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Sizing inventories with readiness-based sparing.

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Introduction

Computing inventories of spares based on some of the more traditional inventory performance metrics--such as fill rate or service level--does not do a good job of delivering mission-ready aircraft.

Fortunately, the US Air Force uses the principles of readiness-based sparing (RBS) to set base levels for reparable spares and repair parts, to guide the procurement and repair of spares, to size air-deployable spares packages (readiness spares packages), and to schedule spares distribution and induction into depot repair. Since the early 1980s, the Office of the Secretary of Defense has fostered the adoption of RBS, and its use is mandated in Department of Defense Directive 4140.1-R, DoD Supply Chain Materiel Management Regulation. (1) In fact, all of the military services have implemented RBS (or elements of RBS, as appropriate) throughout their logistics systems.

Although the mathematical models used in the Air Force applications are complex and can be difficult to understand, the basic idea behind RBS is a simple one: evaluate alternative courses of action about spares--how many to buy, where to put them, and so on--based explicitly on how many mission-ready aircraft that action supports. In this article, we will use the strategic decision of what reparable spares the Air Force should procure for its inventory to illustrate how RBS works and how it is superior to other methods in providing readiness for the least cost.

Background

Aircraft readiness in the Air Force depends on, among other things, having adequate stocks of spare parts--not only just the inexpensive nuts and bolts, filters and washers, but also large and expensive assemblies and subassemblies. Given their cost, many aircraft parts are worth repairing after a failure, whether at an air base, a depot, or a contractor facility.

To keep aircraft in service while failed items are being repaired, the Air Force logistics system maintains an inventory of serviceable spare parts. In this way, a $100M aircraft is not idled for the lack of a much less expensive part. Ideally, the replacement part will be immediately available at the air base, and the aircraft will experience only minimal downtime. If not, a serviceable spare may need to be shipped from a central supply point, resulting in greater aircraft downtime. But this downtime is still less than what would occur if no serviceable spare were available within the logistics system--in which case, the aircraft repair would be delayed while an unserviceable item was repaired or a new spare was procured.

If all spare parts were inexpensive, the best strategy for the Air Force would clearly be to procure and maintain generous quantities of spares at every air base; however, less expensive relative to an aircraft is not at all the same as inexpensive, as we see in Table 1.

Table 1 is just a sampling of Air Force-managed parts, their procurement costs, and end item applications. In fact, the Air Force manages more than 110,000 different types of reparable items and carries a spares inventory valued at more than $28B.

Clearly, there is a need for the Air Force to balance readiness against cost and to optimize its spares inventory to support readiness at the least cost possible. The expense and complexity of these parts (and the relatively small customer base in the aerospace and defense industries) are reflected in the long lead-times needed to procure new spares and in their lengthy repair times. Not only must the Air Force react quickly to a need, it must develop a long-term strategic plan to ensure sufficient spares are in the inventory and accessible when they are needed.

To determine how many spares are needed, the procedure begins with a forecast of future parts demand, (2) which is played out through the future planning horizon (either analytically or with a simulation) to determine when spares will be needed, then deciding whether the parts can be sourced from existing inventory or whether augmentation is needed.

A modern commercial advanced planning and scheduling (APS) system, for example, first simulates the passage of time and deterministically forecasts the occurrence of item demands. It then compares those demands with forecasted asset availability, satisfying the demands with existing serviceable assets whenever possible. If that is not possible, the APS system will determine how best to source assets--either from a repair action, procurement, or redistribution from another location, depending on the system's business rules--and initiates the action (in the simulated world) with the necessary lead-time before the demand is expected. Thus, the serviceable asset appears where it is needed and just in time to satisfy the demand. A record of this simulation provides a demand and supply plan for the future to guide logistics managers.

This is also how the Air Force's D041 Recoverable Consumption Item Requirements System and its successor D200 Requirements Management System have worked since the early 1970s. In fact, this is how the most basic formulations of inventory theory in the 1940s and 1950s proceeded. The use of reorder point triggers or other simplified rules to guide routine activity often obscured these basic principles. That said, they nonetheless underlied the systems of the time and continue to do so today. The essential logic remains the same, and the deterministic simulation develops a plan to satisfy every forecasted demand.

In real life, of course, that perfection is unattainable. Perfect forecasting is simply not possible--item demand rates are variable, sometimes repairs or procurements take longer than expected, to name just a few. Certainly the variability in demand seen by service parts or remanufacturing enterprises (which are driven by random failures in operating equipment) tends to be much higher than what is experienced by a manufacturing operation--especially those working to a production schedule.

While forecasting methods are constantly being improved, a law of diminishing returns seems to be in play. The more responsive the method to changes in item behavior, the more likely it will mistake a blip for a trend and overcorrect, leading to unwarranted inventory growth. To preclude this, it is necessary to address demand forecasting in a more sophisticated, probabilistic manner. Equally critical is to explicitly treat demand as a random variable described by a probability distribution, with a forecasted mean and variance.

Air Force experience certainly confirms this conclusion. When high-technology items are stressed in unforgiving conditions, actual demand rarely matches the forecasted mean because the variance of the probability distribution is not zero--in fact, it tends to be large. Faced with this inconvenient truth, logisticians deal with demand uncertainty (and its associated supply risk) by carrying a safety stock over and above the forecasted mean.

Of course, a difficult and important question remains: What is the best way for the Air Force (or a similar organization, whether military or commercial) to calculate this safety stock requirement? Calculating the amount of inventory needed to satisfy the forecasted mean demand (for example, sufficient assets to fill the replenishment pipeline) is a complex accounting problem but relatively straightforward. (Given similar forecast techniques and business rules, a commercial APS system's computations would be analogous to the DoD Central Secondary Item Stratification [CSIS] and Air Force D200 systems, which compute the replenishment pipeline requirement. (3))

Unfortunately, resolving the safety stock issue is not so cut and dried. The fundamental issues are how much supply risk is the Air Force willing to accept and how to best measure that risk.

Measures of Supply System Performance: Projecting Inventory Performance

To optimally size an inventory, the Air Force requires tools and mathematical expressions that allow it to project inventory performance as a function of spares levels (and, of course, other fixed parameters of the system). We present some of the familiar expressions below. For simplicity, we consider the single location, single indenture, single weapon system case. (4) Suppose component i has a spares level of s at the given location. Demands for component i are generated by a stochastic process, and the number of units x of component i in the resupply pipeline has the probability distribution p(x). So p(0) is the probability that there is no unit in the pipeline, p(1) the probability of 1 unit, and so on. For the purpose of exposition, the exact form of these distributions is not necessary, or we may consider them all to be Poisson, one of the simplest cases. (5) The probabilistic perspective is critical, however--the number of item demands in a time period is variable, and we are trying to effectively manage the risk caused by that variability.

The number of units in the resupply pipeline is a random variable whose pipeline distribution depends on the item demand distribution and the item repair or resupply time. (6) We assume the resupply process begins immediately when a part is demanded from the inventory following an (s-1, s) ordering policy. That is, whenever the inventory position (on-hand minus back orders plus units due-in) falls to s - 1, another unit of item i is ordered. With some reasonable assumptions about independence of demand, we can obtain the following expressions (see Equations 1-4). (7) For clarity's sake, we consolidate our numerical examples into a separate section.

Fill Rate

Fill rate is the proportion of demands that are filled immediately from stock on hand. It is calculated as the number of filled demands divided by the total number of demands over a period (at a specific location or desired level of aggregation). (8) In the Air Force, fill rate is referred to as the issue effectiveness rate, and it is a key statistic on monthly Air Force supply performance management reports. (9)

Statistically, we can express fill rate as the probability that a random demand will be filled immediately. It is given by the probability that there is at least one serviceable spare ready for issue at a random time (when the demand occurs), which is the probability that there are s - 1 or fewer units in resupply, or

Fill rate = [s-1.summation over (x=0)] p(x) (Equation 1)

Service Level

We define service level as the proportion of time that no back orders exist; that is, the ratio of the time with no back orders to the total time. Service level is typically defined by item, and it is usually observed over a period, at specific locations, or for other desired levels of aggregation. (10) It can similarly be expressed as the probability of no back orders existing and is clearly a closely related function to fill rate. Equivalently, service level is the probability that there are s or fewer units in resupply, and is given by

Service level = [s.summation over (x=0)] p(x) (Equation 2)

In this probabilistic framework, service level is sometimes referred to as probability of sufficiency as well as its other alias, ready rate.

Back Orders

The number of unfilled demands, or back orders, provides another snapshot of supply performance, again at a particular location or desired aggregation. However, back-order measures can be a source of confusion because inventory managers may report back orders with or without an associated duration. Inventory managers often report back-order days relative to the total number of days in a period (that is, the number of back orders existing each day summed and divided by the number of days in a given period of interest), or they may just tally and report the occurrence of an unfilled demand, irrespective of how long the back order lasts. For the purposes of this article, we consider back orders as being the former, time-weighted case.

Expected back orders (EBO) is a quantity closely related to aircraft readiness. Defined as the expected number of unfilled demands at a random point in time, it is given by

EBO = [[infinity].summation over (x=s+1)] (x-s)p(x) (Equation 3)

Finally, the aircraft availability (AA) rate of a fleet of N aircraft at a single location is the expected percentage of the fleet not missing a part. It is given by (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (Equation 4)

where EBO(i) is the number of expected back orders of component i, and the product sum is taken over all components i on the weapon system, all of which are line replaceable units (LRUs). (12) If item i has X back orders, the probability of a random aircraft missing one of that item and being unavailable is X/N. The probability the aircraft is not missing the part is 1 - X/N, and the probability over all back-order levels is 1 - EBO(i)/N. Assuming independence of back-order occurrences, AA then equals the product of this term for each item. In reference to standard Air Force reporting systems, we could say that AA is an estimator of 1-TNMCS (Total Not Mission Capable Supply), expressed in percentages. This is the measure of aircraft readiness the Air Force D200 uses; it is the R in RBS.

Differences Among the Measures

All of these inventory measures capture some important facet of inventory performance, which can be useful to inventory managers. For certain types of enterprises, these performance measures are perfectly appropriate and adequate. But for the Air Force, fill rate and service level have serious drawbacks from a strategic point of view, in that they omit some obviously important aspects of operational support.

Fill rate explicitly ignores the time element. It is blind to how long a demand remains unfilled; it addresses only whether it was a fill or not. For example, consider two items: the first with a short procurement lead-time and a second item with a long lead-time. Stocking each item to the same fill rate would have very different consequences in terms of the duration of a back order if one occurs.

Or consider an item that has multiple user locations, but low demand at each location. The best, most economical strategy might be to consolidate supply centrally and fill each demand from the central stock. Although this would result in a delay associated with the order and ship time, it would also minimize the back-order duration. This strategy would balance a lower inventory investment (and lower, even zero, base-level fill rates) with guaranteed short back-order durations at each base against a larger inventory investment to establish stock levels at many locations (with higher base-level fill rates). There are many scenarios in which a centralized stocking strategy provides an acceptable support level at minimal cost; however, a strict focus on fill rate would never result in such a strategy.

In the extreme, if fill rate were used as the only metric to gauge an inventory manager's performance, it could create the perverse incentive to not fill back-ordered demands at all! The late fill would not remove the penalty of the earlier non-fill, and it would use a scarce spare that could prevent a future non-fill. Fill rate is more suited to a commercial retail environment, which is where the concept developed. If Joe's store doesn't have an item, the customer goes down the street and buys it at Mabel's store; Joe loses the sale. In classic inventory theory, this is known as the "lost sales case." In that situation, fill rate can be a valid strategic objective for the enterprise, but it is not the best strategy for the Air Force.

The major drawback of using the service-level measure is its insensitivity to the number (and duration) of unsatisfied demands. It measures only whether there are one or more unfilled demands. Service level could be a perfectly good strategic measure in a manufacturing environment, where a single shortage can idle an assembly line. But a measure that fails to distinguish the differences between having one aircraft down and ten aircraft down is not suitable for the Air Force.

We can see the differences between the related fill rate and service level and the EBO and AA measures clearly in the statistical expressions. From fill rate and service level to EBOs to AA, the expressions capture increasing amounts of complexity in the operating environment. Fill rate and service level ignore information in the tail of the resupply distribution, which is exactly the area of interest for assessing risk when setting spares levels. They lose information about how many unfilled demands exist and whether you are one aircraft short or ten aircraft short of your mission requirement.

The EBO measure and the AA that is derived from it use all the important information in the distribution. They truncate only the first part of the distribution, but this represents time when there are no back orders. The measures only differ in how many serviceable spares remain on the shelf, which is not of primary interest. In particular, the information in the tail of the distribution is the exact information that describes the risk of the corresponding spares level, and this efficient use of information is a great advantage of the EBO and AA measures over the fill-rate and service-level metrics.

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AA is a simplified measure, to be sure. It is intended more to size the inventory than to make the best prediction of aircraft readiness. Its formulation assumes a random scattering of back orders across aircraft, ignoring the possible effects of cannibalization (consolidating back orders on as few aircraft as possible by removing serviceable parts from down aircraft to be used as spares). (13) Incorporating cannibalization and other management adaptations, such as expedited repair or lateral resupply, would improve the forecasts of AA (in terms of readiness), but it would also reduce inventory while institutionalizing hidden costs in terms of maintenance or transportation. In any case, it is clear from a simple inspection that the AA formulation has an obvious advantage over fill rate and service level in terms of the explicit purpose of Air Force inventory--to support mission-ready aircraft.

Numerical Illustration

Suppose that item i has a resupply pipeline of 5 units, which could arise from a demand rate of 2 per day and a local repair time of 2.5 days. If we accept the probability distribution of demand is Poisson, then so is the distribution of the number of items in the pipeline. So the distribution p(x), is Poisson with a mean of 5. Figure 1 shows the probability density function (the p(x) values) for this distribution and the corresponding cumulative distribution function, C(x). (14)

There is a 0.007 probability of no spares in the pipeline, a 0.034 probability of one spare in the pipeline, and a cumulative probability of 0.041 of one or fewer spares in the pipeline, and so on. Suppose we own five spares, enough to fill the pipeline. We note from the chart that this common expression is true only in an average sense. There will frequently be times when there are more than five spares in the pipeline and back orders will still occur. In fact, the cumulative probability of five or fewer spares in the pipeline is 0.616, so the probability of no back orders existing with five spares is 0.616. In other words, the service level is 61.6 percent.

The fill rate with five spares is the probability that there are four or fewer spares in the pipeline; meaning there is at least one spare ready for issue when a demand occurs. This is the cumulative probability 0.440. Note how both these measures use only information from the lower (left-hand) portion of the distribution.

Finding the number of expected back orders when the spares level is five takes a bit of calculation. We have no back orders whenever the number of assets in the pipeline is five or fewer, which has a probability of 0.616. When six assets are in the pipeline, which occurs with a probability of 0.146, we have one back order; having seven assets in the pipeline gives us two back orders with a probability of 0.104, and so on. Using the EBO equation (see Equation 3) gives us the following equation.

EBO(5) = (0 x 0.616) + (1 x 0.146) + (2 x 0.104) + (3 x 0.065)... (Equation 5)

Unfortunately, this is theoretically an infinite sum. (15) We could compute out to 10 or so terms and certainly get close enough, but we could also use a recursive computational approach to get a more exact result. (16)

If this were the only item on a weapon system composing a 100 aircraft fleet, the AA for the weapon system would be 1EBO(5)/100 = 1 - 0.0088 = 0.9912. More realistically, the aircraft would have a mix of several hundred items with different characteristics and spares levels, and the individual contributions to AA would be multiplied together. Suppose the aircraft consisted of 100 items with identical characteristics and spares levels of 5, we would have AA = 0.9912 (100) = 0.413. Note how quickly AA can degrade as the number of components supported increases. Clearly, finding an appropriate inventory mix for a desired AA goal is not a simple task.

Quantitative Comparisons

So how do we best optimize the inventory mix? We can find, with our deterministic simulation, the inventory needed to fill the pipelines and to replace condemnations (unrepairable failures) to maintain that level over our planning horizon; that is, to satisfy the (mean) forecasted demand. The next step is to find a set of optimal safety levels for each item by optimizing to a performance target at the least cost. (17) The question is, What is the best performance measure to use as an objective function in the optimization? Given the discussion from the last section, it would seem that optimizing to an AA and to the closely related EBO objective (either maximizing AA or minimizing EBO) would provide better support. This is certainly true in the case of the Air Force, where fill rate and service level--when used as objective functions for strategic decisions--fall far short of providing optimal support.

Historical Comparisons

Even though EBOs are closely related to availability, using a set number of EBOs (for a single item or for a population of items) as an objective when sizing an inventory is problematic because there is no clear logical target. The question of how many EBOs are acceptable for a fleet of aircraft does not have an intuitively obvious answer. In contrast, managers can reasonably and knowledgeably address the issue of what a valid AA target should be for a weapon system, even though deriving these availability targets from operational planning documents can be a lengthy process.

The Air Force addressed the EBO target definition problem in the early 1970s with its implementation of the Variable Safety Level (VSL) model. VSL was a great advance at the time. It was an EBO-optimizing inventory model that stratified the inventory by weapon system to ensure no weapon systems were shortchanged. VSL finessed the issue of setting an EBO target by using a fill rate target instead. It set spares requirements so that, for each weapon system, the total EBOs for parts on that weapon system were minimized for the cost incurred, while attaining an overall (for the components on each weapon system) 92 percent fill rate. (18)

Sounds like the best of both worlds, doesn't it? But there was a fatal flaw.

Figure 2 is an example of fill rate used in a strategic context; that is, setting spares levels for each weapon system to a 92 percent fill rate target. (19) These spares requirements, as recommended by the VSL model, (20) were then assessed by the Air Force's Aircraft Availability Model (AAM) (21) to determine the AA rates the VSL-computed inventories would likely attain.

Notice how the aircraft availabilities vary widely, even though the systems appear, at first blush, to be equally supported at a 92 percent fill rate.

Also note that the more complex weapon systems suffer the most. This is because of the same probabilistic principle that makes it harder to toss a coin and get five heads in a row rather than getting three. Loosely speaking, if you have a 92 percent chance of having a spare on hand of the first item, and a 92 percent chance of having the second spare, and so on ... the probability of having all the spares you need (in this case an available aircraft) depends on the product of all those 0.92s--which gets smaller and smaller as you have more items and need to multiply more and more terms.

Clearly, VSL was not an efficient way to perform readiness-based sparing. Nor did it ensure budgets could be defended on any sort of reasonable basis, since an explicit linkage to the operational requirement was lacking. In the 1980s, the Air Force agreed that these were indeed serious shortcomings and began to develop an implementation plan for the AAM--the next step forward in RBS inventory management.

Table 2 shows some comparisons made by the Air Force Logistics Command (AFLC, now the Air Force Materiel Command) a few years later, when it was ready to deploy the AAM within its requirements computations system. (22) The VSL columns represent the cost and projected availability rate from the then-standard VSL methodology, computed to a 92 percent fill rate mix of spares. The AAM columns represent the cost to attain the Air Force's desired AA targets for each aircraft.

Note that, in every case, the AAM-recommended spares mix provided an AA that was as good as, or better than, VSL's recommendations. It also represents a smaller inventory investment. Not only was the Air Force able to direct resources to high-priority weapon systems and defend its inventory investment choices on operational grounds, it was also able to generate more aircraft availability for its inventory dollar.

Some Current Comparisons

The following figures further illustrate the superiority of availability-based inventory optimization with analyses using current data. (23) Figure 3 shows curves of the availability and cost produced by spares mixes calculated in three different ways for a unit of HH-60 aircraft at a single base. (24) One set of mixes is built by optimizing on AA. Each point on the corresponding curve shows the AA that results from one such mix and the corresponding cost. Another set of mixes is built by using a suitably modified AAM to develop spares mixes that are optimized on fill rate (labeled FR on the chart). These mixes obtain the best possible fill rate for the associated cost, but we show the resulting AA for that mix on the curve, not the fill rate. A third set of mixes is built similarly, but optimizing on service level or ready rate (labeled RR on the chart). (25) Putting all three curves--all expressed in terms of cost versus AA produced--on a single chart allows us to easily compare the cost-effectiveness of the three optimization goals in producing readiness.

We know, of course, that the AA-optimized mixes will outperform the fill-rate and service-level mixes--that is the definition of optimal. Intuition could lead one to think that optimizing on fill rate, for example, might produce spares mixes that are good enough, even if not mathematically optimal. This is not the case. To achieve an 80 percent AA, the AA-optimal method recommended a spares mix costing slightly under $30M, while the fill-rate method needed a mix costing about $37M, and the service-level method required about $35M.

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Another way to compare the methods is by holding the inventory investment constant. The same $30M that produced an 80 percent AA with AA-optimizing, produced about 57 percent AA using fill-rate optimizing and about 67 percent AA using service-level optimizing. These cost savings and availability differences, in the 15 to 30 percent neighborhood, are very typical of what we have seen whenever RBS applications are implemented.

Figure 4 is a comparable analysis for the F-22. Again we draw similar conclusions. AA outperforms the more traditional optimization methods by a significant margin throughout the entire range of availabilities and costs.

Summary

There are fundamental differences among inventory performance measures and where their use is appropriate. As T. M. Whitin noted in his seminal 1953 work, The Theory of Inventory Management: (26)
 When an "out-of-stock" condition arises for some military item, the
 problem is quite different from that of the private entrepreneur
 ... The military situation is conceptually similar to that of the
 private entrepreneur, but the costs of depletion, aside from the
 greater difficulties of measurement, may be of a much greater
 magnitude than is at all conceivable in private business. If
 important items of equipment are not available when needed, the
 fate of the nation may be at stake.


While the fate of the nation may not be at risk as it was during World War II, it is still true that, coming out of that conflict, logisticians were already looking for a way to move beyond traditional, business-oriented performance measures like fill rate. They wanted a better way to relate inventory resourcing to military readiness. RBS methods provide that capability.

Fill rate and service level can be effective as observational metrics, and a decreasing fill rate or service level can alert managers to potential problems. In the appropriate environment, they can be effective strategic measures, as well. But fill rate's insensitivity to the passage of time is a serious drawback, as is insensitivity to the number of back orders from service level (rather than the binary answer to, Does a back order exist?).

Of course, both metrics are correlated to some degree with readiness. All other things being equal, adding a spare to an inventory because it gives the maximum increase in fill rate or service level would also increase aircraft availability. But it would not do so effectively or efficiently. Add to that the difficulty of relating the availability produced by a given fill rate or service level (without an availability model to perform an assessment). Even if a fill-rate or service-level model could equal the performance of an RBS model (and achieve the same availability for a given inventory investment), it would still lack a straightforward method to set targets, and, thus, know what inventory investment was best.

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We have demonstrated that using fill rate or service level to size spares inventories will unavoidably lead to misallocation of resources across weapon systems and to larger inventories than what is needed to support any given level of readiness. These facts provide clear evidence of the superiority of RBS methods for optimally sizing Air Force spares inventories; neither fill-rate nor service-level methods can provide performance comparable to RBS, nor do either deserve to be considered in place of RBS.

Article Highlights

To optimally size an inventory, the Air Force requires tools and mathematical expressions that allow it to project inventory performance as a function of spares levels.

In "Sizing Inventories with Readiness-Based Sparing" the authors demonstrate that computing inventories of spares based on some of the more traditional inventory performance metrics--such as fill rate or service level--does not do a good job of delivering mission-ready aircraft.

The US Air Force uses the principles of readiness-based sparing (RBS) to set base levels for reparable spares and repair parts, to guide the procurement and repair of spares, to size air-deployable spares packages (readiness spares packages), and to schedule spares distribution and induction into depot repair. Since the early 1980s, the Office of the Secretary of Defense has fostered the adoption of RBS, and its use is mandated in Department of Defense Directive 4140.1-R, DoD Supply Chain Materiel Management Regulation. In fact, all of the military services have implemented RBS (or elements of RBS, as appropriate) throughout their logistics systems.

Although the mathematical models used in the Air Force applications are complex and can be difficult to understand, the basic idea behind RBS is a simple one: evaluate alternative courses of action about spares--how many to buy, where to put them, and so on--based explicitly on how many mission-ready aircraft that action supports. In this article, we will use the strategic decision of what reparable spares the Air Force should procure for its inventory to illustrate how RBS works and how it is superior to other methods in providing readiness for the least cost.

The authors conclude that fill rate and service level can be effective as observational metrics, and a decreasing fill rate or service level can alert managers to potential problems. In the appropriate environment, they can be effective strategic measures, as well. But fill rate's insensitivity to the passage of time is a serious drawback, as is insensitivity to the number of back orders from service level.

Both metrics are correlated to some degree with readiness. All other things being equal, adding a spare to an inventory because it gives the maximum increase in fill rate or service level would also increase aircraft availability. But it would not do so effectively or efficiently. Add to that the difficulty of relating the availability produced by a given fill rate or service level (without an availability model to perform an assessment). Even if a fill-rate or service-level model could equal the performance of an RBS model (and achieve the same availability for a given inventory investment), it would still lack a straightforward method to set targets, and, thus, know what inventory investment was best.

Article Acronyms

AA--Aircraft Availability

AAM--Aircraft Availability Model

AFLC--Air Force Logistics Command

APS--Advanced Planning and Scheduling

CSIS--Central Secondary Item Stratification

DoD--Department of Defense

EBO--Expected Back Orders

FR--Fill Rate

LRU--Line Replaceable Unit

RBS--Readiness-Based Sparing

RR--Ready Rate

TNMCS--Total Not Mission Capable Supply

VSL--Variable Safety Level

Notes

(1.) DoD 4140.1-R, DoD Supply Chain Materiel Management Regulation, 23 May 2003.

(2.) The forecast can be derived in many ways, usually depending on the type of organization. It can be based on a history of demand, engineering estimates, or, in a commercial environment, a retail sales forecast or manufacturing plan. The different forecasts can have widely variable accuracies, and forecasting is a large topic in its own right, but is not of primary interest here. Our focus is on the random recurring demands driven by operational usage, rather than by scheduled overhauls, equipment updates, and the like.

(3.) Most APS systems typically provide item- or location-specific results, whereas the CSIS and D200 provide more aggregate, item-specific results.

(4.) The Air Force supply and maintenance system has multiple echelons, with a central echelon of supply and maintenance functions supporting a lower echelon of operating locations. The maintenance philosophy is one of multiple indentures, with first indenture parts or line replaceable units (LRUs) being removed directly from the aircraft upon failure, second indenture parts being removed from first indenture parts during their repair, and so on. A complete treatment must account for this structure, as the Air Force D200 requirements system does today. Some of our comparisons draw from such complete computations; we use simplified examples to capture the essential character of the measures and allow us to discuss their differences.

(5.) Two references provide excellent discussions on this topic. Craig C. Sherbrooke, Optimal Inventory Modeling of Systems: Multi-Echelon Techniques, New York, New York: Wiley, 1993 and John A. Muckstadt, Analysis and Algorithms for Service Parts Supply Chains, New York, New York: Springer, 2005.

(6.) In the full multi-indenture, multi-echelon (MIME) treatment, the resupply time also depends on delay time because of a lack of lower-indenture spares, transportation time if the item must be shipped for repair at the central depot or parts must be shipped to the base, delay time if spares are not available at the central supply location, and the like.

(7.) Craig C. Sherbrooke, METRIC: A Multi-Echelon Technique for Recoverable Item Control, RAND Memorandum 5078-RM, Santa Monica, California: RAND Corporation, November 1966, G. Hadley and T. M. Whitin, Analysis of Inventory Systems, Englewood Cliffs, New Jersey, Prentice-Hall, 1963.

(8.) These inventory performance measure definitions are based on the APICS Dictionary, 11th ed., APICS--The Operations Management Society, Alexandria, VA, 2005.

(9.) James C. Rainey, ed, Metrics Handbook for Maintenance Leaders, Maxwell AFB, Gunter Annex, Alabama: Air Force Logistics Management Agency, 2001, 44.

(10.) The quantities we define as fill rate and service level are sometimes defined differently in the literature; and the terms fill rate and service level are sometimes used for different quantities. Whatever the nomenclature, they are important and commonly-used measures. As is common practice, we sometimes express these measures as percentages (between 0 and 100) rather than ratios (between 0 and 1) in the obvious way.

(11.) One of the earliest availability formulations of this type is found in J. W. Smith et al., Measurements of Military Essentiality, LMI Report 72-3, August 1972.

(12.) For simplicity's sake, we assume only one unit of item i is installed per aircraft; however, the formulation is readily adaptable to cases with quantities per aircraft greater than one. We also ignore here the effect of shop replaceable units, used in repair of LRUs.

(13.) The Air Force explicitly considers cannibalization in the sizing of its readiness spares packages for deploying units, but its policy for sizing primary operating stock for normal peacetime operations is not to use cannibalization as a source of supply.

(14.) Appendix A contains a table that summarizes the individual p(x) and c(x) values.

(15.) Note how the EBO measure uses all the information in the distribution.

(16.) Notice that, with no spares, the expected number of back orders is exactly the expected number in resupply, EBO(0) = 5. The first spare helps us by eliminating a back order whenever there are one or more assets in the pipeline. The probability of this is 1 - c(0), or 1 - 0.007 = 0.993. So with one spare, we would have EBO(1) = 5 - 0.993 = 4.003. Adding a second spare would eliminate a back order whenever there were two or more assets in the pipeline, which renders a probability of l-c(1) = 1 - 0.04 = 0.96. Thus EBO(2) = EBO(1)-[1c(1)] = 4.003 - 0.96 = 3.043. Exploiting this recursive approach, we can see that EBO(5) = 0.875.

(17.) We are not interested in the mechanics of the optimization, as long as it truly optimizes--provides the greatest return--in terms of the chosen performance measure for the least cost. Obviously, structuring an effective and efficient optimization algorithm for an enterprise as large as the Air Force is a challenge.

(18.) The VSL model used the negative binomial distribution to describe item demand and pipeline processes. A description of the VSL model is in W. B. Fisher et al., Comparison of Aircraft Availability with Variable Safety Level Methods for Budget Program 1500 Allocation, LMI Report AF201, January 1983.

(19.) LMI, Report AF201, 3-6.

(20.) It ran in a multi-echelon mode for a standard requirements determination computation; thus, it considered buys in addition to existing (and already on-order) inventory.

(21.) Actually, an LMI-developed research version of what is now the Air Force AAM.

(22.) Captain Tim Sakulich, "Aircraft Availability," HQ AFLC briefing B 882120, Wright-Patterson AFB, Ohio, 1988.

(23.) The authors gratefully acknowledge the assistance of F. Michael Slay and Julie Castilho in the development of the following examples.

(24.) We've used only LRUs and a single base in the analysis to sharpen the comparison of optimization goals. Introduction of multi-echelon and multi-indenture considerations make the relative performance of FR and RR even worse.

(25.) For both the FR and service-level cases, we used the AAM to assess the AA resulting from their recommended spares mixes.

(26.) T.M. Whitin, The Theory of Inventory Management, Princeton, New Jersey: Princeton University Press, 1953.

He who will not apply new remedies must expect new evils; for time is the greatest innovator.

--Viscount Francis Bacon

The final dictum of history must be that whatever excellence Lee possessed as a strategist or as a tactician, he was the worst Quartermaster General in history, and that, consequently, his strategy had no foundations, with the result that his tactics never once resulted in an overwhelming and decisive victory.

--Major General J. C. Fuller, USA

Defining Logistics

The word logistics entered the American lexicon little more than a century ago. Since that time, professional soldiers, military historians, and military theorists have had a great deal of difficulty agreeing on its precise definition. Even today, the meaning of logistics Can be somewhat fuzzy in spite of its frequent usage in official publications and lengthy definition in Service and Joint regulations. Historian Stanley Falk describes logistics on two levels. First, at the intermediate level:
 Logistics is essentially moving, supplying, and maintaining
 military forces. It is basic to the ability of armies, fleets, and
 air forces to operate-indeed to exist. It involves men and
 materiel, transportation, quarters, depots, communications,
 evacuation and hospitalization, personnel replacement, service, and
 administration.

 Second, at a higher level:

 Logistics is the economics of warfare, including industrial
 mobilization; research and development; funding procurement;
 recruitment and training; testing; and in effect, practically
 everything related to military activities besides strategy and
 tactics.


While there are certainly other definitions of logistics, Falk's encompassing definition and approach provides an ideal backdrop from which to examine and discuss logistics. Today, the term combat support is often used interchangeably with logistics.

The Editors, Air Force Journal of Logistics

The Themes of US Military Logistics

From a historical perspective, ten major themes stand out in modern US military logistics.

* The tendency to neglect logistics in peacetime and expand hastily to respond to military situations or conflict.

* The increasing importance of logistics in terms of strategy and tactics. Since the turn of the century, logistical considerations increasingly have dominated both the formulation and execution of strategy and tactics.

* The growth in both complexity and scale of logistics in the 20th century, Rapid advances in technology and the speed and lethality associated with modern warfare have increased both the complexity and scale of logistics support.

* The need for cooperative logistics to support allied or coalition warfare. Virtually every war involving US forces since World War I has involved providing or, in some cases, receiving logistics support from allies or coalition partners. In peacetime, there has been an increasing reliance on host-nation support and burden sharing.

* Increasing specialization in logistics. The demands of modern warfare have increased the level of specialization among support forces.

* The growing tooth-to-tail ratio and logistics footprint issues associated with modern warfare. Modern, complex, mechanized, and technologically sophisticated military forces, capable of operating in every conceivable worldwide environment, require that a significant portion, if not the majority of it, be dedicated to providing logistics support to a relatively small operational component. At odds with this is the need to reduce the logistics footprint in order to achieve the rapid project of military power.

* The increasing number of civilians needed to provide adequate logistics support to military forces. Two subthemes dominate this area: first, unlike the first half of the 20th century, less reliance on the use of uniformed military logistics personnel and, second, the increasing importance of civilians in senior management positions.

* The centralization of logistics planning functions and a parallel effort to increase efficiency by organizing along functional rather than commodity lines.

* The application of civilian business processes and just-in-time delivery principles, coupled with the elimination of large stocks of spares.

* Competitive sourcing and privatization initiatives that replace traditional military logistics support with support from the private business sector.

The Editors, Air Force Journal of Logistics Integrity is the fundamental premise for military service in a free society. Without integrity, the moral pillars of our military strength, public trust, and self-respect are lost.

--Gen Charles A. Gabriel, USAF

No form of transportation ever really dies out. Every new form is an addition to, and not a substitution for, an old form of transportation.

--Air Marshal Viscount Hugh M. Trenchard, RAF

T. J. O'Malley, LMI

Roger D. Moulder, AFMC/A9A

David K. Peterson, PhD, LMI

T.J. O'Malley is Director Emeritus of LMI's Mathematical Modeling Group. He joined LMI in 1975, and from 1981 until his retirement in 2008 was the director of LMI's mathematical modeling group, managing client research programs in logistics capability assessment and materiel management policy.

Roger D. Moulder is a Lead Operations Research Analyst in the Studies and Analyses Division at Headquarters Air Force Materiel Command (HQ AFMC/A9A), Wright-Patterson AFB, Ohio.

David K, Peterson, PhD, is an LMI senior consultant specializing in readiness-based sparing inventory modeling and its application to Department of Defense logistics support.
Table 1. Aircraft Readiness Depends on a Variety of Expensive,
High-Technology Spare Parts

 Component Price

Hot section module, aircraft $2,909,508
gas-turbine engine

Aileron $1,111,104

Receiver-transmitter group $300,786

Nozzle, turbine, aircraft as-turbine $235,090
engine

Antenna $1,443,661

Gearbox, accessory drive, turbine $136,151
engine

Control-Oscillator $2,143,848

 Component Application

Hot section module, aircraft FlOO-PW229 engine
gas-turbine engine

Aileron C-5A

Receiver-transmitter group Advanced identification-friend
 or foe

Nozzle, turbine, aircraft as-turbine F101 engine
engine

Antenna AN/APQ-170 radar

Gearbox, accessory drive, turbine F-16
engine

Control-Oscillator AN/ALQ-135D(V) countermeasures
 set, transmitter group

Data source: Defense Logistics Agency, Defense Logistics
Information Service, Federal

Logistics Information System web query,
http://www.dlis.dla.mil/WebFlis/pub/pub_search.aspx, accessed
7 May 2010.

Table 2. AFLC Transition to Aircraft Availability (circa 1988)

 VSL AAM

Aircraft Cost (in Cost (in
type millions) Availability millions) Availability

A-10 $136.5 90.0% $119.8 90.0%
E-3 $231.8 52.2% $228.9 82.5%
F-4 $333.0 65.6% $314.5 82.5%
F-15 $857.2 71.1% $797.4 82.5%
F-16C $604.8 74.7% $557.7 82.5%
EF-111 $106.4 59.2% $102.7 82.7%

 Projected
 increase in
Aircraft available
type aircraft due
 to AAM use

A-10 0.1
E-3 8.8
F-4 173.4
F-15 84.9
F-16C 44.2
EF-111 8.5
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Title Annotation:Special Feature
Author:O'Malley, T.J.; Moulder, Roger D.; Peterson, David K.
Publication:Air Force Journal of Logistics
Date:Sep 22, 2011
Words:7619
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