Four Useful Ways to Measure Forecast Error

The Smart Forecaster

 Pursuing best practices in demand planning,

forecasting and inventory optimization

Improve Forecast Accuracy, Eliminate Excess Inventory, & Maximize Service Levels

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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    Improve Forecast Accuracy by Managing Error

    The Smart Forecaster

     Pursuing best practices in demand planning,

    forecasting and inventory optimization

    Improve Forecast Accuracy, Eliminate Excess Inventory, & Maximize Service Levels

    In this video, Dr. Thomas Willemain, co-Founder and SVP Research, talks about improving Forecast Accuracy by Managing Error. This video is the first in our series on effective methods to Improve Forecast Accuracy.  We begin by looking at how forecast error causes pain and the consequential cost related to it. Then we will explain the three most common mistakes to avoid that can help us increase revenue and prevent excess inventory. Tom concludes by reviewing the methods to improve Forecast Accuracy, the importance of measuring forecast error, and the technological opportunities to improve it.

     

    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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      Riding the Tradeoff Curve

      The Smart Forecaster

       Pursuing best practices in demand planning,

      forecasting and inventory optimization

      What We’re Up Against

      As a third-generation Boston Red Sox fan, I’m disinclined to take advice from any New York Yankee ballplayer, even a great one but have to agree that sometimes, you just need to make a decision.   However, wouldn’t it be better if we knew the tradeoffs associated with each decision. Perhaps one road is more scenic but takes longer while the other is more direct but boring. Then you wouldn’t have to simply “take it” but could make an informed decision based on the advantages/disadvantages of each approach.

      In the supply chain planning world, the most fundamental decision is how to balance item availability against the cost of maintaining that availability (service levels and fill rates). At one extreme, you can grossly overstock and never run out until you go broke and have to close up shop from sinking all your cash into inventory that doesn’t sell.  At the other extreme, you can grossly understock and save a bundle on inventory holding costs but go broke and have to close up shop because all your customers took their business elsewhere.

      There is no escaping this fundamental tension. They way to survive and thrive is to find a productive and sustainable balance. To do that requires fact-based tradeoffs based on the numbers. To get the numbers requires software.

      The general drift of things is obvious. If you decide to keep more inventory, you will have more Holding Costs, lower Shortage Costs, and possibly lower Ordering Costs. Whether this costs or saves money is impossible to know without some sophisticated analysis, but usually the result is that the Total Cost goes up. But if you do invest in more inventory, something will be gained, because you will offer your customers higher Service Levels and Fill Rates. How much higher requires, as you might guess, some sophisticated analysis.

      Show Me the Numbers

      This blog lays out what such an analysis looks like. There is no universal solution pointing you to the “right” decision. You might think that the right decision is the one that does best by your bottom line. But to get those numbers, you would need something rarely seen: an accurate model of customer behavior with regard to service level (check out our article “How to choose a target service level”) For example, at what point will a customer walk away and take their business elsewhere?  Will it be after you stock out 1% of the time, 5% of time, 10% of the time? Will you still keep their business as long as you fill back orders quickly?  Will it be after a back order of 1 day, 2 days? 3 weeks? Will it be after this happens one time on one an important part or many times across many parts?  While modeling the precise service level that will allow you to keep your customer while minimizing costs seems like an unapproachable ideal, another type of sophisticated analysis is more pragmatic. 

      Inventory optimization and forecasting software can factor all associated costs such as the cost of stocking out, cost of holding inventory, and cost of ordering inventory in order to prescribe an optimal service level target that yields the lowest total cost. However, even that “optimal” service level is sensitive to changes in the costs making the results potentially questionable.  For example, if you don’t accurately estimate the precise costs (shortage costs are the most difficult) it will be tough to definitely state something like “If I increase my on-hand inventory by an average of one unit for all items in an important product family, my company will see a net gain of $170,500.  That gain increases until I get to 4 units.  At 4 units and higher, the return declines due to excessive holding costs. So, the best decision factoring projected holding, ordering, and stockout is to increase inventory by 3 units to see a net gain of over $500,000.  

      Short of that ideal, you can do something that is simpler yet still extremely valuable: Quantify the tradeoff curve between inventory cost and item availability. While you won’t necessarily know the service level you should target, you will know the costs of varying service levels.  Then you can earn your big bucks by finding a good place to be on that tradeoff curve and communicating where you at risk, where you aren’t, and setting expectations with customers and internal stakeholders.  Without the tradeoff curve to guide you, you are flying blind with no way to rationally modify stocking policy.

      A Scenario to Learn From

      Let’s sketch out a realistic tradeoff curve. We start with a scenario requiring a management decision. The scenario we will use and associated assumptions about demand, lead times, and costs are detailed below:

      Inventory Policy

      • Periodic review – Reorder decisions made every 30 days
      • Order-Up-To-Level (“S”) – Varied from 30 to 60 units
      • Shortage Policy – Allow backorders, no lost orders

      Demand

      • Demand is intermittent
      • Average = 0.8 units per day
      • Standard deviation = 1.2 units per day
      • Largest demand in a year ≈ 9
      • % of days with no demand = 53%

      Lead Time

      • Random at either 7, 14 or 21 days with probabilities 70%, 20% and 10%, respectively

      Cost Parameters

      • Holding cost = $1 per day
      • Ordering Cost = $10 per order without regard to size of order
      • Shortage Cost = $100 per unit not immediately shipped from stock

      We imagine an inventory control policy that is known in the trade as a “periodic review” or (T,S) policy. In this instance, the Review Period (“T”) is 30 days, meaning that every 30 days the inventory position is checked and an ordering decision is made. The order quantity is the difference between the observed number of units on hand and the Order-Up-To Quantity (“S”). So, if the end-of-month inventory is 12 units and S = 20, the order quantity would be S – 12 = 20 -1 2 = 8. The next month, the order quantity is likely to be different. If the inventory ever goes negative (backorders) during a review period, the next order tries to restore equilibrium by ordering more in order to fill those backorders. For example, if the inventory is -5 (meaning 5 units ordered by not available for shipping, the next order would be S – (-5) = S + 5. Details of the hypothetical demand stream, supplier lead times, and cost elements are shown in Figure 1 below. Figure 2 show a sample of daily demand and daily inventory over five review periods. Demand is intermittent, as is often true for spare parts, and therefore difficult to plan for.

      Figure 1: Different choices of inventory policy (order up to), associated costs, and service levels

      Figure 2: Details of five months of system operation given one of the polices

       

      Inventory Planning Software Is Our Friend

      Software encodes the logic of the operation of the (T,S) system, generates many hypothetical but realistic demand scenarios, calculates how each of those scenarios plays out, then looks back on the simulated operation (here, 10 years or 3,650 consecutive days) to calculate cost and performance metrics.

      To reveal the tradeoff curve, we ran several computational experiments in which we varied the Order-Up-To Level, S. The plots Figure 2 show the behavior of the on-hand inventory in “richest” alternative with S = 60. In the snippet shown in Figure 2, the on-hand inventory never comes close to stocking out. You can read that too ways. One, a bit naïve, is to say “Good, we’re well protected.” The other, more aggressive, is to say, “Oh no, we’re bloated. I wonder what would happen if we reduced S.”

      The Tradeoff Curve Revealed

      Figure 3 shows the results of reducing S from 60 down to 30 in steps of 5 units. The table shows that Total Cost is the sum of Holding Cost, Ordering Cost, and Shortage Cost. For the (T,S) policy, the ordering cost is always the same, since an order is placed like clockwork every 30 days. But the other components of cost respond to the changes in S.

      Figure 3: The experimental results and corresponding tradeoff curve showing how changing the Order-Up-To Level (“S”) impacts both Service Level and Total Annual Cost

      Note that the Service Level is always lower than the Fill Rate in these scenarios. As a professor, I always think of this difference in terms of exam grading. Each replenishment cycle is like a test. Service Level is about the probability of a stockout, so it’s a like the grade on pass/fail exam with one question that must be answered perfectly. If there is no stockout in a cycle, that’s an A. If there is a stockout, that’s an F. It doesn’t matter if it’s one unit that’s not supplied or 50 – it’s still an F. But Fill Rate is like a question that is graded with partial credit. So being short one of ten units gets you 90% Fill Rate for that cycle, not 0%. It’s important to understand the difference between these two important metrics for inventory planning – check out this vlog describing service level vs. fill rate via an interactive exercise in Excel.

      The plot in Figure 3 is the real news. It pairs Total Cost and Service Level for various levels of S. If you read the graph right to left, it tells us that there are dramatic cost savings to be had by reducing S with very little penalty in terms of reduced item availability. For instance, reducing S from 60 to 55 saves close to $800 per year on this one item while reducing service level just a bit from (essentially) 100% to a still-impressive 99%. Cutting S some more does the same, though not as dramatically. If you read the graph left to right, you see that moving up from S = 30 to S = 35 costs about $1,000 per year but improves Service Level from an F grade (45%) to at least a C grade (71%). After that, pushing S higher costs progressively more while gaining progressive less.

      The tradeoff curve doesn’t give you an answer to how to set the Order-Up-To Level, but it does let you evaluate the costs and benefits of each possible answer. Take a minute and pretend that this is your problem: Where would you want to be along the tradeoff curve?

      You may object and say you hate your choices and want to change the game. Is there escape from the curve? Not from the general curve, but you might be able to shape a less painful curve. How?

      You may have other cards to play. One avenue is to try to “shape” the demand so that it is less variable. The demand plot in Figure 2 shows a lot of variability. If you could smooth out the demand, the whole tradeoff curve would shift down, making every choice less expensive. A second avenue is to try to reduce the mean and variability of supplier lead times. Achieving either would also shift the curve down to make the choice less painful. Check out our article on how suppliers influence your inventory costs

      Summary

      The tradeoff curve is always with us. Sometimes we may be able to make it more friendly, but we always to pick our spot along it. It is better to know what you’re getting for any choice of inventory policy than to try to guess, and the curve gives you that.  When you have an accurate estimate of that curve, you are no longer flying blind when it comes to inventory planning. 

       

       

       

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        Quantum Inventory Theory?

        The Smart Forecaster

         Pursuing best practices in demand planning,

        forecasting and inventory optimization

        Physicists like my Smart Software co-founder, Dr. Nelson Hartunian, tell us civilians that everything is different when we drill down to the tiniest level of the world. Physics at the quantum level is quite weird – not at all like what we experience in our usual macroscopic life. Among the oddities are “superposition”, “entanglement”, and “quantum foam.”  Weird as these phenomena are, I cannot help seeing analogs in the supposedly different world of supply chain management.

        Consider quantum superposition. Briefly, superposition means any quantum entity can be in two states at once. Schrödinger’s cat is the most famous illustration of this idea. But how many of you readers are also in a state of superposition? Don’t you find yourself being a manager of a team yet a member of your supervisor’s team, a trouble-shooter yet also a forecasting expert or an inventory optimizer and…? And doesn’t all this make you sometimes feel, like that cat, that you are simultaneously both dead and alive? Modern software can ease some of this burden by automating the tasks of demand planning and inventory optimization. The rest is up to you.

        A second quantum analog is entanglement. Briefly, entanglement is the linkage between two elements of a system. They can be light years apart, yet changing one part of an entangled system will instantaneously change the other part. This bugged Albert Einstein, who derided it as “spooky action as a distance.” In our regular world, demand planning and inventory optimization are entangled, since the process of inventory optimization sits on top of the process of demand forecasting. Modern software links the two in an efficient interface.

        Finally, the quantum foam – one of my favorite ideas. As I understand it, quantum foam is a substitute for empty space: there is no empty space, rather a constant bubbling of “vacuum energy” accompanied by a flux of “virtual particles” being born out of nothing and then disappearing back into nothing. In the supply chain world, the analogs of virtual particles are customer orders. Often it seems that they pop up with no warning out of thin air, and sometimes they disappear by cancellation in an equally random and mysterious process. This kind of demand fluctuation is the basis for all the theory of inventory control. Modern software therefore begins with probability models of customer demand. Those models then have implications for such tangible quantities as safety stocks, reorder points, and order quantities.

        Does it really help demand planners and inventory managers to think about these ideas from quantum physics? Well, it’s a bit of fun to see the analogies to our regular world of work. And they do remind us of more macroscopic matters: the basic concepts of the need to deal with more than one task simultaneously, the linkage between forecasting and inventory management, and randomness as the fundamental feature of the supply chain.

         

         

         

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          Stop Leaking Money with Manual Inventory Controls

          The Smart Forecaster

           Pursuing best practices in demand planning,

          forecasting and inventory optimization

          An inventory professional who is responsible for 10,000 items has 10,000 things to stress over every day. Double that for someone responsible for 20,000 items.

          In the crush of business, routine decisions often take second place to fire-fighting: dealing with supplier hiccups, straightening out paperwork mistakes, recovering from that collision between a truck and the loading dock.

          In the meantime, however, your company’s accumulated inventory control policies keep on doing what they do, even if they are leaking money. A good manager will make time to listen to the “background noise” even when he or she hears loud crashing in the warehouse.

          Consider the current settings for your inventory control parameters (e.g., reorder points and order quantities). It’s easy to think of these as “fire and forget” decisions. But these settings usually accumulate over time and end up comprising a mish-mash of forgotten judgement calls that may be misaligned with your current operating environment. Many factors can drift away from their previous levels, such as supplier lead times, ordering costs, or average item demand. These changes can force invisible tradeoffs that are not to your best advantage.

          It’s wise to revisit these control settings now and then to see if it’s possible to align your day-to-day operations with current realities. Of course, it would be infeasible for a busy manager to manually calculate the effects of changing the control settings on, say, 10,000 items. But that’s what modern inventory optimization and demand planning software is for: making large scale analytical tasks feasible. Such software will allow you to automatically process new information and compute adjustments at scale. The result will be easy wins – many of which would otherwise go unrealized.  And continuously saving a little here and there adds up to significant dollars when you are managing thousands of items.

          Consider this example. Company A uses a periodic review inventory system. Every 30 days, they check on-hand inventory for all their items and decide how much replenishment stock to order. Each of their 10,000 items has a specified Order-Up-To Level that determines the size of their replenishment orders.

          For instance, suppose Item 1234 has an Order-Up-To Level of 74, determined by factoring in the average item demand of 1.0 units per day, an average replenishment lead time of 8 days, and a target fill rate of 90% for this item. The choice of 74 as the Order-Up-To Level lets Company A meet its 90% fill rate target for Item 1234, but it also results in an average on hand inventory level of 40 units. At $1,500 per unit, this item alone represents $45,000 of inventory investment.

          Now supposed that average item demand were to drift up from 1.0 to 1.2 units/day. Without anyone noticing, the fill rate for Item 1234 would drop to 82%!

          Now suppose demand were to shift in the other direction and drift down to 0.8 units/day. As with the increase in average demand from 1.0 to 1.2 units/day, this kind of change is difficult to see when looking at a plot (see Figure 1) but can have a significant operational impact. In this case, the fill rate would zoom to a generous 96% but on hand inventory would also zoom: from 40 units to 46. Those six extra units would represent $9,000 in excess inventory.

          Figure 1: Samples of daily demand with two different average values.  The difference in demand is unnoticeable to the naked eye but if not accounted for will have a large operational impact on inventory spend and service levels

          Now imagine similar small shifts happening unnoticed across a full fleet of 10,000 inventory items. The total financial impact of all such shifts would be sufficient to get onto the radar of any CFO.  Trying to keep on top of this turbulence would be impossible if done manually but modern inventory optimization software could calculate the proper adjustments automatically as frequently as your company can handle, even daily helping you realize substantial improvements in service levels, inventory efficiency, while lowering stockout and holding costs!

           

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