Discrete uniform distribution

The uniform distribution is arguably the simplest model of uncertainty, as it assigns the same probability to each outcome:

This makes sense only if there is a finite number n of possible values that the random variable can assume. If they are consecutive integer numbers, we have an integer uniform distribution, which is characterized by the lower bound a and the upper bound b of its support. The condition:

immediately implies p = 1/n. We may think of a uniform distribution as a die with possibly many faces. A uniform distribution characterizes a case where we do not have any reason to believe that any outcome is more likely than the other ones; hence, we might associate a uniform distribution to a situation where we have very little information. There is little to say about this distribution, so it is time to tackle a nontrivial example.

Example 6.9 (Newsvendor problem) In Section 1.2.1 we briefly discussed the problem of purchasing items with a limited time window for sales under demand uncertainty. Now we are able to better understand it and even try a little numerical example. This kind of problem is typically labeled as newsvendor problem, as it resembles the challenge faced by late-nineteenth-century newsboys, who had to purchase newspapers before knowing demand, with considerable danger of scrapping a lot of sold newspaper.¹¹ More generally, the newsvendor problem is a prototype for all decisions involving fashion items, and its relevance is more and more significant as the rate at which products become obsolete is increasing.

To summarize the newsvendor problem:

• We purchase an item with unit cost c; this can also be the unit production cost in the “make” case. The item can be ordered only once, before the beginning of the time window.
• The product is sold at sales price s. The profit margin is s − c; if our order turns out to be small, i.e., if part of demand is not met, we lose profit opportunities.
• Unsold items are marked down and sold for s_u < c. If s_u = 0, then unsold items are just scrapped.
• Demand D is a random variable, whose probability distribution is known.
• The decision to be made is: How many items should we buy (or make)?

To gain a better feeling for the problem, let us consider the following numerical data:

• Unit cost is c = 20.
• Sales price is s = 25.
• Whatever is left, must just be scrapped; hence s_u = 0.
• Demand is uniformly distributed between 5 and 15; since the support consists of 11 values, we immediately see that the probability of each value is ; given the symmetry of the distribution, it is also easy to see that E[D] = 10.

Let us denote our decision variable, the order size, by q. At first sight, whenever we face a problem like this we should just come up with a good forecast of demand and buy that amount, q = E[D] = 10. Actually, this need not be the best strategy. Intuitively, such a naive solution disregards a lot of information, such as demand uncertainty and the involved economics, i.e., cost and price data. Before dwelling on mathematics, let us ask the following questions:

• If we go to a baker’s shop to buy some bread at 6 p.m., are we likely to find what we want, i.e., real bread and not plastic? Experience suggests that it is difficult to find real bread in the evening. Since this is a regular pattern, should we conclude that bakers are dumb demand forecasters?
• On the other hand, if we want to buy winter clothing, we always have the option to wait for markdown sales, since prices are marked down each and every winter, and the same applies to summer seasons. Should we conclude that owners of clothing shops are as dumb as bakers, but of the opposite breed?
• Or maybe none of them are dumb, and there is something we are missing?

It seems that we need more careful analysis, and the first question is: What makes a good solution to our problem? The easy answer is that we would like to maximize profit, but a moment of reflection shows that this is not a good answer and actually makes no sense at all. In fact, profit is random; more precisely it is a function of a random variable. It is our privilege to decide the order quantity q, but profit is a function of the random variable D as well. Formally, profit can be expressed as

In fact, if q ≤ D, we sell everything and the profit is the ordered quantity times profit margin; otherwise, profit is the difference between revenue from selling D items, minus the cost of purchasing q items (this applies to our case, since there is no salvage value associated with unsold items). We observe that our choice of q does not determine profit; rather, by choosing q we choose a probability distribution for profit. But how can we rank probability distributions? To do so, distributions should be associated with a single number. We fully address this issue where we deal with decision making under uncertainty, but the most natural choice is ranking based on expected value of profit. So, our problem can be formalized as

Note that expected profit Π(q) is a function of the decision variable q only, since random demand D is eliminated by expectation; the notation above clarifies that expectation is taken with respect to demand D. In our numerical example, expected profit is

In the sum, we use d to mark the difference between the random variable D and its realization d (a number). The first sum corresponds to demand scenarios where D ≤ q, and the second one corresponds to demand scenarios where D > q; the case in which D = q can be attributed to either sum, without changing the result. Each of the eleven possible outcomes is divided by the uniform probability . Now we may calculate Π(q) for each possible value of q.

Table 6.2 Expected profit for a newsvendor problem.

• There is no point in buying fewer than five items since, according to our model of uncertainty, we are going to sell at least five.
• If q = 5, there is no risk at all, since we will sell everything, anyway, with a profit Π(5) = 5 × 5 = 25.
• If q = 6, things are a little more involved. If it turns out that D = 5, we have a reduction in profit with respect to the previous case, since we lose 20 on one unsold item and profit is just 5. In any other scenario, we earn a profit of 30. To compare this profit distribution with the riskless profit of 25 when choosing q = 5, we compute the expected value:showing that q = 6 is a better decision than q = 5, assuming that we have picked the correct criterion.

Proceeding this way, we get the results given in Table 6.2. We see immediately that the optimal solution is not q = 10, but q = 7. Is this really surprising? Looking at the data, we see that profit margin is 25 − 20 = 5, which is the cost of a stockout occurring if we buy less than necessary. However, the cost of an unsold item is 20. Hence, it is not surprising that we came up with a rather conservative solution. With a large profit margin and/or a higher markdown sales price, the optimal order quantity would increase. Incidentally, this is why bakers are so conservative and fashion shops are so optimistic. Actually, rather than solving a problem by brute force, it would be nice to gain some insight, possibly by an analytical solution; since this is easily done assuming a continuous probability distribution, we defer this to Section 7.4.4.

This example shows a fundamental principle:

A point forecast, i.e., a single number, is typically not enough to come up with a good decision under uncertainty.

Before parting with this little example, it is very (and I stress very) important to understand a fundamental limitation of our modeling framework. We are assuming that our decisions do not influence uncertainty. Granted, there are cases in which this is a sensible assumption, or maybe a necessary one in order to make the problem tractable. Yet, a few histories of financial disasters suggest the opportunity of reflecting a bit on the appropriateness of such an assumption.

• We are assuming that our choice of q does not influence demand. However, if we are too conservative and the problem is repeated over time, a lot of stockouts may erode our customer base.
• If the game is not repeated, then one might question the sensibility of using the expected value, which can be interpreted as a long-run average. Maybe, we should consider a risk measure as well, like standard deviation of profit. Still, if the approach is applied to a large number of items, we might argue that maximizing expected profit should lead to a good solution on average.¹²
• Marketing studies show that the number of items on the shelves may influence demand. Would you buy the last, lonely box of a perishable food item on the shelves?
• We are also assuming that sales price is given, but we might use it as a tool to influence demand and maximize profit. ¹³
• Last but not least, we are also assuming that we can sell all of the leftover items at the markdown price s_u. This should not be taken for granted, since the markdown price could depend on how many items we are left with.

This might suggest opportunities for more complicated modeling, and indeed there are models for dynamic markdown pricing. However, when pulling such mathematical stunts, we should always keep in mind that a more complicated model requires more input data. If these data are not reliable, it is usually wiser to settle for a simpler and more robust model inspired by a parsimony principle.