Recently, Brazil volunteered to host the 2014 World Cup and the 2016 Summer Games in Rio de Janeiro. Personally, I think that those guys that put our name on the candidacy lists are nuts. Indeed, Jerome Valcke, FIFA General Secretary, has been warning us about deadlines since May 2010:
“The red light has been switched on. Brazil is not on the right path. This year there is a presidential election, so almost nothing will happen. Next year, carnival comes along. So is everything going to start only after carnival?”
Nevertheless, since there is no way to go back now, maybe OR can help us planning in advance, reducing costs and avoiding last minute surprises (what didn't happen in the 2007 Pan American Games).
First things first, how can we handle an increasingly tighter schedule of planning and operational activities subject to Brazilian reality?
Let’s start with a very simple example using constraint programming (CP) in the modeling language OPL and then elaborate over it.
Decision variables
Let’s say that we have n activities to be performed, ranging from i=1 to n, and a number of vectors representing properties of those activities, such as earlyStart, lateFinish, minSize and maxSize. With such data, it is quite straightforward to model our main decision variables:
dvar interval activity[i in 1..n]
in earlyStart[i]..lateFinish[i]
size minSize[i]..maxSize[i];
An interval is a special structure representing interrelated variables such as start, end and size. Using indexes, it was quite simple to constrain four properties of each interval of the vector with the input data.
Objective function
What about hushing things up? So far, we know that the size of each activity might vary. Probably, it might cost a bit more if we reduce its size. Thus, if we have a deadline to achieve and a finite budget (as a Brazilian taxpayer, I hope for that!), we must try to minimize such costs of anticipating conclusions. Let’s consider that the cost of reducing the size of one activity in one time unit is given by the vector reductionCost. Our objective function is as follows:
minimize sum(i in 1..n) reductionCost[i]*(maxSize[i]-sizeOf(activity[i]));
OPL is a declarative modeling language, for which reason many mathematical expressions are quite easy to describe, like the cost described in the objective function.
Constraints
The constraints of the problem are surrounded by a “subject to” block. To start with, let’s consider a very simple constraint: the execution of some tasks is constrained by others that must precede them. To represent this in OPL, I will use a tuple for each precedence relation and a set to collect them all:
tuple Precedence {
int previous;
int next;
}
{Precedence} precedencies = {<1,2>, <2, 3>, <3,4>};
Now it is easy to place a constraint among the activities:
subject to {
forall (prec in precedencies) {
endBeforeStart(activity[prec.previous], activity[prec.next]);
}
}
About scheduling and constraint programming
Scheduling is one of the best applications of CP to real-life problems. One of the reasons is that modeling using CP is much more concise than it would be if mathematical programming algorithms were used, and that is crucial if you have a problem with many time slots. On the other hand, such generality implies that the algorithms used for CP are not so efficient in the general case. In fact, when one is using CP, feasibility is much more important than optimality, albeit improvements over the initial solution are remarkable nowadays.
But, how does CP work?
Backtracking, mostly: assign a value to a variable, proceed if the constraints have not been violated and roll-back otherwise. That is done until a solution is found or the problem is found infeasible. However, there are many smart tricks to improve such framework, like predicting if a given assignment will unavoidably lead to a dead-end in the search, recording previous conflicts in order to avoid them and jumping to the variable that must be changed in order to avoiding committing the same mistake again. Most of such mechanisms rely on the special structure of each constraint supported by the language, being that the reason for the name of the technique.
Since OPL is intended to abstract search from user, most people just code their problem and run the CP solver embedded. Nevertheless, understanding how CP works can help people designing code for which the solver finds solutions faster and with a higher quality. If interested, you can find more information about OPL and its solver in the IBM website, where there is an interesting white paper about scheduling with OPL and it is possible to ask for a free academic license.
** This post is my first contribution to the INFORMS Blog Challenge of March: OR and Sports.
I hope to have time for a second part of this post within this month. **