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.

Let’s start by recognizing that increased revenue is a good thing for you, and that increasing the availability of the spare parts you provide is a good thing for your customers.

But let’s also recognize that increasing item availability will not necessarily lead to increased revenue. If you plan incorrectly and end up carrying excess inventory, the net effect may be good for your customers but will definitely be bad for you. There must be some right way to make this a win-win, if only it can be recognized.

To make the right decision here, you have to think systematically about the problem. That requires that you use probabilistic models of the inventory control process.

A Scenario

Let’s consider a specific, realistic scenario. Quite a number of factors have an influence on the results:

• The item: A specific low-volume spare part.
• Demand mean: Averaging 0.1 units per day (so, highly “intermittent”)
• Demand standard deviation: 0.35 units per day (so, highly variable or “overdispersed”).
• Supplier average lead time: 5 days.
• Unit cost: $100.
• Holding cost per year as % of unit cost: 10%.
• Ordering cost per PO cut: $25.
• Stockout consequences: Lost sales (so, a competitive market, no backorders).
• Shortage cost per lost sale: $100.
• Service level target: 85% (so, 15% chance of a stockout in any replenishment cycle).
• Inventory control policy: Periodic-review/Order-up-to (also called at (T,S) policy)

Inventory Control Policy

A word about the inventory control policy. The (T,S) policy is one of several that are common in practice. Though there are other more efficient policies (e.g., they don’t wait for T days to go by before making adjustment to stock), (T,S) is one of the simplest and so it is quite popular. It works this way: Every T days, you check how many units you have in stock, say X units. Then you order S-X units, which appear after the supplier lead time (in this case, 5 days). The T in (T,S) is the “order interval”, the number of days between orders; the S is the “order-up-to level”, the number of units you want to have on hand at the start of each replenishment cycle.

To get the most out of this policy, you must wisely pick values of T and S. Picking wisely means you cannot win by guessing or using simple rule-of-thumb guides like “Keep an average of 3 x average demand on hand.”  Poor choices of T and S hurt both your customers and your bottom line. And sticking too long with choices that were once good can result in poor performance should any of the factors above change significantly, so the values of T and S should be recalculated now and then.

The smart way to pick the right values of T and S is to use probabilistic models encoded in advanced software. Using software is essential when you have to scale up and pick values of T and S that are right for not one item but hundreds or thousands.

Analysis of Scenario

Let’s think about how to make money in this scenario. What’s the upside? If there were no expenses, this item could generate an average of $3,650 per year: 0.1 units/day x 365 days x $100/unit. Subtracted from that will be operating costs, comprised of holding, ordering and shortage costs. Each of those will depend on your choices of T and S.

The software provides specific numbers: Setting T = 321 days and S = 40 units will result in average annual operating costs of $604, giving an expected margin of $3,650 – $604 = $3,046. See Table 1, left column. This use of software is called “predictive analytics” because it translates system design inputs into estimates of a key performance indicator, margin.

Now think about whether you can do better. The service level target in this scenario is 85%, which is a somewhat relaxed standard that is not going to turn any heads. What if you could offer your customers a 99% service level? That sounds like a distinct competitive advantage, but would it reduce your margin? Not if you properly adjust the values of T and S.

Setting T = 216 days and S = 35 units will reduce average annual operating costs to $551 and increase expected margin to $3,650 – $551 = $3,099. See Table 1, right column. Here is the win-win we wanted: higher customer satisfaction and roughly 2% more revenue. This use of the software is called “sensitivity analysis” because it shows how sensitive the margin is to the choice of service level target.

Software can also help you visualize the complex, random dynamics of inventory movements. A by-product of the analysis that populated Table 1 are graphs showing the random paths taken by stock as it decreases over a replenishment cycle. Figure 1 shows a selection of 100 random scenarios for the scenario in which the service level target is 99%. In the figure, only 1 of the 100 scenarios resulted in a stockout, confirming the accuracy of the choice of order-up-to-level.

Summary

Management of spare parts inventories is often done haphazardly using gut instinct, habit, or obsolete rule-of-thumb. Winging it this way is not a reliable and reproducible path to higher margin or higher customer satisfaction. Probability theory, distilled into probability models then encoded in advanced software, is the basis for coherent, efficient guidance about how to manage spare parts based on facts: demand characteristics, lead times, service level targets, costs and the other factors. The scenarios analyzed here illustrate that it is possible to achieve both higher service levels and higher margin. A multitude of scenarios not shown here offer ways to achieve higher service levels but lose margin. Use the software.

Two Inventory Problems

If you both make and sell things, you own two inventory problems. Companies that sell things must focus relentlessly on having enough product inventory to meet customer demand.  Manufacturers and asset intensive industries such as power generation, public transportation, mining, and refining, have an additional inventory concern:  having enough spare parts to keep their machines running. This technical brief reviews the basics of two probabilistic models of machine breakdown. It also relates machine uptime to the adequacy of spare parts inventory.

Modeling the failure of a machine treated as a “black box”

Just as product demand is inherently random, so is the timing of machine breakdowns. Likewise, just as probabilistic modeling is the right way to deal with random demand, it is also the right way to deal with random breakdowns.

Models of machine breakdown have two components. The first deals with the random duration of uptime. The second deals with the random duration of downtime.

The field of reliability theory offers several standard probability models describing the random time until failure of a machine without regard for the reason for the failure. The simplest model of uptime is the exponential distribution. This model says that the hazard rate, i.e., the chance of failing in the next instant of time, is constant no matter how long the system has been operating. The exponential model does a good job at modeling certain types of systems, especially electronics, but it is not universally applicable.

Download the Whitepaper

The next step up in model complexity is the Weibull model (pronounced “WHY-bull”). The Weibull distribution allows the risk of failure to change over time, either decreasing after a burn in period or, more often, increasing as wear and tear accumulate. The exponential distribution is a special case of the Weibull distribution in which the hazard rate is neither increasing nor decreasing.

Figure 1: Three different Weibull survival curves

Figure 1 illustrates the Weibull model’s probability that a machine is still running as a function of how long it has been running. There are three curves corresponding to constant, decreasing and increasing hazard rates. For obvious reasons, these are called survival curves because they plot the probability of surviving for various amounts of time (but they are also called reliability curves). The black curve that starts high and sinks fast (β=3) depicts a machine that wears out with age. The lightest curve in the middle fast (β=1) shows the exponential distribution. The medium-dark curve (β=0.5)  is one that has a high early hazard rate but gets better with age.

Of course, there is another phenomenon that needs to be included in the analysis: downtime. Modeling downtime is where inventory theory enters the picture. Downtime is modeled by a mixture of two different distributions. If a spare part is available to replace the failed part, then the downtime can be very brief, say one day. But if there is no spare in stock, then the downtime can be quite long. Even if the spare can be obtained on an expedited basis, it may be several days or a week before the machine can be repaired. If the spare must be fabricated by a far-away supplier and shipped by sea then by rail then trucked to your plant, the downtime could be weeks or months. This all means that keeping a proper inventory of spares is very important to keeping production humming along.

In this aggregated type of analysis, the machine is treated as a black box that is either working or not. Though ignoring the details of which part failed and when, such a model is useful for sizing the pool of machines needed to maintain some minimum level of production capacity with high probability.

The binomial distribution is the probability model relevant to this problem. The binomial is the same model that describes, for example, the distribution of the number of “heads” resulting from twenty tosses of a coin. In the machine reliability problem, the machines correspond to coins, and an outcome of heads corresponds to having a working machine.

As an example, if

• the chance that any given machine is running on any particular day is 90%
• machine failures are independent (e.g., no flood or tornado to wipe them all out at once)
• you require at least a 95% chance that at least 5 machines are running on any given day

then the binomial model prescribes seven machines to achieve your goal.

Modeling machine failures based on component failures

The Weibull model can also be used to describe the failure of a single part. However, any realistically complex production machine will have multiple parts and therefore have multiple failure modes. This means that calculating the time until the machine fails requires analysis of a “race to failure”, with each part vying for the “honor” of being the first to fail.

If we make the reasonable assumption that parts fail independently, standard probability theory points the way to combining the models of individual part failure into an overall model of machine failure. The time until the first of many parts fails has a poly-Weibull distribution. At this point, though, the analysis can get quite complicated, and the best move may be to switch from analysis-by-equation to analysis-by-simulation.

Simulating machine failure from the details of part failures

Simulation analysis got its modern start as a spinoff of the Manhattan Project to build the first atomic bomb. The method is also commonly called Monte Carlo simulation after the biggest gambling center on earth back in the day (today it would be “Macau simulation”).

A simulation model converts the logic of the sequence of random events into corresponding computer code. Then it uses computer-generated (pseudo-)random numbers as fuel to drive the simulation model. For example, each component’s failure time is created by drawing from its particular Weibull failure time distribution. Then the soonest of those failure times begins the next episode of machine downtime.

Figure 2: A simulation of machine uptime over one year of operation

Figure 2 shows the results of a simulation of the uptime of a single machine. Machines cycle through alternating periods of uptime and downtime. In this simulation, uptime is assumed to have an exponential distribution with an average duration (MTBF = Mean Time Before Failure) of 30 days. Downtime has a 50:50 split between 1 day if a spare is available and 30 days if not. In the simulation shown in Figure 2, the machine is working during 85% of the days in one year of operation.

An approximate formula for machine uptime

Although Monte Carlo simulation can provide more exact results, a simpler algebraic model does well as an approximation and makes it easier to see how the key variables relate.

Define the following key variables:

• MTBF = Mean Time Before Failure (days)
• Pa = Probability that there is a spare part available when needed
• MDTshort = Mean Down Time if there is a spare available when needed
• MDTlong = Mean Down Time if there is no spare available when needed
• Uptime = Percentage of days in which the machine is up and running.

Then there is a simple approximation for the Uptime:

Uptime ≈ 100 x MTBF/(MTBF + MDTshort x Pa + MDTlong x (1-Pa)).    (Equation 1)

Equation 1 tells us that the uptime depends on the availability of a spare. If there is always a spare (Pa=1), then uptime achieves a peak value of about 100 x MTBF/(MTBF + MDTshort). If there is never a spare available (Pa=0), then uptime achieve its lowest value of about 100 x MTBF/(MTBF + MDTlong). When the repair time is about as long as the typical time between failures, uptime sinks to an unacceptable level near 50%. If a spare is always available, uptime can approach 100%.

Relating machine downtime to spare parts inventory

Minimizing downtime requires a multi-pronged initiative involving intensive operator training, use of quality raw materials, effective preventive maintenance – and adequate spare parts. The first three set the conditions for good results. The last deals with contingencies.

Once a machine is down, money is flying out the door and there is a premium on getting it back up pronto. This scene could play out in two ways. The good one has a spare part ready to go, so the downtime can be kept to a minimum. The bad one has no available spare, so there is a scramble to expedite delivery of the needed part. In this case, the manufacturer must bear both the cost of lost production and the cost of expedited shipping, if that is even an option.

If the inventory system is properly designed, spare parts availability will not be a major impediment to machine uptime. By the design of an inventory system, I mean the results of several choices: whether the shortage policy is a backorder policy or a loss policy, whether the inventory review cycle is periodic or continuous, and what reorder points and order quantities are established.

When inventory policies for products are designed, they are evaluated using several criteria. Service Level is the percentage of replenishment periods that pass without a stockout. Fill Rate is the percentage of units ordered that is supplied immediately from stock. Average Inventory Level is the typical number of units on hand.

None of these is exactly the metric needed for spare parts stocking, though they all are related. The needed metric is Item Availability, which is the percentage of days in which there is at least one spare ready for use. Higher Service Levels, Fill Rates, and Inventory Levels all imply high Item Availability, and there are ways to convert from one to the other. (When dealing with multiple machines sharing the same stock of spares, Inventory Availability gets replaced by the probability distribution of the number of spares on any given day. We leave that more complex problem for another day.)

Clearly, keeping a good supply of spares reduces the costs of machine downtime. Of course, keeping a good supply of spares creates its own inventory holding and ordering costs. This is the manufacturer’s second inventory problem. As with any decision involving inventory, the key is to strike the right balance between these two competing cost centers. See this article on probabilistic forecasting for intermittent demand for guidance on striking that balance.

Most demand forecasts are partial or incomplete: They provide only one single number: the most likely value of future demand. This is called a point forecast. Usually, the point forecast estimates the average value of future demand.

Much more useful is a forecast of full probability distribution of demand at any future time. This is more commonly referred to as probability forecasting and is much more useful.

The Average is Not the Answer

The one advantage of a point forecast is its simplicity. If your ERP system is also simple, the point forecast fills in the one number needed by the ERP system to do workforce scheduling or raw material purchases.

The disadvantage of a point forecast is that it is too simple. It ignores additional information in an item’s demand history that can give you a more complete picture of how demand might unfold: a probability forecast.

Going Beyond the Average: Probability Forecasting

While the point forecast provides limited information, e.g., “The most likely demand next month is 15 units”, the probability forecast adds crucial information, e.g., “There is a 20% chance that demand will exceed 28 units and a 10% chance that it will be less than 5 units”.

This information lets you do risk assessment and contingency planning. Contingency planning is necessary because the point forecast usually has only a small chance of actually being correct. A probability forecast may also say “The chance of demand being 15 units is only 10%, even though it is the single most likely value.” In other words, there is a 90% chance that the point forecast is wrong. This kind of error is not a mistake in the forecasting calculations: it is the reality of dealing with demand volatility. It might better be called an “uncertainty” than an “error”.

An operations manager can use the extra information in a probability forecast in both informal and formal ways. Informally, even if an ERP system requires a single-number forecast as input, a wise manager will want to have some clue about the risks associated with that point forecast, i.e., its margin of error. So a forecast of 15 ± 1 unit is a lot safer than a forecast of 15 ± 10. The ± part is a compression of a probabilistic forecast. Figure 1 below shows an item’s demand history (red line), point forecasts for the next 12 months (green line) and their margins of error (cyan lines). The lowest forecast of about 3,300 units occurs in June, but the actual demand might be as much as 800 units higher or lower.

Bonus: Application to Inventory Management

Inventory management requires that you balance item availability against the inventory cost. It turns out that knowing the full probability distribution of demand over a replenishment lead time is essential for setting reorder points (also called mins) on a rational, scientific basis. Figure 2 shows a probability forecast of total demand during the 33 week replenishment lead time for a certain spare part. While the average lead time demand is 3 units, the most likely demand is zero, and a reorder point of 14 is needed to insure that the chance of stocking out is only 1%. Once again, the average is not the answer.

Knowing more is always better than knowing less and the probability forecast provides that extra bit of crucial information. Software has been able to supply a point forecast for over 40 years, but modern software can do better and provide the whole picture.

Figure 1: The red line shows the demand history of a finished good. The green line shows the point forecasts for the next 12 months. The blue lines indicate the margins of error in the 12 point forecasts.

Figure 2: A probabilistic forecast of demand for a spare part over a 33 week replenishment lead time. The most likely demand is zero, the average demand is 3, but a reorder point of 14 units is required to have only a 1% chance of stock out.