By: Jaafar Ballout
Re-posted from: https://www.supplychaindataanalytics.com/linear-programming-in-julia-with-glpk-and-jump/
In previous posts we have covered various linear programming examples in both Python and R. We have e.g. introduced lpSolve in R, PuLP in Python, Gekko in Python, MIP in Python, ortools in Python as well as many other packages and modules for linear programming. In this post I will demonstrate how one can implement linear programming in Julia.
Julia is a flexible dynamic language that is appropriate for scientific and numerical computing. Besides, Julia is an open-source project. Its source code is available on GitHub. Using Julia provides access to many packages already developed for this language. For the linear programming example presented in this post I will use the Julia packages GLPK and JuMP.
Financial engineering example utilizing JuMP and GLPK in Julia
The demonstrated example is about a bank loan policy. This simple example will help us understand the power of linear programming in solving most of the problems we face in supply chain management and the financial fields. Indeed, Julia will be incorporated to solve the optimization problem. The example is adapted from Hamdy Taha’s book available on Amazon.
Type of loan | Interest rate | Bad-debt ratio |
---|---|---|
Personal | 0.14 | 0.1 |
Car | 0.13 | 0.07 |
Home | 0.12 | 0.03 |
Farm | 0.125 | 0.05 |
Commercial | 0.10 | 0.02 |
A bank is in the process of devising a loan policy that involves a maximum of $12 million. The table, shown above, provides the data about the available loans. It is important to note that bad debts are unrecoverable and produce no interest revenue. Competition with other financial institutions dictates the allocation of at least 40% of the funds to farm and commercial loans. To assist the housing industry in the region, home loans must equal at least 50% of the personal, car, and home loans. The bank limits the overall ratio of bad debts on all loans to at most 4%.
Developing a mathematical problem statement
The hardest part of any optimization problem is building the mathematical model. The steps to overcome this hardship are as follows:
- Define the decision variables
- Write down the objective function
- Formulate all the constraints
Decison variables
According to the given in the problem, we need to determine the amount of loan in million dollars. The table shows five categories or types of loans. Thus, we need five decision variables corresponding to each category.
- x1 = personal loans
- x2 = car loans
- x3 = home loans
- x4 = farm loans
- x5 = commercial loans
Objective function
The objective function is to maximize profits (net return). The net return is the difference between the revenues, generated by interest rate, and losses due to the bad-debt ratio.
Then, the objective function is as follows:
Constraints
Total Loans should be less or equal to 12 million dollars
Farm and Commercial loans equal at least 40% of all loans
Home loans should equal at least 50% of the personal, car, and home loans
Bad debts should not exceed 4% of all loans
Non-negativity
Application of linear programming in Julia
The first step is to add the required packages to solve the linear program. This is done using the <Pkg> which is the built-in package manager in Julia. We can add the needed packages using the following commands in Julia REPL. If Julia is installed, typing Julia at the command line, in the computer command prompt, is enough to open REPL.
using Pkg
Pkg.add("JuMP")
Pkg.add("GLPK")
The work presented below is done in a Jupyter notebook with a Julia kernel. JuMP is a modeling language for mathematical optimization in Julia while GLPK is the solver that we will use. After importing the packages in Jupyter, we defined the optimization problem by creating a variable BM that stands for Bank Model. The next step is setting the optimizer by declaring the solver (GLPK) and the model (BM). Moving on, we specified the decision variables, constraints, and objective function.
In [1]:
## Importing the necessary packages ## using JuMP using GLPK
In [2]:
## Defining the model ## BM = Model() # BM stands for Bank Model ## Setting the optimizer ## set_optimizer(BM,GLPK.Optimizer) ## Define the variables ## @variable(BM, x1>=0) @variable(BM, x2>=0) @variable(BM, x3>=0) @variable(BM, x4>=0) @variable(BM, x5>=0) ## Define the constraints ## @constraint(BM, x1+x2+x3+x4+x5<=12) @constraint(BM, 0.4x1+0.4x2+0.4x3-0.6x4-0.6x5<=0) @constraint(BM, 0.5x1+0.5x2-0.5x3<=0) @constraint(BM, 0.06x1+0.03x2-0.01x3+0.01x4-0.02x5<=0) ## Define the objective function ## @objective(BM, Max, 0.026x1+0.0509x2+0.0864x3+0.06875x4+0.078x5) ## Run the optimization ## optimize!(BM)
Declaring the objective function and constraints in Julia is easier because the algebraic equation can be inputted as it is (i.e. no need to show the multiplication operator between the variable and the constant). After running the optimization model, we can view the output of the model as follows:
In [3]:
BM
Out[3]:
A JuMP Model Maximization problem with: Variables: 5 Objective function type: AffExpr `AffExpr`-in-`MathOptInterface.LessThan{Float64}`: 4 constraints `VariableRef`-in-`MathOptInterface.GreaterThan{Float64}`: 5 constraints Model mode: AUTOMATIC CachingOptimizer state: ATTACHED_OPTIMIZER Solver name: GLPK Names registered in the model: x1, x2, x3, x4, x5
In [4]:
objective_value(BM)
Out[4]:
0.99648
In [5]:
objective_sense(BM)
Out[5]:
MAX_SENSE::OptimizationSense = 1
In [6]:
println("x1 = ", value.(x1), "\n","x2 = ",value.(x2), "\n", "x3 = ",value.(x3), "\n", "x4 = ",value.(x4), "\n", "x5 = ",value.(x5))
x1 = 0.0 x2 = 0.0 x3 = 7.199999999999999 x4 = 0.0 x5 = 4.8
Analyzing the output of the model, we can see that the bank should spend 7.2 million dollars on home loans and 4.8 million dollars on commercial loans. This distribution will help the bank maximize its net returns (i.e. profits) to 0.99648 million dollars.
Concluding remarks
In this post, we explored the power of linear programming in the banking sector and used Julia to solve the LP optimization problem. It is nice to see that open-source languages can solve these kinds of problems in an easy way without extensive hours of coding. In the future, we will have an insight into non-linear problems in supply chain management.
A chemical engineer interested in optimization and using open-source software to empower research and academia!
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