# I don't understand the constraints for this scheduling problem

In a flexible job-shop scheduling problem, we are trying to minimize the makespan. We have $n$ jobs that need to run on $m$ machines. Each job $i$ consists of $n_{i}$ operations ($O_{i1},O_{i2},…,O_{in_{i}}$). For each operation $O_{ik}$, processing machine must be from the machine set $A_{ik}$.

Here is the formulation of variables and parameters:

Indices:

$i, h$: job index; $i, h = 1,2.,...,n$

$j$: machine index, $j = 1,2,...,m$

$k,g$: operation index, $k,g= 1,2,...,n_{i}$

Parameters:

$n$: total number of jobs

$m$: total number of machines

$n_{i}$: total number of operations of job $i_{t}$

$t_{ikj}$: processing time of $k$th operation of job $i$

Decision variables :

$c_{ik}$: completed time of $O_{ik}$

$x_{ikj}$: machine $j$ is selected for $O_{ik}$

The last two constraints are intuitive, with the variable $x_{ikj}$ either being wrong or true, and the time needed for each operation always being positive. But I can't easily formulate the remaining three constraints.

The first constraint says that the elapsed time between starting and ending one of the operations ($k$) in a particular job ($i$) is at least the time required for that operation on machine $j$ ($t_{ikj}$) if the operation is done on machine $j$ ($x_{ikj}=1$) and at least 0 if not ($x_{ikj}=0$).
The second constraint prohibits jobs from overlapping on a machine. Note that the middle operator ($\vee$) means "or", so the constraint is satifisfied if either bracketed term is true. Also note that if either $x_{hgj}$ or $x_{ikj}$ is 0, the corresponding bracketed term is 0 ($0\ge 0$) and the constraint is satisfied. Now assume that $x_{hgj}=1=x_{ikj}$, meaning operation $g$ of job $h$ and operation $k$ of job $i$ are both being done on machine $j$. $c_{hg} - c_{ik} - t_{hgj} \ge 0$ means that operation $k$ of job $i$ ended before operation $g$ of job $h$ began. The other bracketed term being true means operation $g$ of job $h$ ended before operation $k$ of job $i$ began. So the only way to satisfy this constraint when the same machine is doing both operations is for the operations not to overlap.