In the product mix problem, we assumed perfect knowledge of future demand, but, unfortunately, exact demand forecasts are a bit of scarce commodity in the true world. Indeed, the standard trouble in supply chain management is purchasing an item for which demand information is quite uncertain. If we order too much, one or more of the following scenarios might occur:
- Finance will suffer, as money is tied up in inventories.
- Items may become obsolete because of fads or product innovation, and money will be lost in inventory writeoffs.
- Perishable items may run out of their shelf life before being sold, and money will be lost again.
On the other hand, if we do not order enough items, we may not be able to meet customer demand and revenue will suffer (as well as our career; life is hard, isn’t it?).
To take our first baby steps, let us consider a relatively simple version of the problem. We are in charge of purchasing an item with a very limited shelf life. Both purchased quantities and demand are given as small integer numbers, which makes sense for a niche product. Items are purchased for delivery at the beginning of each week, and any unsold item is scrapped at the end of the same week; hence, each time we face a brand-new problem, in the sense that nothing is left in inventory from the previous time periods. Demand for the next week is not known, but we do have some information about past demand. The following list shows demand for the past 20 weeks:
The big question is: How many items should we order right now?
When asked this question, most students suggest considering the average demand, which is easily calculated as
Not too difficult, even though this result may leave us a bit uncertain, as we cannot really order fractional amounts of items. Yet, it seems that a reasonable choice could be between 2 and 3.
Other students suggest that we should stock the most likely value of demand. To see what this means exactly, it would be nice to see some more structure in the demand history, maybe by counting the frequency at which each value has occurred in the past. If we sort demand data, we get the following picture:
These numbers provide us with the frequencies at which each value occurred in the observed timespan. If we divide each frequency by the number of observations, we get relative frequencies. For instance, the relative frequency of the value 2 is 
or, in percentage terms, 40%. We may also calculate average demand by using relative frequencies:
Table 1.2 Frequencies (F), relative frequencies (Frel), and cumulated (relative) quencies (Fcum) for demand data.
Not surprisingly, we get the same average as above. We see that average demand is a weighted average of observed values, where weights correspond to relative frequencies. If we believe that the future will reflect the past, relative frequencies provide us with useful information about the likelihood of each demand value in the future.
Frequencies and relative frequencies are tabulated in columns 2 and 3 of Table 1.2. Be sure to note that relative frequencies cannot be negative and add up to 1, or 100%. Frequencies and relative frequencies may also be visualized using a histogram, as shown in Fig. 1.4. The observed values are reported on the horizontal axis (abscissa); the vertical axis (ordinate) may represent frequencies (a) of relative frequencies (b). The two plots are qualitatively the same, as relative frequencies are just obtained by normalizing frequencies with respect to the number of observations. After a quick glance at the graphical representation of relative frequencies, the intuitive idea of a “likelihood measure” of each demand value comes to mind rather naturally. Indeed, it is possible to interpret relative frequencies as probabilities. However, some caution should be exercised and we that probability is not such a trivial concept, as there are alternative interpretations. Still, this intuitive interpretation may be useful in many practical cases.
Looking at Table 1.2, we see that the most likely value (or the most frequent value in the past, to be honest with ourselves) is 2, which is not too different from the average value. In descriptive statistics, the most likely value is called mode. Since we get similar solutions by considering either mean or mode, we could be tricked into believing that we will always make a sensible choice by relying on them. Before we get so overconfident, let us consider the histograms of relative frequencies in Fig. 1.5. In histogram (a), we see that the most likely value is zero, but would we really stock nothing? Probably not. The two histograms in Fig. 1.5 are two examples of asymmetric cases. They are “skewed” into opposite directions, and we probably need a way to characterize skewness. We will deal with this and other summary measures but it is already clear that mean and mode do not tell the whole story and they are not always sufficient to come up with a solution for a decision problem. Lack of symmetry is likely to affect our stocking decisions, but there is still another essential point that we are missing: dispersion. Consider the two histograms in Fig. 1.6. Histogram (a) looks more concentrated, which arguably suggests less uncertainty about future demand with respect to histogram (b). We need some ways to measure dispersion as well, and to figure out how it can affect our choice. Indeed, we need some ways to characterize uncertainty, and this motivates the study of descriptive statistics. This is fine, but it is utterly useless, unless we find a way to use that information to come up with a decision. It is important to realize how many points we are missing, if we just consider relative frequencies.
Fig. 1.4 Histograms visualizing frequencies and relative frequencies for demand data.
Fig. 1.5 Two skewed distributions.
Fig. 1.6 The role of dispersion.
The role of economics. If we have a stockout, i.e., we run out of stock and do not meet the whole customer demand, how much money do we lose? And what if we have an overage, i.e., we stock too much and have to scrap perished or obsolete items? To see the point, consider the following problem. We have to decide how many T-shirts to make (or buy) for an upcoming major sport event. Producing and distributing a T-shirt costs €5; each T-shirt sells for €20, but unsold items at the end of the event must be sold at a markdown price, resulting in a loss.8 Let us assume that the discount on sales after the event is 80%, so that the markdown price is €4. A credible forecast, based on similar events, suggests that the expected value of sales is 12,000 pieces. We will clarify what we mean by expected value exactly, but you may think of it as the “best forecast” given our knowledge. However, demand is quite uncertain. A consultant, considering demand uncertainty and the risk of unsold items, suggests to keep on the safe side and produce just 10,000 pieces. Is this a good idea?
Please! Wait and think about the question before going on.
When we sell a T-shirt, our profit is €15; if we have to mark down, we lose only €1. Given that, most people would probably suggest a more aggressive strategy and buy a bit more than the expected value. Indeed, most fashion stores mark prices down at some time, which means that they tend to overstock. Would you change your idea if profit margin were €2 and the cost of an unsold item were €5? Economics must play a role here, as well as dispersion. Without any information about uncertainty, we cannot specify how much above or below the expected value of demand we should place our order. A plain point forecast, i.e., a single number, is not enough for robust decision making, a point that we will stress again when dealing with regression and time series models for forecasting.
Predictable vs. unpredictable variability. Consider once again the demand data in (1.3), but this time imagine that the time series, in chronological order, is
Mean, mode, etc., are not affected by this reshuffling of data, but should we neglect the clear pattern that we see? There is a trend in demand, which is not captured by simple summary measures. And what should we do with a demand pattern such as the following one?
In this case, we notice a seasonal pattern, with regular up- and downswings in demand. Trend and seasonality contribute to demand variability, but we should set predictable and unpredictable components of variability apart. We describe some simple methods for doing so.
The role of time and intertemporal dependence. The previous point shows that time does play a role, when we can identify partially predictable patterns such as trend and seasonality. Time may also play a role when our assumptions about ordering and shelf life are less restrictive. Assume that the shelf life is longer than the time between the orders we issue to suppliers. In making our decision, we should also consider the inventory level, and this would make the problem dynamic rather than static. A safe guess is that this is no simplification.
An even subtler point must be considered in order to properly represent unpredictable variability. I will illustrate it with a real-life story. A few years ago in Turin, where I live, there was a period of intense rain followed by an impressive flood. A weird thing with such an event is that there is way too much water in the streets, but you do not get any from your water tap at home. In that case, the high level of the main river in the city prevented the pumping stations from working. This problem, as I recall, was solved quickly, but the immediate consequence was a race to buy any bottle of mineral water around (with plenty of amicable exchange of ideas between tactful customers at retail stores). Now, if you were the demand manager for a company selling mineral water, would you interpret that spike in demand as a signal of an increasing market share? Well, not really, I guess. On the contrary, you could expect a period of low demand, when households deplete their unusually high inventories. More generally, if consumption of an item is relatively steady over time, a spike in demand (maybe due to a trade promotion) is likely to be followed by a period of low demand.9 In order to take such issues into account, we need statistical tools to investigate correlation. Correlation is useful in many settings where we ask questions about random variables rather than random events, such as
• If demand for an item has been larger than usual today, can we say something about demand tomorrow? Will it be larger or smaller than usual?
• If the return from a financial investment has been good, does this tell us something about the return of the same investment in the future, or maybe about the return from other investments?
The role of alternative items and competitors. We have just considered one item, disregarding possible interactions with other ones. In practice, items may interact in many ways:
• Shelf space. Limited shelf space at a retail store must be allocated to different items; then, stocking decisions are not independent.
• Substitute products. A stockout on one item may be almost irrelevant, if the customer. switches easily to a substitute item that we sell.
• Complementary products. Stocking out on one item may have a detrimental effect on other items, too; imagine a customer order consisting of several lines, related to different items; the customer could cancel the order if some order line is not satisfied.
• New products. If the assortment is changed by the introduction of a new item, sales of older items are likely to be affected by cannibalization.
By the same token, we should not disregard the role of competitors. If a competitor is about to launch a new product, we should plan in advance a suitable reaction. In such a case, just looking at past sales will be as safe and smart as driving a car by just looking into the rear mirror. Furthermore, pricing is likely to affect sales along multiple dimensions: the price of the item, the price of related items, and the price asked by competitors. We should investigate, among other things, the relationship between price and demand. One way of doing that involves regression models, which are the subject.
The role of sampling uncertainty. When we evaluate summary measures such as the mean of a variable, we look at a limited set of past data. But how reliable is that information? Intuitively, the more data we have, the better. However, looking too far into the past is dangerous, as we might take into account information that is hardly relevant for new market conditions. Would you use information about stock returns before World War II to manage a pension fund? We deal with inferential statistics, which may help us in assessing the degree of confidence we can have in our estimates.
Observables vs. unobservables. Finally, available data may not be what is actually relevant and needed. We have taken for granted that demand data were at our disposal. Unfortunately, in many practical settings we do not really observe demand, but sales. If there is a stockout on an item at our retail store, will the customer inform anyone that her demand was unmet? Not necessarily; maybe she will just go and buy at another retail store. If we are lucky, she will just settle for a product substitute. But even in this case, what we gather is sales data by scanning bar codes at the cash desk or point of sale. Clearly, this can result in an underestimation of actual demand. In other cases, data are available, but they are not gathered because of a wrong business process. Imagine a business to business setting, where a potential customer calls about immediate product availability. When the desired items are not in stock, she tries with another supplier. If the business process is such that only agreed-on orders are entered into the information system, disregarding lost sales, we are underestimating demand again. Often we have to settle for a proxy of what we cannot observe directly, and this might affect decisions.
After this long list of complicating factors,10 you may feel a little overwhelmed, but please don’t: There is a long array of powerful quantitative methods that we may integrate with good old common sense in order to tackle challenging problems. Anyway, if you want to take a short route, you could always try to do just a little better than your competitor. Say that he is able to meet demand completely in 80% of the weeks. The probability of not having a stockout is one of the many ways in which one can measure the service level. By choosing a stocking level S, you are setting the probability of satisfying all customers. For instance, with the data of Table 1.2, if you choose S = 1, demand will be met only when it is 1, which happens with probability 0.15. If S = 2, you meet demand when it is 1 or 2; summing the respective relative frequencies, we see that this happens with probability 0.15 + 0.40 = 0.55. The pattern is clear: The service level for increasing stocking levels is obtained by summing probabilities. This leads us to the concept of cumulative relative frequencies, which are displayed in the last column of Table 1.2. If you want to do a little better than the competitor, setting a 85% service level, you should stock four items, implying a 90% service level.
This last observation leads us to concepts such as cumulative distributions and quantiles. They are fundamental in many areas, such as inferential statistics and risk management, and will be among the most important topics.
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