Automatic forecasting is the most popular and most used feature of SmartForecasts and Smart Demand Planner. Creating Automatic forecasts is easy. But, the simplicity of Automatic Forecasting masks a powerful interaction of a number of highly effective methods of forecasting. In this blog, we discuss some of the theory behind this core feature. We focus on Automatic forecasting, in part because of its popularity and in part because many other forecasting methods produce similar outputs. Knowledge of Automatic forecasting immediately carries over to Simple Moving Average, Linear Moving Average, Single Exponential Smoothing, Double Exponential Smoothing, Winters’ Exponential Smoothing, and Promo forecasting.

Forecasting tournament

Automatic forecasting works by conducting a tournament among a set of competing methods. Because personal computers and cloud computing are fast, and because we have coded very efficient algorithms into the SmartForecasts’ Automatic forecasting engine, it is practical to take a purely empirical approach to deciding which extrapolative forecasting method to use. This means that you can afford to try out a number of approaches and then retain the one that does best at forecasting the particular data series at hand. SmartForecasts fully automates this process for you by trying the different forecasting methods in a simulated forecasting tournament. The winner of the tournament is the method that comes closest to  predicting new data values from old. Accuracy is measured by average absolute error (that is, the average error, ignoring any minus signs). The average is computed over a set of forecasts, each using a portion of the data, in a process known as sliding simulation.

Sliding simulation

The sliding simulation sweeps repeatedly through ever-longer portions of the historical data, in each case forecasting ahead the desired number of periods in your forecast horizon. Suppose there are 36 historical data values and you need to forecast six periods ahead. Imagine that you want to assess the forecast accuracy of some particular method, say a moving average of four observations, on the data series at hand.

At one point in the sliding simulation, the first 24 points (only) are used to forecast the 25th through 30th historical data values, which we temporarily regard as unknown. We say that points 25-30 are “held out” of the analysis. Computing the absolute values of the differences between the six forecasts and the corresponding actual historical values provides one instance each of a 1-step, 2-step, 3-step, 4-step, 5-step, and 6-step ahead absolute forecast error. Repeating this process using the first 25 points provides more instances of 1-step, 2-step, 3-step ahead errors, and so on. The average over all of the absolute error estimates obtained this way provides a single-number summary of accuracy.

Methods used in Automatic forecasting

Normally, there are six extrapolative forecasting methods competing in the Automatic forecasting tournament:

• Simple moving average
• Linear moving average
• Single exponential smoothing
• Double exponential smoothing
• Additive version of Winters’ exponential smoothing
• Multiplicative version of Winters’ exponential smoothing

The latter two methods are appropriate for seasonal series; however, they are automatically excluded from the tournament if there are fewer than two full seasonal cycles of data (for example, fewer than 24 periods of monthly data or eight periods of quarterly data).

These six classical, smoothing-based methods have proven themselves to be easy to understand, easy to compute and accurate. You can exclude any of these methods from the tournament if you have a preference for some of the competitors and not others.

A forecast is a prediction about the value of a time series variable at some time in the future. For instance, one might want to estimate next month’s sales or demand for a product item. A time series is a sequence of numbers recorded at equally spaced time intervals; for example, unit sales recorded every month.

The objectives you pursue when you forecast depend on the nature of your job and your business. Every forecast is uncertain; in fact, there is a range of possible values for any variable you forecast. Values near the middle of this range have a higher likelihood of actually occurring, while values at the extremes of the range are less likely to occur. The following figure illustrates a typical distribution of forecast values.

Illustrating a forecast distribution of forecast values

Point forecasts

The most common use of forecasts is to estimate a sequence of numbers representing the most likely future values of the variable of interest. For instance, suppose you are developing a sales and marketing plan for your company. You may need to fill in 12 cells in a financial spreadsheet with estimates of your company’s total revenues over the next 12 months. Such estimates are called point forecasts because you want a single number (data point) for each forecast period. Smart Demand Planner’ Automatic forecasting feature provides you with these point forecasts automatically.

Interval forecasts

Although point forecasts are convenient, you will often benefit more from interval forecasts. Interval forecasts show the most likely range (interval) of values that might arise in the future. These are usually more useful than point forecasts because they convey the amount of uncertainty or risk involved in a forecast. The forecast interval percentage can be specified in the various forecasting dialog boxes in the Demand Planning Software.  Each of the many forecasting methods (automatic, moving average, exponential smoothing and so on) available in Smart Demand Planner allow you to set a forecast interval.

The default configuration in Smart Demand Planner provides 90% forecast intervals. Interpret these intervals as the range within which the actual values will fall 90% of the time. If the intervals are wide, then there is a great deal of uncertainty associated with the point forecasts. If the intervals are narrow, you can be more confident. If you are performing a planning function and want best case and worst case values for the variables of interest at several times in the future, you can use the upper and lower limits of the forecast intervals for that purpose, with the single point estimate providing the most likely value. In the previous figure, the 90% forecast interval extends from 3.36 to 6.64.

Upper percentiles

In inventory control, your goal may be to make good estimates of a high percentile of the demand for a product item. These estimates help you cope with the tradeoff between, on the one hand, minimizing the costs of holding and ordering stock, and, on the other hand, minimizing the number of lost or back-ordered sales due to a stock out. For this reason, you may wish to know the 99th percentile or service level of demand, since the chance of exceeding that level is only 1%.

When forecasting individual variables with features like Automatic forecasting, note that the upper limit of a 90% forecast interval represents the 95th percentile of the predicted distribution of the demand for that variable. (Subtracting the 5th percentile from the 95th percentile leaves an interval containing 95%-5% = 90% of the possible values.) This means you can estimate upper percentiles by changing the value of the forecast interval. In the figure, “Illustrating a forecast distribution”, the 95th percentile is 6.64.

To optimize stocking policies at the desired service level or to let the system recommend which stocking policy and service level generates the best return, consider using Smart Inventory Optimization.   It is designed to support what-if scenarios that show predicted tradeoffs of varying inventory polices including different service level targets.

Lower percentiles

Sometimes you may be concerned with the lower end of the predicted distribution for a variable. Such cases often arise in financial applications, where a low percentile of a revenue estimate represents a contingency requiring financial reserves. You can use Smart Demand Planner in this case in a way analogous to the case of forecasting upper percentiles. In the figure, “Illustrating a forecast distribution” , the 5th percentile is 3.36.

In conclusion, forecasting involves predicting future values, with point forecasts offering single estimates and interval forecasts providing likely value ranges. Smart Demand Planner automates point forecasts and allows users to set intervals, aiding in uncertainty assessment. For inventory control, the tool facilitates understanding upper (e.g., 99th percentile) and lower (e.g., 5th percentile) percentiles. To optimize stocking policies and service levels, Smart Inventory Optimization supports what-if scenarios, ensuring effective decision-making on how much to stock given the risk of stock out you are willing to accept.

Summary

Smart Software’s advanced supply chain analytics exploits multiple advanced methods. Two of the most important are “statistical bootstrapping” and “Monte Carlo simulation”. Since both involve lots of random numbers flying around, folks sometimes get confused about which is which and what they are good for. Hence, this note. Bottom line up front: Statistical bootstrapping generates demand scenarios for forecasting. Monte Carlo simulation uses the scenarios for inventory optimization.

Bootstrapping

Bootstrapping, also called “resampling” is a method of computational statistics that we use to create demand scenarios for forecasting. The essence of the forecasting problem is to expose possible futures that your company might confront so you can work out how to manage business risks. Traditional forecasting methods focus on computing “most likely” futures, but they fall short of presenting the full risk picture. Bootstrapping provides an unlimited number of realistic what-if scenarios.

Bootstrapping does this without making unrealistic assumptions about the demand, i.e., that it is not intermittent, or that it has a bell-shaped distribution of sizes. Those assumptions are crutches to make the math simpler, but the bootstrap is a procedure,  not an equation, so it doesn’t need such simplifications.

For the simplest demand type, which is a stable randomness with no seasonality or trend, bootstrapping is dead easy. To get a reasonable idea of what a single future demand value might be, pick one of the historical demands at random. To create a demand scenario, make multiple random selections from the past and string them together. Done. It is possible to add a little more realism by “jittering” the demand values, i.e., adding or subtracting a bit of additional randomness to each one, but even that is simple.

Figure 1 shows a simple bootstrap. The first line is a short sequence of historical demand for an SKU. The following lines show scenarios of future demand created by randomly selecting values from the demand history. For instance, the next three demand might be (0, 14, 6), or (2, 3, 5), etc.

Figure 1: Example of demand scenarios generated by a simple bootstrap

Higher frequency operations such as daily forecasting bring with them more complex demand patterns, such as double seasonality (e.g., day-of-week and month-of-year) and/or trend. This challenged us to invent a new generation of bootstrapping algorithms. We recently won a US Patent for this breakthrough, but the essence is as described above.

Monte Carlo Simulation

Monte Carlo is famous for its casinos, which, like bootstrapping, invoke the idea of randomness. Monte Carlo methods go back a long way, but the modern impetus came with the need to do some hairy calculations about where neutrons would fly when an A-bomb explodes.

The essence of Monte Carlo analysis is this: “Our problem is too complicated to analyze with paper-and-pencil equations. So, let’s write a computer program that codes the individual steps of the process, put in the random elements (e.g., which way a neutron shoots away), wind it up and watch it go. Since there’s a lot of randomness, let’s run the program a zillion times and average the results.”

Applying this approach to inventory management, we have a different set of randomly occurring events: e.g., a demand of a given size arrives on a random day, a replenishment of a given size arrives after a random lead time, we cut a replenishment PO of a given size when stock drops to or below a given reorder point. We code the logic relating these events into a program. We feed it with a random demand sequence (see bootstrapping above), run the program for a while, say one year of daily operations, compute performance metrics like Fill Rate and Average On Hand inventory, and “toss the dice” by re-running the program many times and averaging the results of many simulated years. The result is a good estimate of what happens when we make key management decisions: “If we set the reorder point at 10 units and the order quantity at 15 units, we can expect to get a service level of 89% and an average on hand of 21 units.” What the simulation is doing for us is exposing the consequences of management decisions based on realistic demand scenarios and solid math. The guesswork is gone.

Figure 2 shows some of the inner workings of a Monte Carlo simulation of an inventory system in four panels. The system uses a Min/Max inventory control policy with Min=10 and Max=25. No backorders are allowed: you have the good or you lose the business. Replenishment lead times are usually 7 days but sometimes 14. This simulation ran for one year.

The first panel shows a complex random demand scenario in which there is no demand on weekends, but demand generally increases each day from Monday to Friday. The second panel shows the random number of units on hand, which ebbs and flows with each replenishment cycle. The third panel shows the random sizes and timings of replenishment orders coming in from the supplier. The final panel shows the unsatisfied demand that jeopardizes customer relationships. This kind of detail can be very useful for building insight into the dynamics of an inventory system.

Figure 2: Details of a Monte Carlo simulation

Figure 2 shows only one of the countless ways that the year could play out. Generally, we want to average the results of many simulated years. After all, nobody would flip a coin once to decide if it were a fair coin. Figure 3 shows how four key performance metrics (KPI’s) vary from year to year for this system. Some metrics are relatively stable across simulations (Fill Rate), but others show more relative variability (Operating Cost= Holding Cost + Ordering Cost + Shortage Cost). Eyeballing the plots, we can estimate that the choices of Min=10, Max=25 leads to an average Operating cost of around $3,000 per year, a Fill Rate of around 90%, a Service Level of around 75%, and an Average On Hand of about 10

Figure 3: Variation in KPI’s computed over 1,000 simulated years

In fact, it is now possible to answer a higher level of management question. We can go beyond “What will happen if I do such-and-such?” to “What is the best thing I can do to achieve a fill rate of at least 90% for this item at the lowest possible cost?” The mathemagic  behind this leap is yet another key technology called “stochastic optimization”, but we’ll stop here for now. Suffice it to say that Smart’s SIO&P software can search the “design space” of Min and Max values to automatically find the best choice.

1. Forecasting is an art that requires a mix of professional judgment and objective statistical analysis. Successful demand forecasts require a baseline prediction leveraging statistical forecasting methods. Once established, the process can focus on how best to adjust statistical forecasts based on your own insights and business knowledge.

2. The forecasting process is usually iterative. You may need to make several refinements of your initial forecast before you are satisfied. It is important to be able to generate and compare alternative forecasts quickly and easily. Tracking accuracy of these forecasts over time, including alternatives that were not used, helps inform and improve the process.

3. The credibility of forecasts depends heavily on graphical comparisons with historical data.  A picture is worth a thousand words, so always display forecasts via instantly available graphical displays with supporting numerical reports.

4. One of the major technical tasks in forecasting is to match the choice of forecasting technique to the nature of the data. Effective demand forecasting processes employ capabilities that identify the right method to use.  Features of a data series like trend, seasonality or abrupt shifts in level suggest certain techniques instead of others. An automatic selection, which selects and uses the appropriate forecasting method automatically, saves time and ensures your baseline forecast is as accurate as possible.

5. Successful demand forecasting processes work in tandem with other business processes.   For example, forecasting can be an essential first step in financial analysis.  In addition, accurate sales and product demand forecasts are fundamental inputs to a manufacturing company’s production planning and inventory control processes.

6. A good planning process recognizes that forecasts are never exactly correct. Because some error creeps into even the best forecasting process, one of the most useful supplements to a forecast are honest estimates of its margin of error and forecast bias.

Sales forecasts are often inaccurate simply because the sales team is forced to give a number even though they don’t really know what their customer demand is going to be. Let the sales teams sell.  Don’t bother playing the game of feigning acceptance of these forecasts when both sides (sales and supply chain) know it is often nothing more than a WAG.   Do this instead:

• Accept demand variability as a fact of life. Develop a planning process that does a better job account for demand variability.

• Agree on a level of stockout risk that is acceptable across groups of items.

• Once the stockout risk is agreed to, use software to generate an accurate estimate of the safety stock needed to counter the demand variability.

• Get buy-in. Customers must be willing to pay a higher price per unit for you to deliver extremely high service levels.  Salespeople must accept that certain items are more likely to have backorders if they prioritize inventory investment on other items.

• Using a consensus #safetystock process ensures you are properly buffering and setting the right expectations with sales, customers, finance, and supply chain.

When you do this, you free all parties from having to play the prediction game they were not equipped to play in the first place. You’ll get better results, such as higher service levels with lower inventory costs. And with much less finger-pointing.

What’s all the hoopla around the term “probabilistic forecasting?” Is it just a more recent marketing term some software vendors and consultants have coined to feign innovation? Is there any real tangible difference compared to predecessor “best fit” techniques?  Aren’t all forecasts probabilistic anyway?

To answer this question, it is helpful to think about what the forecast really is telling you in terms of probabilities.  A “good” forecast should be unbiased and therefore yield a 50/50 probability being higher or lower than the actual.  A “bad” forecast will build in subjective buffers (or artificially depress the forecast) and result in demand that is either biased high or low.  Consider a salesperson that intentionally reduces their forecast by not reporting sales they expect to close to be “conservative.” Their forecasts will have negative forecast bias as actuals will nearly always be higher than what they predicted.   On the other hand, consider a customer that provides an inflated forecast to their manufacturer.  Worried about stockouts, they overestimate demand to ensure their supply.  Their forecast will have a positive bias as actuals will nearly always be lower than what they predicted. 

These types of one-number forecasts described above are problematic.  We refer to these predictions as “point forecasts” since they represent one point (or a series of points over time) on a plot of what might happen in the future.   They don’t provide a complete picture because to make effective business decisions such as determining how much inventory to stock or the number of employees to be available to support demand requires detailed information on how much lower or higher the actual will be!  In other words, you need the probabilities for each possible outcome that might occur.  So, by itself, the point forecast isn’t probabilistic one.   

To get a probabilistic forecast, you need to know the distribution of possible demands around that forecast.  Once you compute this, the forecast becomes “probabilistic.”  How forecasting systems and practitioners such as demand planners, inventory analysts, material managers, and CFOs determine these probabilities is the heart of the question: “what makes a forecast probabilistic?”     

Normal Distributions
Most forecasts and the systems/software that produce them start with a prediction of demand.  Then they figure out the range of possible demands around that forecast by making incorrect theoretical assumptions about the distribution.  If you’ve ever used a “confidence interval” in your forecasting software, this is based on a probability distribution around the forecast.  The way this range of demand is determined is to assume a particular type of distribution.  Most often this means assuming a bell shaped, otherwise known as a normal distribution.  When demand is intermittent, some inventory optimization and demand forecasting systems may assume the demand is Poisson shaped. 

After creating the forecast, the assumed distribution is slapped around the demand forecast and you then have your estimate of probabilities for every possible demand – i.e., a “probabilistic forecast.”  These estimates of demand and associated probabilities can then be used to determine extreme values or anything in between if desired.  The extreme values at the upper percentiles of the distribution (i.e., 92%, 95%, 99%, etc.) are most often used as inputs to inventory control models.  For example, reorder points for critical spare parts in an electrical utility might be planned based on a 99.5% service level or even higher.  While a non-critical service part might be planned at an 85% or 90% service level.

The problem with making assumptions about the distribution is that you’ll get these probabilities wrong.  For example, if the demand isn’t normally distributed but you are forcing a bell shaped/normal curve on the forecast then how can then the probabilities will be incorrect.  Specifically, you might want to know the level of inventory needed to achieve a 99% probability of not running out of stock and the normal distribution will tell you to stock 200 units.  But when compared to the actual demand, you come to find out that 200 units only filled demand entirely in 40/50 observations.  So, instead of getting a 99% service level you only achieved an 80% service level!  This is a gigantic miss resulting from trying to fit a square peg into a round hole.  The miss would have led you to take an incorrect inventory reduction.

Empirically Estimated Distributions are Smart
To produce a smart (read accurate) probabilistic forecast you need to first estimate the distribution of demand empirically without any naïve assumptions about the shape of the distribution.  Smart Software does this by running tens of thousands of simulated demand and lead time scenarios.  Our solution leverages patented techniques that incorporate Monte Carlo simulation, Statistical Bootstrapping, and other methods.  The scenarios are designed to simulate real life uncertainty and randomness of both demand and lead times.  Actual historical observations are utilized as the primary inputs, but the solution will give you the option of simulating from non-observed values as well.  For example, just because 100 units was the peak historical demand, that doesn’t mean you are guaranteed to peak out at 100 in the future.  After the scenarios are done you will know the exact probability for each outcome. The “point” forecast then becomes the center of that distribution.  Each future period over time is expressed in terms of the probability distribution associated with that period.

Leaders in Probabilistic Forecasting
Smart Software, Inc. was the first company to ever introduce statistical bootstrapping as part of a commercially available demand forecasting software system twenty years ago.  We were awarded a US patent at the time for it and named a finalist in the APICS Corporate Awards of Excellence for Technological Innovation.  Our NSF Sponsored research that led to this and other discoveries were instrumental in advancing forecasting and inventory optimization.    We are committed to ongoing innovation, and you can find further information about our most recent patent here.