https://github.com/javierbarbero/DataEnvelopmentAnalysis.jl
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[{"location":"economic/profitability/#","page":"Profitability Models","title":"Profitability Models","text":"CurrentModule = DataEnvelopmentAnalysis\nDocTestSetup = quote\n using DataEnvelopmentAnalysis\n # Solve nonlinear problem to display Ipopt initial message\n X = [1; 2; 3];\n Y = [1; 1; 1];\n deagdf(X, Y, 0.5, rts = :VRS)\nend","category":"page"},{"location":"economic/profitability/#Profitability-Models-1","page":"Profitability Models","title":"Profitability Models","text":"","category":"section"},{"location":"economic/profitability/#Profitability-Model-1","page":"Profitability Models","title":"Profitability Model","text":"","category":"section"},{"location":"economic/profitability/#","page":"Profitability Models","title":"Profitability Models","text":"The profitabilty function defines as mathrmPleft(mathbfwmathbfpright)=max Big sumlimits_i=1^sp_iy_isumlimits_i=1^mw_ix_i mathbfx geqslant Xmathbflambdamathbfy leqslant Ymathbflambda lambda geqslant mathbf0 Big. Zofío and Prieto (2006) introduced the following program that allows calculating profitability efficiency.","category":"page"},{"location":"economic/profitability/#","page":"Profitability Models","title":"Profitability Models","text":"beginalign\nlabeleqmaxprofit\n undersetmathbfxylambda_jomega mathopmin quad quad quad omega \n textsubject textto nonumber \n quad quad quad quad quad sum_j=1^j lambda^j fracw^j x^jp^j y^j = omega fracw^j x^j_op^j y^j_o nonumber \n quad quad quad quad quad sumnolimits_j=1^nlambda^j=1 nonumber\n\nnonumber \n quad quad quad quad quad mathbflambda ge mathbf0 nonumber \nendalign","category":"page"},{"location":"economic/profitability/#","page":"Profitability Models","title":"Profitability Models","text":"Profitabilty efficiency defines as the ratio between maximum profitabilty and observed profitabilty. Following the duality results introduced by Zofío and Prieto (2006) it is possible to decompose it into technical and allocative efficiencies under constant returns to scale. Profitabilty efficiency can be then decomposed into the generalizaed distance fucntion and the residual ratio corresponding to the allocative profit efficiency. Allocative efficiency defines then as the ratio of profitability at the technically efficient projection on the frontier to maximum profitability. ","category":"page"},{"location":"economic/profitability/#","page":"Profitability Models","title":"Profitability Models","text":"In this example we compute the profitability efficiency measure:","category":"page"},{"location":"economic/profitability/#","page":"Profitability Models","title":"Profitability Models","text":"julia> X = [5 3; 2 4; 4 2; 4 8; 7 9.0];\n\njulia> Y = [7 4; 10 8; 8 10; 5 4; 3 6.0];\n\njulia> W = [2 1; 2 1; 2 1; 2 1; 2 1.0];\n\njulia> P = [3 2; 3 2; 3 2; 3 2; 3 2.0];\n\njulia> deaprofitability(X, Y, W, P)\nProfitability DEA Model\nDMUs = 5; Inputs = 2; Outputs = 2\nalpha = 0.5; Returns to Scale = VRS\n─────────────────────────────────────────────────────────\n Profitability CRS VRS Scale Allocative\n─────────────────────────────────────────────────────────\n1 0.38796 0.636364 0.68185 0.93329 0.609651\n2 1.0 1.0 1.0 1.0 1.0\n3 0.765217 1.0 1.0 1.0 0.765217\n4 0.25 0.25 0.25 1.0 1.0\n5 0.15879 0.26087 0.36 0.724638 0.608696\n─────────────────────────────────────────────────────────","category":"page"},{"location":"economic/profitability/#deaprofitability-Function-Documentation-1","page":"Profitability Models","title":"deaprofitability Function Documentation","text":"","category":"section"},{"location":"economic/profitability/#","page":"Profitability Models","title":"Profitability Models","text":"deaprofitability","category":"page"},{"location":"economic/profitability/#DataEnvelopmentAnalysis.deaprofitability","page":"Profitability Models","title":"DataEnvelopmentAnalysis.deaprofitability","text":"deaprofitability(X, Y, W, P)\n\nCompute profitability efficiency using data envelopment analysis for inputs X, outputs Y, price of inputs W, and price of outputs P.\n\nOptional Arguments\n\nalpha=0.5: alpha to use for the generalized distance function.\n\nExamples\n\njulia> X = [5 3; 2 4; 4 2; 4 8; 7 9.0];\n\njulia> Y = [7 4; 10 8; 8 10; 5 4; 3 6.0];\n\njulia> W = [2 1; 2 1; 2 1; 2 1; 2 1.0];\n\njulia> P = [3 2; 3 2; 3 2; 3 2; 3 2.0];\n\njulia> deaprofitability(X, Y, W, P)\nProfitability DEA Model\nDMUs = 5; Inputs = 2; Outputs = 2\nalpha = 0.5; Returns to Scale = VRS\n─────────────────────────────────────────────────────────\n Profitability CRS VRS Scale Allocative\n─────────────────────────────────────────────────────────\n1 0.38796 0.636364 0.68185 0.93329 0.609651\n2 1.0 1.0 1.0 1.0 1.0\n3 0.765217 1.0 1.0 1.0 0.765217\n4 0.25 0.25 0.25 1.0 1.0\n5 0.15879 0.26087 0.36 0.724638 0.608696\n─────────────────────────────────────────────────────────\n\n\n\n\n\n","category":"function"},{"location":"technical/generalizeddf/#","page":"Generalized Distance Function Models","title":"Generalized Distance Function Models","text":"CurrentModule = DataEnvelopmentAnalysis\nDocTestSetup = quote\n using DataEnvelopmentAnalysis\n # Solve nonlinear problem to display Ipopt initial message\n X = [1; 2; 3];\n Y = [1; 1; 1];\n deagdf(X, Y, 0.5, rts = :VRS)\nend","category":"page"},{"location":"technical/generalizeddf/#Generalized-Distance-Function-Models-1","page":"Generalized Distance Function Models","title":"Generalized Distance Function Models","text":"","category":"section"},{"location":"technical/generalizeddf/#Generalized-Distance-Function-Model-1","page":"Generalized Distance Function Models","title":"Generalized Distance Function Model","text":"","category":"section"},{"location":"technical/generalizeddf/#","page":"Generalized Distance Function Models","title":"Generalized Distance Function Models","text":"Chavas and Cox (1999) introduced a generalized distance function efficiency measure that reescales both inputs and outputs toward the frontier technology.","category":"page"},{"location":"technical/generalizeddf/#","page":"Generalized Distance Function Models","title":"Generalized Distance Function Models","text":"beginalign\nlabeleqrim\n undersetdelta mathbflambda mathopmin quad quad quad delta \n textsubject textto nonumber \n quad quad quad quad quad Xmathbflambda le delta^1 - alpha mathbfx_o nonumber \n quad quad quad quad quad Ymathbflambda ge mathbfy_o delta^alpha nonumber\n quad quad quad quad quad mathbflambda ge mathbf0 nonumber\nendalign","category":"page"},{"location":"technical/generalizeddf/#","page":"Generalized Distance Function Models","title":"Generalized Distance Function Models","text":"The measurement of technical efficiency assuming variable returns to scale, VRS, adds the following condition:","category":"page"},{"location":"technical/generalizeddf/#","page":"Generalized Distance Function Models","title":"Generalized Distance Function Models","text":"sumnolimits_j=1^nlambda_j=1","category":"page"},{"location":"technical/generalizeddf/#","page":"Generalized Distance Function Models","title":"Generalized Distance Function Models","text":"In this example we compute the generalized distance function DEA model under variable returns to scale using 05 for the value of alpha:","category":"page"},{"location":"technical/generalizeddf/#","page":"Generalized Distance Function Models","title":"Generalized Distance Function Models","text":"julia> X = [5 3; 2 4; 4 2; 4 8; 7 9];\n\njulia> Y = [7 4; 10 8; 8 10; 5 4; 3 6];\n\njulia> deagdf(X, Y, 0.5, rts = :VRS, slack = false)\nGeneralized DF DEA Model\nDMUs = 5; Inputs = 2; Outputs = 2\nalpha = 0.5; Returns to Scale = VRS\n─────────────\n efficiency\n─────────────\n1 0.68185\n2 1.0 \n3 1.0 \n4 0.25 \n5 0.36 \n─────────────","category":"page"},{"location":"technical/generalizeddf/#deagdf-Function-Documentation-1","page":"Generalized Distance Function Models","title":"deagdf Function Documentation","text":"","category":"section"},{"location":"technical/generalizeddf/#","page":"Generalized Distance Function Models","title":"Generalized Distance Function Models","text":"deagdf","category":"page"},{"location":"technical/generalizeddf/#DataEnvelopmentAnalysis.deagdf","page":"Generalized Distance Function Models","title":"DataEnvelopmentAnalysis.deagdf","text":"deagdf(X, Y, alpha)\n\nCompute generalized distance function data envelopment analysis model for inputs X, outputs Y, and alpha.\n\nOptional Arguments\n\nrts=:CRS: chooses constant returns to scale. For variable returns to scale choose :VRS.\nslack=true: compute input and output slacks.\nXref=X: Identifies the reference set of inputs against which the units are evaluated.\nYref=Y: Identifies the reference set of outputs against which the units are evaluated.\n\nExamples\n\njulia> X = [5 3; 2 4; 4 2; 4 8; 7 9];\n\njulia> Y = [7 4; 10 8; 8 10; 5 4; 3 6];\n\njulia> deagdf(X, Y, 0.5, rts = :VRS, slack = false)\nGeneralized DF DEA Model\nDMUs = 5; Inputs = 2; Outputs = 2\nalpha = 0.5; Returns to Scale = VRS\n─────────────\n efficiency\n─────────────\n1 0.68185\n2 1.0\n3 1.0\n4 0.25\n5 0.36\n─────────────\n\n\n\n\n\n","category":"function"},{"location":"economic/profit/#","page":"Profit Models","title":"Profit Models","text":"CurrentModule = DataEnvelopmentAnalysis\nDocTestSetup = quote\n using DataEnvelopmentAnalysis\nend","category":"page"},{"location":"economic/profit/#Profit-Models-1","page":"Profit Models","title":"Profit Models","text":"","category":"section"},{"location":"economic/profit/#Profit-Efficiency-Model-with-Directional-Distance-Function-Technical-Efficiency-1","page":"Profit Models","title":"Profit Efficiency Model with Directional Distance Function Technical Efficiency","text":"","category":"section"},{"location":"economic/profit/#","page":"Profit Models","title":"Profit Models","text":"The profit function defines as Pileft(mathbfwmathbfpright)=max Big sumlimits_i=1^sp_iy_i-sumlimits_i=1^mw_ix_i mathbfx geqslant Xmathbflambdamathbfy leqslant Ymathbflambda mathbfmathbfelambda=1 lambda geqslant mathbf0 Big. Calculating maximum profit along with the optimal output and input quantities mathbfy^*and mathbfx^* requires solving: ","category":"page"},{"location":"economic/profit/#","page":"Profit Models","title":"Profit Models","text":"beginalign\nlabeleqmaxprofit\n undersetmathbfxylambda mathopmax quad quad quad Pileft(mathbfwmathbfpright)=mathbfpy^*-wx^* \n textsubject textto nonumber \n quad quad quad quad quad mathbfxge Xmathbflambda=x nonumber \n quad quad quad quad quad mathbfy le Ymathbflambda =y nonumber\n quad quad quad quad quad mathbfelambda=1\nnonumber \n quad quad quad quad quad mathbflambda ge mathbf0 nonumber \nendalign","category":"page"},{"location":"economic/profit/#","page":"Profit Models","title":"Profit Models","text":"Profit efficiency defines as the difference between maximum profit and observed profit. Following the duality results introduced by Chambers, Chung and Färe (1998) it is possible to decompose it into technical and allocative efficiencies under variable returns to scale. Profit efficiency can be then decomposed into the directional distance fucntion and the residual difference corresponding to the allocative profit efficiency. Allocative efficiency defines then as the difference between maximum profit and profit at the technically efficient projection on the frontier. The approach relies on the directional vector to normalize these components, thereby ensuring that their values can be compared across DMUs. ","category":"page"},{"location":"economic/profit/#","page":"Profit Models","title":"Profit Models","text":"In this example we compute the profit efficiency measure under variable returns to scale:","category":"page"},{"location":"economic/profit/#","page":"Profit Models","title":"Profit Models","text":"julia> X = [1 1; 1 1; 0.75 1.5; 0.5 2; 0.5 2; 2 2; 2.75 3.5; 1.375 1.75];\n\njulia> Y = [1 11; 5 3; 5 5; 2 9; 4 5; 4 2; 3 3; 4.5 3.5];\n\njulia> P = [2 1; 2 1; 2 1; 2 1; 2 1; 2 1; 2 1; 2 1];\n\njulia> W = [2 1; 2 1; 2 1; 2 1; 2 1; 2 1; 2 1; 2 1];\n\njulia> GxGydollar = 1 ./ (sum(P, dims = 2) + sum(W, dims = 2));\n\njulia> Gx = repeat(GxGydollar, 1, 2);\n\njulia> Gy = repeat(GxGydollar, 1, 2);\n\njulia> deaprofit(X, Y, W, P, Gx, Gy)\nProfit DEA Model \nDMUs = 8; Inputs = 2; Outputs = 2\nReturns to Scale = VRS\n─────────────────────────────────────\n Profit Technical Allocative\n─────────────────────────────────────\n1 2.0 0.0 2.0 \n2 2.0 -5.41234e-16 2.0 \n3 0.0 0.0 0.0 \n4 2.0 0.0 2.0 \n5 2.0 0.0 2.0 \n6 8.0 6.0 2.0 \n7 12.0 12.0 -1.77636e-15\n8 4.0 3.0 1.0 \n─────────────────────────────────────","category":"page"},{"location":"economic/profit/#deaprofit-Function-Documentation-1","page":"Profit Models","title":"deaprofit Function Documentation","text":"","category":"section"},{"location":"economic/profit/#","page":"Profit Models","title":"Profit Models","text":"deaprofit","category":"page"},{"location":"economic/profit/#DataEnvelopmentAnalysis.deaprofit","page":"Profit Models","title":"DataEnvelopmentAnalysis.deaprofit","text":"deaprofit(X, Y, W, P)\n\nCompute profit efficiency using data envelopment analysis model for inputs X, outputs Y, price of inputs W, and price of outputs P.\n\nExamples\n\njulia> X = [1 1; 1 1; 0.75 1.5; 0.5 2; 0.5 2; 2 2; 2.75 3.5; 1.375 1.75];\n\njulia> Y = [1 11; 5 3; 5 5; 2 9; 4 5; 4 2; 3 3; 4.5 3.5];\n\njulia> P = [2 1; 2 1; 2 1; 2 1; 2 1; 2 1; 2 1; 2 1];\n\njulia> W = [2 1; 2 1; 2 1; 2 1; 2 1; 2 1; 2 1; 2 1];\n\njulia> GxGydollar = 1 ./ (sum(P, dims = 2) + sum(W, dims = 2));\n\njulia> Gx = repeat(GxGydollar, 1, 2);\n\njulia> Gy = repeat(GxGydollar, 1, 2);\n\njulia> deaprofit(X, Y, W, P, Gx, Gy)\nProfit DEA Model\nDMUs = 8; Inputs = 2; Outputs = 2\nReturns to Scale = VRS\n─────────────────────────────────────\n Profit Technical Allocative\n─────────────────────────────────────\n1 2.0 0.0 2.0\n2 2.0 -5.41234e-16 2.0\n3 0.0 0.0 0.0\n4 2.0 0.0 2.0\n5 2.0 0.0 2.0\n6 8.0 6.0 2.0\n7 12.0 12.0 -1.77636e-15\n8 4.0 3.0 1.0\n─────────────────────────────────────\n\n\n\n\n\n","category":"function"},{"location":"economic/revenue/#","page":"Revenue Models","title":"Revenue Models","text":"CurrentModule = DataEnvelopmentAnalysis\nDocTestSetup = quote\n using DataEnvelopmentAnalysis\nend","category":"page"},{"location":"economic/revenue/#Revenue-Models-1","page":"Revenue Models","title":"Revenue Models","text":"","category":"section"},{"location":"economic/revenue/#Revenue-Efficiency-Model-with-Radial-Technical-Efficiency-1","page":"Revenue Models","title":"Revenue Efficiency Model with Radial Technical Efficiency","text":"","category":"section"},{"location":"economic/revenue/#","page":"Revenue Models","title":"Revenue Models","text":"Let us denote by Rleft(mathbfxmathbfpright) the maximum feasible revenue using inputs' levels mathbfx and given the outputs' prices mathbfp: Rleft(mathbfxmathbfpright)=max left sumlimits_i=1^sp_iy_i mathbfx_o geqslant Xmathbflambdamathbfy leqslant Ymathbflambda mathbflambda geqslant mathbf0 right; i.e., considering the output possibility set producible with mathbfx_o. In this case, we calculate maximum revenue along with the optimal output quantities mathbfy^* by solving the following program:","category":"page"},{"location":"economic/revenue/#","page":"Revenue Models","title":"Revenue Models","text":"beginalign\nlabeleqmaxrev\n undersetmathbfy mathbflambda mathopmax quad quad quad Rleft(mathbfx_omathbfpright)=mathbfpy^* \n textsubject textto nonumber \n quad quad quad quad quad mathbfx_oge Xmathbflambda nonumber \n quad quad quad quad quad Ymathbflambda ge mathbfy nonumber \n quad quad quad quad quad mathbflambda ge mathbf0 nonumber \nendalign","category":"page"},{"location":"economic/revenue/#","page":"Revenue Models","title":"Revenue Models","text":"The measurement of revenue efficiency assuming variable returns to scale, VRS, adds the following condition:","category":"page"},{"location":"economic/revenue/#","page":"Revenue Models","title":"Revenue Models","text":"sumnolimits_j=1^nlambda_j=1","category":"page"},{"location":"economic/revenue/#","page":"Revenue Models","title":"Revenue Models","text":"Revenue efficiency defines as the ratio of observed revenue to maximum revenue: RE=mathbfpy_oRleft(mathbfxmathbfpright) Duality results presented in *Shephard (1953)* from an output perspective allow us to decompose RE into the output oriented technical efficiency measure and the residual difference corresponding to the allocative revenue efficiency. Allocative efficiency defines as the ratio of revenue at the technically efficient projection of the observation to maximum revenue.","category":"page"},{"location":"economic/revenue/#","page":"Revenue Models","title":"Revenue Models","text":"In this example we compute the revnue efficiency measure under variable returns to scale:","category":"page"},{"location":"economic/revenue/#","page":"Revenue Models","title":"Revenue Models","text":"julia> X = [5 3; 2 4; 4 2; 4 8; 7 9.0];\n\njulia> Y = [7 4; 10 8; 8 10; 5 4; 3 6.0];\n\njulia> P = [3 2; 3 2; 3 2; 3 2; 3 2.0];\n\njulia> dearevenue(X, Y, P)\nRevenue DEA Model\nDMUs = 5; Inputs = 2; Outputs = 2\nOrientation = Output; Returns to Scale = VRS\n──────────────────────────────────\n Revenue Technical Allocative\n──────────────────────────────────\n1 0.644444 0.777778 0.828571\n2 1.0 1.0 1.0\n3 1.0 1.0 1.0\n4 0.5 0.5 1.0\n5 0.456522 0.6 0.76087\n──────────────────────────────────","category":"page"},{"location":"economic/revenue/#dearevenue-Function-Documentation-1","page":"Revenue Models","title":"dearevenue Function Documentation","text":"","category":"section"},{"location":"economic/revenue/#","page":"Revenue Models","title":"Revenue Models","text":"dearevenue","category":"page"},{"location":"economic/revenue/#DataEnvelopmentAnalysis.dearevenue","page":"Revenue Models","title":"DataEnvelopmentAnalysis.dearevenue","text":"dearevenue(X, Y, P)\n\nCompute revenue efficiency using data envelopment analysis for inputs X, outputs Y and price of outputs P.\n\nOptional Arguments\n\nrts=:VRS: chooses variable returns to scale. For constant returns to scale choose :CRS.\ndisposal=:Strong: chooses strong disposal of inputs. For weak disposal choose :Weak.\n\nExamples\n\njulia> X = [5 3; 2 4; 4 2; 4 8; 7 9.0];\n\njulia> Y = [7 4; 10 8; 8 10; 5 4; 3 6.0];\n\njulia> P = [3 2; 3 2; 3 2; 3 2; 3 2.0];\n\njulia> dearevenue(X, Y, P)\nRevenue DEA Model\nDMUs = 5; Inputs = 2; Outputs = 2\nOrientation = Output; Returns to Scale = VRS\n──────────────────────────────────\n Revenue Technical Allocative\n──────────────────────────────────\n1 0.644444 0.777778 0.828571\n2 1.0 1.0 1.0\n3 1.0 1.0 1.0\n4 0.5 0.5 1.0\n5 0.456522 0.6 0.76087\n──────────────────────────────────\n\n\n\n\n\n","category":"function"},{"location":"productivity/malmquist/#","page":"Malmquist Index","title":"Malmquist Index","text":"CurrentModule = DataEnvelopmentAnalysis\nDocTestSetup = quote\n using DataEnvelopmentAnalysis\nend","category":"page"},{"location":"productivity/malmquist/#The-Malmquist-index-1","page":"Malmquist Index","title":"The Malmquist index","text":"","category":"section"},{"location":"productivity/malmquist/#The-Malmquist-Productivity-Index-1","page":"Malmquist Index","title":"The Malmquist Productivity Index","text":"","category":"section"},{"location":"productivity/malmquist/#","page":"Malmquist Index","title":"Malmquist Index","text":"The Malmquist index introduced by Caves, Christensen and Diewert(1982) measures the change in productivity of the observation under evaluation by comparing its relative performance with respect to reference technologies corresponding to two different time periods.","category":"page"},{"location":"productivity/malmquist/#","page":"Malmquist Index","title":"Malmquist Index","text":"Following Fare, Grosskopf, Norris and Zhang (1994) productivity change can be decomposed into efficiency change and technical change under the assumption of a constant returns to scale techncology.","category":"page"},{"location":"productivity/malmquist/#","page":"Malmquist Index","title":"Malmquist Index","text":"In this example we compute the Malmquist productivity index:","category":"page"},{"location":"productivity/malmquist/#","page":"Malmquist Index","title":"Malmquist Index","text":"julia> X = Array{Float64,3}(undef, 5, 1, 2);\n\njulia> X[:, :, 1] = [2; 3; 5; 4; 4];\n\njulia> X[:, :, 2] = [1; 2; 4; 3; 4];\n\njulia> Y = Array{Float64,3}(undef, 5, 1, 2);\n\njulia> Y[:, :, 1] = [1; 4; 6; 3; 5];\n\njulia> Y[:, :, 2] = [1; 4; 6; 3; 3];\n\njulia> malmquist(X, Y)\nMamlmquist DEA Model\nDMUs = 5; Inputs = 1; Outputs = 1; Time periods = 2\nOrientation = Input; Returns to Scale = CRS\nReferene period = Geomean\n─────────────────────────\n M EC TC\n─────────────────────────\n1 2.0 1.33333 1.5\n2 1.5 1.0 1.5\n3 1.25 0.833333 1.5\n4 1.33333 0.888889 1.5\n5 0.6 0.4 1.5\n─────────────────────────\nM = Malmquist Productivity Index\nEC = Efficiency Change\nTC = Technological Change","category":"page"},{"location":"productivity/malmquist/#malmquist-Function-Documentation-1","page":"Malmquist Index","title":"malmquist Function Documentation","text":"","category":"section"},{"location":"productivity/malmquist/#","page":"Malmquist Index","title":"Malmquist Index","text":"malmquist","category":"page"},{"location":"productivity/malmquist/#DataEnvelopmentAnalysis.malmquist","page":"Malmquist Index","title":"DataEnvelopmentAnalysis.malmquist","text":"malmquist(X, Y)\n\nCompute the Malmquist productivity index using data envelopment analysis for inputs X and outputs Y.\n\nOptional Arguments\n\norient=:Input: chooses between input oriented radial model :Input or output oriented radial model :Output.\nrefperiod=:Geomean: chooses reference period for technological change: :Base, :Comparison or :Geomean.\nrts=:CRS: chooses constant returns to scale. For variable returns to scale choose :VRS.\n\nExamples\n\njulia> X = Array{Float64,3}(undef, 5, 1, 2);\n\njulia> X[:, :, 1] = [2; 3; 5; 4; 4];\n\njulia> X[:, :, 2] = [1; 2; 4; 3; 4];\n\njulia> Y = Array{Float64,3}(undef, 5, 1, 2);\n\njulia> Y[:, :, 1] = [1; 4; 6; 3; 5];\n\njulia> Y[:, :, 2] = [1; 4; 6; 3; 3];\n\njulia> malmquist(X, Y)\nMamlmquist DEA Model\nDMUs = 5; Inputs = 1; Outputs = 1; Time periods = 2\nOrientation = Input; Returns to Scale = CRS\nReferene period = Geomean\n─────────────────────────\n M EC TC\n─────────────────────────\n1 2.0 1.33333 1.5\n2 1.5 1.0 1.5\n3 1.25 0.833333 1.5\n4 1.33333 0.888889 1.5\n5 0.6 0.4 1.5\n─────────────────────────\nM = Malmquist Productivity Index\nEC = Efficiency Change\nTC = Technological Change\n\n\n\n\n\n","category":"function"},{"location":"technical/directional/#","page":"Directional Distance Function Models","title":"Directional Distance Function Models","text":"CurrentModule = DataEnvelopmentAnalysis\nDocTestSetup = quote\n using DataEnvelopmentAnalysis\nend","category":"page"},{"location":"technical/directional/#Directional-Distance-Function-Models-1","page":"Directional Distance Function Models","title":"Directional Distance Function Models","text":"","category":"section"},{"location":"technical/directional/#Directional-Distance-Function-Model-1","page":"Directional Distance Function Models","title":"Directional Distance Function Model","text":"","category":"section"},{"location":"technical/directional/#","page":"Directional Distance Function Models","title":"Directional Distance Function Models","text":"Chambers, Chung and Fare (1996) introduced a measure of efficiency that projects observation left( mathbfx_omathbfy_o right) in a pre-assigned direction mathbfg= left(-mathbfg_x^-mathbfg^+_y right)neqmathbf0_m+s, mathbfg^-_xmathbbin R^m and mathbfg^+_ymathbbin R^s, in a proportion beta. The associated linear program is:","category":"page"},{"location":"technical/directional/#","page":"Directional Distance Function Models","title":"Directional Distance Function Models","text":"beginalign\nlabeleqddf\n undersetbeta mathbflambda mathopmax quad quad quad quad beta \n textsubject textto nonumber\n quad quad quad quad quad Xlambdale mathbfx_o -betamathbfg^-_x nonumber\n quad quad quad quad quad Ymathbflambda ge mathbfy_o+beta mathbfg^+_y nonumber\n quad quad quad quad quad mathbflambda ge mathbf0nonumber quad nonumber\nendalign","category":"page"},{"location":"technical/directional/#","page":"Directional Distance Function Models","title":"Directional Distance Function Models","text":"The measurement of technical efficiency assuming variable returns to scale, VRS, adds the following condition:","category":"page"},{"location":"technical/directional/#","page":"Directional Distance Function Models","title":"Directional Distance Function Models","text":"sumnolimits_j=1^nlambda_j=1","category":"page"},{"location":"technical/directional/#","page":"Directional Distance Function Models","title":"Directional Distance Function Models","text":"In this example we compute the directional distance function DEA model under constant returns to scale using ones as directions for both inputs and outputs:","category":"page"},{"location":"technical/directional/#","page":"Directional Distance Function Models","title":"Directional Distance Function Models","text":"julia> using DataEnvelopmentAnalysis\n\njulia> X = [5 13; 16 12; 16 26; 17 15; 18 14; 23 6; 25 10; 27 22; 37 14; 42 25; 5 17];\n\njulia> Y = [12; 14; 25; 26; 8; 9; 27; 30; 31; 26; 12];\n\njulia> deaddf(X, Y, ones(size(X)), ones(size(Y)))\nDirectional DF DEA Model \nDMUs = 11; Inputs = 2; Outputs = 1\nReturns to Scale = CRS\n─────────────────────────────────────────────────────\n efficiency slackX1 slackX2 slackY1\n─────────────────────────────────────────────────────\n1 -3.43053e-16 0.0 0.0 0.0\n2 3.21996 -3.21359e-15 0.0 0.0\n3 2.12169 0.0 -4.80367e-15 0.0\n4 0.0 -8.03397e-16 0.0 0.0\n5 6.73567 -2.41019e-15 0.0 0.0\n6 1.94595 10.9189 0.0 0.0\n7 0.0 0.0 0.0 0.0\n8 3.63586 6.42718e-15 0.0 0.0\n9 1.83784 4.75676 0.0 0.0\n10 10.2311 6.12173e-15 0.0 0.0\n11 0.0 0.0 4.0 0.0\n─────────────────────────────────────────────────────","category":"page"},{"location":"technical/directional/#","page":"Directional Distance Function Models","title":"Directional Distance Function Models","text":"To compute the variable returns to scale model, we simply set the rts parameter to :VRS:","category":"page"},{"location":"technical/directional/#","page":"Directional Distance Function Models","title":"Directional Distance Function Models","text":"julia> deaddf(X, Y, ones(size(X)), ones(size(Y)), rts = :VRS)\nDirectional DF DEA Model \nDMUs = 11; Inputs = 2; Outputs = 1\nReturns to Scale = VRS\n────────────────────────────────────────────────────\n efficiency slackX1 slackX2 slackY1\n────────────────────────────────────────────────────\n1 -3.43053e-16 0.0 0.0 0.0 \n2 1.41887 0.0 0.0 7.41268e-15\n3 0.0 0.0 0.0 0.0 \n4 0.0 -8.03397e-16 0.0 0.0 \n5 4.06792 0.0 0.0 0.0 \n6 -1.81673e-16 2.70127e-16 0.0 3.78178e-16\n7 0.0 0.0 0.0 0.0 \n8 0.0 0.0 0.0 0.0 \n9 0.0 0.0 0.0 0.0 \n10 5.0 0.0 6.0 0.0 \n11 0.0 0.0 4.0 4.78849e-16\n────────────────────────────────────────────────────","category":"page"},{"location":"technical/directional/#","page":"Directional Distance Function Models","title":"Directional Distance Function Models","text":"Estimated efficiency scores are returned with the efficiency function:","category":"page"},{"location":"technical/directional/#","page":"Directional Distance Function Models","title":"Directional Distance Function Models","text":"julia> deaddfvrs = deaddf(X, Y, ones(size(X)), ones(size(Y)), rts = :VRS);\n\njulia> efficiency(deaddfvrs)\n11-element Array{Float64,1}:\n -3.4305304041327586e-16\n 1.4188679245283022 \n 0.0 \n 0.0 \n 4.067924528301886 \n -1.816728585750256e-16 \n 0.0 \n 0.0 \n 0.0 \n 5.000000000000003 \n 0.0 ","category":"page"},{"location":"technical/directional/#","page":"Directional Distance Function Models","title":"Directional Distance Function Models","text":"The optimal peers, λ, are returned with the peers function and are returned as a SparseArrays.SparseMatrixCSC{Float64,Int64} object:","category":"page"},{"location":"technical/directional/#","page":"Directional Distance Function Models","title":"Directional Distance Function Models","text":"peers(deaddfvrs)","category":"page"},{"location":"technical/directional/#deaddf-Function-Documentation-1","page":"Directional Distance Function Models","title":"deaddf Function Documentation","text":"","category":"section"},{"location":"technical/directional/#","page":"Directional Distance Function Models","title":"Directional Distance Function Models","text":"deaddf","category":"page"},{"location":"technical/directional/#DataEnvelopmentAnalysis.deaddf","page":"Directional Distance Function Models","title":"DataEnvelopmentAnalysis.deaddf","text":"deaddf(X, Y, Gx, Gy)\n\nCompute data envelopment analysis directional distance function model for inputs X and outputs Y, using directions Gx and Gy.\n\nOptional Arguments\n\nrts=:CRS: chooses constant returns to scale. For variable returns to scale choose :VRS.\nslack=true: computes input and output slacks.\nXref=X: Identifies the reference set of inputs against which the units are evaluated.\nYref=Y: Identifies the reference set of outputs against which the units are evaluated.\n\nExamples\n\njulia> X = [5 13; 16 12; 16 26; 17 15; 18 14; 23 6; 25 10; 27 22; 37 14; 42 25; 5 17];\n\njulia> Y = [12; 14; 25; 26; 8; 9; 27; 30; 31; 26; 12];\n\njulia> deaddf(X, Y, ones(size(X)), ones(size(Y)))\nDirectional DF DEA Model\nDMUs = 11; Inputs = 2; Outputs = 1\nReturns to Scale = CRS\n────────────────\n efficiency\n────────────────\n1 -3.43053e-16\n2 3.21996\n3 2.12169\n4 0.0\n5 6.73567\n6 1.94595\n7 0.0\n8 3.63586\n9 1.83784\n10 10.2311\n11 0.0\n────────────────\n\n\n\n\n\n","category":"function"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"CurrentModule = DataEnvelopmentAnalysis\nDocTestSetup = quote\n using DataEnvelopmentAnalysis\nend","category":"page"},{"location":"technical/additive/#Additive-Models-1","page":"Additive Models","title":"Additive Models","text":"","category":"section"},{"location":"technical/additive/#Weighted-Additive-Model-1","page":"Additive Models","title":"Weighted Additive Model","text":"","category":"section"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"The additive model measures technical efficiency based solely on input excesses and output shortfalls, and characterizes efficiency in terms of the input and output slacks: mathbfs^-mathbbin R^m and mathbfs^+mathbbin R^s, respectively. . The package implements the weighted additive formulation of Cooper and Pastor (1995) and Pastor, Lovell and Aparicio (2011), whose associated linear program is:","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"beginalign\nlabeleqadd\n undersetmathbflambda mathbfs^-mathbfs^+mathopmax quad quad quad quad omega =mathbfrho_x^-mathbfs^mathbf-+mathbfrho_y^+mathbfs^+ \n textsubject textto nonumber\n quad quad quad quad quad quad Xmathbflambda +mathbfs^mathbf-= mathbfx_o nonumber\n quad quad quad quad quad quad Ymathbflambda -mathbfs^+= mathbfy_o nonumber\n quad quad quad quad quad quad mathbfelambda=1 nonumber\n quad quad quad quad quad quad mathbflambda ge mathbf0 mathbfs^mathbf-ge 0mathbfs^+ge 0 nonumber\nendalign","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"where (mathbfrho_x^- mathbfrho_y^+) mathbbin R^m_+times mathbbR_+^s are the inputs and outputs weight vectors whose elements can vary across DMUs.","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"In this example we compute the additive DEA model with all weights equal to one:","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"julia> using DataEnvelopmentAnalysis\n\njulia> X = [5 13; 16 12; 16 26; 17 15; 18 14; 23 6; 25 10; 27 22; 37 14; 42 25; 5 17];\n\njulia> Y = [12; 14; 25; 26; 8; 9; 27; 30; 31; 26; 12];\n\njulia> deaadd(X, Y)\nWeighted Additive DEA Model\nDMUs = 11; Inputs = 2; Outputs = 1\nWeights = Ones; Returns to Scale = VRS\n────────────────────────────────────────────────────\n efficiency slackX1 slackX2 slackY1\n────────────────────────────────────────────────────\n1 0.0 0.0 0.0 0.0\n2 7.33333 4.33333 0.0 3.0\n3 0.0 0.0 0.0 0.0\n4 -8.03397e-16 -8.03397e-16 0.0 0.0\n5 18.0 13.0 1.0 4.0\n6 6.48305e-16 2.70127e-16 0.0 3.78178e-16\n7 0.0 0.0 0.0 0.0\n8 0.0 0.0 0.0 0.0\n9 0.0 0.0 0.0 0.0\n10 35.0 25.0 10.0 0.0\n11 4.0 0.0 4.0 4.78849e-16\n────────────────────────────────────────────────────","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"The same model is computed with:","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"deaadd(X, Y, :Ones)","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"The additive DEA model can be computed under constant returns to scale setting the rts parameter to :CRS:","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"deaadd(X, Y, :Ones, rts = :CRS)","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"The package can compute a wide class of different DEA models known as general efficiency measures (GEMs):","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"The measure of inefficiency proportions (MIP).\nThe normalized weighted additive DEA model.\nThe range adjusted measure (RAM).\nThe bounded adjusted measure (BAM).","category":"page"},{"location":"technical/additive/#Measure-of-Inefficiency-Proportions-(MIP)-1","page":"Additive Models","title":"Measure of Inefficiency Proportions (MIP)","text":"","category":"section"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"The measure of inefficiency proportions (MIP), Charnes et al. (1987) and Cooper et al. (1999), use the weights:","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"(mathbfrho_x^- mathbfrho_y^+)=(1mathbfx_o1mathbfy_o)","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"julia> deaadd(X, Y, :MIP)\nWeighted Additive DEA Model\nDMUs = 11; Inputs = 2; Outputs = 1\nWeights = MIP; Returns to Scale = VRS\n─────────────────────────────────────────────────────\n efficiency slackX1 slackX2 slackY1\n─────────────────────────────────────────────────────\n1 0.0 0.0 0.0 0.0\n2 0.507519 0.0 0.0 7.10526\n3 0.0 0.0 0.0 0.0\n4 -4.72586e-17 -8.03397e-16 0.0 0.0\n5 2.20395 0.0 0.0 17.6316\n6 1.31279e-16 8.10382e-16 0.0 8.64407e-16\n7 0.0 0.0 0.0 0.0\n8 0.0 0.0 0.0 0.0\n9 0.0 0.0 0.0 0.0\n10 1.04322 17.0 15.0 1.0\n11 0.235294 0.0 4.0 0.0\n─────────────────────────────────────────────────────","category":"page"},{"location":"technical/additive/#Normalized-Weighted-Additive-Model-1","page":"Additive Models","title":"Normalized Weighted Additive Model","text":"","category":"section"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"The normalized weighted additive DEA model, Lovell and Pastor (1995), use the weights:","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"(mathbfrho_x^- mathbfrho_y^+)=(1mathbfσ^-1mathbfσ^+)","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"where mathbfσ^-and mathbfσ^+ are the standard deviations of inputs and outputs respectively.","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"julia> deaadd(X, Y, :Normalized)\nWeighted Additive DEA Model\nDMUs = 11; Inputs = 2; Outputs = 1\nWeights = Normalized; Returns to Scale = VRS\n──────────────────────────────────────────────────────────\n efficiency slackX1 slackX2 slackY1\n──────────────────────────────────────────────────────────\n1 0.0 0.0 0.0 0.0\n2 0.804925 0.0 0.65 6.25\n3 0.0 0.0 0.0 0.0\n4 -9.79609e-17 0.0 -6.09909e-16 0.0\n5 2.01497 0.0 2.95 13.75\n6 4.81529e-16 2.49057e-15 0.0 2.37658e-15\n7 0.0 0.0 0.0 0.0\n8 0.0 0.0 0.0 0.0\n9 0.0 0.0 0.0 0.0\n10 3.98989 17.0 15.0 1.0\n11 0.642462 0.0 4.0 0.0\n──────────────────────────────────────────────────────────","category":"page"},{"location":"technical/additive/#Range-Adjusted-Measure-(RAM)-1","page":"Additive Models","title":"Range Adjusted Measure (RAM)","text":"","category":"section"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"The range adjusted measure (RAM), Cooper et al. (1999), use the weights::","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"(mathbfrho^- mathbfrho^+)=(1(m+s)R^-(1(m+s)R^+)","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"where R^-and R^+are the inputs and outputs variables' ranges.","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"julia> deaadd(X, Y, :RAM)\nWeighted Additive DEA Model\nDMUs = 11; Inputs = 2; Outputs = 1\nWeights = RAM; Returns to Scale = VRS\n──────────────────────────────────────────────────────────\n efficiency slackX1 slackX2 slackY1\n──────────────────────────────────────────────────────────\n1 0.0 0.0 0.0 0.0\n2 0.102975 0.0 0.0 7.10526\n3 0.0 0.0 0.0 0.0\n4 -1.01651e-17 0.0 -6.09909e-16 0.0\n5 0.25553 0.0 0.0 17.6316\n6 5.68808e-17 2.49057e-15 0.0 2.37658e-15\n7 0.0 0.0 0.0 0.0\n8 0.0 0.0 0.0 0.0\n9 0.0 0.0 0.0 0.0\n10 0.417646 17.0 15.0 1.0\n11 0.0666667 0.0 4.0 0.0\n──────────────────────────────────────────────────────────","category":"page"},{"location":"technical/additive/#Bounded-Adjusted-Measure-(BAM)-1","page":"Additive Models","title":"Bounded Adjusted Measure (BAM)","text":"","category":"section"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"The bounded adjusted measure (BAM), Cooper et al. (2011), use the weights:::","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"(mathbfrho_x^- mathbfrho_y^+)=(1(m+s)(mathbfx_o-mathbfunderlinex)(1(m+s)(mathbfoverliney - mathbfy_o)","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"where mathbfunderlinex and mathbfoverliney are the minimum and maximum observed values of inputs and outputs respectively.","category":"page"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"julia> deaadd(X, Y, :BAM)\nWeighted Additive DEA Model\nDMUs = 11; Inputs = 2; Outputs = 1\nWeights = BAM; Returns to Scale = VRS\n─────────────────────────────────────────────────\n efficiency slackX1 slackX2 slackY1\n─────────────────────────────────────────────────\n1 0.0 0.0 0.0 0.0\n2 0.199894 6.59649 0.0 0.0\n3 0.0 0.0 0.0 0.0\n4 -3.78838e-17 0.0 0.0 -5.68256e-16\n5 0.432971 13.0 1.0 4.0\n6 1.11554e-17 0.0 0.0 7.36254e-16\n7 0.0 0.0 0.0 0.0\n8 0.0 0.0 0.0 0.0\n9 0.0 0.0 0.0 0.0\n10 0.571361 5.0 11.0 5.0\n11 0.121212 0.0 4.0 0.0\n─────────────────────────────────────────────────","category":"page"},{"location":"technical/additive/#deaadd-Function-Documentation-1","page":"Additive Models","title":"deaadd Function Documentation","text":"","category":"section"},{"location":"technical/additive/#","page":"Additive Models","title":"Additive Models","text":"deaadd","category":"page"},{"location":"technical/additive/#DataEnvelopmentAnalysis.deaadd","page":"Additive Models","title":"DataEnvelopmentAnalysis.deaadd","text":"deaadd(X, Y, model)\n\nCompute related data envelopment analysis weighted additive models for inputs X and outputs Y.\n\nModel specification:\n\n:Ones: standard additive DEA model.\n:MIP: Measure of Inefficiency Proportions. (Charnes et al., 1987; Cooper et al., 1999)\n:Normalized: Normalized weighted additive DEA model. (Lovell and Pastor, 1995)\n:RAM: Range Adjusted Measure. (Cooper et al., 1999)\n:BAM: Bounded Adjusted Measure. (Cooper et al, 2011)\n:Custom: User supplied weights.\n\nOptional Arguments\n\nrts=:VRS: chosse between constant returns to scale :CRS or variable returns to scale :VRS.\nwX: matrix of weights of inputs. Only if model=:Custom.\nWY: matrix of weights of outputs. Only if model=:Custom.\nXref=X: Identifies the reference set of inputs against which the units are evaluated.\nYref=Y: Identifies the reference set of outputs against which the units are evaluated.\n\nExamples\n\njulia> X = [5 13; 16 12; 16 26; 17 15; 18 14; 23 6; 25 10; 27 22; 37 14; 42 25; 5 17];\n\njulia> Y = [12; 14; 25; 26; 8; 9; 27; 30; 31; 26; 12];\n\njulia> deaadd(X, Y, :MIP)\nWeighted Additive DEA Model\nDMUs = 11; Inputs = 2; Outputs = 1\nWeights = MIP; Returns to Scale = VRS\n─────────────────────────────────────────────────────\n efficiency slackX1 slackX2 slackY1\n─────────────────────────────────────────────────────\n1 0.0 0.0 0.0 0.0\n2 0.507519 0.0 0.0 7.10526\n3 0.0 0.0 0.0 0.0\n4 -4.72586e-17 -8.03397e-16 0.0 0.0\n5 2.20395 0.0 0.0 17.6316\n6 1.31279e-16 8.10382e-16 0.0 8.64407e-16\n7 0.0 0.0 0.0 0.0\n8 0.0 0.0 0.0 0.0\n9 0.0 0.0 0.0 0.0\n10 1.04322 17.0 15.0 1.0\n11 0.235294 0.0 4.0 0.0\n─────────────────────────────────────────────────────\n\n\n\n\n\n","category":"function"},{"location":"technical/radial/#","page":"Radial Models","title":"Radial Models","text":"CurrentModule = DataEnvelopmentAnalysis\nDocTestSetup = quote\n using DataEnvelopmentAnalysis\nend","category":"page"},{"location":"technical/radial/#Radial-Models-1","page":"Radial Models","title":"Radial Models","text":"","category":"section"},{"location":"technical/radial/#Radial-Input-Oriented-Model-1","page":"Radial Models","title":"Radial Input Oriented Model","text":"","category":"section"},{"location":"technical/radial/#","page":"Radial Models","title":"Radial Models","text":"Based on the data matrix (XY), we calculate the input oriented efficiency of each observation o by solving n times the following linear programming problem – known as the Charnes, Cooper, and Rhodes (1978), CCR, model:","category":"page"},{"location":"technical/radial/#","page":"Radial Models","title":"Radial Models","text":"beginalign\nlabeleqrim\n undersettheta mathbflambda mathopmin quad quad quad theta \n textsubject textto nonumber \n quad quad quad quad quad Xmathbflambda le theta mathbfx_o nonumber \n quad quad quad quad quad Ymathbflambda ge mathbfy_o nonumber\n quad quad quad quad quad mathbflambda ge mathbf0 nonumber\nendalign","category":"page"},{"location":"technical/radial/#","page":"Radial Models","title":"Radial Models","text":"The measurement of technical efficiency assuming variable returns to scale, VRS, as introduced by Banker, Charnes and Cooper (1984) – known as the Banker, Charnes and Cooper, BCC, model – adds the following condition:","category":"page"},{"location":"technical/radial/#","page":"Radial Models","title":"Radial Models","text":"sumnolimits_j=1^nlambda_j=1","category":"page"},{"location":"technical/radial/#","page":"Radial Models","title":"Radial Models","text":"In this example we compute the radial input oriented DEA model under constant returns to scale:","category":"page"},{"location":"technical/radial/#","page":"Radial Models","title":"Radial Models","text":"julia> using DataEnvelopmentAnalysis\n\njulia> X = [5 13; 16 12; 16 26; 17 15; 18 14; 23 6; 25 10; 27 22; 37 14; 42 25; 5 17];\n\njulia> Y = [12; 14; 25; 26; 8; 9; 27; 30; 31; 26; 12];\n\njulia> dea(X, Y, orient = :Input, rts = :CRS)\nRadial DEA Model \nDMUs = 11; Inputs = 2; Outputs = 1\nOrientation = Input; Returns to Scale = CRS\n──────────────────────────────────────────────────\n efficiency slackX1 slackX2 slackY1\n──────────────────────────────────────────────────\n1 1.0 0.0 0.0 0.0\n2 0.62229 -4.41868e-15 0.0 0.0\n3 0.819856 0.0 8.17926e-15 0.0\n4 1.0 -8.03397e-16 0.0 0.0\n5 0.310371 1.80764e-15 0.0 0.0\n6 0.555556 4.44444 0.0 0.0\n7 1.0 0.0 0.0 0.0\n8 0.757669 1.60679e-15 0.0 0.0\n9 0.820106 1.64021 0.0 0.0\n10 0.490566 9.68683e-15 0.0 0.0\n11 1.0 0.0 4.0 0.0\n──────────────────────────────────────────────────","category":"page"},{"location":"technical/radial/#","page":"Radial Models","title":"Radial Models","text":"To compute the variable returns to scale model, we simply set the rts parameter to :VRS:","category":"page"},{"location":"technical/radial/#","page":"Radial Models","title":"Radial Models","text":"julia> dea(X, Y, orient = :Input, rts = :VRS)\nRadial DEA Model \nDMUs = 11; Inputs = 2; Outputs = 1\nOrientation = Input; Returns to Scale = VRS\n───────────────────────────────────────────────────────\n efficiency slackX1 slackX2 slackY1\n───────────────────────────────────────────────────────\n1 1.0 0.0 0.0 0.0 \n2 0.869986 0.0 0.0 0.0 \n3 1.0 0.0 2.56789e-13 0.0 \n4 1.0 -8.03397e-16 0.0 0.0 \n5 0.71164 0.0 0.0 2.69841 \n6 1.0 2.70127e-16 0.0 3.78178e-16\n7 1.0 0.0 0.0 0.0 \n8 1.0 0.0 -1.27018e-14 0.0 \n9 1.0 0.0 0.0 0.0 \n10 0.493121 3.90444e-15 0.0 0.0 \n11 1.0 0.0 4.0 4.78849e-16\n───────────────────────────────────────────────────────","category":"page"},{"location":"technical/radial/#","page":"Radial Models","title":"Radial Models","text":"Estimated efficiency scores are returned with the efficiency function:","category":"page"},{"location":"technical/radial/#","page":"Radial Models","title":"Radial Models","text":"julia> deaiovrs = dea(X, Y, orient = :Input, rts = :VRS);\n\njulia> efficiency(deaiovrs)\n11-element Array{Float64,1}:\n 1.0\n 0.8699861687413553\n 1.0000000000000002\n 1.0\n 0.7116402116402116\n 1.0\n 1.0\n 0.9999999999999999\n 1.0\n 0.4931209268645909\n 1.0","category":"page"},{"location":"technical/radial/#","page":"Radial Models","title":"Radial Models","text":"The optimal peers, λ, are returned with the peers function and are returned as a SparseArrays.SparseMatrixCSC{Float64,Int64} object:","category":"page"},{"location":"technical/radial/#","page":"Radial Models","title":"Radial Models","text":"peers(deaiovrs)","category":"page"},{"location":"technical/radial/#Radial-Output-Oriented-Model-1","page":"Radial Models","title":"Radial Output Oriented Model","text":"","category":"section"},{"location":"technical/radial/#","page":"Radial Models","title":"Radial Models","text":"It is possible to calculate the output oriented technical efficiency of each observation by solving the following linear program:","category":"page"},{"location":"technical/radial/#","page":"Radial Models","title":"Radial Models","text":"beginalign\nlabeleqrom\n undersetphi mathbflambda mathopmax quad quad quad quad phi \n textsubject textto nonumber\n quad quad quad quad quad Xlambdale mathbfx_o nonumber\n quad quad quad quad quad Ymathbflambda ge phi mathbfy_o nonumber\n quad quad quad quad quad mathbflambda ge mathbf0nonumber quad nonumber\nendalign","category":"page"},{"location":"technical/radial/#","page":"Radial Models","title":"Radial Models","text":"with the following condition when assuming variable returns to scale:","category":"page"},{"location":"technical/radial/#","page":"Radial Models","title":"Radial Models","text":"sumnolimits_j=1^nlambda_j=1","category":"page"},{"location":"technical/radial/#","page":"Radial Models","title":"Radial Models","text":"In this example we compute the radial output oriented DEA model under variable returns to scale:","category":"page"},{"location":"technical/radial/#","page":"Radial Models","title":"Radial Models","text":"julia> dea(X, Y, orient = :Output, rts = :VRS)\nRadial DEA Model \nDMUs = 11; Inputs = 2; Outputs = 1\nOrientation = Output; Returns to Scale = VRS\n──────────────────────────────────────────────────\n efficiency slackX1 slackX2 slackY1\n──────────────────────────────────────────────────\n1 1.0 0.0 0.0 0.0 \n2 1.50752 5.78599e-15 0.0 0.0 \n3 1.0 0.0 0.0 0.0 \n4 1.0 -8.03397e-16 0.0 0.0 \n5 3.20395 -3.38377e-15 0.0 0.0 \n6 1.0 2.70127e-16 0.0 3.78178e-16\n7 1.0 0.0 0.0 0.0 \n8 1.0 0.0 0.0 0.0 \n9 1.0 0.0 0.0 0.0 \n10 1.19231 5.0 11.0 0.0 \n11 1.0 0.0 4.0 4.78849e-16\n──────────────────────────────────────────────────","category":"page"},{"location":"technical/radial/#dea-Function-Documentation-1","page":"Radial Models","title":"dea Function Documentation","text":"","category":"section"},{"location":"technical/radial/#","page":"Radial Models","title":"Radial Models","text":"dea","category":"page"},{"location":"technical/radial/#DataEnvelopmentAnalysis.dea","page":"Radial Models","title":"DataEnvelopmentAnalysis.dea","text":"dea(X, Y)\n\nCompute the radial model using data envelopment analysis for inputs X and outputs Y.\n\nOptional Arguments\n\norient=:Input: chooses the radially oriented input mode. For the radially oriented output model choose :Output.\nrts=:CRS: chooses constant returns to scale. For variable returns to scale choose :VRS.\nslack=true: computes input and output slacks.\nXref=X: Identifies the reference set of inputs against which the units are evaluated.\nYref=Y: Identifies the reference set of outputs against which the units are evaluated.\ndisposalX=:Strong: chooses strong disposal of inputs. For weak disposal choose :Weak.\ndisposalY=:Strong: chooses strong disposal of outputs. For weak disposal choose :Weak.\n\nExamples\n\njulia> X = [5 13; 16 12; 16 26; 17 15; 18 14; 23 6; 25 10; 27 22; 37 14; 42 25; 5 17];\n\njulia> Y = [12; 14; 25; 26; 8; 9; 27; 30; 31; 26; 12];\n\njulia> dea(X, Y)\nRadial DEA Model\nDMUs = 11; Inputs = 2; Outputs = 1\nOrientation = Input; Returns to Scale = CRS\n──────────────────────────────────────────────────\n efficiency slackX1 slackX2 slackY1\n──────────────────────────────────────────────────\n1 1.0 0.0 0.0 0.0\n2 0.62229 -4.41868e-15 0.0 0.0\n3 0.819856 0.0 8.17926e-15 0.0\n4 1.0 -8.03397e-16 0.0 0.0\n5 0.310371 1.80764e-15 0.0 0.0\n6 0.555556 4.44444 0.0 0.0\n7 1.0 0.0 0.0 0.0\n8 0.757669 1.60679e-15 0.0 0.0\n9 0.820106 1.64021 0.0 0.0\n10 0.490566 9.68683e-15 0.0 0.0\n11 1.0 0.0 4.0 0.0\n──────────────────────────────────────────────────\n\n\n\n\n\n","category":"function"},{"location":"economic/cost/#","page":"Cost Models","title":"Cost Models","text":"CurrentModule = DataEnvelopmentAnalysis\nDocTestSetup = quote\n using DataEnvelopmentAnalysis\nend","category":"page"},{"location":"economic/cost/#Cost-Models-1","page":"Cost Models","title":"Cost Models","text":"","category":"section"},{"location":"economic/cost/#Cost-Efficiency-Model-with-Radial-Technical-Efficiency-1","page":"Cost Models","title":"Cost Efficiency Model with Radial Technical Efficiency","text":"","category":"section"},{"location":"economic/cost/#","page":"Cost Models","title":"Cost Models","text":"Let us denote by Cleft(mathbfymathbfwright) the minimum cost of producing the output level mathbfy given the input price vector mathbfw: Cleft(mathbfymathbfwright)=min left sumlimits_i=1^mw_ix_i mathbfx geqslant Xmathbflambda mathbfy_o leqslant Ymathbflambda mathbflambda geqslant mathbf0 right, which considers the input possibility set capable of producing mathbfy_o. For the observed outputs levels we can calculate minimum cost and the associated optimal quantities of inputs mathbfx^* consistent with the production technology by solving the following program:","category":"page"},{"location":"economic/cost/#","page":"Cost Models","title":"Cost Models","text":"beginalign\nlabeleqmincost\n undersetmathbfx mathbflambda mathopmin quad quad quad Cleft(mathbfy_mathbfwright)=mathbfwx^* \n textsubject textto nonumber \n quad quad quad quad quad mathbfxge Xmathbflambda nonumber \n quad quad quad quad quad Ymathbflambda ge mathbfy_o nonumber \n quad quad quad quad quad mathbflambda ge mathbf0 nonumber \nendalign","category":"page"},{"location":"economic/cost/#","page":"Cost Models","title":"Cost Models","text":"The measurement of cost efficiency assuming variable returns to scale, VRS, adds the following condition:","category":"page"},{"location":"economic/cost/#","page":"Cost Models","title":"Cost Models","text":"sumnolimits_j=1^nlambda_j=1","category":"page"},{"location":"economic/cost/#","page":"Cost Models","title":"Cost Models","text":"Cost efficiency defines as the ratio of minimum cost to observed cost: CE=Cleft(mathbfymathbfwright)mathbfwx_o. Thanks to duality results presented by Shephard (1953) , and following Farrell (1957), cost efficiency can be decomposed into the radially input oriented technical efficiency measure and the residual difference corresponding to allocative cost efficiency. Allocative efficiency defines as the ratio between minimum cost to production cost at the technically efficient projection of the unit under evaluation.","category":"page"},{"location":"economic/cost/#","page":"Cost Models","title":"Cost Models","text":"In this example we compute the cost efficiency measure under variable returns to scale:","category":"page"},{"location":"economic/cost/#","page":"Cost Models","title":"Cost Models","text":"julia> X = [5 3; 2 4; 4 2; 4 8; 7 9.0];\n\njulia> Y = [7 4; 10 8; 8 10; 5 4; 3 6.0];\n\njulia> W = [2 1; 2 1; 2 1; 2 1; 2 1.0];\n\njulia> deacost(X, Y, W)\nCost DEA Model\nDMUs = 5; Inputs = 2; Outputs = 2\nOrientation = Input; Returns to Scale = VRS\n──────────────────────────────────\n Cost Technical Allocative\n──────────────────────────────────\n1 0.615385 0.75 0.820513\n2 1.0 1.0 1.0\n3 1.0 1.0 1.0\n4 0.5 0.5 1.0\n5 0.347826 0.375 0.927536\n──────────────────────────────────","category":"page"},{"location":"economic/cost/#deacost-Function-Documentation-1","page":"Cost Models","title":"deacost Function Documentation","text":"","category":"section"},{"location":"economic/cost/#","page":"Cost Models","title":"Cost Models","text":"deacost","category":"page"},{"location":"economic/cost/#DataEnvelopmentAnalysis.deacost","page":"Cost Models","title":"DataEnvelopmentAnalysis.deacost","text":"deacost(X, Y, W)\n\nCompute cost efficiency using data envelopment analysis for inputs X, outputs Y and price of inputs W.\n\nOptional Arguments\n\nrts=:VRS: chooses variable returns to scale. For constant returns to scale choose :CRS.\ndisposal=:Strong: chooses strong disposal of outputs. For weak disposal choose :Weak.\n\nExamples\n\njulia> X = [5 3; 2 4; 4 2; 4 8; 7 9.0];\n\njulia> Y = [7 4; 10 8; 8 10; 5 4; 3 6.0];\n\njulia> W = [2 1; 2 1; 2 1; 2 1; 2 1.0];\n\njulia> deacost(X, Y, W)\nCost DEA Model\nDMUs = 5; Inputs = 2; Outputs = 2\nOrientation = Input; Returns to Scale = VRS\n──────────────────────────────────\n Cost Technical Allocative\n──────────────────────────────────\n1 0.615385 0.75 0.820513\n2 1.0 1.0 1.0\n3 1.0 1.0 1.0\n4 0.5 0.5 1.0\n5 0.347826 0.375 0.927536\n──────────────────────────────────\n\n\n\n\n\n","category":"function"},{"location":"bibliography/#","page":"Bibliography","title":"Bibliography","text":"CurrentModule = DataEnvelopmentAnalysis\nDocTestSetup = quote\n using DataEnvelopmentAnalysis\nend","category":"page"},{"location":"bibliography/#Bibliography-1","page":"Bibliography","title":"Bibliography","text":"","category":"section"},{"location":"bibliography/#","page":"Bibliography","title":"Bibliography","text":"Banker, R., Charnes, A., and Cooper, W.W. (1984). \"Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis.\" Management Science, 30(9), 1078–1092.\nCaves D.W., Christensen L.R., Diewert, W.E. (1982). \"The Economic Theory of Index Numbers and the Measurement of Input, Output, and Productivity.\" Econometrica, 50(6), 1393–1414.\nChambers, R.G., Chung, Y., and Färe R. (1996). \"Benefit and Distance Functions.\" Journal of Economic Theory, 70(2), 407 – 419.\nChambers, R.G., Chung, Y., and Färe R. (1998). \"Profit, Directional Distance Functions, and Nerlovian Efficiency.” Journal of Optimization Theory and Applications 98(2), 351-364.\nCharnes, A., Cooper, W.W., and Rhodes, E. (1978). \"Measuring the efficiency of decision making units.\" European Journal of Operational Research, 2(6), 429–444.\nCharnes, A., Cooper, W.W., Rousseau, J. , and Semple, J. (1987). \"Data Envelopment Analyses and Axiomatic Notions of Efficiency and Reference Sets.\" Research Report, Center for Cybernetic Studies, The University of Texas at Austin.\nChavas J., and Cox, T. (1999). \"A Generalized Distance Function and the Analysis of Production Efficiency.\" Southern Economic Journal, 66(2), 294-318.\nCooper, W.W., Park, K.S. and Pastor, J.T. (1999). \"RAM: A Range Adjusted Measure of Inefficiency for Use with Additive Models, and Relations to Other Models and Measures in DEA\" Journal of Productivity Analysis 11(1), 5-42. \nCooper, W.W., and Pastor, J.T. (1995). \"Global Efficiency Measurement in DEA.” Working Paper, Depto Est e Inv. Oper. Universidad Alicante, Alicante, Spain.\nCooper, W.W., Pastor, J.T., Borras, F., Aparicio, J. and Pastor, D. (2011). \"BAM: a bounded adjusted measure of efficiency for use with bounded additive models.\" Journal of Productivity Analysis, 35(2), 85-94.\nFare, R., Grosskopf, S., Norris, M. and Zhang, Z. (1994). \"Productivity Growth, Technical Progress, and Efficiency Change in Industrialized Countries.\" American Economic Review, 84(1), 66–83.\nFarrell, M. J. (1957). \"\"The Measurement of Productive Efficiency of Production.\" Journal of the Royal Statistical Society, Series A, 120(III), 253-281.\nLovell, C. A. K., and Pastor, J. T. (1995). \"Units Invariant and Translation Invariant DEA Models.\" Operations Research Letters, 18, 147–151.\nPastor, J.T., Lovell, C.A.K., and Aparicio, J. (2012). \"Families of linear efficiency programs based on Debreu’s loss function.\" Journal of Productivity Analysis 38(2), 109-120.\nShephard, R.W. (1953). Cost and production functions. Princeton University Press, Princeton, New Jersey.\nZofío, J.L. and Prieto, A.M. (2006). \"Return to Dollar, Generalized Distance Function and the Fisher Productivity Index\" Spanish Economic Review, 8(2), 113-138.","category":"page"},{"location":"#DataEnvelopmentAnalysis-Documentation-1","page":"Home","title":"DataEnvelopmentAnalysis Documentation","text":"","category":"section"},{"location":"#","page":"Home","title":"Home","text":"DataEnvelopmentAnalysis.jl is a Julia package that provides functions for efficiency and productivity measurement using Data Envelopment Analysis (DEA). Particularly, it implements a variety of technical efficiency models, economic efficiency models and productivity change models.","category":"page"},{"location":"#","page":"Home","title":"Home","text":"The package is being developed for Julia 1.0 and above on Linux, macOS, and Windows.","category":"page"},{"location":"#","page":"Home","title":"Home","text":"The packes uses internally the JuMP modelling language for mathematicall optimization with solvers GLPK and Ipopt.","category":"page"},{"location":"#Installation-1","page":"Home","title":"Installation","text":"","category":"section"},{"location":"#","page":"Home","title":"Home","text":"The package can be installed with the Julia package manager:","category":"page"},{"location":"#","page":"Home","title":"Home","text":"julia> using Pkg; Pkg.add(\"DataEnvelopmentAnalysis\")","category":"page"},{"location":"#Available-models-1","page":"Home","title":"Available models","text":"","category":"section"},{"location":"#","page":"Home","title":"Home","text":"Technical efficiency DEA models:","category":"page"},{"location":"#","page":"Home","title":"Home","text":"Pages = [\"technical/radial.md\", \"technical/directional.md\", \"technical/additive.md\", \"technical/generalizeddf.md\"]\nDepth = 2","category":"page"},{"location":"#","page":"Home","title":"Home","text":"Economic efficiency DEA models:","category":"page"},{"location":"#","page":"Home","title":"Home","text":"Pages = [\"economic/cost.md\", \"economic/revenue.md\", \"economic/profit.md\", \"economic/profitability.md\"]\nDepth = 1","category":"page"},{"location":"#","page":"Home","title":"Home","text":"Productivity change models:","category":"page"},{"location":"#","page":"Home","title":"Home","text":"Pages = [\"productivity/malmquist.md\"]\nDepth = 1","category":"page"},{"location":"#Authors-1","page":"Home","title":"Authors","text":"","category":"section"},{"location":"#","page":"Home","title":"Home","text":"DataEnvelopmentAnalysis.jl is being developed by Javier Barbero and José Luís Zofío.","category":"page"}]
}
