*sn*port”

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### Table of Contents

# A JuMP tutorial for GAMS users

JuMP (Julia for Mathematical Optimization) is an Algebraic Modelling Language (AML) that allows to write optimisation problems using a concise mathematical formulation, acting as interface to the specific solver engine API. For non-linear optimisation problems it allows to keep a high-level approach that doesn't require the modeller to compute the Jacobian or the Hessian.

It is developed by the MIT Operations Research Center and appeared in 2013 as an open source package of the relatively new Julia programming language.

GAMS (The General Algebraic Modeling System) does more or less the same things and appeared in the '70s as a project of the World Bank. GAMS is hence a very mature project (maybe *too* mature) with a lot of followers in the economic domain, where it is used mainly to solve equilibria problems.

This mini-tutorial is intended for gams users that want to try JuMP. There may be two reasons for someone to with to use JuMP instead of GAMS.

The most obvious one, even if often it isn't the key driver, is that GAMS is a commercial software while JuMP being open-source is free both as freedom and as a free beer.

While for GAMS a licence for the underlying solver engine is often included with a particular version of GAMS, JuMP would still require the user to buy a licence to use a specific commercial solvers. However JuMP interfaces with both GLPK (for linear and mixed-integer programming) and IPOPT (for non-linear optimisation) open-source solvers, both of which are top on their classes, leaving the necessity to acquire a licence for a commercial solver to niche cases.

The second reason (and, to me, the most important one) resides in the language features and in the availability of development environments. GAMS uses a VERY ODD syntax, somehow derived from the Cobol language, that is very distant from any programming language in use nowadays. For example a macro mechanism to provide an elementary way to structure the code in reusable components has been introduced only in GAMS 22.9.
Its own editor is also very terrible, but as most text editors do not provide a GAMS syntax highlighting, it's still the most common way to code in GAMS.

JuMP, at the opposite, is both open source and it allow to write the model in a powerful general-purpose language like Julia

You have plenty of development environment to choose from (e.g. Jupiter, Juno), a clear modern language, the possibility to interface your model with third party libraries.. all of this basically for free.

It is also, at least for my user case, much faster than GAMS. Aside the preparation of the model to pass to the solver, where it is roughly equivalent, in the solver execution I can benefit of having on my system a version of IPOPT compiled with the much more performing ma27 linear solver, while for GAMS I would have to rely on the embedded version that is compiled with the MUMPS linear solver. That's part of the flexibility you gain in using JuMP in place of GAMS.
That's said, for people that don't need such flexibility, the package automatically install a local pre-compiled version of the solver, so just adding the package relative to the solver is enough to start writing the model. Even more, for people that doesn't care too much about performances, there is a service on JuliaBox.com that allows to run Julia/JuMP scripts for free in the browser, without anything to install on the local computer.

So let's start. We will see how to code the trasnport.gms problem, the one that ship as default example in GAMS^{1)}, using JuMP. For a fictions product, there are three canning plants and three markets and the objective of the model is to find the optimal allocation of products between plants and markets that minimises the (transport) costs.

GAMS equivalent code is inserted as single-dash comments. The original GAMS code needs slightly different ordering of the commands and it's available at http://www.gams.com/mccarl/trnsport.gms

## Installation

**Step 1:**

- Option a: Get an account on JuliaBox.com to run julia/JuMP script without installing anything on the local computer
- Option b: Install Julia for your platform (http://julialang.org/downloads/)

**Step 2:**

Run, only once, the following code to install JuMP language and a couple of open source solvers:

Pkg.update() # To refresh the list of newest packages Pkg.add("JuMP") # The mathematical optimisation library Pkg.add("GLPKMathProgInterface") # A lineaqr and MIP solver Pkg.add("Ipopt") # A non-linear solver Pkg.add("DataFrames") # A library to deal with dataframes (R-like tabular data)

## Model components

### Importing the libraries

You will need to import as a minima the `JuMP`

module. If you wish to specify a solver engine rather than letting JuMP select a suitable one, you will need to import also the module relative to the solver, e.g. `Ipopt`

or `GLPKMathProgInterface`

# Import of the JuMP and DataFrames modules (the latter one just to import the data from a header-based table, as in the original trasnport example in GAMS using JuMP, DataFrames

### Defining the "sets"

JuMP doesn't really have a concept of sets, but it uses the native containers available in the core Julia language\\Variables, parameters and constraints can be indexed using these containers.

While many works with position-based lists, I find more readable using dictionaries instead. So the “sets” are represented as lists, but then everything else is a dictionary with the elements of the list as keys.

One note: it seems that Julia/JuMP don't like much the “-” symbol, so I replaced it to “_”.

## Define sets ## # Sets # i canning plants / seattle, san-diego / # j markets / new-york, chicago, topeka / ; plants = ["seattle","san_diego"] # canning plants markets = ["new_york","chicago","topeka"] # markets

### Definition of the "parameters"

Capacity of plants and demand of markets are directly defined as dictionaries, while the distance is first read as a DataFrame from a white-space separated table and then it is converted in a “(plant, market) ⇒ value” dictionary.

## Define parameters ## # Parameters # a(i) capacity of plant i in cases # / seattle 350 # san-diego 600 / a = Dict( # capacity of plant i in cases "seattle" => 350, "san_diego" => 600, ) # b(j) demand at market j in cases # / new-york 325 # chicago 300 # topeka 275 / ; b = Dict( # demand at market j in cases "new_york" => 325, "chicago" => 300, "topeka" => 275, ) # Table d(i,j) distance in thousands of miles # new-york chicago topeka # seattle 2.5 1.7 1.8 # san-diego 2.5 1.8 1.4 ; d_table = wsv""" plants new_york chicago topeka seattle 2.5 1.7 1.8 san_diego 2.5 1.8 1.4 """ d = Dict( (r[:plants],m) => r[Symbol(m)] for r in eachrow(d_table), m in markets) # Here we are converting the table in a "(plant, market) => distance" dictionary # r[:plants]: the first key, row field using a fixed header # m: the second key # r[Symbol(m)]: the value, the row field with a dynamic header # Scalar f freight in dollars per case per thousand miles /90/ ; f = 90 # freight in dollars per case per thousand miles # Parameter c(i,j) transport cost in thousands of dollars per case ; # c(i,j) = f * d(i,j) / 1000 ; # We first declare an empty dictionary and then we fill it with the values c = Dict() # transport cost in thousands of dollars per case ; [ c[p,m] = f * d[p,m] / 1000 for p in plants, m in markets]

The above code take advantage of List Comprehensions, a powerful feature of the Julia language that provides a concise way to loop over a list. If we take the creation of the d dictionary as example, without List Comprehensions we would have had to write a nested for loop like:

d = Dict() for r in eachrow(d_table) for m in markets d = (r[:plants],m) => r[Symbol(m)] end end

Using List Comprehension is however quicker to code and more readable.

### Declaration of the model

Here we declare a JuML optimisation model and we give it a name. This name will be then passed as first argument to all the subsequent operations, like creation of variables, constraints and objective function.

We can, if we wish, works with several models at the same time.

If we do not specify a solver, we let JuML use a suitable solver for the type of problem. Aside to specify the solver, we can also pass it solver-level options, e.g.:
`mymodel = Model(solver=IpoptSolver(print_level=0))`

# Model declaration trmodel = Model() # transport model

### Declaration of the model variables

Variables can have multiple-dimensions - that is, being indexed under several indexes -, and bounds are given at the same time as their declaration.

Differently from GAMS, we don't need to define the variable that is on the left hand side of the objective function.

## Define variables ## # Variables # x(i,j) shipment quantities in cases # z total transportation costs in thousands of dollars ; # Positive Variable x ; @variables trmodel begin x[p in plants, m in markets] >= 0 # shipment quantities in cases end

### Declaration of the model constraints

As in GAMS, each constraint can actually be a “family” of constraints:

## Define contrains ## # supply(i) observe supply limit at plant i # supply(i) .. sum (j, x(i,j)) =l= a(i) # demand(j) satisfy demand at market j ; # demand(j) .. sum(i, x(i,j)) =g= b(j); @constraints trmodel begin supply[p in plants], # observe supply limit at plant p sum(x[p,m] for m in markets) <= a[p] demand[m in markets], # satisfy demand at market m sum(x[p,m] for p in plants) >= b[m] end

### Declaration of the model objective

Contrary to constraints and variables, the objective is always a unique function. Note that it is at this point that we specify the direction of the optimisation.

# Objective @objective trmodel Min begin sum(c[p,m]*x[p,m] for p in plants, m in markets) end

### Human-readable visualisation of the model (optional)

If we wish we can get the optimisation model printed in a human-readable fashion, so we can expect all is like it should be

print(trmodel) </code ==== Resolution of the model ==== It is at this point that the solver is called and the model is passed to the solver engine for its solution. The return value is the status of the optimisation (":Optimal" if all went fine) <code julia> status = solve(trmodel)

### Visualisation of the results

While you can do any fancy output you may wish after you retrieve the optimal value of the variables with `getvalue(var_name)`

, you can just `println(getvalue(x))`

to get a basic output.

Notice that you can also easily retrieve the dual value associated to the constraint with `getdual(constraint_name)`

.

if status == :Optimal println("Objective value: ", getobjectivevalue(trmodel)) println(getvalue(x)) println("Shadow prices of supply:") [println("$p = $(getdual(supply[p]))") for p in plants] println("Shadow prices of demand:") [println("$m = $(getdual(demand[m]))") for m in markets] else println("Model didn't solved") println(status) end

### Editing and running the script

Differently from GAMS you can use whatever editor environment you wish to code a JuMP script. If you don't need debugging features, a simple text editor like Notepad++ (in windows), gedit or kate (in Linux) will suffice. They already have syntax highlight for Julia.

If you want advanced features and debugging capabilities you can use a dedicated Julia IDE, like e.g. Juno.

If you are using instead the Julia console, you can run the script as `julia transport.jl`

.

## Further help

Documentation of JuMP is available from this page. However if you want to do serious things with juMP, it is most likely that you will have to either look at the source code or consult the discussion forum.

Happy modelling with JuMP

## Complete script

Here is the complete script:

#= Transposition in JuMP of the basic transport model used in the GAMS tutorial This problem finds a least cost shipping schedule that meets requirements at markets and supplies at factories. - Original formulation: Dantzig, G B, Chapter 3.3. In Linear Programming and Extensions. Princeton University Press, Princeton, New Jersey, 1963. - Gams implementation: This formulation is described in detail in: Rosenthal, R E, Chapter 2: A GAMS Tutorial. In GAMS: A User's Guide. The Scientific Press, Redwood City, California, 1988. - JuMP implementation: Antonello Lobianco =# using JuMP, DataFrames # Sets plants = ["seattle","san_diego"] # canning plants markets = ["new_york","chicago","topeka"] # markets # Parameters a = Dict( # capacity of plant i in cases "seattle" => 350, "san_diego" => 600, ) b = Dict( # demand at market j in cases "new_york" => 325, "chicago" => 300, "topeka" => 275, ) # distance in thousands of miles d_table = wsv""" plants new_york chicago topeka seattle 2.5 1.7 1.8 san_diego 2.5 1.8 1.4 """ d = Dict( (r[:plants],m) => r[Symbol(m)] for r in eachrow(d_table), m in markets) f = 90 # freight in dollars per case per thousand miles c = Dict() # transport cost in thousands of dollars per case ; [ c[p,m] = f * d[p,m] / 1000 for p in plants, m in markets] # Model declaration trmodel = Model() # transport model # Variables @variables trmodel begin x[p in plants, m in markets] >= 0 # shipment quantities in cases end # Constraints @constraints trmodel begin supply[p in plants], # observe supply limit at plant p sum(x[p,m] for m in markets) <= a[p] demand[m in markets], # satisfy demand at market m sum(x[p,m] for p in plants) >= b[m] end # Objective @objective trmodel Min begin sum(c[p,m]*x[p,m] for p in plants, m in markets) end print(trmodel) status = solve(trmodel) if status == :Optimal println("Objective value: ", getobjectivevalue(trmodel)) println("Shipped quantities: ") println(getvalue(x)) println("Shadow prices of supply:") [println("$p = $(getdual(supply[p]))") for p in plants] println("Shadow prices of demand:") [println("$m = $(getdual(demand[m]))") for m in markets] else println("Model didn't solved") println(status) end # Expected result: # obj= 153.675 #['seattle','new-york'] = 50 #['seattle','chicago'] = 300 #['seattle','topeka'] = 0 #['san-diego','new-york'] = 275 #['san-diego','chicago'] = 0 #['san-diego','topeka'] = 275

^{1)}

## Discussion

You can use a normal `if`/`else` statement or use the more concise ? ternary operator:

aVariable = CONDITION ? ExprIfTrue : ExprIfFalse

Obviously if the condition is whitin a model to optimise, the model will need to be declared nonlinear.