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Goldilocks Inventory Levels

You may remember the story of Goldilocks from your long-ago youth. Sometimes the porridge was too hot, sometimes it was too cold, but just once it was just right. Now that we are adults, we can translate that fairy tale into a professional principle for inventory planning: There can be too little or too much inventory, and there is some Goldilocks level that is “just right.” This blog is about finding that sweet spot.

To illustrate our supply chain fable, consider this example. Imagine that you sell service parts to keep your customers systems up and running. You offer a particular service part that costs you \$100 to make but sells for a 20% markup. You can make \$20 on each unit you sell, but you don’t get to keep the whole \$20 because of the inventory operating costs you bear to be able to sell the part. There are holding costs to keep the part in good repair while in stock and ordering costs to replenish units you sell. Finally, sometimes you lose revenue from lost sales due to stockouts.

These operating costs can be directly related to the way you manage the part in inventory. For our example, assume you use a (Q,R) inventory policy, where Q is the replenishment order quantity and R is the reorder point. Assume further that the reason you are not making \$30 per unit is that you have competitors, and customers will get the part from them if they can’t get it from you.

Both your revenue and your costs depend in complex ways on your choices for Q and R. These will determine how much you order, when and therefore how often you order, how often you stock out and therefore how many sales you lose, and how much cash you tie up in inventory. It is impossible to cost out these relationships by guesswork, but modern software can make the relationships visible and calculate the dollar figures you need to guide your choice of values for Q and R. It does this by running detailed, fact-based, probabilistic simulations that predict costs and performance by averaging over a large number of realistic demand scenarios.

With these results in hand, you can work out the margin associated with (Q,R) values using the simple formula

Margin = (Demand – Lost Sales) x Profit per unit sold – Ordering Costs – Holding Costs.

In this formula, Lost Sales, Ordering Costs and Holding Costs are dependent on reorder point R and order quantity Q.

Figure 1 shows the result of simulations that fixed Q at 25 units and varied R from 10 to 30 in steps of 5. While the curve is rather flat on top, you would make the most money by keeping on-hand inventory around 25 units (which corresponds to setting R = 20). More inventory, despite a higher service level and fewer lost sales, would make a little less money (and ties up a lot more cash), and less inventory would make a lot less.

Figure 1: Showing that there can be too little or too much inventory on hand

Without relying on the inventory simulation software, we would not be able to discover

• a) that it is possible to carry too little and too much inventory
• b) what the best level of inventory is
• c) how to get there by proper choices of reorder point R and order quantity Q.

Without an explicit understanding of the above, companies will make daily inventory decisions relying on gut feel and averaging based rule of thumb methods. The tradeoffs described here are not exposed and the resulting mix of inventory yields a far lower return forfeiting hundreds of thousands to millions per year in lost profits.  So be like Goldilocks.  With the right systems and software tools, you too can get it just right!

Related Posts

## Goldilocks Inventory Levels

You may remember the story of Goldilocks from your long-ago youth. Sometimes the porridge was too hot, sometimes it was too cold, but just once it was just right. Now that we are adults, we can translate that fairy tale into a professional principle for inventory planning: There can be too little or too much inventory, and there is some Goldilocks level that is “just right.” This blog is about finding that sweet spot.

## Call an Audible to Proactively Counter Supply Chain Noise

You know the situation: You work out the best way to manage each inventory item by computing the proper reorder points and replenishment targets, then average demand increases or decreases, or demand volatility changes, or suppliers’ lead times change, or your own costs change.

## An Example of Simulation-Based Multiechelon Inventory Optimization

Managing the inventory across multiple facilities arrayed in multiple echelons can be a huge challenge for any company. The complexity arises from the interactions among the echelons, with demands at the lower levels bubbling up and any shortages at the higher levels cascading down.

An Example of Simulation-Based Multiechelon Inventory Optimization

Managing the inventory in a single facility is difficult enough, but the problem becomes much more complex when there are multiple facilities arrayed in multiple echelons. The complexity arises from the interactions among the echelons, with demands at the lower levels bubbling up and any shortages at the higher levels cascading down.

If each of the facilities were to be managed in isolation, standard methods could be used, without regard to interactions, to set inventory control parameters such as reorder points and order quantities. However, ignoring the interactions between levels can lead to catastrophic failures. Experience and trial and error allow the design of stable systems, but that stability can be shattered by changes in demand patterns or lead times or by the addition of new facilities. Coping with such changes is greatly aided by advanced supply chain analytics, which provide a safe “sandbox” within which to test out proposed system changes before deploying them. This blog illustrates that point.

The Scenario

To have some hope of discussing this problem usefully, this blog will simplify the problem by considering the two-level hierarchy pictured in Figure 1. Imagine the facilities at the lower level to be warehouses (WHs) from which customer demands are meant to be satisfied, and that the inventory items at each WH are service parts sold to a wide range of external customers.

Figure 1: General structure of one type of two-level inventory system

Imagine the higher level to consist of a single distribution center (DC) which does not service customers directly but does replenish the WHs. For simplicity, assume the DC itself is replenished from a Source that always has (or makes) sufficient stock to immediately ship parts to the DC, though with some delay. (Alternatively, we could consider the system to have retail stores supplied by one warehouse).

Each level can be described in terms of demand levels (treated as random), lead times (random), inventory control parameters (here, Min and Max values) and shortage policy (here, backorders allowed).

The Method of Analysis

The academic literature has made progress on this problem, though usually at the cost of simplifications necessary to facilitate a purely mathematical solution. Our approach here is more accessible and flexible: Monte Carlo simulation. That is, we build a computer program that incorporates the logic of the system operation. The program “creates” random demand at the WH level, processes the demand according to the logic of a chosen inventory policy, and creates demand for the DC by pooling the random requests for replenishment made by the WHs. This approach lets us observe many simulated days of system operation while watching for significant events like stockouts at either level.

An Example

To illustrate an analysis, we simulated a system consisting of four WHs and one DC. Average demand varied across the WHs. Replenishment from the DC to any WH took from 4 to 7 days, averaging 5.15 days. Replenishment of the DC from the Source took either 7, 14, 21 or 28 days, but 90% of the time it was either 21 or 28 days, making the average 21 days. Each facility had Min and Max values set by analyst judgement after some rough calculations.

Figure 2 shows the results of one year of simulated daily operation of this system. The first row in the figure shows the daily demand for the item at each WH, which was assumed to be “purely random”, meaning it had a Poisson distribution. The second row shows the on-hand inventory at the end of each day, with Min and Max values indicated by blue lines. The third row describes operations at the DC.  Contrary to the assumption of much theory, the demand into the DC was not close to being Poisson, nor was the demand out of the DC to the Source. In this scenario, Min and Max values were sufficient to keep item availability was high at each WH and at the DC, with no stockouts observed at any of the five facilities.

Figure 2 – Simulated year of operation of a system with four WHs and one DC.

Now let’s vary the scenario. When stockouts are extremely rare, as in Figure 2, there is often excess inventory in the system. Suppose somebody suggests that the inventory level at the DC looks a bit fat and thinks it would be good idea to save money there. Their suggestion for reducing the stock at the DC is to reduce the value of the Min at the DC from 100 to 50. What happens? You could guess, or you could simulate.

Figure 3 shows the simulation – the result is not pretty. The system runs fine for much of the year, then the DC runs out of stock and cannot catch up despite sending successively larger replenishment orders to the Source. Three of the four WHs descend into death spirals by the end of the year (and WH1 follows thereafter). The simulation has highlighted a sensitivity that cannot be ignored and has flagged a bad decision.

Figure 3 – Simulated effects of reducing the Min at the DC.

Now the inventory managers can go back to the drawing board and test out other possible ways to reduce the investment in inventory at the DC level. One move that always helps, if you and your supplier can jointly make it happen, is to create a more agile system by reducing replenishment lead time. Working with the Source to insure that the DC always gets its replenishments in either 7 or 14 days stabilizes the system, as shown in Figure 4.

Figure 4 – Simulated effects of reducing the lead time for replenishing the DC.

Unfortunately, the intent of reducing the inventory at the DC has not been achieved. The original daily inventory count was about 80 units and remains about 80 units after reducing the DC’s Min and drastically improving the Source-to-DC lead time. But with the simulation model, the planning team can try out other ideas until they arrive at a satisfactory redesign. Or, given that Figure 4 shows the DC inventory starting to flirt with zero, they might think it prudent to accept the need for an average of about 80 units at the DC and look for ways to trim inventory investment at the WHs instead.

The Takeaways

1. Multiechelon inventory optimization (MEIO) is complex. Many factors interact to produce system behaviors that can be surprising in even simple two-level systems.
2. Monte Carlo simulation is a useful tool for planners who need to design new systems or tweak existing systems.

Related Posts

## Goldilocks Inventory Levels

You may remember the story of Goldilocks from your long-ago youth. Sometimes the porridge was too hot, sometimes it was too cold, but just once it was just right. Now that we are adults, we can translate that fairy tale into a professional principle for inventory planning: There can be too little or too much inventory, and there is some Goldilocks level that is “just right.” This blog is about finding that sweet spot.

## Call an Audible to Proactively Counter Supply Chain Noise

You know the situation: You work out the best way to manage each inventory item by computing the proper reorder points and replenishment targets, then average demand increases or decreases, or demand volatility changes, or suppliers’ lead times change, or your own costs change.

## An Example of Simulation-Based Multiechelon Inventory Optimization

Managing the inventory across multiple facilities arrayed in multiple echelons can be a huge challenge for any company. The complexity arises from the interactions among the echelons, with demands at the lower levels bubbling up and any shortages at the higher levels cascading down.

Fact and Fantasy in Multiechelon Inventory Optimization

For most small-to-medium manufacturers and distributors, single-level or single-echelon inventory optimization is at the cutting edge of logistics practice. Multi-echelon inventory optimization (“MEIO”) involves playing the game at an even higher level and is therefore much less common. This blog is the first of two. It aims to explain what MEIO is, why standard MEIO theories break down, and how probabilistic modeling through scenario simulation can restore reality to the MEIO process. The second blog will show a particular example.

Definition of Inventory Optimization

An inventory system is built on a set of design choices.

The first choice is the policy for responding to stockouts: Do you just lose the sale to a competitor, or can you convince the customer to accept a backorder? The former is more common with distributors than manufacturers, but this may not be much of a choice since customers may dictate the answer.

The second choice is the inventory policy. These divide into “continuous review” and “periodic review” policies, with several options within each type. You can link to a video tutorial describing several common inventory policies here.  Perhaps the most efficient is known to practitioners as “Min/Max” and to academics as (s, S) or “little S, Big S.” We use this policy in the scenario simulations below. It works as follows: When on-hand inventory drops to or below the Min (s), an order is placed for replenishment. The size of the order is the gap between the on-hand inventory and the Max (S), so if Min is 10, Max is 25 and on-hand is 8, it’s time for an order of 25-8 = 17 units.

The third choice is to decide on the best values of the inventory policy “parameters”, e.g., the values to use for Min and Max. Before assigning numbers to Min and Max, you need clarity on what “best” means for you. Commonly, best means choices that minimize inventory operating costs subject to a floor on item availability, expressed either as Service Level or Fill Rate. In mathematical terms, this is a “two-dimensional constrained integer optimization problem”. “Two-dimensional” because you have to pick two numbers: Min and Max. “Integer” because Min and Max have to be whole numbers. “Constrained” because you must pick Min and Max values that give a high-enough level of item availability such as service levels and fill rates. “Optimization” because you  want to get there with the lowest operating cost (operating cost combines holding, ordering and shortage costs).

Multiechelon Inventory Systems

The optimization problem becomes more difficult in multi-echelon systems. In a single-echelon system, each inventory item can be analyzed in isolation: one pair of Min/Max values per SKU. Because there are more parts to a multiechelon system, there is a bigger computational problem.

Figure 1 shows a simple two-level system for managing a single SKU. At the lower level, demands arrive at multiple warehouses. When those are in danger of stocking out, they are resupplied from a distribution center (DC). When the DC itself is in danger of stocking out, it is supplied by some outside source, such as the manufacturer of the item.

The design problem here is multidimensional: We need Min and Max values for 4 warehouses and for the DC, so the optimization occurs in 4×2+1×2=10 dimensions. The analysis must take account of a multitude of contextual factors:

• The average level and volatility of demand coming into each warehouse.
• The average and variability of replenishment lead times from the DC.
• The average and variability of replenishment lead times from the source.
• The required minimum service level at the warehouses.
• The required minimum service level at the DC.
• The holding, ordering and shortage costs at each warehouse.
• The holding, ordering and shortage costs at the DC.

As you might expect, seat-of-the-pants guesses won’t do well in this situation. Neither will trying to simplify the problem by analyzing each echelon separately. For instance, stockouts at the DC increase the risk of stockouts at the warehouse level and vice versa.

This problem is obviously too complicated to try to solve without help from some sort of computer model.

Why Standard Inventory Theory is Bad Math

With a little looking, you can find models, journal articles and book about MEIO. These are valuable sources of information and insight, even numbers. But most of them rely on the expedient of over-simplifying the problem to make it possible to write and solve equations. This is the “Fantasy” referred to in the title.

Doing so is a classic modeling maneuver and is not necessarily a bad idea. When I was a graduate student at MIT, I was taught the value of having two models: a small, rough model to serve as a kind of sighting scope and a larger, more accurate model to produce reliable numbers. The smaller model is equation-based and theory-based; the bigger model is procedure-based and data-based, i.e., a detailed system simulation. Models based on simple theories and equations can produce bad numerical estimates and even miss whole phenomena. In contrast, models based on procedures (e.g., “order up to the Max when you breach the Min”) and facts (e.g., the last 3 years of daily item demand) will require a lot more computing but give more realistic answers. Luckily, thanks to the cloud, we have a lot of computing power at our fingertips.

Perhaps the greatest modeling “sin” in the MEIO literature is the assumption that demands at all echelons can be modeled as purely random Poisson processes. Even if it were true at the warehouse level, it would be far from true at the DC level. The Poisson process is the “white rat of demand modeling” because it is simple and permits more paper-and-pencil equation manipulation. Since not all demands are Poisson shaped, this results in unrealistic recommendations.

Scenario-based Simulation Optimization

To get realism, we must get down into the details of how the inventory systems operate at each echelon. With few limits except those imposed by hardware such as size of memory, computer programs can keep up any level of complexity. For instance, there is no need to assume that each of the warehouses faces identical demand streams or has the same costs as all the others.

A computer simulation works as follows.

1. The real-world demand history and lead time history are gathered for each SKU at each location.
2. Values of inventory parameters (e.g., Min and Max) are selected for trial.
3. The demand and replenishment histories are used to create scenarios depicting inputs to the computer program that encodes the rules of operation of the system.
4. The inputs are used to drive the operation of a computer model of the system with the chosen parameter values over a long period, say one year.
5. Key performance indicators (KPI’s) are calculated for the simulated year.
6. Steps 2-5 are repeated many times and the results averaged to link parameter choices to system performance.
7.

Inventory optimization adds another “outer loop” to the calculations by systematically searching over the possible values of Min and Max. Among those parameter pairs that satisfy the item availability constraint, further search identifies the Min and Max values that result in the lowest operating cost.

Figure 1: General structure of one type of two-level inventory system

Stay Tuned for our next Blog

COMING SOON. To see an example of a simulation of the system in Figure 1, read the second blog on this topic

Related Posts

## Goldilocks Inventory Levels

You may remember the story of Goldilocks from your long-ago youth. Sometimes the porridge was too hot, sometimes it was too cold, but just once it was just right. Now that we are adults, we can translate that fairy tale into a professional principle for inventory planning: There can be too little or too much inventory, and there is some Goldilocks level that is “just right.” This blog is about finding that sweet spot.

## Call an Audible to Proactively Counter Supply Chain Noise

You know the situation: You work out the best way to manage each inventory item by computing the proper reorder points and replenishment targets, then average demand increases or decreases, or demand volatility changes, or suppliers’ lead times change, or your own costs change.

## An Example of Simulation-Based Multiechelon Inventory Optimization

Managing the inventory across multiple facilities arrayed in multiple echelons can be a huge challenge for any company. The complexity arises from the interactions among the echelons, with demands at the lower levels bubbling up and any shortages at the higher levels cascading down.

Inventory Planning Becomes More Interesting

# forecasting and inventory optimization

Taiichi Ohno of Toyota is credited with inventing Just-In-Time (JIT) manufacturing in the 1950s. JIT ensures that a manufacturer produces only what is needed, only when required, and only in the necessary amount. That innovation has since had major impacts, some good, some less so.

A recent New York Times article “How the World Ran out of Everything” describes some of the “less so” impacts.  For example, JIT has kept inventory costs very low improving return on assets.  This in turn is rewarded by Wall Street, so many companies have spent the last few decades reducing their inventories dramatically. Focused as they were on financials, many companies ignored the risks inherent in reducing inventories to the point that “lean” began to border on “emaciated.” Combined with increased globalization and new risks of supply interruption, stock-outs have abounded.

Some industries have gone too far, leaving them exposed to disruption. In a competition to get to the lowest cost, companies have inadvertently concentrated their risk, been interrupted by shortages of raw materials or components, and sometimes forced to halt assembly lines. Wall Street does not look kindly on production halts.

We all know that random events have added to the problem. First among them has been the Covid pandemic. As the pandemic has hindered factory operations and spread disarray in global shipping, many economies worldwide have been tormented by shortages of an immense range of goods — from computer chips to lumber to clothing.

The damage is compounded when more unexpected things go wrong. The Suez Canal Blockage is a prime example, obstructing the main trade route between Europe and Asia. Recently, cyberattacks have added another layer of disruption.

The reaction creates its own problems, just as the cyberattack on the Colonial Pipeline created gas shortages through panic buying. Suppliers start filling orders more slowly than usual. Manufacturers and distributors reverse course and increase inventories and diversify their suppliers to avoid future stockouts. Simply expanding warehouses may not deliver the solution, and the need to determine how much inventory to keep is more urgent every day.

So how can you execute a real-world plan for JIT inventory amidst all this risk and uncertainty? The foundation of your response is your corporate data. Uncertainty has two sources: supply and demand. You need the facts for both.

On the supply side, exploit the data you have on recent supplier lead times, which reflect the current turbulence. Don’t use average values when you can use probability distributions that reflect the full range of contingencies. Consider this comparison. Supplier A is now reliably filling orders in exactly 10 days. Supplier B also averages 10 days but does with a 78%/22% mix of 7 and 21 days. Both A and B have an average replenishment delay of 10 days, but the operational results they provide will be very different. You can only recognize this if you use probability models of inventory performance.

On the demand side, similar considerations apply. First, recognize that there may have been a major shift in the character of item demand (statisticians call this a “regime change”), so purge from your analysis any data that represent the “good old days.” Then, again, stop thinking in terms of averages. While the average demand is important, it is not a sufficient descriptor of the problem you face. Equally important is the volatility of demand. Volatility is the reason you keep inventory in the first place. If demand were completely predictable, you would have neither stockouts nor excess inventory. Just as you need to estimate the full probability distribution of replenishment lead times, you need the full distribution of demand values.

Once you understand the range of variability in both supply and demand, probabilistic forecasting will allow you to account for disruptions and unusual events. Software will convert your data on demand and lead times into huge numbers of scenarios representing how your next planning period might play out. Given those scenarios, the software can determine how best to meet your goals for such metrics as inventory costs and stockout rates. Using solutions such as Smart Inventory Optimization , you will confidently plan based on your targeted stockout risk with minimal inventory carrying cost. You may also consider letting the solution prescribe optimal service level targets by assessing the costs of additional inventory vs. stockout cost.

In inventory planning, as in science, we cannot escape the reality of uncertainty and the impact of unusual events. We must plan accordingly: using inventory optimization software helps you identify the least-cost service level. This creates a coherent, company-wide effort that combines visibility into current operations with mathematically correct assessments of future risks and conditions.

Inventory planning has become more “interesting” and requires a greater degree of risk awareness and agility. The right software can help.

Related Posts

## Call an Audible to Proactively Counter Supply Chain Noise

You know the situation: You work out the best way to manage each inventory item by computing the proper reorder points and replenishment targets, then average demand increases or decreases, or demand volatility changes, or suppliers’ lead times change, or your own costs change.

## Four Ways to Optimize Inventory

Inventory optimization has become an even higher priority in recent months for many of our customers.  Some are finding their products in vastly greater demand; more have the opposite problem. In either case, events like the Covid19 pandemic are forcing a reexamination of standard operating conditions, such as choices of reorder points and order quantities.

## Top 3 Most Common Inventory Control Policies

To make the right decision, you’ll need to know how demand forecasting supports inventory management, choice of which policy to use, and calculation of the inputs that drive these policies.The process of ordering replenishment stock is sufficiently expensive and cumbersome that you also want to minimize the number of purchase orders you must generate.

#### Recent Posts

• Smart Software to Present at Epicor Insights 2022
Smart Software CEO will present at this year's Epicor Insights event in Nashville. If you plan to attend this year, please join us at booth #9 and #705, and learn more about Epicor Smart Inventory Planning and Optimization. […]
• Smart Software and Arizona Public Service to Present at WERC 2022
Smart Software CEO and APS Inventory Manager to present WERC 2022 Studio Session on implementing Smart IP&O in 90 Days and achieve significant savings by optimizing reorder points and order quantities for over 250,000 spare parts. […]

#### Blog Categories

Increasing Revenue by Increasing Spare Part Availability

# forecasting and inventory optimization

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.

Related Posts

## Call an Audible to Proactively Counter Supply Chain Noise

You know the situation: You work out the best way to manage each inventory item by computing the proper reorder points and replenishment targets, then average demand increases or decreases, or demand volatility changes, or suppliers’ lead times change, or your own costs change.

## Four Ways to Optimize Inventory

Inventory optimization has become an even higher priority in recent months for many of our customers.  Some are finding their products in vastly greater demand; more have the opposite problem. In either case, events like the Covid19 pandemic are forcing a reexamination of standard operating conditions, such as choices of reorder points and order quantities.

## Top 3 Most Common Inventory Control Policies

To make the right decision, you’ll need to know how demand forecasting supports inventory management, choice of which policy to use, and calculation of the inputs that drive these policies.The process of ordering replenishment stock is sufficiently expensive and cumbersome that you also want to minimize the number of purchase orders you must generate.

#### Recent Posts

• Smart Software to Present at Epicor Insights 2022
Smart Software CEO will present at this year's Epicor Insights event in Nashville. If you plan to attend this year, please join us at booth #9 and #705, and learn more about Epicor Smart Inventory Planning and Optimization. […]
• Smart Software and Arizona Public Service to Present at WERC 2022
Smart Software CEO and APS Inventory Manager to present WERC 2022 Studio Session on implementing Smart IP&O in 90 Days and achieve significant savings by optimizing reorder points and order quantities for over 250,000 spare parts. […]

#### Blog Categories

Maximize Machine Uptime with Probabilistic Modeling

# 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.

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.

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