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