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.

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.

What is the difference between Demand planning and Inventory optimization ? 

The Smart Demand Planning app (SDP) provides demand forecasts. The SDP forecasting engine is also the core of the Smart Inventory Optimization app (SIO), which stress-tests various inventory policies using a number of demand scenarios to find optimal inventory policy settings.

Service Level Driven Planning (SLDP) is an approach to inventory planning. It prescribes optimal service level targets continually identifies and communicates trade-offs between service and cost that are at the root of all wise inventory decisions. When an organization understands this relationship, they can communicate where they are at risk, where they are not, and effectively wield their inventory assets.  SLDP helps expose inventory imbalances and enables informed decisions on how best to correct them.  To implement SLDP, you’ll need to look beyond traditional planning approaches such as arbitrary service level targeting (all of my A items should get 99% service level, B items 95%, C items 80%, etc.) and demand forecasting that unrealistically attempts to predict exactly what will happen and when. SLDP unfolds in 4 steps: Benchmark, Collaborate, Plan, and Track.

Step 1. Benchmark Performance

All participants in the inventory planning and investment process must hold a common understanding of how current policy is performing across an agreed upon set of inventory metrics. Metrics should include historically achieved service levels and fill rates, delivery time to customers, supplier lead time performance, inventory turns, and inventory investment. Once these metrics have been benchmarked and can be reported on daily, the organization will have the information it needs to begin prioritize planning efforts. For example, if inventory has increased but service levels have not, this would indicate that the inventory is not being properly allocated across SKUs.  Reports should be generated within mouse-clicks enabling planners to focus on analysis instead of time intensive report generation.   Past performance isn’t a guarantee of future performance since demand variability, costs, priorities, and lead times are always changing. So SLDP enables predictive benchmarking that estimates what performance is likely to be in the future. Inventory optimization software utilizing probability forecasting can be used to estimate a realistic range of potential demands and replenishment cycles stress testing your planning parameters helping uncover how often and which items to expect stockouts and excess.

Step 2. “What if” Planning & Collaboration

“What if” inventory modeling and collaboration is at the heart of SLDP. The historical and predictive benchmarks should first be shared with all relevant stakeholders including sales, finance, and operations. Efforts should be placed on answering the following questions:

– Are both the current performance and investment acceptable?
– If not, how should they be improved?
– Which SKUs are likely to be demanded next and in what quantities?
– Where are we willing to take more stock out risk?
– Where must stock-out risk be minimized?
– What are the specific stock out costs?
– What business rules and constraints must we adhere to (customer service level agreements, inventory thresholds, etc.)

Once the above questions are answered, new inventory planning policies can be developed.  Inventory Optimization software can reconcile all costs associated with managing inventory including stockout costs to generate the right set of planning parameters (min/max, safety stock, reorder points, etc.) and prescribed service levels.  The optimal policy can be compared to the current policy and modified based on constraints and business rules. For example, certain items might be targeted at a target service level in order to conform to a customer service level agreement.   Various “what if” inventory planning scenarios can be developed and shared with key stakeholders. For example, you might model how shorter lead times impacts inventory costs. Once consensus has been achieved and the risks and costs are clearly communicated,  the modified policies can be uploaded to the ERP system to drive inventory replenishment.

Step 3. Continually Plan and Manage by Exception

SLDP continually reforecasts optimized planning parameters based on changing demands, lead times, costs, and other factors. This means that service levels and inventory value have the potential to change.  For example, the prescribed service level target of 95% might increase to 99% the next planning period if the stock-out costs on that item increased suddenly. This is also true if opting to arbitrarily target a given service level or fix planning parameters to a specific unit quantity. For example, a target service level of 95% might require $1,000 in inventory today but $2,000 next month if lead times spiked.  Similarly, a reorder point of 10 units might get 95% service today and only 85% service next month in response to increased demand variability. Inventory Optimization software will identify which items are forecasted to have significant changes in service level and/or inventory value and which items aren’t being ordered according to the consensus plan. Exception lists are automatically produced making it easy for you to review these items and decide how to manage them moving forward. Prescriptive Analytics can help identify whether the root cause of the change is a demand anomaly, change in overall demand variability, change in lead time, or change in cost helping you fine tune the policy accordingly.

Step 4. Track Ongoing Performance

SLDP processes regularly measure historical and current operational performance.   Results must be monitored to ensure that service levels are improving and inventory levels are decreasing when compared to the historical benchmarks determined in Step 1.  Track metrics such as turns, aggregate and item specific service levels, fill rates, out-of-stocks, and supplier lead time performance.  Share results across the organization and identify root causes to operational inefficiencies.  SLDP processes makes performance tracking easy by providing tools that automatically generate the necessary reports rather than placing this burden on planners to manage in Excel. Doing so enables the organization to uncover operational issues impacting performance and provide feedback on what is working and what should be improved.

Conclusion

The SLDP framework is a way to rationalize the inventory planning process and generate a significant economic return. Its organizing principle is that customer service levels and inventory costs associated with the chosen policy should be understood, tracked, and continually refined. Utilizing inventory optimization software helps ensure that you are able to identify the least-cost service level.  This creates a coherent, company-wide effort that combines visibility into current operations with scientific assessments of future risks and conditions. It is realized by a combination of executive vision, staff subject matter expertise, and the power of modern inventory planning and optimization software.

See how Smart Inventory Optimization Supports Service Level Driven Planning and download the product sheet here: https://smartcorp.com/inventory-optimization/

Now More than Ever

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.

Even in quieter times, inventory control parameters like Mins and Maxes may be set far from their best values. We may ask “Why is the reorder point for SKU_1234 set at 20 units and the order quantify set at 35?” Those choices were probably the ossified result of years of accumulated guesses. A little investigation may show that the choices of 20 and 35 are no longer properly aligned with current demand level, demand volatility, supplier lead time and item costs.

The nagging feeling that “We should re-think all these choices” is often followed by “Oh no, we have to figure this out for all 10,000 items in inventory?” The savior here is advanced software that can scale up the process and make it not only desirable but feasible.  The software uses sophisticated algorithms to translate changes in inventory parameters such as reorder points into key performance indicators such as service levels and operating costs (defined as the sum of holding costs, ordering costs, and shortage costs).

This blog describes how to gain the benefits of inventory optimization by outlining 4 approaches with varying degrees of automation.

Four Approaches to Inventory Optimization

Hunt-and Peck

The first way is item-specific “hunt and peck” optimization. That is, you isolate one inventory item at a time and make “what if” guesses about how to manage that item. For instance, you may ask software to evaluate what happens if you change the reorder point for SKU123 from 20 to 21 while leaving the order quantity fixed at 35. Then you might try leaving 20 alone and reducing 35 to 34. Hours later, because your intuitions are good, you may have hit on a better pair of choices, but you don’t know if there is an even better combination that you didn’t try, and you may have to move on to the next SKU and the next and the next… You need something more automated and comprehensive.

There are three ways to get the job done more productively. The first two combine your intuition with the efficiency of treating groups of related items. The third is a fully automatic search.

Service-level Driven Optimization

1. Identify items that you want to all have the same service level. For instance, you might manage hundreds of “C” items and wonder whether their service level target should be 70%, or more, or less.
2. Input a potential service level target and have the software predict the consequences in terms of inventory dollar investment and inventory operating cost.
3. If you don’t like what you see, try another service level target until you are comfortable. Here the software does group-level predictions of the consequences of your choices, but you are still exploring your choices.

Optimization by Reallocation from a Benchmark

1. Identify items that are related in some way, such as “all spares for undercarriages of light rail vehicles.”
2. Use the software to assess the current spectrum of service levels and costs across the group of items. Usually, you will discover some items to be grossly overstocked (as indicated by service levels unreasonably high) and others grossly understocked (service levels embarrassingly low).
3. Use the software to calculate the changes needed to lower the highest service levels and raise the lowest. This adjustment will often result in achieving two goals at once: increasing average service level while simultaneously decreasing average operating costs.

Fully automated, Item-Specific Optimization

1. Identify items that all require service levels above a certain minimum. For instance, maybe you want all your “A” items to have at least a 95% service level.
2. Use the software to identify, for each item, the choice of inventory parameters that will minimize the cost of meeting or exceeding the service level minimum. The software will efficiently search the “design space” defined by pairs of inventory parameters (e.g., Min and Max) for designs (e.g., Min=10, Max=23) that satisfy the service level constraint. Among those, it will identify the least cost design.

This approach goes farthest to shift the burden from the planner to the program. Many would benefit from making this the standard way they manage huge numbers of inventory items. For some items, it may be useful to put in a little more time to make sure that additional considerations are also accounted for. For instance, limited capacity in a purchasing department may force the solution away from the ideal by requiring a decrease in the frequency of orders, despite the price paid in higher overall operating costs.

Going Forward

Optimizing inventory parameters has never been more important, but it has always seemed like an impossible dream: it was too much work, and there were no good models to relate parameter choices to key performance indicators like service level and operating cost. Modern software for supply chain analytics has changed the game. Now the question is not “Why would we do that?” but “Why are we not doing that?” With software, you can connect “Here’s what we want” to “Make it so.”

Inventory control policies and strategies for a more profitable business

In this Video Dr. Thomas Willemain, co–Founder and SVP Research, defines and compares the three most used inventory control policies. These policies are divided into two groups, periodic review and continuous review. There is also a fourth policy called MRP logic or forecast based inventory planning which is the subject of a separate video blog that you can see here. These videos explain each policy, how they are used in practice and the pros and cons of each approach.

We frequently encounter confusion about the process of setting safety stock levels. This blog hopes to clarify the issue.

Safety stock is a critical component in any system of inventory management. Indeed, some inventory software treats safety stock as the key decision variable in the quest to balance inventory cost against item availability. Unfortunately, that approach is not the best way to strike the balance.

First, realize that safety stock is part of a general equation:

Inventory Target = Average Lead Time Demand + Safety Stock.

Average Lead Time Demand is defined as the average units demanded multiplied by the average replenishment lead time. Example: If daily demand averages 2 units and the average lead time is 7 days, then the average lead time demand is 2 x 7= 14 units. Keeping 14 units on hand suffices to handle typical demand.

But we all know that demand is random, so keeping enough stock on hand to cover the average lead time demand invites stockouts. As we like to say, “The average is not the answer.” The smart answer is to add in some safety stock to accommodate any random spikes in demand. But how much?

There’s the problem. If you try to guesstimate a number for the safety stock, you are on thin ice. How do you know what the “right” number is?  You may think that you don’t have to worry about that because you have a good-enough answer now, but that answer has a sell-by date. Lead times change. So do demand patterns. So do company priorities. That means today’s good answer may become tomorrow’s blunder.

Some companies try to wing it using a crude rule of thumb approach. For instance, they may say something like “Set safety stock at an additional two weeks of average demand.” This approach is seductive: It only needs simple math, and it is clear.  But for the reasons listed in the previous paragraph, it’s foolish. Better to get a good answer than a convenient answer.

You need a principled, objective way to answer the question that takes account of the mathematics of randomness.  More than that, you need an answer that is linked to the key performance indicators (KPI’s) of the system: inventory cost and item availability.

Simple logic gives you some sense of the answer, but it doesn’t provide the number you need. You know that more safety stock increases both cost and availability, while less safety stock decreases both. But without knowing how much those metrics will change if you change the safety stock, you have no way to align the safety stock decision with management’s intent for striking the balance between cost and availability.

Rather than flying blind, you can back into the choice of safety stock by first finding the right choice for inventory target. Once you’ve done that, the safety stock pops out by a simple subtraction:

 Safety Stock = Inventory Target – Average Lead Time Demand.

Often times, companies will state that they don’t carry safety stock because the safety stock field in their ERP system is blank. Nearly always, safety stock is built into the targeted inventory level they have established.  So, using the above formula to “back out” how much safety stock you are building into the plan is quite helpful.  The key is not just to know how much safety stock you are carrying but the link between your inventory target, safety stocks, and its corresponding KPI’s.

For instance, suppose you can tolerate only a 5% chance of stocking out while waiting for replenishment (inventory texts call this interval the “period of risk.”). Software can examine the demand history of each item and work out the odds of stockout based on the thousands of different demand scenarios that can occur during the lead time. Then the right answer for the inventory target is the choice that leads to no more than a 5% stockout risk. Given that target and knowing the average lead time demand, the appropriate safety stock value falls right out by subtraction. You also get to know the average holding, ordering and shortage costs.

That’s what we mean by “backing into the safety stock.” Start with company objectives, determine the appropriate inventory target, then derive the safety stock as the last step. Don’t start with a guess about safety stock and hope for the best.