Machine Scheduling

Here [1] is a problem with multiple machines:

• There are 3 machines and 4 jobs in this example
• Not all machines can do all jobs
• We have a few precedence constraints
• Given known processing times, find a schedule that minimizes the makespan (the time the last job is finished)

Binary variables
\[\color{DarkRed}{\mathit{assign}}_{j,m} = \begin{cases} 1 & \text{if job $j$ is placed on machine $m$}\\ 0 & \text{otherwise}\end{cases}\] \[\color{DarkRed}{\mathit{after}}_{j_1,j_2} = \begin{cases} 1 & \text{if job $j_1$ is executed after job $j_2$ when placed on the same machine}\\ 0 & \text{if job $j_1$ is executed before job $j_2$ when placed on the same machine}\end{cases}\]

Note that the variable \(\mathit{after}_{j_1,j_2}\) will be used only for \(j_1 \lt j_2\). This is to avoid double checking the same pair.
We also have a few continuous variables:

Continuous variables
\[\begin{align} & \color{DarkRed}{\mathit{start}}_{j} \ge 0 && \text{start time of job $j$}\\ & \color{DarkRed}{\mathit{finish}}_{j} \ge 0 && \text{finish time of job $j$}\\ & \color{DarkRed}{\mathit{makespan}} \ge 0 && \text{last finish time} \end{align}\]

Constraints

The model can look like:

Mixed Integer Programming Model

\[\begin{align} \min\> & \color{DarkRed}{\mathit{makespan}} \\ & \color{DarkRed}{\mathit{makespan}} \ge \color{DarkRed}{\mathit{finish}}_j && \forall j \\ & \color{DarkRed}{\mathit{finish}}_j = \color{DarkRed}{\mathit{start}}_j + \color{DarkBlue}{\mathit{proctime}}_j && \forall j\\ & \sum_m \color{DarkRed}{\mathit{assign}}_{j,m} = 1 && \forall j\\ & \color{DarkRed}{\mathit{start}}_{j_1} \ge \color{DarkRed}{\mathit{finish}}_{j_2} - \color{DarkBlue}T (1-\color{DarkRed}{\mathit{after}}_{j_1,j_2}) - \color{DarkBlue}T (1-\color{DarkRed}{\mathit{assign}}_{j_1,m})- \color{DarkBlue}T (1-\color{DarkRed}{\mathit{assign}}_{j_2,m}) && \forall m, j_1 \lt j_2 \\ & \color{DarkRed}{\mathit{start}}_{j_2} \ge \color{DarkRed}{\mathit{finish}}_{j_1} - \color{DarkBlue}T \color{DarkRed}{\mathit{after}}_{j_1,j_2} - \color{DarkBlue}T (1-\color{DarkRed}{\mathit{assign}}_{j_1,m})- \color{DarkBlue}T (1-\color{DarkRed}{\mathit{assign}}_{j_2,m}) && \forall m, j_1 \lt j_2 \\ & \color{DarkRed}{\mathit{start}}_{j_2} \ge \color{DarkRed}{\mathit{finish}}_{j_1} && \forall \color{DarkBlue}{\mathit{prec}}_{j_1,j_2} \\ & \color{DarkRed}{\mathit{assign}}_{j,m} = 0 && \forall \text{ not } \color{DarkBlue}{\mathit{ok}}_{j,m} \end{align}\]

The no-overlap constraints are the most complicated in this model. If the problem was a single machine scheduling problem, the no-overlap constraints would look like: \[\begin{align} & \mathit{start}_{j_1} \ge \mathit{finish}_{j_2} - T (1-{\mathit{after}}_{j_1,j_2}) && \forall j_1 \lt j_2 \\ & \mathit{start}_{j_2} \ge \mathit{finish}_{j_1} - T {\mathit{after}}_{j_1,j_2} && \forall j_1 \lt j_2 \end{align}\] i.e. job \(j_1\) executes before or after job \(j_2\), but not in parallel. When we have multiple machines, we only want to keep this constraint active when jobs \(j_1\) and \(j_2\) are assigned to the same machine. That is: jobs are allowed to execute in parallel when they are on different machines. In a sense, we have built some nested big-\(M\) constraints.

Results

After running this model we get as results:




How should we interpret the variable \(\mathit{after}\)? Only jobs 1 and 2 are on the same machine and for this combination we have \(\mathit{after}_{1,2}=0\). This means job1 is not after job2, and thus: job2 must be after job1. This is indeed what is happening. The values for \(\mathit{after}\) for jobs not on the same machine are basically just random: they are not used in active constraints. This means: the printed value \(\mathit{after}_{1,3}=1\) is not relevant as jobs 1 and 3 are on different machines.


Graphically this looks like:


We see that this solution obeys the precedence constraints (jobs 2 and 4 after job 1). Furthermore, it only assigns jobs to allowed machines (e.g. job 4 can only run on machine 3).

Larger data set

The above data set is very small. Such a data set is great during development. A beginners mistake I often see is using a large data set during model development. It is much more convenient and efficient using a very small data set when things are in flux, and you do many runs just to get the model right.

Now, it is important to see how the model behaves with a large data set. I used random data with 8 machines and 50 jobs. For this problem we only needed a few seconds to solve to guaranteed optimality. The model had about 20k constraints, 1,700 variables (of which 1,400 binary variables).

solution of 8 machine, 50 job problem with random data

The performance seems to be quite reasonable. The main problem is that for large models the number of constraints becomes big, and the big-\(M\) constraints may not be as tight as we want. But for a first formulation, not so bad.

References