Supply Chain

JHU Carey Business School | BU_610.760_SUPPLY CHAIN ANALYTICS_G.AYDIN_MODULE 4_VIDEO 3

INSTRUCTOR: Let's now put these ideas together around inventory pooling. I'm switching to inventory pooling at this point because this is going to be one of the central ideas about distribution networks, the idea of inventory pooling. What do we mean by inventory pooling?

Going with my smaller example, and we're going to generalize from here eventually, but going with my smaller

example, if ABC decides to pool all of its inventory in one location, I'm missing the word one there on that slide, so in one location. They are essentially saying I'll take the East and West warehouses, I'll combine them in one

location. Let's call it the central warehouse. That's-- at this very basic level, is inventory pooling. I'll take two

locations at which are used to keep inventory, I'll put them in one and the same location.

So, when we do this, we can assume that the week demand at the central warehouse will be the sum of the

weekly demands at the East and West warehouses. Whatever demand was coming to the East and the West will now be channeled to the central warehouse. And let's also assume that the lead time from the plants to the

central warehouse will still be three weeks. Remember, when we had two separate warehouses, it was taking

three weeks for the plant to send the shipment to the East or to the West. Let's say to the central warehouse also, it takes three weeks to send the shipment from the plant.

Let's also assume that ABC decided to maintain the same total safety stock as before because in the last part of the example, we calculated if I want 95% cycle service level, here is the safety stock I need in the East. Here is

the safety stock I needed in the West, right? So now I'm saying when they replace the East and the West with

one warehouse called the central warehouse, they decided to keep the same safety stock. So they will add the

two safety stocks together, 65 from the East, 98 from the West, a total of 164 units of safety stock in this new

central warehouse.

The question I'm asking at the end is, what is the cycle service level ABC will achieve at this central warehouse, and how does it compare to the 95% cycle service level ABC was achieving before pooling when it had the East and the West separately? So let's answer this question. What is the cycle service level ABC will achieve at the

central warehouse? To be able to answer it, I'm going to have to remind you again a couple of things from your

statistics classes, probability classes, and so on.

But the key thing we need to figure out before we can answer that question is, what is the weekly demand at the

central warehouse? What do we know about that? Well, what we know about the weekly demand at the central warehouse, let me back up by one slide. It's the sum of the demands in the East and the West. I know the

demand in the East, it's some normal distribution. I know the demand in the West. It's some normal distribution. The demand in the central warehouse is the sum of the two. So it's adding two normal distributions together.

How do we get that resulting demand distribution for the central? So, recall from your statistics courses that if I have x and y and both of them are normally distributed variables and let's say they are statistically independent, meaning there is no correlation between them. Let's say the mean of x is given by mu sub x as notation, and

sigma sub x is the standard deviation of x. So the variance of x would be sigma x squared. Let's say the mean

and standard deviation of y are mu sub y and sigma y. So the variance of y is sigma y squared.

Then here's what I know about when I add x and y together. Here's what I know about x plus y. x plus y would

also be normally distributed. It's mean would be the mean of x plus the mean of y. Makes sense, right? Take the

mean of x, take the mean of y, add the two together, you get the mean of x plus y. And its variance would be the

variance of x plus the variance of y. Again, hopefully that makes sense. Take the variance of x, take the variance

of y, add the two together, you get the variance of x plus y.

By the way, if this is the variance of x plus y, then you know that standard deviation is a square root of variance. So the standard deviation of x plus y would be the square root of that. So we can now apply this to ABC. The

weekly demand at the central warehouse, it's the sum of the East and West demands, right? East demand is

normal, West demand is normal. Therefore, the weekly demand at the central warehouse also has a normal distribution. Its mean would be the mean of East plus the mean of West. Mean of East is 100, mean of West is

100, that's 200. So I know the mean of weekly demand in central warehouse.

Its variance is, OK, that's the variance of East plus the variance of West. What is the variance of East? Well, back

up a couple of slides here. We know the standard deviation in the East. The standard deviation in the East is 20. Therefore, its variance is 20 squared. So the variance of East is 20 squared. Plus, the variance of y. What is the

variance of y? Its standard deviation is 30, if you remember from the example, the variance of the West warehouse. Multiply-- take the square of that. So the variance of the central warehouse demand is going to be if you do the algebra there, that algebra comes out to 1,300. And therefore, the standard deviation at the central warehouse is the square root of 1,300, which gives me 36.1.

So there is my answer to the question about what is the weekly demand at the central warehouse? It's normally

distributed. It's mean is 200, its standard deviation is 36.1. And now that I have that, I can go back and calculate

the cycle service level we are achieving at the central warehouse, and this gives me one more opportunity to

repeat once more the mechanics of going from an order-up-to level to going from an order-up-to level or safety

stock to a cycle service level. So at the central warehouse, here's what I know is my input data. I calculate the

safety stock that I must have at the central warehouse, right? It was the safety stock of the East plus the safety

stock of the West. So we calculated that earlier to be 164 units.

We know that the lead time there is three weeks, and we know that we just calculated the standard deviation of weekly demand at the central warehouse is 36.1 units. So what is sigma sub L? Again, remember what sigma

sub L means. If sigma is standard deviation of per-period demand, the standard deviation of weekly demand, let's say, sigma sub L is standard deviation of total demand during L plus 1 periods. And that is square root of L

plus 1 times the sigma itself.

L is 3 plus 1 times sigma, we calculated earlier 36.1. If you do the algebra there, that comes out to be 72.2. Now I can calculate z, what we're calling the safety factor. The formula for safety factor is here. Take the safety stock

divided by sigma sub L. Safety stock is here, 164.5 divided by sigma sub L is here, 72.2. That comes out to be

2.27. And finally, the cycle service level, I can calculate that by plugging into this Excel function NORMSDIST, short for normal standard distribution. Plug the z value in there. It will give you 98.8%.

Now notice the result. At the central warehouse, the cycle service there will be 98.8%. When we were keeping

the inventory separately in the East and the West, the cycle service level in each was 95%. So by putting the two

together, we improved our cycle service level even though we did not increase our overall safety stock. We just combined it in one place and the formulas tell us that the cycle service level will improve. Why is this happening?

Well, because in general, inventory pooling, we expect will improve customer service level by combining safety

stocks in one location. The intuition behind this is similar to diversifying your portfolio of investments. You don't put all your money in GameStop, because if GameStop goes down, then you lose all your money. You don't put all of your money in crypto, because if crypto goes down, then you lose all your money. Instead, you diversify

your portfolio of investments so you're not too vulnerable to changes in one stock only.

The idea is similar here. When we pool our inventory to meet the demands from multiple sources-- in this case, East and the West-- there will be times when the East demand is high and the West demand is low, and there will be times when the West demand is high and East demand is low. Now you add the two together, those are-- the

sum is sort of curves, the peaks and valleys that you see in each. It leads to a more stable demand pattern

overall, and when you have a more stable pattern overall, it becomes easier to offer better service to your

customers.

So that's the intuition behind why the cycle service level went from 95% when we had two separate locations, to

98% when we combine those two in the same location. But, of course, inventory pooling can have lots of disadvantages that our small example did not really account for. So, for example, you might have some

customers in the far side of the East and the far side of the West, and they might say, you know what? You took

away my warehouse and you put it in the center. It's too far away from me now, so I'll stop buying from you. So

that would be one possible disadvantage of consolidating two warehouses from-- in the center.

It's also true that you're now going to be transporting over larger distances by shipping to customers. When you

had East and the West, East was serving only East customers, West was serving only West customers. Now

central will have to serve customers both in the East and the West. So your customers, your shipments from your

warehouse to your customers, will travel longer distances which might mean that your transportation costs will go up. So those are some possible disadvantages of inventory pooling. And we'll see some of those in our next two cases.

But at this point, the fundamental trade-off in designing distribution networks, generally, we expect that when we

pool the inventory at different locations, it helps us serve our customers better with even less safety stock. So

that's a good thing. Let's say stocks, better service, win-win, by pooling inventory in fewer locations. But also, it puts us further away from our customers if we have fewer locations. And that could have disadvantages. So how

do we manage this trade-off? This is some of what we will see as we move on to cases to look at more elaborate

examples of this inventory pooling idea and distribution networks.