A[j, ], type = type[j], rhs = 0) } set.rhs(lpmodel, b = c(1, rep(0, s + m))) obj <- as.numeric((-1/M) * (1/base[i, (s + 1):(s + m)])) if (!is.null(fixed)) { obj[index.fixed.x - s] <- 0 } obj <- c(1, rep(0, n + s), obj) set.objfn(lpmodel, obj = obj) x <- solve(lpmodel) if (print.status == TRUE) { solution.status[k] <- x } re[k, ] <- c(get.objective(lpmodel), tail(get.primal.solution(lpmodel), n + s + m)/get.primal.solution(lpmodel)[1 + 1 + s + m + 1]) if (x != 0) { re[k, ] <- rep(NA, ncol(re)) } } } if (rts == 1 & !is.null(bound)) { index <- which(colSums(bound) != 0) nrows <- length(index) A.bound <- matrix(0, nrow = nrows, ncol = n + s + m) kk <- 0 for (i in index) { kk <- kk + 1 A.bound[kk, n + index[kk]] <- 1 } A.bound <- cbind(0, A.bound) for (i in whichDMUs) { k <- k + 1 lpmodel <- make.lp(nrow = 0, ncol = 1 + n + s + m) A <- cbind(-t(base)[, i], A.aux) A.bound[, 1] <- as.numeric(-bound[i, index]) A.bound[A.bound[, 1] == 0, ] <- 0 A <- rbind(A, A.bound) xt <- as.numeric((1/S) * (1/base[i, 1:s])) if (!is.null(fixed)) { xt[index.fixed.y] <- 0 } xt <- c(1, rep(0, n), xt, rep(0, m)) add.constraint(lpmodel, xt = xt, type = "=", rhs = 0) for (j in 1:(s + m)) { add.constraint(lpmodel, xt = A[j, ], type = type[j], rhs = 0) } for (l in 1:kk) { add.constraint(lpmodel, xt = A[s + m + l, ], type = "<=", rhs = 0) } set.rhs(lpmodel, b = c(1, rep(0, s + m + l))) obj <- as.numeric((-1/M) * (1/base[i, (s + 1):(s + m)])) if (!is.null(fixed)) { obj[index.fixed.x - s] <- 0 } obj <- c(1, rep(0, n + s), obj) set.objfn(lpmodel, obj = obj) x <- solve(lpmodel) if (print.status == TRUE) { solution.status[k] <- x } re[k, ] <- c(get.objective(lpmodel), tail(get.primal.solution(lpmodel), n + s + m)/get.primal.solution(lpmodel)[1 + 1 + s + m + sum(colSums(bound) != 0) + 1]) if (x != 0) { re[k, ] <- rep(NA, ncol(re)) } } } if (!is.null(fixed)) { re <- re[, -(1 + n + fixed)] } if (print.status == TRUE) { reList <- list() reList[[1]] <- re reList[[2]] <- solution.status names(reList[[2]]) <- whichDMUs re <- reList } return(re) } ```

dea.sbm function