Optimization in Julia

Julia is a relatively young technical computing language. One of its features is JuMP (Julia for Mathematical Programming), a powerful package that allows mathematical models to be built and solved within the Julia environment. My first blog post offered a simple model for scheduling the Big Ten conference in a round-robin tournament. To complete the Circle of Life, this final blog post will revisit the model by implementing it in Julia/JuMP.

First things first! Let’s tell Julia we need to use the JuMP package for creating a model and the CPLEX package for accessing the CPLEX solver.

using JuMP, CPLEX

Previously, we defined a set M of teams in the Big Ten and a set D of allowable days for play. To avoid dealing with byes on a 14 team schedule, we specified there to be 13 days. We’ll create an array of days storing the numbers 1 through 13.

M = ["Ind" "UMD" "UMich" "MSU" "OSU" "Penn" "Rtgrs" "Ill" "Iowa" "UMN" "UNL" "NU" "Purd" "UW"]
D = collect(1:13)

We now create a JuMP model object with CPLEX as the solver. This model will have variables, constraints, and an objective added to it. CPLEX options can be passed as arguments into the CplexSolver object; e.g., CPX_PARAM_THREADS=4.

model = Model(solver=CplexSolver())

The binary decision variables will be x_{ijt}=1 if team i\in M hosts team j\in M on day t\in D, and 0 otherwise. Let’s add these variables to our model. This is accomplished with the @variable macro. JuMP is even kind enough to allow us to index our variables with our team names!

@variable(model, x[M,M,D], Bin)

Obviously, a team can’t play itself. Let’s set all binary variables of the form x_{iit} equal to 0, where i\in M and t\in D. Constraints are added to JuMP models using the @constraint macro.

for i in M
	for t in D
		@constraint(model, x[i,i,t] == 0)
	end
end

Our original model included the constraint that each team can play at most once per day, whether they are home or away. The mathematical formulation of this constraint is below.

\displaystyle\sum_{\substack{j\in M: \\ j\neq i}} \left( x_{ijt} + x_{jit} \right) \leq 1 \phantom{XXXXX}\forall i\in M, t\in D

The following code in Julia adds this exact constraint to our model. We add this constraint for every combination of teams (M) and days (D).

for i in M
	for t in D
		@constraint(model, sum{x[i,j,t] + x[j,i,t], j in setdiff(M,[i])} <= 1)
	end
end

The sum function sums the expression x_{ijt} + x_{jit} over the indices j\in M\setminus \{i\}. The Julia command for referencing this set of indices is setdiff(M,[i]).

Because we want a round-robin schedule, each team must play every other team exactly once. In the first blog post, we saved ourselves from adding redundant constraints by adding a constraint for all i\in M and j\in M such that i<j, not i\neq j.

\displaystyle\sum_{t\in D} \left( x_{ijt} + x_{jit} \right) = 1\phantom{XXXXX}\forall i\in M, j\in M : i<j

We add this constraint to the JuMP model for all i\in M and j\in M satisfying i<j.

for i in 1:length(M)
	for j in M[i+1:end]
		@constraint(model, sum{(x[M[i],j,t] + x[j,M[i],t]), t in D} == 1)
	end
end

So far, there’s nothing in our model that prevents a team from playing all home or all away games. Let’s make sure each team plays either 6 or 7 home games, as well as 6 or 7 away games. These can be accomplished together by stipulating that all teams play at least 6 home games. Our mathematical constraint was

\displaystyle\sum_{\substack{j\in M: \\ j\neq i}} \displaystyle\sum_{t\in D} x_{ijt} \geq \left \lceil{\frac{|D| - 1}{2}}\right \rceil \phantom{XXXXX}\forall i\in M.

In Julia, we will implement this constraint by simultaneously summing over the two indices mentioned above: j\in M\setminus\{i\} and t\in D.

for i in M
	@constraint(model, sum{x[i,j,t], j in setdiff(M,[i]), t in D} >= ceil((length(D) - 1)/2))
end

The final constraints explicitly provided in the first blog post limited the number of consecutive home and away games. This is important to facilitate a healthy, “mixed” schedule. These constraints stated that a team cannot play more than two home games within any consecutive three-game window. Similarly, teams cannot play more than two away games within any consecutive three-game window.

\displaystyle\sum_{\substack{j\in M: \\ j\neq i}} \displaystyle\sum_{\substack{s\in D: \\ t\leq s \leq t+2}} x_{ijs} \leq 2 \phantom{XXXXX} \forall i\in M, t\in D: t\leq |D| - 2

\displaystyle\sum_{\substack{j\in M: \\ j\neq i}} \displaystyle\sum_{\substack{s\in D: \\ t\leq s \leq t+2}} x_{jis} \leq 2 \phantom{XXXXX} \forall i\in M, t\in D: t\leq |D| - 2

The JuMP analog of these constraints again sums over two indices simultaneously.

for i in M
	for t in 1:length(D)-2
		@constraint(model, sum{x[i,j,s], j in M, s in t:t+2} >= 2)
		@constraint(model, sum{x[j,i,s], j in M, s in t:t+2} >= 2)
	end
end

These are all of the constraints we added to the original model. All that’s left is to solve the model.

status = solve(model)

The solve command will return a symbol, stored in the status variable, that lets us know how the model solved. It could take on a value of :Optimal, :Unbounded, or :Infeasible, among other values. Fortunately for us, this model returned an “optimal” value. Remember that at this point, we are just searching for a schedule feasible for our constraints. After tidying up the output, we have a complete schedule.

big_ten_schedule_1

However, this may not be the perfect schedule. Notice that there are 23 instances of teams playing back-to-back away games. Let’s say we wanted to reduce this number as much as possible. How would we do it?

One way to minimize the total number of back-to-back away games is to introduce a binary variable y_{it} to be equal to 1 if team i\in M has back-to-back away games beginning on day t\in D\setminus \{|D|\}, and 0 otherwise.

@variable(model, y[M,1:length(D)-1], Bin)

These variables should be triggered to equal 1 if a team plays two away games in a row. This can be accomplished with the following constraint, courtesy of Felix the Cat’s Slide of Tricks from ISyE 323.

\displaystyle\sum_{\substack{j\in M: \\ j\neq i}} \displaystyle\sum_{\substack{s\in D: \\ t\leq s \leq t+1}} x_{jis} \leq 1 + y_{it} \phantom{XXXXX} \forall i\in M, t\in D: t\leq |D| - 1

The JuMP constraint looks very similar to the previous “window” constraints.

for i in M
	for t in 1:length(D)-1
		@constraint(model, sum{x[j,i,s], j in M, s in t:t+1} <= 1 + y[i,t])
	end
end

The last step is to add an objective function to our JuMP model. We want to minimize the total number of y_{it} variables that are equal to 1.

\min \displaystyle\sum_{i\in M} \displaystyle\sum_{\substack{t\in D: \\ t\leq |D|-1}} y_{it}

Objective functions are added to JuMP models with the @objective macro. We declare the sense of the objective function to be Min because we are minimizing our linear objective. Summations in the objective function behave the same way as they do in the constraints.

@objective(model, Min, sum{y[i,t], i in M, t in 1:length(D)-1})

Our model is complete! After executing the solve(model) command once more, we have a new solution. This new model is very difficult to solve, so we present an improved solution obtained after running the solver for a few minutes.

big_ten_schedule_2

This schedule features 7 instances of back-to-back away games. This is an enormous improvement to our earlier solution!

JuMP is also capable of more advanced solving techniques, such as solver callbacks for lazy constraints and user cuts. If you’re interested in trying out Julia/JuMP, you can download Julia here. Packages are incredibly easy to add to your Julia distribution (Pkg.add("JuMP")). Happy optimizing!

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