Probabilistic vs. Deterministic Order Planning

The Smart Forecaster

Man with a computer in a warehouse best practices in demand planning, forecasting and inventory optimization

Consider the problem of replenishing inventory. To be specific, suppose the inventory item in question is a spare part. Both you and your supplier will want some sense of how much you will be ordering and when. And your ERP system may be insisting that you let it in on the secret too.

Deterministic Model of Replenishment

The simplest way to get a decent answer to this question is to assume the world is, well, simple. In this case, simple means “not random” or, in geek speak, “deterministic.” In particular, you pretend that the random size and timing of demand is really a continuous drip-drip-drip of a fixed size coming at a fixed interval, e.g., 2, 2, 2, 2, 2, 2… If this seems unrealistic, it is. Real demand might look more like this: 0, 1, 10, 0, 1, 0, 0, 0 with lots of zeros, occasional but random spikes.

But simplicity has its virtues. If you pretend that the average demand occurs every day like clockwork, it is easy to work out when you will need to place your next order, and how many units you will need.  For instance, suppose your inventory policy is of the (Q,R) type, where Q is a fixed order quantity and R is a fixed reorder point. When stock drops to or below the reorder point R, you order Q units more. To round out the fantasy, assume that the replenishment lead time is also fixed: after L days, those Q new units will be on the shelf ready to satisfy demand.

All you need now to answer your questions is the average demand per day D for the item. The logic goes like this:

  1. You start each replenishment cycle with Q units on hand.
  2. You deplete that stock by D units per day.
  3. So, you hit the reorder point R after (Q-R)/D days.
  4. So, you order every (Q-R)/D days.
  5. Each replenishment cycle lasts (Q-R)/D + L days, so you make a total of 365D/(Q-R+LD) orders per year.
  6. As long as lead time L < R/D, you will never stock out and your inventory will be as small as possible.

Figure 1 shows the plot of on-hand inventory vs time for the deterministic model. Around Smart Software, we refer to this plot as the “Deterministic Sawtooth.” The stock starts at the level of the last order quantity Q. After steadily decreasing over the drop time (Q-R)/D, the level hits the reorder point R and triggers an order for another Q units. Over the lead time L, the stock drops to exactly zero, then the reorder magically arrives and the next cycle begins.

Figure 1 Deterministic model of on-hand inventory

Figure 1: Deterministic model of on-hand inventory

 

This model has two things going for it. It requires no more than high school algebra, and it combines (almost) all the relevant factors to answer the two related questions: When will we have to place the next order? How many orders will we place in a year?

Probabilistic Model of Replenishment

Not surprisingly, if we strip away some of the fantasy from the deterministic model, we get more useful information. The probabilistic model incorporates all the messy randomness in the real-world problem: the uncertainty in both the timing and size of demand, the variation in replenishment lead time, and the consequences of those two factors: the chance of stock on hand undershooting the reorder point, the chance that there will be a stockout, the variability in the time until the next order, and the variable number of orders executed in a year.

The probabilistic model works by simulating the consequences of uncertain demand and variable lead time. By analyzing the item’s historical demand patterns (and excluding any observations that were recorded during a time when demand may have been fundamentally different), advanced statistical methods create an unlimited number of realistic demand scenarios. Similar analysis is applied to records of supplier lead times. Combining these supply and demand scenarios with the operational rules of any given inventory control policy produces scenarios of the number of parts on hand. From these scenarios, we can extract summaries of the varying intervals between orders.

Figure 2 shows an example of a probabilistic scenario; demand is random, and the item is managed using reorder point R = 10 and order quantity Q=20. Gone is the Deterministic Sawtooth; in its place is something more complex and realistic (the Probabilistic Staircase). During the 90 simulated days of operation, there were 9 orders placed, and the time between orders clearly varied.

Using the probabilistic model, the answers to the two questions (how long between orders and how many in a year) get expressed as probability distributions reflecting the relative likelihoods of various scenarios. Figure 3 shows the distribution of the number of days between orders after ten years of simulated operation. While the average is about 8 days, the actual number varies widely, from 2 to 17.

Instead of telling your supplier that you will place X orders next year, you can now project X ± Y orders, and your supplier knows better their upside and downside risks. Better yet, you could provide the entire distribution as the richest possible answer.

Figure 2 A probabilistic scenario of on-hand inventory

Figure 2 A probabilistic scenario of on-hand inventory

 

Figure 3 Distribution of days between orders

Figure 3: Distribution of days between orders

 

Climbing the Random Staircase to Greater Efficiency

Moving beyond the deterministic model of  inventory opens up new possibilities for optimizing operations. First, the probabilistic model allows realistic assessment of stockout risk. The simple model in Figure 1 implies there is never a stockout, whereas probabilistic scenarios allow for the possibility (though in Figure 2 there was only one close call around day 70). Once the risk is known, software can optimize by searching  the “design space” (i.e., all possible values of R and Q) to find a design that meets a target level of stockout risk at minimal cost. The value of the deterministic model in this more realistic analysis is that it provides a good starting point for the search through design space.

Summary

Modern software provides answers to operational questions with various degrees of detail. Using the example of the time between replenishment orders, we’ve shown that the answer can be calculated approximately but quickly by a simple deterministic model. But it can also be provided in much richer detail with all the variability exposed by a probabilistic model. We think of these alternatives as complementary. The deterministic model bundles all the key variables into an easy-to-understand form. The probabilistic model provides additional realism that professionals expect and supports effective search for optimal choices of reorder point and order quantity.

 

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In this video, Dr. Thomas Willemain, co-Founder and SVP Research, talks about improving forecast accuracy by measuring forecast error. We begin by overviewing the various types of error metrics: scale-dependent error, percentage error, relative error, and scale-free error metrics. While some error is inevitable, there are ways to reduce it, and forecast metrics are necessary aids for monitoring and improving forecast accuracy. Then we will explain the special problem of intermittent demand and divide-by-zero problems. Tom concludes by explaining how to assess forecasts of multiple items and how it often makes sense to use weighted averages, weighting items differently by volume or revenue.

 

Four general types of error metrics 

1. Scale-dependent error
2. Percentage error
3. Relative error
4 .Scale-free error

Remark: Scale-dependent metrics are expressed in the units of the forecasted variable. The other three are expresses as percentages.

 

1. Scale-dependent error metrics

  • Mean Absolute Error (MAE) aka Mean Absolute Deviation (MAD)
  • Median Absolute Error (MdAE)
  • Root Mean Square Error (RMSE)
  • These metrics express the error in the original units of the data.
    • Ex: units, cases, barrels, kilograms, dollars, liters, etc.
  • Since forecasts can be too high or too low, the signs of the errors will be either positive or negative, allowing for unwanted cancellations.
    • Ex: You don’t want errors of +50 and -50 to cancel and show “no error”.
  • To deal with the cancellation problem, these metrics take away negative signs by either squaring or using absolute value.

 

2. Percentage error metric

  • Mean Absolute Percentage Error (MAPE)
  • This metric expresses the size of the error as a percentage of the actual value of the forecasted variable.
  • The advantage of this approach is that it immediately makes clear whether the error is a big deal or not.
  • Ex: Suppose the MAE is 100 units. Is a typical error of 100 units horrible? ok? great?
  • The answer depends on the size of the variable being forecasted. If the actual value is 100, then a MAE = 100 is as big as the thing being forecasted. But if the actual value is 10,000, then a MAE = 100 shows great accuracy, since the MAPE is only 1% of the actual.

 

3. Relative error metric

  • Median Relative Absolute Error (MdRAE)
  • Relative to what? To a benchmark forecast.
  • What benchmark? Usually, the “naïve” forecast.
  • What is the naïve forecast? Next forecast value = last actual value.
  • Why use the naïve forecast? Because if you can’t beat that, you are in tough shape.

 

4. Scale-Free error metric

  • Median Relative Scaled Error (MdRSE)
  • This metric expresses the absolute forecast error as a percentage of the natural level of randomness (volatility) in the data.
  • The volatility is measured by the average size of the change in the forecasted variable from one time period to the next.
    • (This is the same as the error made by the naïve forecast.)
  • How does this metric differ from the MdRAE above?
    • They do both use the naïve forecast, but this metric uses errors in forecasting the demand history, while the MdRAE uses errors in forecasting future values.
    • This matters because there are usually many more history values than there are forecasts.
    • In turn, that matters because this metric would “blow up” if all the data were zero, which is less likely when using the demand history.

 

Intermittent Demand Planning and Parts Forecasting

 

The special problem of intermittent demand

  • “Intermittent” demand has many zero demands mixed in with random non-zero demands.
  • MAPE gets ruined when errors are divided by zero.
  • MdRAE can also get ruined.
  • MdSAE is less likely to get ruined.

 

Recap and remarks

  • Forecast metrics are necessary aids for monitoring and improving forecast accuracy.
  • There are two major classes of metrics: absolute and relative.
  • Absolute measures (MAE, MdAE, RMSE) are natural choices when assessing forecasts of one item.
  • Relative measures (MAPE, MdRAE, MdSAE) are useful when comparing accuracy across items or between alternative forecasts of the same item or assessing accuracy relative to the natural variability of an item.
  • Intermittent demand presents divide-by-zero problems which favor MdSAE over MAPE.
  • When assessing forecasts of multiple items, it often makes sense to use weighted averages, weighting items differently by volume or revenue.
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    Forecast error can be consequential

    Consider one item of many

    • Product X costs $100 to make and nets $50 profit per unit.
    • Sales of Product X will turn out to be 1,000/month over the next 12 months.
    • Consider one item of many

    What is the cost of forecast error?

    • If the forecast is 10% high, end the year with $120,000 of excess inventory.
    • 100 extra/month x 12 months x $100/unit
    • If the forecast is 10% low, miss out on $60,000 of profit.
    • 100 too few/month x 12 months x $50/unit

     

    Three mistakes to avoid

    1. Ignoring error.

    • Unprofessional, dereliction of duty.
    • Wishing will not make it so.
    • Treat accuracy assessment as data science, not a blame game.

    2. Tolerating more error than necessary.

    • Statistical forecasting methods can improve accuracy at scale.
    • Improving data inputs can help.
    • Collecting and analyzing forecast error metrics can identify weak spots.

    3. Wasting time and money going too far trying to eliminate error.

    • Some product/market combinations are inherently more difficult to forecast. After a point, let them be (but be alert for new specialized forecasting methods).
    • Sometimes steps meant to reduce error can backfire (e.g., adjustment).
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      A Primer on Probabilistic Forecasting

      The Smart Forecaster

       Pursuing best practices in demand planning,

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      If you keep up with the news about supply chain analytics, you are more frequently encountering the phrase “probabilistic forecasting.” If this phrase is puzzling, read on.

      You probably already know what “forecasting” means. And you probably also know that there seem to be lots of different ways to do it. And you’ve probably heard pungent little phrases like “every forecast is wrong.” So you know that some kind of mathemagic might calculate that “the forecast is you will sell 100 units next month”, and then you might sell 110 units, in which case you have a 10% forecast error.

      You may not know that what I just described is a particular kind of forecast called a “point forecast.” A point forecast is so named because it consists of just a single number (i.e., one point on the number line, if you recall the number line from your youth).

      Point forecasts have one virtue: They are simple. They also have a flaw: They give rise to snarky statements like “every forecast is wrong.” That is, in most realistic cases, it is unlikely that the actual value will exactly equal the forecast. (Which isn’t such a big deal if the forecast is close enough.)

      This gets us to “probabilistic forecasting.” This approach is a step up, because instead of producing a single-number (point) forecast, it yields a probability distribution for the forecast. And unlike traditional extrapolative models that rely purely on the historical data, probabilistic forecasts have the ability to simulate future values that aren’t anchored to the past.

      “Probability distribution” is a forbidding phrase, evoking some arcane math that you may have heard of but never studied. Luckily, most adults have enough life experience to have an intuitive grasp of the concept.  When broken down, it’s quite straightforward to understand.

      Imagine the simple act of flipping two coins. You might call this harmless fun, but I call it a “probabilistic experiment.” The total number of heads that turn up on the two coins will be either zero, one or two. Flipping two coins is a “random experiment.” The resulting number of heads is a “random variable.” It has a “probability distribution”, which is nothing more than a table of how likely it is that the random variable will turn out to have any of its possible values. The probability of getting two heads when the coins are fair works out to be ¼, as is the probability of no heads. The chance of one head is ½.

      The same approach can describe a more interesting random variable, like the daily demand for a spare part.  Figure 2 shows such a probability distribution. It was computed by compiling three years of daily demand data on a certain part used in a scientific instrument sold to hospitals.

       

      Probabilistic demand forecast 1

      Figure 1: The probability distribution of daily demand for a certain spare part

       

      The distribution in Figure 1 can be thought of as a probabilistic forecast of demand in a single day. For this particular part, we see that the forecast is very likely to be zero (97% chance), but sometimes will be for a handful of units, and once in three years will be twenty units. Even though the most likely forecast is zero, you would want to keep a few on hand if this part were critical (“…for want of a nail…”)

      Now let’s use this information to make a more complicated probabilistic forecast. Suppose you have three units on hand. How many days will it take for you to have none? There are many possible answers, ranging from a single day (if you immediately get a demand for three or more) up to a very large number (since 97% of days see no demand).  The analysis of this question is a bit complicated because of all the many ways this situation can play out, but the final answer that is most informative will be a probability distribution. It turns out that the number of days until there are no units left in stock has the distribution shown in Figure 2.

      Probabilistic demand forecast 2

      Figure 2: Distribution of the number of days until all three units are gone

       

      The average number of days is 74, which would be a point forecast, but there is a lot of variation around the average. From the perspective of inventory management, it is notable that there is a 25% chance that all the units will be gone after 32 days. So if you decided to order more when you were down to only three on the shelf, it would be good to have the supplier get them to you before a month has passed. If they couldn’t, you’d have a 75% chance of stocking out – not good for a critical part.

      The analysis behind Figure 2 involved making some assumptions that were convenient but not necessary if they were not true. The results came from a method called “Monte Carlo simulation”, in which we start with three units, pick a random demand from the distribution in Figure 1, subtract it from the current stock, and continue until the stock is gone, recording how many days went by before you ran out. Repeating this process 100,000 times produced Figure 2.

      Applications of Monte Carlo simulation extend to problems of even larger scope than the “when do we run out” example above. Especially important are Monte Carlo forecasts of future demand. While the usual forecasting result is a set of point forecasts (e.g., expected unit demand over the next twelve months), we know that there are any number of ways that the actual demand could play out. Simulation could be used to produce, say, one thousand possible sets of 365 daily demand demands.

      This set of demand scenarios would more fully expose the range of possible situations with which an inventory system would have to cope. This use of simulation is called “stress testing”, because it exposes a system to a range of varied but realistic scenarios, including some nasty ones. Those scenarios are then input to mathematical models of the system to see how well it will cope, as reflected in key performance indicators (KPI’s). For instance, in those thousand simulated years of operation, how many stockouts are there in the worst year? the average year? the best year? In fact, what is the full probability distribution of the number of stockouts in a year, and what is the distribution of their size?

      Figures 3 and 4 illustrate probabilistic modeling of an inventory control system that converts stockouts to backorders. The system simulated uses a Min/Max control policy with Min = 10 units and Max = 20 units.

      Figure 3 shows one simulated year of daily operations in four plots. The first plot shows a particular pattern of random daily demand in which average demand increases steadily from Monday to Friday but disappears on weekends. The second plot shows the number of units on hand each day. Note that there are a dozen times during this simulated year when inventory goes negative, indicating stockouts. The third plot shows the size and timing of replenishment orders. The fourth plot shows the size and timing of backorders.  The information in these plots can be translated into estimates of inventory investment, average units on hand, holding costs, ordering costs and shortage costs.

      Probabilistic demand forecast 3

      Figure 3: One simulated year of inventory system operation

       

      Figure 3 shows one of one thousand simulated years. Each year will have different daily demands, resulting in different values of metrics like units on hand and the various components of operating cost. Figure 4 plots the distribution of 1,000 simulated values of four KPI’s. Simulating 1,000 years of imagined operation exposes the range of possible results so that planners can account not just for average results but also see best-case and worst-case values.

      Probabilistic demand forecast 4

      Figure 4: Distributions of four KPI’s based on 1,000 simulations

       

      Monte Carlo simulation is a low-math/high-results approach to probabilistic forecasting: very practical and easy to explain. Advanced probabilistic forecasting methods employed by Smart Software expand upon standard Monte Carlo simulation, yielding extremely accurate estimates of required inventory levels.

       

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