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

Author	Observation
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 Q_P and Q_R:

Eq 2

 

                                            

Eq 3

where:

d          = demand rate (units per time unit t)

r           = recovery rate of failed units

(1-r)     = scrap rate of failed units

A_P        = fixed cost per procurement order

A_R        = 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.

Summary

Certainly, F. W. Harris understood the fundamental changes that his mathematically-driven lot-sizing approach implied for early 20^th 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.

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[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.

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[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 www.laipla.net/wp-content/uploads/2013/05/Harris-EOQ-Centennial.pdf‎ 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.

[xx] Nahmias, Steven. "Managing repairable item inventory systems: a review." TIMS Studies in the Management Sciences, Vol. 16: pp. 253-277. (1981)

“You say repairable, and I say reparable. You say rotable, and I say recoverable … So let’s call the thing exchangeable!” *



“Engineman Fireman Recruit Megan Cotrell conducts maintenance on an engine in one of the main spaces aboard the amphibious transport dock ship USS New Orleans (LPD 18).”[i] (Photo courtesy U.S. Navy)

 

There’s a term for everything in logistics. The problem is, the specific meaning of a term is often nuanced and depends on where you work or where you “grew up” as a logistician. Since every logistics organization has its own dialect, we take a closer look at several terms for spare parts, especially those parts managed by readiness-based sparing (RBS).

 

A closer look at terminology

Table 1 includes some of the more common terms and definitions for spare parts.  Use this table as a convenient point of reference...or translation; investigate the citations further to better understand the specific intent behind each usage.

Table 1 General Definitions

General term	Reference definition
Repair part	“Part required to return a vehicle to operational condition, necessary to perform its intended purpose. This does not include maintenance supplies; such as, wiping rags, antifreeze, radiator flush and stop-leak, solvents, grease, etc.”[ii]
Spare part	“An individual part, sub-assembly or assembly supplied for the maintenance or repair of systems or equipment.”[iii]
Service parts	“Those modules, components, and elements that are planned to be used without modification to replace an original part.”[iv]

 

Table 2 provides some additional specificity and definitions for spare parts based upon the item’s ultimate usage.

Table 2 Specific Definitions

The item is consumed or loses its identity in use.
Specific term	Reference definition
Consumable item	“An item of supply (except explosive ordnance and major end items of equipment) that is normally expended or used up beyond recovery in the use for which it is designed or intended.”[v]
Expendable item	“Item [that] is consumed in use or [that] loses its original identity during periods of use by incorporation into, or attachment upon, another assembly.”[vi]
Shop supplies	“Expendable items consumed in operation and maintenance (waste, oils, solvents, tape, packing, flux, welding rod).”[vii]
The item is not consumed and retains its identity while in use.
Specific term	Reference definition
Capital spare	“[C]apital spares are not consumed or used to destruction on the basis that they can be repaired and so become ‘rotable items’.”[viii]
Exchangeable item	“Recoverable components, such as pumps, electric motors, carburetors, and fuel controls.”[ix]
Recoverable item	“A spare part [that] normally is not expended in use and [that] can be reused after recovery and repair.”[x]
Repairable item	·       “An item that can be reconditioned or economically repaired for reuse when it becomes unserviceable.”[xi] ·       “A component, module, assembly, subassembly or equipment determined by the inventory manager to be economically repairable when it becomes unserviceable.”[xii]
Reparable item	“An item of supply subject to economical repair, and for which the repair (at either depot or field level) of unserviceable assets is considered in satisfying computed requirements at any inventory level.”[xiii]
Rotable item	“[R]otable spare parts are materials and supplies … that are acquired for installation on a unit of property, removable from that unit of property, generally repaired or improved, and either reinstalled on the same or other property or stored for later installation.”[xiv]
Rotable pool	“Supply's inventory of repairable items is commonly referred to as the ‘rotable pool’ or just the ‘pool’."[xv]
Note: Two definitions include qualifying statements.  The USAF definition for reparable includes this addendum: “This term suggests the logistics status rather than the condition of an item.”[xvi] The IRS definition for a rotable is further explained as: “flight equipment rotable spare parts and assemblies are tangible property for which depreciation is allowable while expendable flight equipment spare parts are materials and supplies.”[xvii]

Summary

In short, two broadly applicable definitions will work well in most logistics conversations regarding spare parts:[xviii]

1.     Consumable items are typically low-cost items that are consumed in use. Consumable items also lose their identity when installed on higher assemblies (you do not distinguish the part from the whole). It is usually physically or economically infeasible to repair failed consumable items—better to replace them. Common consumables include nuts, bolts, paint, wire, solder, memory chips, and brackets.

2.     Reparable items are high-cost items that are not consumed in use. It is often mechanically and economically feasible to repair these items. Examples of reparables (which retain their identity when in use) include items such as radios, radar units, engine components, or landing gear.

Readiness-based sparing methodologies are commonly applied to the management of inventories of reparable items and select (typically higher-cost) consumable items.

If you are still confused, don’t fret, you are not alone. The very etymologies of reparable and repairable differ.  Reparable comes from the Latin reparabilis meaning “able to be restored or regained,” while repair (and repairable) comes from the Latin reparare meaning “to restore, put back in order.”[xix]  Obviously, the need for repairing and maintaining equipment goes back a long, long time!



“Staff Sgt. Christopher Matthews, an aerospace propulsion systems craftsman with the 52nd Component Maintenance Squadron, inspects an F-16 Fighting Falcon exhaust nozzle March 16, 2010, at Spangdahlem Air Base, Germany.”[xx] (Photo courtesy of the U.S. Air Force)

* With a sincere thanks and acknowledgment to George and Ira Gershwin for their timeless music and corruptible lyrics. Also, a special thanks to Maggie Wise for her review and very helpful suggestions regarding this posting.

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[i] U.S. Navy Photograph by Mass Communication Specialist 2nd Class Dominique Pineiro/Released, USS New Orleans (LPD 18), downloaded from http://www.navy.mil/management/photodb/photos/120124-N-PB383-789.jpg on 6 Mar 2013. 

[ii] Department of the Air Force.  The USAF Supply Manual.  AFM 23-110, Vol I, Pt. 1, Chap 1, “General and Administrative,” Atch 1A-1 (Definitions and Supporting Information), Washington: HQ USAF, 31 Mar 2005. (p. 1-102)

[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. 637)

[iv] APICS, APICS Dictionary, definition downloaded from http://www.apics.org/industry-content-research/publications/apics-dictionary on 10 March 2013.

[v] Office of the Deputy Under Secretary of Defense for Logistics and Materiel Readiness (DUSD[L&MR]), DoD Supply Chain Materiel Management Regulation, DoD 4140.1-R, 23 May 2003. (p. 192)

[vi] Department of the Air Force, 2005. (p. 1-73)

[vii] McCann (p. 627)

[viii] United Kingdom Ministry of Defence. Defence Logistics Support Chain Manual.  JSP 886, Vol 2, Part 6 “Financial Accounting for Inventory,” Ver. 1-5, 5 February 2013, p. 8.  Downloaded from https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/73215/jsp886_vol2_pt6_fin_acct_for_Inv_v1-5.pdf on 22 Apr 2013.

[ix] McCann (p. 273)

[x] McCann (p. 571)

[xi] Department of Defense.  Department of Defense Dictionary of Military and Associated Terms, Joint Publication 1-02, 8 November 2010 (As Amended Through 15 October 2011). (p. 288)

[xii] Department of the Navy.  Supply Appendices.  NAVSUP P-485 Volume II - Supply Appendices, Revision 4, Glossary of Supply Abbreviations, Acronyms and Terms, Part B: Terms, Naval Supply Systems Command, NAVSUP HQ, 18 May 2009. (p. G-86)

[xiii] Department of the Army.  Logistics Provisioning of U.S. Army Equipment.  Army Regulation 700–18.  Washington, DC, Headquarters, 20 September 2009. (Page 31)  Downloaded from http://www.apd.army.mil/pdffiles/r700_18.pdf on 6 March 2012.

[xiv] U.S. Government, Internal Revenue Service.  Internal Revenue Bulletin: 2012-14, T.D. 9564 Guidance Regarding Deduction and Capitalization of Expenditures Related to Tangible Property, April 2, 2012.  Downloaded from http://www.irs.gov/irb/2012-14_IRB/ar05.html  on 14 Dec 2013.

[xv] Wirwille, James W. and William T. Ainsworth.  Analysis of a Proposal to Consolidate Aircraft Intermediate Maintenance Capabilities.  MS Thesis, Naval Postgraduate School, Monterey CA, December 1991 (AD-A246187). (p. 22)

[xvi] Department of the Air Force, 2005. (p. 1-102)

[xvii] U. S. Government.  Federal Register. Guidance Regarding Deduction and Capitalization of Expenditures Related to Tangible Property - A Rule by the Internal Revenue Service on 09/19/2013.  Downloaded from https://www.federalregister.gov/articles/2013/09/19/2013-21756/guidance-regarding-deduction-and-capitalization-of-expenditures-related-to-tangible-property on 14 Dec 2013.

[xviii] Pankonin, Captain George C. and Captain David K. Peterson. A Spares Stockage Algorithm for Low-Density Equipment. MS thesis, AFIT/GLM/33-82.  School of Systems and Logistics, Air Force Institute of Technology (AU), Wright-Patterson AFB OH, September 1982 (ADA123709). (pgs. 8-9).

[xix] Harper, Douglas. Online Etymology Dictionary.  http://www.etymonline.com/index.php?allowed_in_frame=0&search=repair&searchmode=none.  Accessed 14 December 2013.

[xx] U.S. Air Force Photograph by Airman 1st Class Nick Wilson, downloaded from http://www.af.mil/News/Photos.aspx?igphoto=2000293449 on 15 December 2013.