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.

 

Margins vs Inventory Level Business

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!    

 

 

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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.

Fact and Fantasy in Multiechelon Inventory Optimization

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

 

 

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Inventory Planning Becomes More Interesting

The Smart Forecaster

 Pursuing best practices in demand planning,

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.Manager In Warehouse With Inventory Management Software

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.

 

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

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      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:

      1. You start each replenishment cycle with Q units on hand.
      2. You deplete that stock by D units per day.
      3. So, you hit the reorder point R after (Q-R)/D days.
      4. So, you order every (Q-R)/D days.
      5. Each replenishment cycle lasts (Q-R)/D + L days, so you make a total of 365D/(Q-R+LD) orders per year.
      6. 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

      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 2 A probabilistic scenario of on-hand inventory

       

      Figure 3 Distribution of days between orders

      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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      The Cost of Spreadsheet Planning

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      Increasing Revenue by Increasing Spare Part Availability

      The Smart Forecaster

       Pursuing best practices in demand planning,

      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.

      Scenarios with different service level targets

      Stock on hand during one replenishment cycle

       

       

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