How do I write the following constraint?

Question

I have defined the index for time to be T = [1,2,3,4,5,6]. I want that that the demand that to be addressed twice in a given a time period, when the difference between the time the demand is addressed the first time and the demand is addressed the second time is 2. Suppose the demand at a location is is given by $D_i$, where $i \in I$ represents the location of the demand and $\omega_{ijt}$ are the number of people who are meant to be located from location $j$ to location $i$ in a time period $t\in T$. Currently I have that $$\sum_{j \in J} \sum_{t \in T} \omega_{ijt} = D_i $$, which will ensure that the demand is allocated to location $j$ in time t. However, how can introduce the idea of having the same demand reintroduced in my model, when the difference between the demand first addressed and the the model time is $2$.

It will help me greatly, if someone could please help me formulate this idea.

Edit: If I have a demand of 40 people at a, 50 people at b, 60 people at c. I want that all of these people at locations $a,b,c \in I$ visit location $j \in J$ within a certain time period $t < T$, where T can be $6$ weeks and $t$ can be 3 weeks . This means that all of the demand can be addressed within 3 weeks. Next, I want that this demand may only visit the facility again if difference between the time that the demand first visited and next visit will be 3 weeks. I do not know how to model this.

Thanks.

Answer 1

When you reason about sums of people moving from $a$ to $b$ and sums of people being at $a$ at time point $t$ you can't reason at a level of the individual. In your formulation you can't express constraints on the position in time of the individual as you currently don't describe the position where an individual is in your model.

To track the position of individuals you need to introduce a boolean array of $\text{is_at}_{p, l, t}$ where $p$ is the unique number of the person and $l$ is it's location at time point $t$.

Since you probably don't want one person to be at multiple locations at the same time you need introduce some constraint like $\forall p \in \text{Persons}, \forall t \in \text{Time}: \sum_{l \in L} \text{is_at}_{p, l, t} \leq 1$ meaning one person can be at most at one location at a time.

On this state it is possible to express that a certain individual needs to revisit the same place 2 time steps later:

$$ (\text{is_at}_{p, l, t} \land \neg \text{is_at}_{p, l, t-2}) \implies (\text{is_at}_{p, l, t+2}) $$

which reads as if Individual $p$ visited $l$ at time point $t$ but not at $t-2$ then $p$ needs to visit $l$ at $t+2$. You might have noticed that this is not in a MILP form but it can be converted into one using these equivalences. Note that around the boundaries of your time interval you can simplify this formulation as $t+2$ or $t-2$ might not exist and you need to define what that means for your model. This formulation will also be a lot bigger then your current one as you are keeping track of individuals and not just sums of people.

You can also modify this formulation to allow individuals to visit after 2 or 3 or 4 days if he never visited this place before: $$ (\text{is_at}_{p, l, t} \land \neg ( \bigvee\limits_{i=1}^{t-1} \text{is_at}_{p, l, i})) \implies (\text{is_at}_{p, l, t+2} \lor \text{is_at}_{p, l, t+3} \lor \text{is_at}_{p, l, t+4}) $$ Or do another large number of simple modifications that might be able to capture what you want.