## Tuesday, April 21, 2020

### A CP Model for Toasting Bread

A question on Mathematics Stack Exchange deals with a problem (apparently from the book "Thinking Mathematically") about toasting bread on a grill. The grill can hold two slices at a time and can only toast one side of each at a time. You have three slices to toast, and the issue is to figure out how to do it in the minimum possible amount of time (what operations management people refer to as the makespan).

The questioner had a solution that I was able to prove is optimal, using a constraint programming (CP) model that I coded using the Java API to IBM's CPOptimizer (part of the CPLEX Studio product). I won't swear my model is elegant or efficient, since I'm pretty new to CPO, but I think it is correct. If anyone getting started with CPO and the Java API wants to see the source code, it is available in my repository. I'll describe a few key aspects below.

I assumed in my model that there is a single cook. The fundamental components of the model are CPO "interval variables" (instances of IloIntervalVar) for each task (inserting a slice, toasting one side, removing a slice, flipping a slice) along with a dummy task for being done and a placeholder task I called "reversing". Interval variables represent time spans during which tasks are done.

In the problem, there are two ways to get from toasting the front of a slice to toasting the back: you can leave it on the grill and flip it; or you can remove it and (later) replace it with the other side up. Since I didn't know a priori which slices will be handled which way, I created interval variables for removing each slice after the first side, replacing each slice with the second side up, and flipping each slice. Those variables are declared optional, meaning each interval may or may not show up in the solution. For each slice, there is an interval variable for the "reversing" task that is not optional. Each slice has to be reversed, one way or the other. The tasks for replacing a slice (after removing it) and for flipping the slice are listed as alternatives for the reversing task for that slice, which means exactly one of flipping or replacing must be in the solution. Separate constraints ensure that a slice is reinserted if and only if it was removed after the first side toasted. Those constraints use the IloCP.presenceOf function to test whether the remove and reinsert intervals are present, and set the results equal (so both present or neither present).

The sequencing of operations (insert, toast first side, reverse, toast second side, remove) is enforced through a bunch of constraints that use IloCP.endBeforeStart (which says the first argument has to end before the second argument starts). The dummy "done" task is sequenced to start only after each slice has been removed for the final time. I'm pretty sure the objective value (the time everything is done) could be handled other ways, but I tend to think of completion as a zero-length task.

The cook can only do one thing at a time. This is handled using the IloCP.noOverlap function. It is passed a list of every interval that requires the cook's attention (everything but the actual toasting tasks), and prevents any of those intervals from overlapping with any other.

Finally, I need to prevent more than two slices from occupying the grill at any one time. The noOverlap function is no help here. Instead, I use an instance of IloCumFunctionExpr, which represents a function over time that changes as intervals begin and end. In this case, the function measures occupancy.  This is handled by treating the usage as a combination of step functions (IloCP.stepAtStart and IloCP.stepAtEnd). Usage steps up by one at the start of the task of inserting a slice and steps down by one at the end of the task of removing a slice. (Toasting and flipping have no effect on occupancy.) The Javadoc for the relevant functions a bit, um, sparse, essentially saying nothing about the step height argument. Thus I discovered the hard way (through error messages) that adding a step with height -1 when a slice was removed was not acceptable. Instead, I had to subtract a step of height +1.

Although it was not really necessary on a problem this small, I removed some symmetry resulting from the arbitrary ordering of the slices by setting precedence constraints saying that slice 1 is started before slice 2, which in turn is started before slice 3.

It is possible to model the problem as an integer program, and in fact I initially leaned that direction. The IP model, however, would be bulkier and less expressive (which would make it more prone to logic errors), and quite possibly would be slower to solve. CPOptimizer is designed specifically with scheduling problems in mind, so it is the better tool for this particular job.