Saturday, December 31, 2016

Viewing Reparable Supply Chains as Echeloned Support Networks

Figure 1 – An illustrative aviation logistics network for a reparable item
The phrase supply chain is a business term that has made its way into the popular vernacular—but with many different connotations. More formally, however, APICS defines a supply chain as:

“The global network used to deliver products and services from raw materials to end customers through an engineered flow of information, physical distribution, and cash.” [i]

Short but very encompassing.

What are supply chains?

This post explores the physical structure of supply chains, viewing them as echeloned support networks. As APICS notes, designing and deploying these support networks requires understanding the
“sourcing, manufacturing, and distribution facilities and information flows to meet the organization’s strategic goals… [and] includes determining the best locations, numbers, sizes, capacities, capabilities, and ownership models of facilities to support these goals.” [ii]

Modern descriptions of supply chains and their associated activities are descended from the post-World War II concept of logistics:

“those aspects of military operations which deal with… design and development, acquisition, storage, movement, distribution, maintenance, evacuation, and disposition of materiel…” [iii]

As the illustrative aviation logistics network in Figure 1 shows, reparable item support networks comprise both a forward movement of serviceable spares as well as a reverse flow of repairable (non-serviceable) spares. Typically, these flows pass through several echelons of supply and maintenance activities.

Can We View These Supply Chains as Echelons of Support Networks?

Business writers frequently critique the use of the term “supply chain” because it evokes a very linear system—one without the complexity of real-world support activities. Figure 2 is a more realistic portrayal of consumable and reparable asset flows in an aviation supply network by Andrew Clark, from a 1958 Rand report.[iv]

Figure 2 – Diagram of the Air Force Supply System (Clark, p. 2)
Clark describes the system’s operation saying:

“Real-world supply systems are often more complex… since many contain several supply echelons and regenerate failed parts through repair. In the Air Force… there may be many operational bases which stock a particular item, and a depot which replenishes base stocks in accordance with base ordering policies. The depot, in turn, is resupplied from a factory or from a repair facility that repairs those failed items it economically can and scraps the rest.
“While [Figure 2] is representative of the supply structure for many items in the Air Force inventory, the structure for other items may be much more complex. For example, some failed items may be repaired at the base itself and turned back into the serviceable stock. There may also be several depots, each supporting a complex of bases; more than one depot-level repair facility; and even more than one factory producing the item. Furthermore, there may be a wide variety of bases or ‘customers’ with different usage patterns and maintenance capability.
“It is evident that supply decisions in such complex systems should be closely interrelated… the main decisions at any given time are how much to ship to each base, how much to repair, and how much to procure from the factory. The dynamic element in the problem is accentuated by the various bases phasing into and out of operation at different times, and experiencing changing failure patterns while in operation.” [v] [1]
Slightly over four decades later, David Simchi-Levi, Philip Kaminsky and Edith Simchi-Levi echoed Clark’s comments with these observations about the complexity of supply chain management:

“First, supply chain management takes into consideration every facility that has an impact on cost and plays a role in making the product conform to customer requirements; from supplier and manufacturing facilities through warehouses and distribution centers to retailers and stores…
“Second, the objective of supply chain management is to be efficient and cost-effective across the entire system; total systemwide costs, from transportation and distribution to inventories of raw materials, work in process, and finished goods…
“Finally, because supply chain management revolves around efficient integration of suppliers, manufacturers, warehouses, and stores, it encompasses the firm’s activities at many levels, from the strategic level through the tactical to the operational level.” [vi]
So, it does seem that viewing supply chains as echeloned support networks may allow a better portrayal of modern support network complexity.

However, lest you think that understanding echeloned support networks is only relevant for aviation support system managers, let’s consider the Washington Metro – the rapid transit system servicing the Washington DC metropolitan area (Figure 3).  Metro inventory managers are faced with a very complex supply chain. As of November 2016, Metro was operating a fleet of over 1200 railcars composed of six major design series[vii] from five different major manufacturers [viii]. Although externally similar in appearance, the different railcar series share varying degrees of component commonality. Metro stores and maintains their railcar fleet at nine rail yards located throughout the system,[ix] and component overhauls are performed either by Metro-internal overhaul shops or by outside vendors.[x] Metro’s Office of Procurement and Materials “manages an inventory of tens of thousands of items at the {Rail] Car Maintenance operating locations as well as the [central] Metro Supply Facility.”[xi] Clearly, Metro’s support structure (and challenges) fits very well with the echeloned support network paradigm in Figure 2!

Figure 3 – One of Washington Metro’s 6000-series railcars (photo by author, November 2015)
In reality, these echeloned support networks are used by many different industries to sustain technologically complex, high cost, and mission critical end-items.

So, what does all this complexity mean for reparable item sparing?

Although business analytical tools and capabilities have significantly progressed since the 1950’s, the challenges of managing complex support activities still remain and warrant careful consideration by inventory managers.

What your support network looks like; the organizations and activities included; and decisions on where to keep supplies, the levels and location of maintenance, and types of distribution activities all significantly influence the amount of systemwide inventory requirements. To the inventory practitioner, what this means is that reparable spares inventories must be sufficient to cover the number of spares contained within, and moving back and forth between echelons of the supply network. As Slyman and Culosi note:

“Within the computation of [inventory] levels, the term "single echelon" refers to an approach that considers the supply activity as a single, autonomous level of supply… [C]ustomers place demands on the activity, the activity fills as many of the demands as possible with issues from its stock and establishes backorders for the rest to be issued when stocks are replenished. The activity replenishes its stocks with orders placed on its sources of supply.
“In contrast, the terms ‘multiple echelon’ and ‘multi-echelon’ refer to an approach in which a chain of two or more supply activities is considered. [Figure 1] illustrates a two-echelon system. Customers place demands on the first activity in the chain and that activity fills as many of the demands as possible and either backorders or refers the rest to the next activity. The first activity replenishes its stocks with requisitions placed on the next activity in the chain, and that activity… fills as many of the requisitions and referrals as possible and either backorders or refers the rest to the next activity. The last activity in the chain fills both requisitions and referrals from lower echelons and replenishes stocks from its sources of supply. An activity in the chain may choose not to stock an item and refer all requisitions for that item to the next higher echelon or in the case of the last activity in the chain, the source of supply.” [xii]
Sounds simple, right? Actually, efficiently and effectively managing reparable item inventories requires a thorough understanding your supply chain! Often, this necessitates viewing the supply chain as an interrelated series of echeloned support networks. As John Muckstadt notes:
“The underlying echelon or network resupply structure will have a substantial impact on the amount of inventory needed. There are clearly many possible structures. However, for each one, there is usually a well-defined resupply plan…
“There are many variations on this theme: some systems have many more echelons, some have fewer. Nonetheless, they are similar in structure.” [xiii]
In future posts, we’ll expand on this structurally-oriented theme and take a closer look at the asset and information flows within a reparable supply chain.

End Notes

[i] Pittman, Paul H., PhD and J. Brian Atwater, PhD, ed. APICS Dictionary, Fifteenth Edition.  Chicago, IL, 2016. (p. 183)
[ii] Pittman and Atwater (p. 117)
[iii] McCann, Colonel John A., USAF (Ret.), ed. Compendium of Authenticated Systems and Logistics Terms, Definitions, and Acronyms.  AU-AFIT-LS-3-71. School of Systems and Logistics, AFIT, Wright-Patterson AFB OH, 1981. (p. 401)
[iv] Clark, A. J., A Dynamic, Single-Item, Multi-Echelon Inventory Model, Rand Memorandum RM-2297, December 8, 1958, downloaded from on 26 December 2016.
[v] Clark (pp. 1-3)
[vi] Simchi-Levi, David, et al. Designing and Managing the Supply Chain: Concepts, Strategies, and Case Studies. Boston: Irwin McGraw-Hill, 2000. (p. 2)
[vii] Washington Metropolitan Area Transit Authority (WMATA) Office of Planning, “10-Year Capital Needs: Inventory and Prioritization, CY 2017- 2026 Needs,” November 2016, downloaded from on 28 December 2016. (p. 1-15)
[viii] “Washington Metro,” from Wikipedia, downloaded from on 31 December 2016.
[ix] WMATA Office of Planning (p. 1-19)
[x] WMATA Office of Operations Planning and Administrative Support, “MetroRail Revenue Vehicle Fleet Management Plan,” revised May 2007, downloaded from on 28 December 2016. (p. 60)
[xi] Transportation Resource Associates, “WMATA Triennial On-Site Safety Review Final Report,” 13 November 2007, downloaded from on 30 December 2016. (p. 98)
[xii] Slyman and Culosi (p. 5-8)
[xiii] Muckstadt, John A. Analysis and Algorithms for Service Parts Supply Chains. New York: Springer, 2005. (p. 4)


[1] In an historically-interesting comment, Clark (p. 1) observes: “The word ‘echelon’ is used rather than ‘level’ to avoid confusion with stock levels, and rather than ‘stage’… because the term ‘multi-stage problems’ has recently been used to designate problems in which time is divided into discrete decision-intervals or stages.”

Sunday, May 11, 2014

Translating a System’s Physical Structure into a Readiness-Based Sparing Analytical Structure

Figure 1 – Starboard Pump Module Location on the International Space Station (ISS) (photo courtesy of NASA)[i]

In mid-December 2013, “the pump module on one of the space station’s two external cooling loops automatically shut down when it reached pre-set temperature limits. These loops circulate ammonia outside the station to keep both internal and external equipment cool.”[ii] During a pre-Christmas news interview, astronaut Doug Wheelock said that on “Christmas morning we [hope to] find a new pump module under the Christmas tree on the Space Station.”[iii]  Finally, on Christmas Eve, NASA reported “Following two spacewalks to replace a degraded pump module on the truss … of the International Space Station, flight controllers in the Mission Control Center at NASA's Johnson Space Center in Houston successfully restarted the new pump….”[iv] Clearly, the need for a spare part can happen to any system, anywhere, at any time!

Portraying physical items in technical data

So, how did ISS engineers know what spare parts might be required to perform an emergency on-orbit cooling system repair?
In a previous posting, reparable items were defined as “high-cost items that are not consumed in use. It is often mechanically and economically feasible to repair these items … [which] retain their identity when in use….” The ammonia pump module that the ISS astronauts needed certainly qualifies as a reparable item.
A reparable item exists and operates in a 3-dimensional environment; it has a length, width and height. Further, you can disassemble a reparable item into its constituent components which can also be characterized dimensionally.
The technical data for complex systems (such as the ISS cooling system) are traditionally stored and maintained in some combination of paper and digital formats. An illustrated parts breakdown (IPB) is typically used to portray an item and its components, and “contains information required for ordering parts, including stock numbers, and for identification of parts and arrangements of parts in assemblies.”[v] Figure 2 is an example of a reparable item and its components as they would be represented in an IPB.[vi]


Figure 2 – Example of a Reparable Item as Portrayed in an Illustrated Parts Breakdown

What is a bill of material?

The IPB essentially portrays an item’s 3-dimensional physical form as a technical drawing. The challenge, of course, is in translating an item’s technical drawing into an analytical data structure that is better suited to readiness-based sparing (RBS) computations. Figure 3 is a very simple example of the beginning of this process where a 3-dimensional cube is translated (essentially “unfolded”) into a 2-dimensional perspective.


Figure 3 – Example of translating a 3-dimensional object into a 2-dimensional perspective
A bill of material (BOM) is used to represent a physical item in an analytical database which lists “all [of] the subassemblies, intermediates, parts, and raw materials that go into a parent assembly showing the quantity of each required to make an assembly.”[vii] Figure 4 is an example of a BOM for the cube in Figure 3, listing the end-item (indenture 0), sub-assemblies (indenture 1), and components (indenture 2); along with any multi-unit quantities (in parens).


Figure 4 – Example of a Bill of Material

So why is knowing the indenture important?

Let’s consider the term indenture a little more carefully.  An indenture is “used to indicate an order of dependence when items are broken down into assemblies, subassemblies, components, and parts. A lower indenture item is a part of the next higher assembly.”[viii] In the logistics vernacular, indenture level 0 is the end-item (or system), level 1 items are referred to as line replaceable units (LRUs), and level 2 items are referred to as shop replaceable units (SRUs). The terms LRU and SRU derive from where items at these indenture levels are used. For example, failed LRUs are removed and replaced by flight line maintenance activities (working directly on the end-item) while failed SRUs are removed and replaced by back shop maintenance activities (conducting repairs of LRUs).[ix]
In readiness-based sparing, the term indenture is also used to describe how many BOM levels are explicitly considered in an inventory optimization. Muckstadt [x] is recognized as developing an early, if not the first, reparable inventory model that accounted for the differential system impacts between LRU and SRU shortages while optimizing spares levels.
If an RBS model has a multi-indenture capability, its selection process considers the position of the item in the system or end item's configuration. A first indentured item goes directly on the end item, while a second-indentured item goes on a first-indentured item. If an item is at the top of the hierarchy (i.e., is a first-indentured item), its impact on operational availability is more direct and consequently greater than if it is lower in the hierarchy (i.e., second-, third-, etc., indentured items). For example, a spare for a second-indentured item will reduce the repair time for its associated first-indentured item and that will, in turn, reduce the time the system or end item is not available when the first-indentured item fails. On the other hand, a spare for the first-indentured item will directly reduce that time.[xi]
When a readiness-based sparing model optimizes spares stockage at just the LRU-level, it is referred to as a single-indenture optimization.  However, if the optimization extends beyond the LRU to include the SRU-level (or even indentures below that), the model is referred to as a multi-indenture optimization.
This brings us back to NASA, the pump, and the ISS ammonia pump module.  For NASA, its LRU-equivalent is called an orbital replacement unit (ORU).  “Orbital replacement units are parts of the main systems and subsystems of the external elements of the ISS…. ORUs, can be hardware such as radiators, or simply batteries or communication antennas. Essentially any element [on the ISS] that can readily be removed and replaced when required.”[xii]
As noted in a U.S. Government Accountability Office report, the “ISS is composed of about 1,000,000 pounds of hardware brought to orbit over the course of more than a decade.” [xiii] This mass can be divided into two general types of hardware:
(1) primary structures, i.e., the external trusses which serve as the backbone of the station and the pressurized modules that are occupied by the ISS crew, and (2) functional systems composed of orbital replacement units (ORUs), i.e., system components modularized to support simple on-orbit replacement.[xiv]
Figure 5 shows a replacement pump module in storage prior to its retrieval and installation during an August 2010 spacewalk.


Figure 5 – Spare pump module in storage. (photo courtesy of NASA) [xv]


The tendency of systems is to evolve towards ever greater complexity (and specialization) over time—be it aircraft, automobiles, ships, computers, or the like. RBS needs to make this complexity analytically-tractable; that is, translating complex, 3-dimensional physical items into a 2-dimensional representation called the BOM.
The accuracy of a BOM is dependent not only on its completeness in terms of capturing the many components and sub-components of a system, but also on how well it identifies the quantities of each item as well as their position with regard to the system’s indentured relationships.  As we’ll see in later postings, each of these dimensions of BOM accuracy is critical to RBS computations and their resulting stockage recommendations.
In future postings, we’ll take a comparable look at another important RBS model characteristic – how to represent reparable supply chains and account for their echelon support structures.


On 7 May 2014, I attended the Charles A. Lindbergh Memorial Lecture at the Smithsonian Institution’s National Air and Space Museum in Washington, DC.[xvi] Prior to Mr. Norman Augustine’s formal lecture, the IMAX® film Hubble 3D was shown. This is an excellent film which documents the space shuttle’s Hubble Space Telescope servicing mission STS-125 in 2009. The film shows astronauts going on several spacewalks to replace ORUs on the Hubble Space Telescope and gives viewers a good appreciation for the challenges of maintaining complex equipment in space. The movie trailer provides a nice summary of the show and can be found at

[i] NASA, “Starboard Pump Module Location.” Downloaded from, on 12 December 2013.
[ii] NASA, “Space Station - Update on Space Station Cooling System,” 11 December 2013.  Downloaded from, on 11 December 2013.
[iii] Doug Wheelock Interview, ABC World News Tonight, 22 Dec 2013
[iv] NASA, “Station's Replacement Pump Successfully Restarted,” 24 December 2013.  Downloaded from, on 24 December 2013.
[v] McCann, Colonel John A., USAF (Ret.), ed. Compendium of Authenticated Systems and Logistics Terms, Definitions, and Acronyms.  AU-AFIT-LS-3-71.  School of Systems and Logistics, AFIT, Wright-Patterson AFB OH, 1981. (p. 336)
[vi] MIL-DTL-38807D, “Detail Specification Technical Manuals - Illustrated Parts Breakdown,” Department of the Air Force, 17 September 2012.  Downloaded from  on 24 February 2014. (p. 22)
[vii] APICS, APICS Dictionary, definition downloaded from|dictionary on 24 February 2014.
[viii] McCann (1981), p. 341.
[ix] In the U.S. Navy, indenture level 1 and 2 items are referred to as weapon replaceable assemblies (WRAs) and shop replaceable assemblies (SRAs), respectively. See Department of the Navy.  Operational Availability of Equipments and Weapons Systems and Operational Availability Handbook: A Practical Guide for Military Systems, Sub-Systems, and Equipment . OPNAV INSTRUCTION 3000.12A, Office of the Chief of Naval Operations N40, 2 September 2003. (pp. 71-2) Downloaded from
[x] Muckstadt, John A., “A Model for a Multi-Item, Multi-Echelon, Multi-Indenture Inventory System,” Management Science, Vol. 20, No. 4, Part I, December 1973.
[xi] Slyman, George L. and Dennis L. Zimmerman.  Improved Inventory Models for the United States Coast Guard Requirements Determination Process.  LMI Report CG201RD6.  Bethesda, MD: Logistics Management Institute, October 1993.  (Page A-4) Downloaded from on 8 March 2012. 
[xii] Wikipedia, “Orbital Replacement Unit.”, 5 March 2014.
[xiii] United States Government Accountability Office, “International Space Station (ISS) – Ongoing Assessments for Life Extension Appear to be Supported,” Washington DC, April 11, 2011, (p. 4)
[xiv] United States Government Accountability Office, (2011), p. 4.
[xv]    NASA, “Spare Pump Module.” Downloaded from on 5 March 2014.
[xvi] The video of this lecture is available at

Friday, January 3, 2014

The Economic Order Quantity – An Analytical Centennial

Figure 1 Graphical Derivation of an Optimal Order Quantity (Q*)

As 2013 drew to a close, it was a time for reflecting on the year just past and on the year just beginning. With that frame of mind, I recently reread several articles by Donald Erlenkotter. As documented in these articles, Ford Whitman Harris first published the basic tenets of the economic order quantity (EOQ) inventory model in February 1913. Acknowledging and celebrating EOQ’s analytical centennial and its relationship to readiness-based sparing is the subject of this post.

What’s the to-do about EOQ?

Before examining the life of F. W. Harris, let’s take a quick look at what experts have written about the EOQ model over the years.
Table 1 Some Observations about the EOQ Model
T. M. Whitin (1954)
“It is encouraging that even extremely simplified formulations have found their counterpart in the world of reality and have been successfully applied in spite of (or perhaps because of) their superficiality. Considerable time must elapse before business practice catches up with the more complex models, but with a continuance of the present interchange between theoreticians and practitioners, rapid progress will be made.”[i]
H. E. Scarf (1963)
“The model on which this calculation is based is highly simplified and neglects a good number of the important reasons for maintaining inventories. On the other hand, the formula … provides a remarkably good approximation to ‘optimal policies’ in considerably more elaborate and realistic models.”[ii]
G. Hadley and T. M. Whitin (1963)
“[T]he results obtained from these models yield, qualitatively, the proper sort of behavior—even when the deterministic demand assumption is removed.”[iii]
E. S. Buffa (1969)
“In practice, the formula itself is not used often; rather, charts, graphs, and tables based on the formula are used to minimize computations.”[iv]
H. M. Wagner (1975)
“Clearly, you can rarely be certain that demand behaves in so precise a manner…. Nevertheless, many industrial firms have been able to employ these models and have thereby realized substantial cost savings. To do so, however, the models … are usually modified so that demand is treated probabilistically.”[v]
B. S. Blanchard (1981)
“The EOQ model is generally applicable in instances where there are relatively large quantities of common spares and repair parts.”[vi]
G. W. Plossl (1985)
“[T]hese fairly sophisticated techniques of inventory management had very little application. Perhaps this was because the 1930s and 1940s were not years that encouraged scientific management. For most companies during the depression of the 1930s, the most important objective was survival…. During the 1940s, when pent-up demand provided a ready market for every article that could be produced, the objectives of inventory control… were not important in most business operations.”[vii]
R. J. Tersine        (1988)
“The robustness of the EOQ … helps justify their widespread use. When deterministic models are insensitive to parameter changes, they provide an excellent approximation to real-world phenomena.”[viii]
D. Erlenkotter (1989)
“The familiar square-root formula for the optimal economic order quantity (EOQ) in simple inventory models is a result so fundamental to management science and operations research that it appears in every elementary textbook.”[ix]
D. Erlenkotter (1990)
“Today the EOQ model is so well known that we accept its basic structure as obvious. In 1913, however, it was a modeling achievement of classical elegance.”[x]
S. Axsäter    (2006)
“The most well-known result in the whole inventory control area may be the classical economic order quantity formula. This simple result has had and still has an enormous number of practical applications.”[xi]

It’s worth noting that Whitin began his survey of inventory control research with a discussion of the EOQ model (and that this was also the very first issue of Management Science)! Further, consider some of the phrases used above to describe EOQ—“successfully applied,” “a remarkably good approximation,” “the proper sort of behavior,” “generally applicable,” robustness,” and “classical elegance.” Clearly, for an analytical technique to remain referenced and in use for over a century it must have some redeeming characteristics! But what do we know about the man behind the model?

A Glimpse of F. W. Harris

In his articles, Erlenkotter presents a compelling discussion of EOQ’s provenance, however, even more interesting is Erlenkotter’s portrait of F. W. Harris—the man.
Ford Whitman Harris … had a long and distinguished career as an engineer and a patent attorney. His career is the more remarkable in that he received no formal education after the age of 17: he was self-educated in the broadest sense of the term; he received more than 100 patents for inventions; he was admitted to practice before the U.S. Supreme Court; and, according to his daughter, he knew French and could recite from memory passages from Milton.[xii]
Further, F. W. Harris (not unlike many of us) was faced with a career crossroads when the family moved from Pittsburgh, PA to Los Angeles, CA in 1912.
[A]t the age of 35 Harris was faced with the need to retool his career…. He had little in the way of formal educational credentials. But he did have an engineering background and experience in a major industrial corporation. It appears, then, that he began writing and publishing work on industrial management topics in 1913 to help establish his credentials in this broader field.[xiii]
The timeline in Figure 2 highlights some key events in Harris’ career as described by Erlenkotter (1990). In reality, Harris had two primary occupations—engineering and patent law. Interestingly, the vast majority of Harris’ professional career involved patent law. Erlekotter provides this quip by Harris that helps to explain this dramatic mid-career change: “I made a precarious living as an engineer for a considerable period before I broke down the fence into what I thought was a greener pasture.”[xiv]
Surprisingly, Harris’ obituary in the Los Angeles Times only makes this brief reference to his engineering expertise saying that “he was a self-educated engineer.”[xv]

Figure 2 F. W. Harris' Career Timeline

So how does EOQ relate to reparable items and RBS?

Some would say that the EOQ methodology is ubiquitous in the inventory literature and it would be hard to counter that claim. In fact, the basic EOQ formulation has been extended in a number of different dimensions—quantity discounts, production lot-sizing, probabilistic demand, transportation costing, and even to reparable items.[1]
In 1967, Schrady[xvi] published a variation on the deterministic EOQ model tailored to reparable items. The key to this formulation is acknowledging that reparable items in a supply system move between two physical states—as fully serviceable ready for issue (RFI) assets or as failed (but repairable) not ready for issue (NRFI) assets.
Schrady observes that there is a trade-off between holding stock in the RFI and the NRFI conditions. [T]he cost of this resource, NRFI items, is less than the cost of the RFI resource by at least the cost of repair labor and replacement parts. Thus, if inventory is to be held in the system it would be better held in NRFI condition than in RFI condition.[xvii]  Realizing that some fraction of carcasses will exceed repair capabilities, the business rules for operating such a system are then described:
100 percent of demand [is supplied] from repaired items until the supply of NRFI items decreases to a point where there are insufficient carcasses on hand to induct another batch. At this time, a procurement quantity is received, and O&R [overhaul and repair] inductions are suspended. While the procurement quantity lasts, carcasses are accumulated at the O&R. Inductions are resumed a repair leadtime before the procurement quantity is exhausted…. Note that the repair trigger is in the RFI inventory and the procurement trigger is dependent upon the NFRI inventory….[xviii]
Ultimately, Schrady develops an expression for the reparable item’s inventory total costs per unit time: [xix]

Eq 1

and determines values for the item’s optimal procurement quantity (Q*P) and repair batch size (Q*R) by setting the respective partial derivatives of the total cost equation equal to zero and solving for QP and QR:
Eq 2


Eq 3

d          = demand rate (units per time unit t)
r           = recovery rate of failed units
(1-r)     = scrap rate of failed units
AP        = fixed cost per procurement order
AR        = fixed cost per repair batch induction
h1        = RFI holding cost
h2        = NRFI holding cost
Nahmias commented on the applicability of models like this—“[D]eterministic models [such as Schrady’s formulation] can often be useful in pointing out potential underlying relationships in the system that can be generalized to [cases with] random demand.[xx] This comment seems rather prescient, if you look up Schrady’s paper in Google Scholar, his paper has been well-cited over the years by a number of authors extending it to multi-item probabilistic demand and exploring such diverse topics as reverse logistics, remanufacturing, green supply chains, hazardous materials management, and lean supply chains.


Certainly, F. W. Harris understood the fundamental changes that his mathematically-driven lot-sizing approach implied for early 20th century inventory management. However, it’s interesting to speculate whether or not F. W. Harris had any sense of how long his EOQ model would be influencing modern inventory management. And it would have been even more unlikely for him to have imagined the diversity of extensions to the original EOQ model.
Schrady’s extension of the deterministic EOQ model to reparable item management was an early attempt at seeking how to properly balance the number of serviceable and repairable units of an item within an inventory system. However, it was a deterministic, single-item, single echelon, single indenture optimization.  In future postings, we’ll take a closer look at each of these reparable item inventory model characteristics and their relationship to readiness-based sparing.

[1] Recall that in an earlier posting, we defined reparable items as high-cost items that are not consumed in use and are often mechanically and economically feasible to repair. Examples of reparables (which retain their identity when in use) include items such as radios, radar units, engine components, or landing gear.

[i] Whitin, T. M. “Inventory Control Research: A Survey,” Management Science, Vol. 1, No. 1: pp. 32-40 (1954).
[ii] Scarf, Herbert E. “A Survey of Analytic Techniques in Inventory Theory,” in Multistage Inventory Models and Techniques. Ed. Herbert E. Scarf et al.  Stanford CA: Stanford University Press, 1963. (p. 192)
[iii]Hadley, G. and T.M. Whitin. Analysis of Inventory Systems. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1963. (p. 29)
[iv]Buffa, Elwood S. Modern Production Management (Third Edition). New York: John Wiley and Sons, Inc., 1969. (p. 519)
[v]Wagner, Harvey M. Principles of Operations Research: With Applications to Managerial Decisions (Second Edition). Englewood Cliffs, NJ: Prentice-Hall, Inc., 1975. (pp. 813-4)
[vi] Blanchard, Benjamin S. Logistics Engineering and Management (Second Edition). Englewood Cliffs, NJ: Prentice-Hall, Inc., 1981. (p. 61)
[vii] Plossl, George W. Production and Inventory Control: Principles and Techniques (Second Edition). Englewood Cliffs, NJ: Prentice-Hall, Inc., 1985. (p. 3)
[viii] Tersine, Richard J. Principles of Inventory and Materials Management (Third Edition). New York: Elsevier Science Publishing Co., Inc., 1988. (p. 142)
[ix] Erlenkotter, Donald. “An Early Classic Misplaced: Ford W. Harris’s Economic Order Quantity Model of 1915*,” Management Science, Vol. 35, No. 7: pp. 898-900 (July 1989).
[x] Erlenkotter, Donald. “Ford Whitman Harris and the Economic Order Quantity Model,” Operations Research, Vol. 38, No. 6: pp. 937-946 (Nov-Dec 1990).
[xi] Axsäter, Sven. Inventory Control (Second Edition), New York: Springer Science+Business Media, LLC, 2006. (p. 52)
[xii] Erlenkotter (1990) p. 941.
[xiii] Erlenkotter, Donald. “Ford Whitman Harris’s Economical Lot Size Model,” downloaded from‎ on 23 Dec 2013. (Note – this paper has been accepted for an upcoming special issue of International Journal of Production Economics focusing on EOQ). 
[xiv] Erlenkotter (1990) p. 942. 
[xv] “Ford Harris, Pioneer Patent Attorney, Dies.” Los Angeles Times, 29 October 1962, Part I, p. 28.
[xvi] Schrady, David A.  “A Deterministic Inventory Model for Reparable Items,” Naval Research Logistics Quarterly, Vol. 14, Is. 3: pp. 391-398. (1967)
[xvii] Schrady (1967). p. 393.
[xviii] Schrady (1967), p. 393.
[xix] Schrady (1967) p. 396.
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