Modeling contiguity of machine processing in a flow shop environment via a MIP I'm working on a Mixed-Integer-Programing (MIP) formulation for a flow shop scheduling problem. One of the requirements/wishes is that for each machine $i$, processing should be contiguous, or at least contiguous to a certain extent. What I mean with contiguous processing is visualized below for a single machine:
High contiguity :  Low contiguity: 
Some important details of this problem:

*

*Not all jobs visit all machines. For each machine $i$, the jobs that visit it are a member of the set $\mathcal{V}_i$.

*Operations are not allowed to overlap on a machine.

I would like to influence the contiguity somehow by including it as a part of my cost function. My approach so far was:

*

*For each operation $(i,j)$ of job $j$ on machine $i$, I do have a decision variable $c_{i,j}$ for its completion time. What's more, binary decision variable $x_{i,j,k}$ equals 1 IFF job $j$ precedes job $k$ on machine $i$ (not necessarily directly).

*I introduce a continuous variable $\gamma_{i,j,k}$ that captures the difference between the completion time of operation $(i,j)$ and the start of operation $(i,k)$. That is
$$\gamma_{i,j,k} = \quad (c_{i,k} - p_{i,k}) - c_{i,j}\quad \forall i; j\neq k\in\mathcal{V}_i\times \mathcal{V}_i,$$ where $p_{i,k}$ is the processing time of operation $(i,k)$ so that the first term denotes the corresponding starting time. Note that $\gamma_{i,j,k}$ is negative when $k$ precedes $j$ on machine $i$.

*I introduce a binary variable $b_{i,j,k}$ that equals 1 IFF $\gamma_{i,j,k}\leq 0$. I model this via big-M formulations with sufficiently large constant $Q$:
$$Q\cdot\big(1-b_{i,j,k}\big)-\gamma_{i,j,k}  \geq \quad 0 \quad \forall i; j\neq k\in\mathcal{V}_i\times \mathcal{V}_i$$
$$Q\cdot\big(1-b_{i,j,k}\big)-\gamma_{i,j,k} \leq Q-\epsilon \quad \forall i; j\neq k\in\mathcal{V}_i\times \mathcal{V}_i$$

*I then add the following term to my cost function over which I minimize:
$$-\sum_i\sum_{j\neq k\in\mathcal{V}_i\times \mathcal{V}_i} x_{i,j,k}\cdot b_{i,j,k}.$$ By multiplying $b_{i,j,k}$ with $x_{i,j,k}$ I make sure that only those $b_{i,j,k}$ values contribute where job $j$ actually precedes job $k$ on machine $i$. In other words, for each machine $i$, for each pair of jobs $j,k$ ($j$ precedes $k$ on machine $i$), I look at the time inbetween their completion and start and add $1$ as penalty whenever this time is positive, that is, whenever these jobs do not start immediately after each other.

Note: whenever I use the indices $j,k$ simultaneously, I assume $j\neq k$.
Question:
Not sure if I can ask this here, but I was wondering if I could get some feedback on this. I observe that including this in my cost function dramatically slows up the solving time. Maybe someone has an idea for improvement, or a whole different approach to obtain the same?
Edit:
One issue here is that I do not know which jobs directly succeed each other. If I knew, it would be much easier to solve the problem. Then I would only have to iterate over all pairs of directly consecutive jobs, instead of all possible pairs for every machine. However, given the mentioned decision variables, I don't see how I can determine this.
 A: If I understand the question correctly (not a certainty), you want to reduce the number of times a machine $i$ falls idle. To start, suppose that we redefine $x_{i,j,k}$ to be 1 iff job $j$ immediately precedes job $k$ on machine $i$. That should be a fairly easy change to implement in the overall model, and the number of binary variables remains the same.
Now we add binary variable $z_{i,j}$ for all machines $i$ and all jobs $j\in \mathcal{V}_i$, taking value 1 iff machine $i$ falls idle for a time longer than some threshold $\epsilon > 0$ immediately after processing job $j$. Assuming you are going to penalize $z_{i,j}=1$, we only need to enforce the "if" side; the solver will not set $z_{i,j}=1$ unless it has to. To define $z_{i,j}$, we add the constraints $$s_{i,k} - c_{i,j} \le \epsilon + M(z_{i,j} + 1 - x_{i,j,k})$$ for all machines $i$ and all $j,k \in \mathcal{V_i}$ with $j\neq k$, where $M$ is the usual "large enough" constant and $s_{i,k}$ is the start time for job $k$ on machine $i$. The inequality is potentially binding only if $x_{i,j,k} = 1$ and $z_{i,j}=0$ (meaning $k$ immediately follows $j$ on $i$ with at most $\epsilon$ slack time).
Addendum: To address a comment, here is how to model using immediate precedence. We start with three constraints: $$\sum_k x_{i,j,k} \le 1 \quad \forall i,j$$$$\sum_j x_{i,j,k} \le 1 \quad \forall i,k$$ $$\sum_{j,k} x_{i,j,k} = \vert \mathcal{V_i} \vert - 1.$$These specify that every job has at most one immediate successor and at most one immediate predecessor, and that every job but one has a successor. To that we add the following constraint on timing:$$ s_{i,k} \ge c_{i,j} - M(1-x_{i,j,k}) \quad \forall i,j,k.$$This enforces the rule that the immediate successor of a task cannot start until the predecessor ends. (This is basically the Miller-Tucker-Zemlin TSP formulation with the start and end times taking on the role served by their counting variables (subtour elimination).
