So what happens during this period of uncertainty? So during this period of uncertainty, we are experiencing some demand, and that demand is going to have some probability distribution. It is common to assume that this probability distribution is the Gaussian distribution, or what's more commonly called Normal Demand Distribution. You may have seen the figure that's shown here as the bell curve in popular literature. Now, a normal demand distribution is used commonly because of what's called the central limit theorem, which allows us to say that as long as there are a number of different influences, independent influences on something that we are studying, then it's very likely that the thing that we are studying will tend to have a normal distribution. If you look at the sidebar about normal distribution, you will see a little bit more detail about the normal distribution and how to work with the normal distribution. The normal distribution is usually denoted by N for normal, and then it has two parameters, the mean and the variance. So N with the mean and variance specifies what normal distribution we are concerned with. The picture that we have shows in normal 3.3,1 demand distribution, where the 3.3 is the mean and 1 is the variance. Now, remember that the variance is the square of the standard deviation. So if I want the standard deviation of this normal distribution, I take the variance and take the square root of that. So in this particular case. The square root of 1 is equal to 1, so the standard deviation is also going to be equal to 1. So how do we figure out what the reorder point is once we know what the demand distribution is? The way we do this is that to find the average demand, we look at the mean of this demand distribution. So during that period of uncertainty, we experience an average demand and now we have to add to it a safety stock. How much safety stock we add depends on what tolerance I have for stockouts. So for example, if I want to have stockouts no more than 5 percent of the time, so if I say that the probability of a stockout is no more than 0.05. In that case, I calculate what is known as the z-value and for a stockout probability of 0.05, which means the probability of not having a stockout is 95 percent or 0.95, I have to use a z-value of 1.645. So using the z-value, I can calculate what is the reorder point as follows. I take 3.3 the mean, plus the z-value multiplied by the standard deviation, so 3.3 plus 1.645 times 1 gives me 4.945. If on the other hand I want to be even more service conscious, and I decide that I want the probability of stockout to be no more than two and a half percent, in that case I have to use a z-value of 1.96, and that gives me correspondingly a reorder point of 5.26, which is larger than 4.945, which means that I'm carrying extra safety stock to make sure that I have stockouts less frequently. Let's figure out how we are going to calculate what our safety stock is, and let's do this in the form of an algorithm. So let's start out by assuming that we have been given the demand distribution during the period of uncertainty. For our lecture, we are going to assume that this demand is normally distributed. So the first step we have to do is we have to decide what is our tolerance to stockout, so we are going to decide the probability of a stockout, and we're going to call this probability Alpha. This is the Greek letter Alpha. Now, based on that, we get the probability of not having a stockout, which we call our service level, as 1 minus the probability of stockout, or 1 minus Alpha. So if I have an Alpha equals 0.05, then my service level will be 0.95, or I have a service level of 95 percent. Given this service level, I have to find the corresponding z-value, which I can do using either a standard normal table or I can use using Microsoft Excel and using the function NORM.INV, where INV stands for inverse. The side bar shows you how to actually use this both this normal table as well as the Excel function. So once we've figured out what the z-value is, the reorder point is simply the mean plus the z-value that we just found, multiplied by the standard deviation for our demand distribution for the period of uncertainty, and from there, we can calculate the safety stock as simply being the reorder point minus the mean or during that period of uncertainty. Let's take an example. Suppose I have a daily demand which is normally distributed, and we have a mean of 5 for the daily demand, and a standard deviation of daily demand equal to 3. Let's say that we have a period of uncertainty of seven days. So during the seven days, the mean demand that I'm going to experience is going to be 7 times 5, and the variance of demand that I'm going to experience is going to be 7 times the variance for the daily demand, which is 3 squared or 9, so 7 times 9. So I'm going to have a mean during the period of uncertainty of 35, and a standard deviation of 7.94, and my demand distribution is going to be normally distributed. So let's now say I decide that I want a stockout probability of 0.05, which then gives me a service level of 0.95 or 95 percent. I go to my z-tables, and my z-tables give me a z-value of 1.645 for a service level of 0.95. So now I calculate my reorder point. My mean is 35, my z-value is 1.645, and my standard deviation is 7.94. I do the calculation, and it gives me a reorder point of 48. In turn, this now gives me a safety stock of 48 minus 35 equal to 30. So far, we've considered what happens when we have uncertainty in the demand during the leadtime. So demand is uncertain, but what happens if the leadtime itself is also uncertain. Most of us have waited for packages to be shipped from Amazon, and we wait for our package, we've been told it's going to come in three days, but occasionally, it comes in two days and occasionally it comes in four days. So even though we have an average time of three days, it could be sometimes less and sometimes more, and so leadtime itself can be a variable. So since leadtime is variable, let's say that the mean leadtime happens to be L as before, but now we also have a standard deviation of leadtime which we're calling standard deviation of L. So we have a mean demand during this time of D and a standard deviation during this time SD of D, and then we also have a leadtime which has a mean L and a standard deviation of SD with subscript L. So if I have a period of uncertainty which is not only uncertain because of the leadtime, but is even more uncertain because of the uncertainty in the leadtime itself, I now can calculate the mean demand during the period of uncertainty as being L multiplied by D. The standard deviation during this period of uncertainty do becomes a little more complicated. It has two components, one component looks at the variability caused by the demand uncertainty. The second component looks at the variability caused by the leadtime uncertainty. Put together, we get this complicated looking formula which is square root of mean leadtime multiplied by the variance of the demand, plus square of the mean demand multiplied by the variance of the leadtime uncertainty. Let's look at an example to see how we work with this. Suppose I have a daily demand of 5, which has a standard deviation of 1. Let's say my average leadtime happens to be 7, and let's say the standard deviation of leadtime happens to be 2. Then the mean demand during the period of uncertainty is going to be L multiplied by D, which is 35, and the standard deviation during the period of uncertainty, using the formula that we gave before, turns out to be 7 multiplied by 1 squared plus 5 squared, multiplied by 2 squared, which turns out to be 10.334. If I assume a service level of 95 percent, I get a z-value of 1.645, which then in turn gives me a reorder point of 35, which was my mean during the period of uncertainty plus the z-value of 1.64 multiplied by the standard deviation during the period of uncertainty of 10.334, which is approximately 52. Notice my reorder point now is 52, which is higher than what I had of 48 previously when there was no leadtime uncertainty. So I've had to increase the amount of safety stock to take into account the fact that I have leadtime uncertainty.