Solving large scale DEA problems



Copyright © 2017 by Wen-Chih Chen. Released under the MIT License.

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FrontierEfficiencyAnalysis.jl is a package for Frontier Efficiency Analysis (aka Data Envelopment Analysis, DEA) computation. It is embedded in the Julia programming language, and is an extension to the JuMP modeling language. It is particularly designed to enhance large-scale DEA computation and to solve DEA problems by size-limited solvers.

Disclaimer : FrontierEfficiencyAnalysis is not developed or maintained by the JuMP developers.


In Julia, call Pkg.add("FrontierEfficiencyAnalysis") to install FrontierEfficiencyAnalysis.


DEA is a linear program (LP)-based method used to determine a firm’s relative efficiency. Users can use JuMP to model and solve the DEA problems (special LP problems). Rather than solve the LPs by calling JuMP.solve(), FrontierEfficiencyAnalysis.jl can solve the large-scale problems more efficiently and/or by a solver with size limitation (e.g. 300 variables).

Please refer to Quick Start Guide of JuMP for modeling details. What needed is to call our FrontierEfficiencyAnalysis.jl function:


instead of calling



# The example determine the efficiencies for all DMUs based on the CRS input-oriented model (CCR model)
using JuMP
using Gurobi # Gurobi is used as the LP solver
using FrontierEfficiencyAnalysis

data = readcsv("example.csv") # input User's (.csv) data path
scale, dimension = size(data) # scale is the number of DMU, dimension is the total number of inputs and outputs

for t = 1 : scale
    ### Modeling section
    # Here is the CRS input-oriented model (CCR model) to evaluate DMU t
    model = Model(solver = GurobiSolver()) # Gurobi is used as the LP solver here. Users can choose their favorite solver.
    @variable(model, Lambda[1:scale] >= 0)
    @variable(model, Theta)
    @objective(model, Min, Theta)
    @constraint(model, inputCon[i=1:2], sum(Lambda[r]*data[r,i] for r = 1:scale) <= Theta*data[t,i])
    @constraint(model, outputCon[j=3:dimension], sum(Lambda[r]*data[r,j] for r = 1:scale) >= data[t,j])
    # uncomment to add the convexity constraint for the VRS model
    # @constraint(model, sum(Lambda[r] for r = 1:scale) == 1)

    ### Problem solving

    ### Display
    println("Results for DMU $t")
    println("The efficicncy: $(getobjectivevalue(model))")
    println("The efficicncy: $(getvalue(Theta))")
    println("The lambdas: $(getvalue(Lambda)))")

    ## Use the following if returning dual values and slacks is needed

    ### Problem solving
    # duals, slacks = solveDEA(model)
    ### Display
    # println("Results for DMU $t")
    # println("The efficicncy: $(getobjectivevalue(model))")
    # println("The efficicncy: $(getvalue(Theta))")
    # println("The lambdas: $(getvalue(Lambda)))")

    # #println("The dual values (weights): $duals") # a vector associated with the constraints you define from the top to the bottom
    # println("the slack values: $slacks") # a vector associated with the constraints you define from the top to the bottom


incrementSize : the incremental size to expand the sample ( default value: 100 ).

solveDEA(model, incrementSize = 200) # set the incremental size to 200

tol : the solution tolerance for solving DEA problem (default value: 1e-6). It also resets the dual feasibility tolerance in the solver to the given value.

solveDEA(model, tol = 10^-4) # set the solution tolerance to 1e-4

lpUB : the size limit of the LP, i.e. the limitation of number of variables in the LP (default value: Inf).

solveDEA(model, lpUB = 300) # set the LP size limitation to 300 variables

extremeValueSetFlag : to enable (=1) or disable (=0) performing initial sampling by selecing extreme value in each input/output dimension (default value: 0).

solveDEA(model, extremeValueSetFlag = 1) # enable


If you find FrontierEfficiencyAnalysis useful in your work, we kindly request that you cite the following papers

author = {Wen-Chih Chen and Sheng-Yung Lai},
title = {Determining radial efficiency with a large data set by solving small-size linear programs},
journal = {Annals of Operations Research},
volume = {250},
number = {1},
pages = {147-166},
year = {2017},
doi = {10.1007/s10479-015-1968-4},


Author = {Wen-Chih Chen and Yueh-Shan Chung},
Title = {A generalized non-radial efficiency measure and its application in DEA computation},
Year = {2017},
Eprint = {http://dx.doi.org/10.2139/ssrn.2496847},


FrontierEfficiencyAnalysis has been developed under the financial support of the Ministry of Science and Technology, Taiwan (Grant No. 104-2410-H-009-026-MY2). The contributors include Yueh-Shan Chung and Hao-Yun Chen.

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