JuMP4DEA.jl

JuMP4DEA.jl is a package for solving large-scale data envelopment analysis (DEA) problems. Benefited from the development of JuMP, a modeling language in Julia, and cutting-edge solvers, JuMP4DEA is user-friendly, flexible and computationally efficient to solve different DEA problems, including radial, non-radial and different returns to scale assumptions.

Installation

julia> Pkg.clone("git://github.com/henry8527/JuMP4DEA.jl.git")

Efficiency

We are going to Demo the Timing compared with JuMP4DEA.solve() and JuMP.solve() at the belowing form.

Solver : GurobiSolver
CPU : Intel Core i7 - 4790
RAM : 16GB
OS SYSTEM : Windows 8 64bits

Data Scale = 25K

Note : JuMP4DEA (sec) / JuMP (sec)

Dimension \ Density	1%	10%	25%
2 - 3	579.16 / 1669.34	581.42 / 1685.77	560.71 / 1671.79
5 - 5
7 - 8
10 - 10

Data Scale = 50K

Note : JuMP4DEA (sec) / JuMP (sec)

Dimension \ Density	1%	10%	25%
2 - 3	2755.49 / 7487.93	2560.43 / 7434.29	2577.92 / 7550.34
5 - 5
7 - 8
10 - 10

Data Scale = 75K

Note : JuMP4DEA (sec) / JuMP (sec)

Dimension \ Density	1%	10%	25%
2 - 3	6050.52 / 18384.48
5 - 5
7 - 8
10 - 10

Quick Start Guide

Creating Model

Models are Julia objects. They are created by calling the constructor:

m = model()

All variables and constraints are associated with a Model object. Usually, you’ll also want to provide a solver object here by using the solver= keyword argument

In our example code , default using solver = GurobiSolver()

Defining Variables

Variables are also Julia objects, and are defined using the @variable macro. The first argument will always be the Model to associate this variable with. In the examples below we assume m is already defined. The second argument is an expression that declares the variable name and optionally allows specification of lower and upper bounds.

@variable(m, x )              # No bounds
@variable(m, x >= lb )        # Lower bound only (note: 'lb <= x' is not valid)
@variable(m, x <= ub )        # Upper bound only
@variable(m, lb <= x <= ub )  # Lower and upper bounds

In our example code , we define varibles `CLambda` and `Ctheta` for solving DEA problems.

@variable(crs,cLambda[1:scale] >= 0)
@variable(crs,cTheta) 

Objective and Constraints

JuMP allows users to use a natural notation to describe linear expressions. To add constraints, use the @constraint() and @objective() macros

@constraint(m, x[i] - s[i] <= 0)  # Other options: == and >=
@constraint(m, sum(x[i] for i=1:numLocation) == 1)
@objective(m, Max, 5x + 22y + (x+y)/2) # or Min

In our example code , We define `Min` objective , and add constraints for product input and output.

@objective(crs,Min,cTheta)
@constraint(crs, C1[j=1:2], sum{data[i,j]*cLambda[i], i = 1:scale} >= data[t,j])
@constraint(crs, C2[j=3:5], sum{data[i,j]*cLambda[i], i = 1:scale} <= cTheta*data[t,j])

Solve problem

call our JuMP4DEA function solveDEA

JuMP4DEA.solveDEA(model)

Parameter and its default value in solveDEA

incrementSize : smallLP increment size when its size under lpUP for resampling

JuMP4DEA.solveDEA(model, incrementSize = 100)

Tol : the accuracy for solving DEA problem

JuMP4DEA.solveDEA(model, Tol = 10^-6)

lpUB : the limit size of LP

JuMP4DEA.solveDEA(model, lpUB = Inf)

extremeValueSetFlag : first sampling take extremeValueSet or not ( 0 = turn off, 1 = turn on )

JuMP4DEA.solveDEA(model, extremeValueSetFlag = 0)

Getting Answer from solveDEA

# Get DEA problem answer 
#--------------------------
#println("the real data to test: $benchmark")
#println("lambdas: $(getvalue(cLambda)))")
#println("Objective value: $(getobjectivevalue(crs))")
#--------------------------

Extra Require For Getting " Duals " and " Slack " in solveDEA

You also can call solveDEA with return value for solving Duals value and Slack Value

duals, slack = JuMP4DEA.solveDEA(model)

and then can get more answer what you want

# Get DEA problem answer (extra version)
#--------------------------
#println("the real data to test: $benchmark")
#println("lambdas: $(getvalue(cLambda)))")
#println("Objective value: $(getobjectivevalue(crs))")
#println("Duals value: $duals")
#println("Slack value: $slack")
#--------------------------	

Citing JuMP4DEA

If you find JuMP4DEA useful in your work, we kindly request that you cite the following paper

@article{ChenLai2017,
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},
}