A Primer on Probabilistic Forecasting

The Smart Forecaster

Pursuing best practices in demand planning,

forecasting and inventory optimization

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.

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.

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.

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.

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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Undershoot is Sabotaging your Service Level!

Undershoot means that the lead time begins not at the reorder point but below it. Undershoot happens every time the demand that breached the reorder point took the stock down below (not down to) the reorder point. Undershoot picks your pocket before you even begin to roll the dice. It deludes the inventory professional into thinking his or her reorder points are sufficient to achieve their targets, whereas actual performance will not make the grade.

How to Choose a Target Service Level

When setting a target service level, make sure to take into account factors like current service levels, replenishment lead times, cost constraints, the pain inflicted by shortages on you and your customers, and your competitive position.

Reveal Your Real Inventory Planning and Forecasting Policy by Answering These 10 Questions

In this blog, we review 10 specific questions you can ask to uncover what’s really happening with the inventory planning and demand forecasting policy at your company. We detail the typical answers provided when a forecasting/inventory planning policy doesn’t really exist, explain how to interpret these answers, and offer some clear advice on what to do about it.

Inventory Optimization for Manufacturers, Distributors, and MRO

Managing Spare Parts Inventory: Best Practices
In this blog, we’ll explore several effective strategies for managing spare parts inventory, emphasizing the importance of optimizing stock levels, maintaining service levels, and using smart tools to aid in decision-making. Managing spare parts inventory is a critical component for businesses that depend on equipment uptime and service reliability. Unlike regular inventory items, spare parts often have unpredictable demand patterns, making them more challenging to manage effectively. An efficient spare parts inventory management system helps prevent stockouts that can lead to operational downtime and costly delays while also avoiding overstocking that unnecessarily ties up capital and increases holding costs. […]
Innovating the OEM Aftermarket with AI-Driven Inventory Optimization
The aftermarket sector provides OEMs with a decisive advantage by offering a steady revenue stream and fostering customer loyalty through the reliable and timely delivery of service parts. However, managing inventory and forecasting demand in the aftermarket is fraught with challenges, including unpredictable demand patterns, vast product ranges, and the necessity for quick turnarounds. Traditional methods often fall short due to the complexity and variability of demand in the aftermarket. The latest technologies can analyze large datasets to predict future demand more accurately and optimize inventory levels, leading to better service and lower costs. […]
Future-Proofing Utilities: Advanced Analytics for Supply Chain Optimization
Utilities in the electrical, natural gas, urban water, and telecommunications fields are all asset-intensive and reliant on physical infrastructure that must be properly maintained, updated, and upgraded over time. Maximizing asset uptime and the reliability of physical infrastructure demands effective inventory management, spare parts forecasting, and supplier management. A utility that executes these processes effectively will outperform its peers, provide better returns for its investors and higher service levels for its customers, while reducing its environmental impact. […]
Centering Act: Spare Parts Timing, Pricing, and Reliability
In this article, we'll walk you through the process of crafting a spare parts inventory plan that prioritizes availability metrics such as service levels and fill rates while ensuring cost efficiency. We'll focus on an approach to inventory planning called Service Level-Driven Inventory Optimization. Next, we'll discuss how to determine what parts you should include in your inventory and those that might not be necessary. Lastly, we'll explore ways to enhance your service-level-driven inventory plan consistently. […]

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Stay the course

I’ve stood in front of thousands of students. They’ve been more or less young, more or less technical, more or less experienced – and more or less interested. I’ve done this as a university faculty member since 1972, first at Massachusetts Institute of Technology, then at Harvard University, finally in the School of Engineering at Rensselaer Polytechnic Institute. Between Harvard and RPI I dropped out of academia temporarily to co-found Smart Software with Charlie Smart and Nelson Hartunian. So since then, I’ve also been busy training business users to exploit the power of advanced analytics for forecasting and inventory optimization.

As I write this, I’ve just returned to my office at RPI after introducing first-year Industrial Engineering students to the basic concepts of inventory management. If they stick with the program, they will go on to take required courses in supply chain, system simulation, statistical analysis, and optimization. I told them stories about how useful they will be to their companies should they decide to make a career in the world of supply chain. If I’d had more time, I would have mentioned how capable they will be when they graduate relative to many of their corporate peers. These freshmen and ready and willing to stay the course, soaking up all the techniques and theories we can throw at them, and honing their practical skills in summer jobs or coop assignments.

What I didn’t tell them is that many of them will have to work to keep their intensity when they are on the job. It’s a sad truth that, for whatever reason, many inventory practitioners settle into a kind of stasis that impedes their companies’ ability to exploit the latest technologies, such as cloud-based advanced demand forecasting and inventory optimization. Gather enough of such people in one place and agility and improved efficiency go out the window.

I think one of the factors that dulls people is that the process of implementation frequently feels painfully incremental and prolonged. It often begins with a sobering inventory of relevant data, its correctness, and its currency. Then it moves to an often-awkward discovery that there really is no systematic process in place and the subsequent need to design a good one going forward. Next is the need to learn to use a new software suite. That step involves learning new vocabulary, some level of probabilistic thought, an ability to interpret new graphs and tables, not to mention a new software interface. All this takes time and effort.

We’ve found that a few things help new customers stay the course. One is having a champion among management, an executive sponsor, who can vouch for the commercial importance of a successful implementation while ensuring the users are supported with continuing education. A second is identifying and training a super-user or two having unusual combinations of technical and communication skills. A third is breaking the training into bite-sized chunks and testing for comprehension after each chunk and repeating this process until it is clear that the new concepts, vocabulary, and process are fully absorbed. But all those maneuvers will come to naught without management being all-in and ready to stay the course. Inventory planning practices in place for many years are not going to be replaced entirely over a three-month implementation process. You’ve got to want it to get it.

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Are You Playing the Inventory Guessing Game?

Some companies invest in software to help them manage their inventory, whether it’s spare parts or finished goods. But a surprising number of others play the Inventory Guessing Game every day, trusting to an imagined “Golden Gut” or to plain luck to set their inventory control parameters. But what kind of results do you expect with that approach?

Finding Your Spot on the Tradeoff Curve

Managing inventory, like managing anything, involves balancing competing priorities. Do you want a lean inventory? Yes! Do you want to be able to say “It’s in stock” when a customer wants to buy something? Yes!
But can you have it both ways? Only to a degree. If you lean into leaning your inventory too aggressively, you risk stockouts. If you stamp out stockouts, you create inventory bloat. You are forced to find a satisfactory balance between the two competing goals of lean inventory and high item availability.

Direct to the Brain of the Boss – Inventory Analytics and Reporting

In this blog, the spotlight is cast on the software that creates reports for management, the silent hero that translates the beauty of furious calculations into actionable reports. Watch as the calculations, intricately guided by planners utilizing our software, seamlessly converge into Smart Operational Analytics (SOA) reports, dividing five key areas: inventory analysis, inventory performance, inventory trending, supplier performance, and demand anomalies.

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:

You start each replenishment cycle with Q units on hand.
You deplete that stock by D units per day.
So, you hit the reorder point R after (Q-R)/D days.
So, you order every (Q-R)/D days.
Each replenishment cycle lasts (Q-R)/D + L days, so you make a total of 365D/(Q-R+LD) orders per year.
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

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 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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Are You Playing the Inventory Guessing Game?

Finding Your Spot on the Tradeoff Curve

Direct to the Brain of the Boss – Inventory Analytics and Reporting

Coping with Surging Demand During the Rebound

The Smart Forecaster

Pursuing best practices in demand planning,

forecasting and inventory optimization

Many of our customers that saw demand dry up during the pandemic are now seeing demand return. Some are seeing a significant demand surge. Other customers in critical industries like plastics, biotech, semiconductors and electronics saw demand surges starting as far back as last April. For suggestions about how to cope with these situations, please read on.

Surging demand usually creates two problems: inability to fill orders and inability to get replenishment due to supplier overload. This situation requires changes in the way you use your advanced planning software. Here are three tips to help you cope.

Tip #1: Narrow your temporal focus

In normal times (remember those?), more data implied better results. Nowadays, old data poison your calculations, since they represent conditions that no longer apply. You should base forecasts and other calculations on data from the current situation. Where to cut off past data may be obvious from a plot of the data, or you may decide to set a “reasonable” cutoff date based on a consensus of colleagues. Smart Software has developed machine learning algorithms that automatically identify how much historical data should be optimally fed to the forecast model. Be on the lookout for these enhancements to the software that will be rolling out soon. In the meantime, conduct accuracy tests using held-out actuals using different historical start dates. Smart’s forecast vs. actual feature will support this automatically.

Smart Demand Planner forecasts vs. actual report

Tip #2: Increase your planning tempo

When operations are stable, you can set your inventory policies and trust them to be appropriate for a long time. When times are turbulent, it is important to increase the frequency of your planning cycles to keep old policy settings from drifting too far away from optimality. More frequent recalibration of your stocking policies and forecasts means that you’ll be quicker to catch trends that will surprise your competition and always keep you steps ahead. With software capable of automatically selecting optimal values, all that work can be done in one shot by the software. You should review those changes and possibly tweak them, but it makes sense to let the software do the bulk of the work.

Tip #3: Do more What-If planning

In turbulent times, you might expect even more turbulence in the future. Using your software for what-if planning helps you prepare for changes that may be coming. For example, suppose you’ve been in touch with a key supplier who hints that they may be raising prices or may have to slip their delivery schedules. By feeding the software different inputs, you can do contingency planning. If prices go up, you can see how responding by changing order quantities would impact your inventory operating costs and inventory investment. If lead times go up, you can see what the impact would be on item availability. This foreknowledge helps you figure out what your counter-moves would be before the crisis hits.

If there ever was a time when we could cruise on automatic pilot, it’s in the rear-view mirror. Your organization, coping with explosive growth, has many challenges. Old answers are obsolete; new answers have to come from somewhere, fast. Advanced software that leverages probabilistic forecasting can help, along with changes in planning processes.

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Undershoot is Sabotaging your Service Level!

How to Choose a Target Service Level

Reveal Your Real Inventory Planning and Forecasting Policy by Answering These 10 Questions

Inventory Optimization for Manufacturers, Distributors, and MRO

Managing Spare Parts Inventory: Best Practices
In this blog, we’ll explore several effective strategies for managing spare parts inventory, emphasizing the importance of optimizing stock levels, maintaining service levels, and using smart tools to aid in decision-making. Managing spare parts inventory is a critical component for businesses that depend on equipment uptime and service reliability. Unlike regular inventory items, spare parts often have unpredictable demand patterns, making them more challenging to manage effectively. An efficient spare parts inventory management system helps prevent stockouts that can lead to operational downtime and costly delays while also avoiding overstocking that unnecessarily ties up capital and increases holding costs. […]
Innovating the OEM Aftermarket with AI-Driven Inventory Optimization
The aftermarket sector provides OEMs with a decisive advantage by offering a steady revenue stream and fostering customer loyalty through the reliable and timely delivery of service parts. However, managing inventory and forecasting demand in the aftermarket is fraught with challenges, including unpredictable demand patterns, vast product ranges, and the necessity for quick turnarounds. Traditional methods often fall short due to the complexity and variability of demand in the aftermarket. The latest technologies can analyze large datasets to predict future demand more accurately and optimize inventory levels, leading to better service and lower costs. […]
Future-Proofing Utilities: Advanced Analytics for Supply Chain Optimization
Utilities in the electrical, natural gas, urban water, and telecommunications fields are all asset-intensive and reliant on physical infrastructure that must be properly maintained, updated, and upgraded over time. Maximizing asset uptime and the reliability of physical infrastructure demands effective inventory management, spare parts forecasting, and supplier management. A utility that executes these processes effectively will outperform its peers, provide better returns for its investors and higher service levels for its customers, while reducing its environmental impact. […]
Centering Act: Spare Parts Timing, Pricing, and Reliability
In this article, we'll walk you through the process of crafting a spare parts inventory plan that prioritizes availability metrics such as service levels and fill rates while ensuring cost efficiency. We'll focus on an approach to inventory planning called Service Level-Driven Inventory Optimization. Next, we'll discuss how to determine what parts you should include in your inventory and those that might not be necessary. Lastly, we'll explore ways to enhance your service-level-driven inventory plan consistently. […]

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Maximize Machine Uptime with Probabilistic Modeling

The Smart Forecaster

Pursuing best practices in demand planning,

forecasting and inventory optimization

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.

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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.

Inventory Planning for Manufacturers MRO SAAS

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.

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Inventory Optimization for Manufacturers, Distributors, and MRO

Managing Spare Parts Inventory: Best Practices
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Innovating the OEM Aftermarket with AI-Driven Inventory Optimization
The aftermarket sector provides OEMs with a decisive advantage by offering a steady revenue stream and fostering customer loyalty through the reliable and timely delivery of service parts. However, managing inventory and forecasting demand in the aftermarket is fraught with challenges, including unpredictable demand patterns, vast product ranges, and the necessity for quick turnarounds. Traditional methods often fall short due to the complexity and variability of demand in the aftermarket. The latest technologies can analyze large datasets to predict future demand more accurately and optimize inventory levels, leading to better service and lower costs. […]
Future-Proofing Utilities: Advanced Analytics for Supply Chain Optimization
Utilities in the electrical, natural gas, urban water, and telecommunications fields are all asset-intensive and reliant on physical infrastructure that must be properly maintained, updated, and upgraded over time. Maximizing asset uptime and the reliability of physical infrastructure demands effective inventory management, spare parts forecasting, and supplier management. A utility that executes these processes effectively will outperform its peers, provide better returns for its investors and higher service levels for its customers, while reducing its environmental impact. […]
Centering Act: Spare Parts Timing, Pricing, and Reliability
In this article, we'll walk you through the process of crafting a spare parts inventory plan that prioritizes availability metrics such as service levels and fill rates while ensuring cost efficiency. We'll focus on an approach to inventory planning called Service Level-Driven Inventory Optimization. Next, we'll discuss how to determine what parts you should include in your inventory and those that might not be necessary. Lastly, we'll explore ways to enhance your service-level-driven inventory plan consistently. […]

The Smart Forecaster

Pursuing best practices in demand planning,

forecasting and inventory optimization

Undershoot is Sabotaging your Service Level!

How to Choose a Target Service Level

Reveal Your Real Inventory Planning and Forecasting Policy by Answering These 10 Questions

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Inventory Optimization for Manufacturers, Distributors, and MRO

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The Smart Forecaster

The Smart Forecaster

Pursuing best practices in demand planning,

forecasting and inventory optimization

Tip #1: Narrow your temporal focus

Tip #2: Increase your planning tempo

Tip #3: Do more What-If planning

Undershoot is Sabotaging your Service Level!

How to Choose a Target Service Level

Reveal Your Real Inventory Planning and Forecasting Policy by Answering These 10 Questions

Recent Posts

Inventory Optimization for Manufacturers, Distributors, and MRO

Blog Categories

The Smart Forecaster

Pursuing best practices in demand planning,

forecasting and inventory optimization

Two Inventory Problems

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

Modeling machine failures based on component failures

Simulating machine failure from the details of part failures

An approximate formula for machine uptime

Relating machine downtime to spare parts inventory

Undershoot is Sabotaging your Service Level!

How to Choose a Target Service Level

Reveal Your Real Inventory Planning and Forecasting Policy by Answering These 10 Questions

Recent Posts

Inventory Optimization for Manufacturers, Distributors, and MRO

Blog Categories