So, now that we are considering uncertain demand, we need to go and revisit how we figure out what other quantity we need to order. Until now, we've made the assumption that we need to calculate the EOQ and that's the quantity that we are. However, it's not unusual for demand to be lumpy, and what does that mean? It simply means that instead of demand coming in for single units of whatever we are looking at, demand might come in for multiples of the units at the same time. For example, if I'm selling shirts, it's possible that, let's assume that we are looking at white shirts, in fact, the particular brand that I'm wearing. It's possible that most people come in and order one shirt at a time, but there might be someone who might come in and order five shirts at the same time, and so my demand becomes lumpy because of this. Now, let's look at the case where we are observing the demand and right before we cross the reorder point, we are about the reorder point and then somebody comes in and orders a multiple, say five shirts. So, all of a sudden, we drop down below the reorder point. Now, immediately, the computer system is going to trigger an order and under our normal ordering policy, would have ordered an EOQ, but notice that we are lower than the reorder point by four now because there was an order for five instead of an order for one. So, when we get our order shipped to us, we will end up with an inventory level that is not as high as we would have wanted it to be. It's four less than where we wanted it to be. To overcome this, we do what's called an order up to policy. So, what's an order of policy? Instead of ordering the EOQ quantity, what we do is we calculate an order up to level which is the reorder point plus the EOQ quantity. Now, this establishes the high level of inventory that we are going to ever experience. So, this is the high watermark for inventory for us. Now, the order quantity becomes the order up to level that we just calculated minus the inventory position at the time that we order, and let's look at an example to see how this works. Let's assume that our reorder point is 48 and let's have an economic order quantity of 274. So, our order up to level is going to be 274 plus 48 which is 322. Now, suppose right around the time that we were supposed to order, we got a bulk order, which dropped our inventory position down to 43. In that case, our order quantity is going to be 322 minus 43 which is equal to 279. Notice, this is higher than the EOQ which was to 274 because we are now ordering five extra units so that we can top up with the order up to level. Now, we've set up our inventory system whereby we know how much to order, we know when to order. We have tried to minimize the costs associated with ordering and carrying inventory. We have tried to minimize the probabilities of stock-outs by providing a pre-determined service level. Now, suppose we start executing this inventory system that we've constructed, and we would like to measure the performance of this inventory system. Now, remember how well this inventory system performs, one of the measures of that is what kind of service level we are able to provide. We selected that service level initially and we used it to calculate safety stock etc. However, there's a problem with figuring out whether our inventory system is actually meeting that service level. If you recall, service level is the probability of stock out occurring during the lead time. Now, if I want stock outs to occur infrequently, as I normally do, my service levels are going to be high, 95 percent even 99 percent. Which means the chance of actually seeing a stock out is relatively small. So, to be able to get a good feel for whether I have and whether I'm meeting my service level as we defined it before, I may have to observe perhaps 100 cycles to see one stock out if my service level was 99 percent. Now, an inventory cycle might be several weeks. Let's say an inventory cycle is four weeks. In that case, to observe 100 cycles, I have to wait for 400 weeks. That's roughly 50 years for me to know whether I'm meeting my service level. Now, that doesn't seem like a very practical thing to do. So it takes way too many cycles to calculate it. Secondly, service level is only concerned about what happens during the lead time. Does a stock out occur during the lead time? Because remember, we cannot have a stock out during the rest of the cycle because that's how we've constructed our policy that we watch and when we reach the reorder point, that's when reorder. So, until then, obviously there cannot be a stock out because the reorder point always is a positive inventory. Then lastly, service level has a third problem, which is we get credit for having provider service, if you do not have a stock or during the lead time and get no credit, if we have a stock out during that period. So, we either have a stock out or we don't have a stock out and so we either get credit or we don't get credit. Now, during that lead time, we might have taken care of most of the demand during lead time, but the fact that we had a stock out right at the end means that we get deemed for the whole period. So, for all these various reasons, service level is not a very practical thing for us to actually look at when we are measuring performance. So, accordingly, there is a different measure that's used and this is what's commonly called the fill rate. So, the fill rate is the fraction of demand that is satisfied or filled from inventory. This is a much easier thing to both understand and to measure, because I can take any period of time. I can take a week, a day, I can take a year and I can look at how much demand was requested by the customer. Then, I can look at how much of that demand was I able to satisfy from inventory and that allows me then to calculate the fill rate. There tends to be some confusion sometimes in practice because people often talk loosely about service being provided. So, when they talk about the service level that they are achieving, they're actually talking about the fill rate that's being achieved. In a broad sense, fill rate does measure the service that is being offered, but it's different from the service level. Now, to complicate matters further, there are different ways to measure the fill rate. The first measure is what's called the unit fill rate. So, for example, if during a period of time, let's say during a week, if customers came requesting 100 units of a particular item and we were able to provide them 90 of that item, then the unit fill rate is 90 divided by 100. Now, in large companies, we don't deal with a single item we deal with hundreds of thousands of items and when customers order, they don't just order one thing from us, they order several things. So, suppose a customer orders three items from us and we were able to satisfy completely two of those items from inventory and the third we only have enough to send a partial amount, then from that order, there were three lines and we are only able to fill two of those lines. So, we say that the line fill rate is two out of three. Then, we also have something called the order fill rate which goes across customers. So, I might look at all the orders that came in and I might look at how many of those orders was I able to satisfy completely. So, I look at the total orders that I successfully filled divided by the total orders that were requested and that's called the order fill rate. Let's take an example. Here is an example where we have two customer orders, and the first order is for three items and the second order is for four items. Let's look at the different quantities. So, if I look at the first order, order number 1980-23, I notice that I had three items that were ordered. The total of all items that were ordered happens to be 280, which is the 150 plus the 80 plus the 50. So, that's 280 items were ordered. I was able to ship 110 for the first item, 80 for the second item and 40 for the third item. So, I was able to ship 230 items out of the 280 requested and that gives me a unit fill rate for that particular order equal to 82.1 percent. Now, as you can see in the last column, of the three items that were ordered, I was not able to satisfy two of them and satisfy one of them. That gives me a line fill rate of one out of three, which means 33.3 percent. I can similarly do that for order number 9350-38, and in this particular case, the total number of items ordered, if I sum up the quantities ordered, turns out to be 485. The quantity shipped turns out to be 464 and that gives me a unit fill rate of 95.67. So, if I look at the unit fill rate that's quite a high unit for rate. However, when I look at the line fill rate, none of those four items was I able to ship the complete amount requested by the customer. So, I was not able to fill a single line completely, and so my line fill rate happen to be zero percent. So, the line fill rate is a much stricter measure than the unit fill rate and therefore, you expect line fill rates to be much lower than unit fill rates. Let's look at the overall performance across the two orders. So, now I look at the sum across all of them, and so, if I look at the unit fill rate, I get a unit fill rate of 90.85 percent, I get a line fill rate since I was only able to satisfy one line of the two orders together out of the seven lines that were requested, I get a line fill rate of 14.29 percent, and since neither order was completely filled, I get an order fill rate of zero or two or zero percent. So, the line fill rate tends to be much more severe than the unit fill rate, and the order fill rate tends to be an even stricter measure than the line fill rate. Now, how does the fill rate compare with the service level? So, if you'll recall, the service level was an all or nothing measure. The fill rate gives me partial credit for at least shipping some of the order. The service level was only concerned about the lead time. The fill rate does not distinguish between inverting the cycle I am. So, I'm looking at the entire cycle any order that comes any time during the entire cycle is part of the fill rate. The service level was hard to measure, we talked about how it might take us four years to figure out whether we are meeting our service level if we are aiming for a fairly high service level and we have longs order cycles. The fill rate is much more intuitive, it's what part of the demand have we been able to satisfy. The service level required us to look at multiple order cycles to be able to do this. The fill rate, I can do for any time period, I can do it even for a single order. So, the fill rate is a much more easier thing to calculate and so accordingly is the way people measure the service provided by an inventory system. However, to figure out what the safety stock is, they still usually use a service level. So, both are important for us. One, to design the system, the other one to measure the performance of the system.
