Now let's talk about multiple time period models.
Our standard practice, and what we've done thus far,
is that we model a single bucket of time, typically a year.
We want to find what the annual plan is.
And why do we do this?
It fits a typical planning cycle.
Most companies think in terms of annual cycles.
We can always run "What if" analysis or "What if" scenarios
if we want to increase the demand by 50%, 100%,
and do some sensitivity analysis.
And we showed how to do that.
But what if we want to go and see how the demand changes
or how the supply changes over each,
say, month within that year?
What if we want to develop a model that understands what
happens in January impacts what I do in February,
and in March, and so on.
Well, to do this we would have to say,
well does change happen over the course of a year?
And, of course, it does.
So demand, there's a lot of seasonality
that happens to different products
over the course of a year.
Supply, especially in agriculture, a lot of products
are only made at certain times of the year.
And so there's a lot of seasonality to supply as well.
Cost.
There can be different labor rates
over the course of a year based on seasonality, based
on temperature, or weather.
And also the capacity of different plants
in different facilities are also impacted by the weather, plant
shutdowns, and other things.
So there is change over time.
Typically, when I'm doing a strategic network design,
I still look at it over the course of a year.
But it might make sense to consider multiple time periods.
And that's primarily when I'm looking at a tactical plan.
So it's the end, I'm trying to plan for next year
how to flow things.
I'm getting a little more tactical.
I'm not making decisions now of what plants
to open, what DC's open, but how do I flow through my existing
ones?
And so when I want to consider these multiple time periods,
it's almost like adding multiple commodities.
And one similar aspect, it makes the problem very big,
it gets huge.
But it also means adding time indicates that other decisions
are now frozen.
Like the example I just gave.
You're not going to make a decision to open or close
a DC every month.
So if I'm looking at the tactical monthly plan of flow,
I'm going to assume that all the DC's are open.
I've already determined which distribution centers
will be open.
And by having all these different time periods,
you're building a lot more assumptions
and it requires a lot more data.
But let's talk about it.
Let's talk about what the main difference is.
And the primary difference between a single time period
model and a multiple time period model
is that you have conservation of flow now in two dimensions.
We already know about conservation of flow
over space.
So that's where if I have no j, I
know that everything coming in minus everything going out
has to be equal to zero.
Product doesn't disappear.
Well now, I've got to think about it in terms of time
as well.
So we can accumulate inventory now at a node.
And so I have to introduce a new variable, I_jt,
which is equal to the inventory at node j, at time t.
And now each node has a time index instead
of just a spatial index.
Here, this was just node j.
It's the DC in Atlanta, or Boston, or Madrid, or Sydney.
It had just a location.
Now I have a time.
It's the distribution center in Boston
in January, Boston in February.
So they have a time element as well.
So therefore our flow is going to be
a little more complicated.
I have everything coming into it during that time period t.
I've everything going out of it during that time period t.
But then I've also got to consider what was there
before that time period?
So was there at node j,t the time period before.
So that's j,t - 1, the period before.
And then I know what I have in my inventory at node j
at the end of that time period.
So each node, each node time period, has a conservation.
And so the flow in minus the flow
out during that time period, plus the inventory
I started with, minus the inventory I end with,
has to equal zero.
The inventory, either if it comes in,
it either somewhere else or it sits for a time period
in my same location.
And it's captured in this I_j,t term.
So what does a formulation look like?
Not that different.
That's the cool thing about these models.
When you look at the formulation,
you'll see some very common things.
I'm looking at all my variable cost and my flows.
The only difference here is I've got a subscript t
because every time period now counts.
I need to know how much am I flowing from A to B in time
period one, two, and three.
Because it'll differ.
That's why I'm doing multiple time periods.
But I still have my supply constraints, my demand
constraints, and they're all indexed by t now,
so it has to be specific to each time period.
But then I've got something interesting.
I've got this flow, conservation of flow constraint,
which we just talked about.
That the flow into a node minus the flow out of the node
during that time period, plus what
was in that node in the prior time period,
minus what I end up with, is equal to zero.
So that's that conservation of flow we just talked about.
But by introducing this i term, I'm
introducing this second term here in the objective function.
Which is allow me to penalize or cost out a holding
charge for the inventory.
Because if inventory is sitting there,
I'm getting charged for it.
It's a holding cost.
So what I have here for the first time in these network
models, inventory is explicitly being costed out.
You might have noticed that we haven't talked about it too
much up to this point.
And so that's kind of like the hidden secret.
So what do you note about this?
We can charge for inventory that accumulates at the nodes.
That's with this term is doing.
Because if I have inventory sitting there,
I have to charge for it because it's capital that's tied up.
But this allows me to trade off production, movement,
and holding costs.
So we get to see, does it make sense to produce in period one
all the product you need for a year, which means I will incur
a lot of holding cost?
Or is it better to produce a little bit each month
to satisfy each month, which means
my holding cost will go down, but my transportation
costs will go up.
So now we're getting some more, some richer kind of trade offs.
This also sets a production plan for each time period
and it makes the problem really big,
right, because I'm introducing a new term for every time period.
And so you can just think about how big
this would get if it's multi-commodity, multi-time
period.
So this is a very powerful model,
and it's widely used for tactical planning,
for production planning, and that's where we'll see it next.
But it's very similar, when you look at it,
to all the different models that we've done.
You still have supply constraints,
you have demand constraints.
The same things.