, Daniel Gulde [aut]| Title: | Hyperbolic DEA Estimation |
|---|---|
| Description: | Implements Data Envelopment Analysis (DEA) with a hyperbolic orientation using a non-linear programming solver. It enables flexible estimations with weight restrictions, non-discretionary variables, and a generalized distance function. Additionally, it allows for the calculation of slacks and super-efficiency scores. The methods are detailed in Öttl et al. (2023), <doi:10.1016/j.dajour.2023.100343>. Furthermore, the package provides a non-linear profitability estimation built upon the DEA framework. |
| Authors: | Alexander Öttl [cre, aut]
, Daniel Gulde [aut] |
| Maintainer: | Alexander Öttl <[email protected]> |
| License: | MIT + file LICENSE |
| Version: | 1.0.2 |
| Built: | 2024-11-25 15:14:27 UTC |
| Source: | CRAN |
Cost DEA model optimizing the input allocation with given prices. It returns the estimated lambdas as well as the optimal values for inputs and a cost efficiency score that is the ratio of optimal costs over observed costs.
costDEA(X, Y, pX, RTS = "crs")costDEA(X, Y, pX, RTS = "crs")
X |
Vector, matrix or dataframe with DMUs as rows and inputs as columns |
Y |
Vector, matrix or dataframe with DMUs as rows and outputs as columns |
pX |
Vector, matrix or dataframe with prices for each DMU and input. Therefore it must have the same dimensions as X. |
RTS |
Character string indicating the returns-to-scale, e.g. "crs", "vrs". |
A list object containing the following:
lambdas |
Estimated values for the composition of the respective Benchmarks. The lambdas are stored in a matrix with dimensions nrow(X) x nrow(X), where the row is the DMU under observation and the columns are the peers used for the Benchmark. |
opt_value |
Optimal inputs. |
cost_eff |
Cost efficiency as the ratio of the optimal cost to the observed cost. |
Alexander Öttl
[Benchmarking::cost.opt] for a similar function
X <- matrix(c(1,2,3,3,2,1,2,2), ncol = 2) Y <- matrix(c(1,1,1,1), ncol = 1) pX <- matrix(c(2,1,2,1,2,1,1,2), ncol = 2, byrow = TRUE) cost_eff_input <- costDEA(X,Y,pX)X <- matrix(c(1,2,3,3,2,1,2,2), ncol = 2) Y <- matrix(c(1,1,1,1), ncol = 1) pX <- matrix(c(2,1,2,1,2,1,1,2), ncol = 2, byrow = TRUE) cost_eff_input <- costDEA(X,Y,pX)
Hyperbolic DEA implementation including weight restrictions, non-discretionary variables, gerenralized distance function, external referencing, estimation of slacks and super-efficiency scores. The mathematical and theoretical foundations of the code are presented in the paper "Data Envelopment Analysis and Hyperbolic Efficiency Measures: Extending Applications and Possiblities for Between-Group Comparisons" (2023) by Alexander Öttl, Mette Asmild, and Daniel Gulde.
hyperbolicDEA( X, Y, RTS = "vrs", WR = NULL, SLACK = FALSE, ACCURACY = 1e-10, XREF = NULL, YREF = NULL, SUPEREFF = FALSE, NONDISC_IN = NULL, NONDISC_OUT = NULL, PARALLEL = 1, ALPHA = 0.5 )hyperbolicDEA( X, Y, RTS = "vrs", WR = NULL, SLACK = FALSE, ACCURACY = 1e-10, XREF = NULL, YREF = NULL, SUPEREFF = FALSE, NONDISC_IN = NULL, NONDISC_OUT = NULL, PARALLEL = 1, ALPHA = 0.5 )
X |
Vector, matrix or dataframe with DMUs as rows and inputs as columns |
Y |
Vecotr, matrix or dataframe with DMUs as rows and outputs as columns |
RTS |
Character string indicating the returns-to-scale, e.g. "crs", "vrs", "ndrs", "nirs", "fdh" |
WR |
Matrix with one row per homogeneous linear weight restriction in standard form. The columns are ncol(WR) = ncol(Y) + ncol(X). Hence the first ncol(Y) columns are the restrictions on outputs and the last ncol(X) columns are the restrictions on inputs. |
SLACK |
Boolean variable indicating whether an additional estimation of slacks shall be performed when set to 'TRUE'. Be aware that SLACK estimation can change the lambda values. |
ACCURACY |
Accuracy value for non-linear programm solver. |
XREF |
Vector, matrix or dataframe with firms defining the technology as rows and inputs as columns |
YREF |
Vector, matrix or dataframe with firms defining the technology as rows and outputs as columns |
SUPEREFF |
Boolean variable indicating whether super-efficiencies shall be estimated |
NONDISC_IN |
Vector containing column indices of the input matrix that are non-discretionary variables e.g. c(1,3) so the first and the third input are non-discretionary |
NONDISC_OUT |
Vector containing column indices of the output matrix that are non-discretionary variables e.g. c(1,3) so the first and the third output are non-discretionary |
PARALLEL |
Integer of amount of cores that should be used for parallel computing (Check availability of computer) |
ALPHA |
ALPHA can be chosen between [0,1]. It indicates the relative weights given to the distance function to both outputs and inputs when approaching the frontier. More weight on the input orientation is set by alpha < 0.5. Here, the input efficiency score is estimated in the package. To receive the corresponding output efficiency score, estimate: e^((1-alpha)/alpha). Vice versa for an output weighted model alpha > 0.5. The output efficiency is given and the input efficiency can be recovered with: e^(alpha/(1-alpha)) |
A list object containing the following information:
eff |
Are the estimated efficiency scores for the DMUs under observation stored in a vector with the length nrow(X). |
lambdas |
Estimated values for the composition of the respective Benchmarks. The lambdas are stored in a matrix with the dimensions nrow(X) x nrow(X), where the row is the DMU under observation and the columns the peers used for the Benchmark. |
mus |
If WR != NULL, the estimated decision variables for the imposed weight restrictions are stored in a matrix with the dimensions nrow(X) x nrow(WR), where the rows are the DMUs and columns the weight restrictions. If the values are positive, the WR is binding for the respective DMU. |
slack |
If SLACK = TRUE, the slacks are estimated and stored in a matrix with the dimensions nrow(X) x (ncol(X) + ncol(Y)). Showing the Slack of each DMU (row) for each input and output (column). |
Alexander Öttl
Daniel Gulde
X <- c(1,1,2,4,1.5,2,4,3) Y <- c(1,2,4,4,0.5,2.5,3.5,4) # we now impose linked weght restrictions. We assume outputs decrease by # four units when inputs are reduced by one. And we assume that outputs can # can be increased by one when inputs are increased by four WR <- matrix(c(-4,-1,1,4), nrow = 2, byrow = TRUE) hyperbolicDEA(X,Y,RTS="vrs", WR = WR) # Another example having the same data but just estimate the results for DMU 1 # using XREF YREF and and a higher focus on inputs adjusting the ALPHA towards 0. # Additionally, slacks are estimated. hyperbolicDEA(X[1],Y[1],RTS="vrs", XREF = X, YREF = Y, WR = WR, ALPHA = 0.1, SLACK = TRUE)X <- c(1,1,2,4,1.5,2,4,3) Y <- c(1,2,4,4,0.5,2.5,3.5,4) # we now impose linked weght restrictions. We assume outputs decrease by # four units when inputs are reduced by one. And we assume that outputs can # can be increased by one when inputs are increased by four WR <- matrix(c(-4,-1,1,4), nrow = 2, byrow = TRUE) hyperbolicDEA(X,Y,RTS="vrs", WR = WR) # Another example having the same data but just estimate the results for DMU 1 # using XREF YREF and and a higher focus on inputs adjusting the ALPHA towards 0. # Additionally, slacks are estimated. hyperbolicDEA(X[1],Y[1],RTS="vrs", XREF = X, YREF = Y, WR = WR, ALPHA = 0.1, SLACK = TRUE)
Linear profit DEA model optimizing the difference of cost to revenue. It returns the estimated lambdas as well as the optimal values for inputs and outputs. The linear profit estimation does not account for scale differences and also has issues with negative profits. Therefore, it is recommended to use the non-linear profit model.
lprofitDEA(X, Y, pX, pY, RTS = "crs")lprofitDEA(X, Y, pX, pY, RTS = "crs")
X |
Vector, matrix or dataframe with DMUs as rows and inputs as columns |
Y |
Vector, matrix or dataframe with DMUs as rows and outputs as columns |
pX |
Vector, matrix or dataframe with prices for each DMU and input. Therefore it must have the same dimensions as X. |
pY |
Vector, matrix or dataframe with prices for each DMU and output. Therefore it must have the same dimensions as Y. |
RTS |
Character string indicating the returns-to-scale, e.g. "crs", "vrs". |
A list object containing the following:
lambdas |
Estimated values for the composition of the respective Benchmarks. The lambdas are stored in a matrix with dimensions nrow(X) x nrow(X), where the row is the DMU under observation and the columns are the peers used for the Benchmark. |
opt_value |
Optimal inputs and outputs. |
profit_eff |
Profit efficiency as the ratio of the differences between the observed and optimal difference of cost to revenue. |
Alexander Öttl
[Benchmarking::profit.opt] for a similar function
X <- matrix(c(1,2,3,3,2,1,2,2), ncol = 2) Y <- matrix(c(1,1,1,1), ncol = 1) pX <- matrix(c(2,1,2,1,2,1,1,2), ncol = 2, byrow = TRUE) pY <- matrix(c(1,1,1,1), ncol = 1) max_prof_lin<- lprofitDEA(X,Y,pX,pY)X <- matrix(c(1,2,3,3,2,1,2,2), ncol = 2) Y <- matrix(c(1,1,1,1), ncol = 1) pX <- matrix(c(2,1,2,1,2,1,1,2), ncol = 2, byrow = TRUE) pY <- matrix(c(1,1,1,1), ncol = 1) max_prof_lin<- lprofitDEA(X,Y,pX,pY)
This function implements a non-linear profit DEA model that optimizes the ratio of cost over revenue given the prices for a DMU. It returns the estimated lambdas, optimal values for inputs and outputs, and a profit efficiency score. The profit efficiency score is calculated as the square root of the ratio of the observed revenue-cost ratio to the optimal revenue-cost ratio.
nlprofitDEA(X, Y, pX, pY, RTS = "crs")nlprofitDEA(X, Y, pX, pY, RTS = "crs")
X |
Vector, matrix or dataframe with DMUs as rows and inputs as columns. |
Y |
Vector, matrix or dataframe with DMUs as rows and outputs as columns. |
pX |
Vector, matrix or dataframe with prices for each DMU and input. It must have the same dimensions as X. |
pY |
Vector, matrix or dataframe with prices for each DMU and output. It must have the same dimensions as Y. |
RTS |
Character string indicating the returns-to-scale, e.g. "crs", "vrs". |
A list object containing the following:
lambdas |
Estimated values for the composition of the respective Benchmarks. The lambdas are stored in a matrix with dimensions nrow(X) x nrow(X), where the row is the DMU under observation and the columns are the peers used for the Benchmark. |
opt_value |
Optimal inputs and outputs. |
profit_eff |
New profit efficiency score that accounts for simultaneous adjustments in inputs and outputs. |
Alexander Öttl
'deaprofitability()' function in the Julia package BenchmarkingEconomicEfficiency.jl.
X <- matrix(c(1,2,3,3,2,1,2,2), ncol = 2) Y <- matrix(c(1,1,1,1), ncol = 1) pX <- matrix(c(2,1,2,1,2,1,1,2), ncol = 2, byrow = TRUE) pY <- matrix(c(1,1,1,1), ncol = 1) max_prof_nolin <- nlprofitDEA(X,Y,pX,pY)X <- matrix(c(1,2,3,3,2,1,2,2), ncol = 2) Y <- matrix(c(1,1,1,1), ncol = 1) pX <- matrix(c(2,1,2,1,2,1,1,2), ncol = 2, byrow = TRUE) pY <- matrix(c(1,1,1,1), ncol = 1) max_prof_nolin <- nlprofitDEA(X,Y,pX,pY)
Linear DEA estimation including the possibility of trade-off weight restrictions, external referencing, and super-efficiency scores. Furthermore, in a second stage slacks can be estimated. The function returns efficiency scores and adjusted lambdas according to the imposed weight restrictions and slack estimation. Additionally, mus are returned if weight restrictions are imposed that highlight binding restrictions for DMUs and the absolute slack values if slack-based estimation is applied.
wrDEA( X, Y, ORIENTATION = "out", RTS = "vrs", WR = NULL, XREF = NULL, YREF = NULL, SUPEREFF = FALSE, SLACK = FALSE, ECONOMIC = FALSE )wrDEA( X, Y, ORIENTATION = "out", RTS = "vrs", WR = NULL, XREF = NULL, YREF = NULL, SUPEREFF = FALSE, SLACK = FALSE, ECONOMIC = FALSE )
X |
Vector, matrix or dataframe with DMUs as rows and inputs as columns |
Y |
Vector, matrix or dataframe with DMUs as rows and outputs as columns |
ORIENTATION |
Character string indicating the orientation of the DEA model, e.g. "in", "out" |
RTS |
Character string indicating the returns-to-scale, e.g. "crs", "vrs", "ndrs", "nirs", "fdh" |
WR |
Matrix with one row per homogeneous linear weight restriction in standard form. The columns are ncol(WR) = ncol(Y) + ncol(X). Hence the first ncol(Y) columns are the restrictions on outputs and the last ncol(X) columns are the restrictions on inputs. |
XREF |
Vector, matrix or dataframe with firms defining the technology as rows and inputs as columns |
YREF |
Vector, matrix or dataframe with firms defining the technology as rows and outputs as columns |
SUPEREFF |
Boolean variable indicating whether super-efficiencies shall be estimated |
SLACK |
Boolean variable indicating whether slack-based estimation should be applied |
ECONOMIC |
Boolean variable indicating whether economic efficiency (cost or revenue) is estimated using trade-off weight restrictions. Therefore mus can be positive and negative when set to TRUE and you have a multi-stage optimization where the first stage is the efficiency estimation and the second stage is the minimization of the mus. You have to use single trade-off, hence a 1 to 1 substitution of either inputs or outputs given by the focus on either cost or revenue efficiency. The data must be value data. |
A list object containing the following information:
eff |
Are the estimated efficiency scores for the DMUs under observation stored in a vector with the length nrow(X). |
lambdas |
Estimated values for the composition of the respective Benchmarks. The lambdas are stored in a matrix with the dimensions nrow(X) x nrow(X), where the row is the DMU under observation and the columns the peers used for the Benchmark. NOTE: Lambdas are automatically slack optimized. |
mus |
If WR != NULL, the estimated decision variables for the imposed weight restrictions are stored in a matrix with the dimensions nrow(X) x nrow(WR), where the rows are the DMUs and columns the weight restrictions. If the values are positive, the WR is binding for the respective DMU. |
slack |
If SLACK = TRUE, the slacks are estimated and stored in a matrix with the dimensions nrow(X) x (ncol(X) + ncol(Y)). Showing the Slack of each DMU (row) for each input and output (column). |
Alexander Öttl
X <- c(1,1,2,4,1.5,2,4,3) Y <- c(1,2,4,4,0.5,2.5,3.5,4) # Two weight restrictions in standard form first on output then input. # The first WR shows the trade-off that inputs can be reduced by one unit # which reduces outputs by four units. The second WR shows that outputs can # be increased by one unit when inputs are increased by four units. WR <- matrix(c(-4,-1,1,4), nrow = 2, byrow = TRUE) wrDEA(X, Y, ORIENTATION = "in", RTS="vrs", WR = WR) # For an estimation just focusing on one DMU one can for example use # XREF and YREF to define the technology and then estimate the efficiency for # the DMU under observation (here DMU 1). Let's additionally estimate the slacks. wrDEA(X[1], Y[1], ORIENTATION = "in", RTS="vrs", XREF = X, YREF = Y, SLACK = TRUE, WR = WR)X <- c(1,1,2,4,1.5,2,4,3) Y <- c(1,2,4,4,0.5,2.5,3.5,4) # Two weight restrictions in standard form first on output then input. # The first WR shows the trade-off that inputs can be reduced by one unit # which reduces outputs by four units. The second WR shows that outputs can # be increased by one unit when inputs are increased by four units. WR <- matrix(c(-4,-1,1,4), nrow = 2, byrow = TRUE) wrDEA(X, Y, ORIENTATION = "in", RTS="vrs", WR = WR) # For an estimation just focusing on one DMU one can for example use # XREF and YREF to define the technology and then estimate the efficiency for # the DMU under observation (here DMU 1). Let's additionally estimate the slacks. wrDEA(X[1], Y[1], ORIENTATION = "in", RTS="vrs", XREF = X, YREF = Y, SLACK = TRUE, WR = WR)