"And Then There Were None," by Harvey M. Wagner
University of North Carolina
Chapel Hill, North Carolina 27514
hmwagner@email.unc.edu.
A half century ago, several scholars, some of whom subsequently received Nobel
prizes in economics, developed inventory models primarily in response to the
needs of American military service branches and a few large corporations.
Significant government waste was attributed to mismanagement of weapon systems
assets. There were readiness-debilitating shortages in the presence of available
supplies of weapon components that were crisscrossing the globe. Big
corporations also recognized a potential for profit improvement from getting
more bang from a buck of inventory investment. Given that management
information systems 50 years ago employed punched card processing in most
organizations, it was impossible for a typical company to adopt the new
mathematical inventory models. Fast forward to year 2002. Today we are
frustrated when we repeatedly find a favorite brand out of stock at the local
grocery or drug store. We are dismayed at experiencing third-world customer
service levels only a few blocks from home. We surmise that the stock shortages
are rooted in poor management and not in poor systems. This essay focuses on why
we continue to and empty shelves where we do business despite a half century of
impressive research in inventory modeling, augmented by high-priced
multiplatform supply-chain management software. We observe companies that have
poor customer service despite excessive inventories. Good inventory modeling and
advanced inventory control systems are supposed to eliminate such things. Our
story unfolds as follows: The central theme of this reflection is inventory
theory in the service of practice— not theory for the sake of theory. First, I
will review several formative research findings published between 1950 and 1965.
I will make a case for why these research contributions remain relevant today.
The references at the end of the article are publications that appear no later
than 1965, with only a few exceptions. Next I will segue to an arguable
proposition (with all deference to Ron Howard) that nothing is as impractical as
a good theory of inventory. I will explain why what has been truly good
inventory research for more than 35 years has not done much to advance the
practice of industrial inventory control. Finally, I will suggest that
developments in information technology renew the opportunity to improve practice
provided that particular avenues of inventory systems research are pursued.
Silver et al. (1998) published their third edition of an extraordinary text
dealing with inventory management. It cites and thematically organizes the
findings of more than 1,600 researchers—its topical coverage is encyclopedic
(after 1965). In my opinion, it reaches a superlative level of achievement,
notwithstanding any of my comments below, and it is readily accessible.
Consequently, I am going to refer to it occasionally to back up my propositions;
I will use the abbreviation SPP3 whenever I refer to the book.
1. IN SEARCH OF FULFILLMENT The earliest publications on inventory modeling date
back to the 1920s—the lot size (square root EOQ) model is the most notable
example, with its commercial context of stocks held in businesses. Before 1950,
macroeconomists also wrote about fluctuations in inventory levels in the context
of classic business cycles. In the early 1950s, a few influential research
contributions emerged from this primordial condition. Arrow et al. (1951)
analyzed probabilistic inventory models. Simon (1952) wrote about servo theory
applied to production. Dvoretzky et al. (1952a, 1952b) discussed inventory
models and statistical processes. All of these contributions were published in
Econometrica. Whitin (1953) published a book that was devoted solely to
inventory themes, and Bellman, at the RAND Corporation, wrote a monograph on
dynamic programming in 1953; see Bellman (1957). The United States Air Force,
Navy, and Army funded research efforts aimed at improving the performance of
logistics systems. The research impetus driving inventory modeling was in full
force by 1954. These scholarly endeavors encompassed probability modeling
(especially renewal and queuing processes), feedback systems, statistical
decision theory, microeconomics, and multi-period optimization. All of these
themes are lively today and continue to influence advanced inventory research.
Subject classifications: Forecasting: inventory system effectiveness.
Inventory/production: impact of forecasts on. Professional: comments on. Area of
review: Anniversary Issue (Special). 0030-364X/02/5001-0217 $05.00 1526-5463
electronic ISSN 217 Operations Research ? 2002 INFORMS Vol. 50, No. 1,
January–February 2002, pp. 217–226 218 / Wagner the planning horizon. These
subtleties are taken for granted today. Inventory research is patently indebted,
even now, to the path-breaking modeling of the early 1950s.
2. POINTS OF LIGHT For a decade beginning in the mid-1950s, the core ideas above
were pursued vigorously. With the notable exception of Brown (1959),
publications assumed that either future demand values are given or that their
underlying distribution is completely known. Most researchers continue to make
these assumptions. I will summarize a few of the pivotal themes from this
ten-year span. Wagner and Whitin (1957, 1958), and Manne (1958) extended the
classic deterministic lot-size model with stationary demand to accommodate known
demand that fluctuates from period to period. The dynamic lot-size approach was
eventually recognized to be a deterministic version of a renewal model, and
equivalent to finding a shortest route in an acyclic network. Nevertheless, even
sophisticated texts like SPP3 explain it in a complicated way, despite the fact
that the dynamic lot-size model is a much simpler acyclic network than a typical
critical path, which is standard material in OM textbooks. The network
characterization relies on the concave property of the objective function.
Bowman (1956) and Johnson (1957) explored alternative convex objective function
formulations, and established the consequent optimality of a myopic decision
process. The idea is closely related to the notion of a greedy solution process
(optimize incrementally and never revise a prior decision). The concept of a ?nite
planning horizon determined by the data emerged from the lot-size modeling
above. Conditions were discovered that ensured an optimal solution for a ?nite
horizon remained optimal for an extended horizon; further, these conditions did
not require full knowledge about the longer horizon. Complementary to renewal
theory research, Morse (1958) examined steady-state stochastic replenishment
systems using queuing theory. He viewed inventory items as analogous to servers
in a queuing system, server busy time as tantamount to replenishment lead time,
and a waiting line as comparable to customer demand backlog. The mathematical
form of optimal multiperiod decision rules in the presence of stochastic demand
was explored in different ways. Holt et al. (1956, 1960) demonstrated that under
certain assumptions, an optimal rule is linear in the parameters of the demand
distribution. These assumptions, however, did not turn out to be suf?ciently
appealing to motivate much further research. It was known by the mid-1950s that
the form of an optimal policy is sensitive to whether the objective function
contains a setup cost and a smooth cost function associated with less than
perfect service, the delivery lead time is greater than one period, and unfilled
demand is fully backlogged. The noteworthy research achievements mentioned next
address these technical challenges. In the early 1950s, it was difficult to
obtain historical demand data for individual items (even for weapons systems
components stocked at military bases), and computing capacity to automate
replenishment formulas was extremely limited. There was little opportunity to
empirically test emergent inventory theory. There were, however, fundamental
insights from these early research publications. The new literature revealed the
appropriate form (architecture) of replenishment rules, the tractability of
discrete vis-à-vis continuous time modeling, the interdependence between the
reorder point and reorder quantity, the convenience of particular demand
distributions (Poisson, exponential, and normal), and the usefulness of the
principle of optimality. Since there was little available empirical demand data
at the level of a stock keeping unit (SKU), assuming a Poisson or exponential
distribution versus a normal distribution was a matter of mathematical elegance
versus flexibility in being able to specify both a mean and standard deviation.
Square-root-type formulas using continuous time models were computationally more
practical than renewal recursions. And mathematically derived replenishment
policies that explicitly accounted for imprecision associated with observational
data seemed too elegant, if not esoteric, to implement. Early 1950s research
also shed light on technical details regarding felicitous model formulations. It
was helpful to assume that the economic impact of customer service level is
included in the model’s objective function to be optimized (rather than to
express it as a side condition), unmet demand is fully backlogged, lead time is
knowable and deterministic, demand is iid for an individual SKU (which does not
deteriorate or become obsolete), the time horizon is a single period or
unbounded, and if the latter, all parameter values are stationary. Even what
seems to be slight departures from these assumptions were known to create
serious analytic challenges. The aforementioned stochastic inventory modeling
research was complemented by deterministic multiperiod multiproduct
linear-programming formulations. In the early 1950s, however, performing
linear-programming optimization (by the simplex method) meant using punch-card
computers. By necessity these models were toy-sized. Only later did it become
feasible to think about deterministic multiperiod linear-programming models as a
way to consider multiproduct time-phased production and inventory decisions. An
important legacy of the early 1950s is an insight about the architecture of
inventory control solutions. Optimizing dynamic stochastic models implies
finding a strategy or policy—in other words, a rule in which all future
decisions are contingent on future states of the system. These states are
determined in part by the future outcomes of random events. Consequently, future
decisions are described probabilistically. In contrast, optimizing deterministic
linear-programming models implies using an algorithm that yields numeric values
for all future decisions— therefore all of these values are knowable at the
outset of Wagner . Using reasonable assumptions, Scarf (1960) established
the optimality of s S policies: When inventory on hand plus inventory due in
less backlog falls below s, order enough to bring it up to S—else, do not order.
(It was known that under alternative plausible model assumptions, this simple
policy was not optimal.) Roberts (1962) provided a way to compute approximately
optimal policies. Veinott and Wagner (1965), and subsequently other researchers,
investigated effective computational methods for obtaining optimal policies, and
later, approximately optimal policies. Veinott (1965) showed for an important
class of situations that a myopic replenishment policy is optimal, and thereby
established that an unbounded horizon model can be decoupled so as to yield
near-term optimal decisions using only near-term parameter values. Further, the
work established a sound basis for what has become an important decision rule in
practice: Replenish what you sell. This is an elementary example of a so-called
pull system. Clark and Scarf (1960) formulated a seminal and tractable model for
inventory replenishment in an environment comprised of several echelons that
hold inventory and where ?nal demand is uncertain. Operations research textbooks
at that time did not make sharp distinctions among different inventory
management settings. Inventory scholars realized by then, however, that
inventory theory approximates reality well when a stocked item is ordered from
an outside vendor, but not so well when an item is manufactured by the
enterprise as a direct result of a replenishment decision. Today this underlying
distinction is evident and re?ected in ?nite capacity scheduling models.
Expository habits are hard to break, nevertheless, and even SPP3 discusses EOQ
in terms that allow for both interpretations of purchasing from an outside
vendor and manufacturing the SKU from within the enterprise. By the early 1960s,
it was clear that linear programming is a conceptually well-suited alternative
for addressing a combination of multi-item, multilocation, multiperiod issues in
a capacity-constrained environment, even given its limitations. In contrast, it
was hard to imagine that anything practical would result from an extension of
stochastic inventory theory models in a pull context to multi-item situations
(that is, to something beyond applying single-item analysis to each SKU
individually). The reason is that it would be a heroic task to use historic data
to establish multivariate demand distributions. The fundamental distinction
between multi-item and single-item models corresponds closely to the split in
practice between push and pull inventory control systems. In a multi-item
manufacturing environment, pull systems, which aggregate orders for all items
that are requested, may lead to infeasible production schedules as well as
periods with excess capacity; thus push systems tend to be the rule in practice.
In a stock-replenishment environment, push systems, which keep capacity fully
utilized, may lead to overstocking as well as excessive obsolescence; thus pull
/ 219 systems tend to be the rule in practice, although transportation
constraints sometimes countervail. Forrester (1958, 1961) published an article
and later a book on what he called industrial dynamics, and thereby created a
computational engine that exempli?es economists’ traditional business-cycle
logic. The approach views a SKU’s inventory level as a time series
mathematically created by the difference between cumulative production and
cumulative demand, layered on a base of safety stock. The industrial dynamics
model facilitates visualizing the imposition of boundary conditions on the time
series (and their slopes), along with the imposition of servomechanisms
(feedback). Despite industrial dynamics’ broad sweep, it does not adequately
address probabilistic uncertainty, which ultimately has limited its contribution
to inventory control systems (in particular, how to set safety stocks). Wagner
(1962) published a research monograph under the Operations Research Society’s
sponsorship that introduced statistical issues of importance to senior
management. The research theme focuses on how corporate management can use
aggregate economic measures to ascertain whether lower echelons of a supply
chain are adhering to automated inventory replenishment logic. Recognizing
statistical uncertainty is essential in assessing the aggregate measures. By the
mid-1960s, inventory modeling was technically sophisticated. The easy (and some
not-so-easy) wins were already won. Most of the restrictive assumptions used in
the prior 10 years had been relaxed at least in exploratory research. In the
decades to follow, the technical horizons continued to expand, although the
managerial scope of the theory remained much the same. Inventory theory analyses
rarely merged with strategic management deliberations. Analytic inventory models
usually take as given what strategic thinking views as choices. For example,
product line breadth, a strategic choice but implicitly given in inventory
modeling, influences demand for stocked items.
3. SHOW ME THE DATA By the early 1950s, the theory and practice of inventory
control faced a fundamental issue: The specification of demand and lead-time
processes cannot be done with much precision. It was presumed, however, that
once it was possible to obtain timely historic data on a continuing basis,
sophisticated replenishment formulas could be easily applied by using
appropriate statistical methods. Brown’s Statistical Forecasting for Inventory
Control (1959) was groundbreaking. He suggested specific statistical methods to
use, particularly the approach that he named as exponential smoothing. (In 1956,
Brown presented his ideas at an ORSA conference, and in 1957, Holt wrote an
Office of Naval Research report discussing exponential weighted moving
averages.) In his book, Brown illustrates the calculations with hand-drawn
worksheets. The book appears to encompass both manual systems and automated
calculations; clearly, this was a transitional moment policy that is applicable
(possibly) over an unbounded horizon. An OR model can be implemented in practice
by using repeatedly updated statistical estimates of the parameter values in an
assumed demand probability distribution. In the discussion that follows, I
designate the approach of using point forecasts and observed variation in
forecast error by PFErr. I designate the alternative approach of using an OR
inventory model populated with statistically estimated parameters of an assumed
demand distribution by OREst. (You will find it helpful to jot down the
definitions of these two ad hoc abbreviations.) What we want to explore further
is whether one of the two approaches is significantly more effective than the
other in practice. It is unfortunate that the term forecast has been identified
with the data analysis suggested by Brown and others. To a manager, a forecast
refers to the demand value that actually will be observed; since it is hard to
perfectly anticipate future sales, a point forecast is not to be believed
(unless one has a crystal ball). The output of exponential smoothing and similar
calculations is an estimate of mean future demand, which is a concept—the mean
itself is never actually observed. Likewise, the term forecast error is
interpreted ex post by managers as a mistake, possibly a misjudgment, whereas it
is only one observation from a distribution of forecast errors. The distribution
purportedly reveals inherent uncertainty about future demand and can be used to
determine a value for safety stock. Academic readers may and this niggling over
terminology only mildly amusing, but the misunderstanding of this terminology in
common use has been the downfall of many practitioners. Managers do not grasp
what they are going to get. Brown’s point of view is reflected in today’s
commercial supply-chain software packages: Historical data are utilized to make
point forecasts of future demand. Further, uncertainty in future demand is
formulated by an estimate of the standard deviation of forecast error; lead-time
uncertainty is finessed by some other approximation (I will explain further
below). Practitioners bought into the statistical point of view right away,
whereas most inventory theorists gave it short shrift. The chapter organization
in the book SPP3 lends support for the preceding discussion. Most of SPP3 is
devoted to mathematical models, that is, the results of some 1,600 theorists;
the cited work provides decision-support tools in supply chains. With few
exceptions, these models assume that the demand and lead-time uncertainties are
described completely by known (or postulated) probability distributions with
precisely specified parameter values. SPP3 makes it quite clear in a single long
and comprehensive chapter devoted to forecasting that a supply chain is often
comprised of nonstationary uncertainties. Nowhere is there an explicit
consideration of the impact of statistical noise (from having only finite data
from a nonstationary environment) on the performance of the mathematical models.
Yet dealing with this impact is one of the major challenges facing a
practitioner. with respect to computing power. Brown points out that exponential
smoothing does not require keeping long files of prior demands, viewed as an
advantage at that time but irrelevant, if not legally impossible, today. Brown
expected his readers to know the fundamentals of inventory control. It is not
easy, however, to spot any fully developed mathematical replenishment models in
Brown’s book. But Brown does set forth a stock-replenishment point of view by
posing two central questions in automated inventory control (a) Is it now time
to replenish inventory? (b) What should be the order quantity? His approach to
these questions is rooted in the requirements of practice and not in the
underpinnings of inventory theory. This is evident from the way the questions
are posed—the operative word in (a) is now. We review the implied architecture
below. To make the replenishment-order decision (a) above, assume that
replenishment review occurs at the start of each period. Practical
considerations rise when we attempt to ascertain a SKU’s current stock-status
level, defined as inventory on hand plus inventory due in, less backlog.
Frequently, the stock-status level is hard to calculate, given the organization
of corporate MIS systems—typically the components of stock status are in
different transaction systems. We determine whether the stock-status level gives
enough service protection for demand over lead time plus a review period, taking
into account uncertainty in demand and perhaps lead time. We order now if the
stock-status level does not provide enough service protection; else, we wait for
another period and test again. One of two situations arises. If bona ?de
probability distributions for the uncertain quantities are available, the above
determination about service can be made using a mathematical inventory model. We
directly apply a formula to calculate s (reorder point). But if there is only
historic data about these quantities, then we use the data to forecast demand
over lead time plus a review period, and prospectively take account of
accompanying forecast error so as to hedge against demand and lead-time
uncertainties. The hedge often is called safety stock, and how to calculate it
is the challenge. A similar dichotomy occurs in answering the
replenishment-quantity decision (b) above. If the relevant probability
distributions are known, we calculate S (the order-up-to point), and order
enough to bring the stock status level to S. But if there is only historic data,
the alternative is to forecast what will be sold over a span of time that is
expected to elapse until the next replenishment, possibly adjusting the amount
upward if the stock-status level is exceptionally low. A statistical approach is
a far cry from operations research logic, such as that underlying a classic s S
policy. You can spot the difference immediately by observing that a statistical
approach produces two numbers based on history as of now, whereas an OR model
produces a reorder.
4. IT’S THE DATA’S FAULT With the passage of 50 years, one thing is true by now.
There is no reason to complain anymore about scarcity of data. All businesses in
the United States have loads of sales data. Usually this data is accessible for
at least two years back, and often further back than that (although the earlier
data may not be online). The agony that is felt today reflects that the demand
data are dirty. This is not quite the same as what statisticians call missing
observations and errors of measurement, although dirty data may include both.
Here are some typical examples of data challenges. ? Demand is not level across
a year, so a twelve-week running average of demand fluctuates considerably. ? A
SKU is new and being phased in, or is old, obsolete, uncompetitive and being
phased out; in either case, the span of historical data is limited. ? A SKU was
new last year and introduced without enough supply to meet demand, so last
year’s sales understate potential demand. ? A supplier failed to ship orders and
consequently the SKU was out of stock, so historic sales understate demand for a
span and overstate demand in the weeks immediately after. ? A SKU received
special promotions (maybe under a different SKU number, and with a different set
of promotion dates, depending on the physical location of demand), so weekly
sales data show spikes and subsequent valleys, and these differ by location. ?
Several times last year, demand was unusually large due to the unexpected orders
of a few large customers, so the mean and standard deviation of historic demand
may not be appropriate for inventory control. ? The SKU’s product specifications
changed (color, flavor, package size, quality, etc.), so there is discontinuity
in the pattern of historic demand. ? Customers’ returns are netted out of a
current week’s sales, so raw sales may overstate demand in one time period and
understate it later. ? Some holidays fall on different days and weeks from one
year to the next, so sales data have to be adjusted from one year to the next to
obtain comparability. ? Weather impacts demand, so historic sales re?ect special
conditions that may not recur. ? A competitor introduced a new product that
altered market share, so historic demand declined—the size of the reduction
varies, re?ecting the number of weeks since the new product introduction. ? A
new SKU was introduced and thereby cannibalized the sales of several other SKUs.
This list is long enough to convey why the phrase garbage-in-garbage-out is
often used in discussions about demand forecasting and the distribution of
forecast errors, and why automated replenishment systems commonly under-perform
relative to manager’s expectations. The preceding examples give rise to distinct
classes of data problems. Some of the issues cited are solvable by / 221
suturing data streams; this involves identifying holidays, promotional weeks,
and other special events. But other data issues are fundamental and reflect
business practices—an important example is when a SKU’s life span is less than
two years. Some companies have invested in computer systems that assist in
cleaning up dirty data. (These are almost never off-the-shelf software packages;
the system requires considerable hands-on guidance.) Consider a company that has
an effective MIS system so that the cleaned sales data reasonably state what the
relevant historic demand is for each SKU. This setting makes it easier to
compare alternative architectures for automated inventory control. Note in
passing that systems for cleaning historic data rarely clean previous forecasts.
Hence, automated replenishment logic based on the distribution of historical
forecast error is problematic when data cleaning is implemented. This is another
reason to be circumspect about adopting an automated replenishment architecture
that relies on forecast errors. It is unlikely that in practice there is much
difference between the effectiveness of PFErr and OREst in deciding what should
be the order quantity? Here is the reason. Suppose that the real situation is so
simple that the stationary EOQ formula applies. Then early research established
that EOQ is often a good approximation to the optimal value of S s in an s
S model. Further, the actual value of the objective function is insensitive to
an order quantity that differs considerably from EOQ. In most real situations,
however, there are other factors that influence the order quantity, such as pack
size, discount pricing, transportation minimums and maximums, etc. The classic
lot-size model is formulated to characterize the order decision as a quantity of
goods. An equivalent alternative is to characterize the decision as a time
interval over which the imminent order should last, such as weeks of coverage.
This perspective yields an EOI (economic order interval). Point forecasts of
future demand over EOI can be effective and integrated equally as well into a
PFErr or OREst system. Note that when the EOI is a single period, the
order-quantity decision comes down to how close the replenishment quantity
should be to what was sold last period. An analytic challenge with respect to
order quantity is recognizing in advance when to make the final order for a SKU,
and then choosing how much. It always is clearer in retrospect which order is
responsible for unsalable leftover stock. Our comparison of PFErr and OREst
systems rests primarily then on how to determine is it now time to replenish
inventory? Given the current stock-status level, the question is whether
postponing a reorder for at least another period gives acceptable customer
service over the interval from now until an order placed next period would be
delivered. This is a prospective assessment and essentially uses probabilistic
thinking. PFErr and OREst differ essentially comprised of statistical estimates
of mean demand, mean lead time, variance of forecast error, and variance of lead
time. Typically the statistical estimate for the variance of lead time is
seriously in error. Commercially available supply-chain software that purports
to do these calculations sometimes documents the algorithms. But often the
software architecture makes it practically impossible for users to verify the
underlying logic by testing numeric examples. In simple terms, these systems
operate in real time and do not easily accommodate what if explorations.
Consequently, when unintuitive forecasts and resupply recommendations occur, the
user must take the results on faith. Suppose that the PFErr system is
technically sound. The final step above produces a fraction—for illustrative
purposes, assume that the number is .93, which represents 93% service over the
relevant time slice. If .93 is below management’s targeted service level for
this time span, a replenishment occurs. Otherwise it does not. In any case, the
calculations are repeated with updated forecasts and stock status at the start
of the next period. Practitioners naively assume that .93 (in our illustration)
is an accurate evaluation of expected service. Rarely, if ever, does such a
value give a sufficiently accurate evaluation, and typically it overstates
service performance noticeably. No wonder many companies are dissatisfied with
the service performance of automatic replenishment software. They expect to get
high service at an acceptable level of inventory investment, and actual service
falls short of expectation. To conserve space, I omit a review of all the
underlying assumptions in PFErr that in practice are seriously violated and
contribute to the illusion that a replenishment system will actually obtain a
service level close to what is targeted. The main point in this discussion is
that PFErr does not deliver what it promises. OREst has comparable problems.
When OREst employs historic data to estimate the mean and standard deviation of
a postulated demand distribution, the calculated service measure may well be an
overestimate due to the postulated distribution being inappropriate and the
parameter estimates from data being inaccurate. The underlying mathematical
reason is that the statistical error in an OR inventory model impacts service
asymmetrically and nonlinearly; in other words, the improvement in service from
an overestimate of a SKU’s mean or variability of demand is not the same
magnitude as the degradation in service from an underestimate by the same
amount. In realistic settings the amount of historic demand information is not
sufficient to provide much confidence that the shape of the demand distribution
(which is unknown) is well represented with respect to that part of the assumed
distribution that is most critical, the right tail. The above discussion brings
us to a realization about our objective of answering is PFErr or OREst
significantly more effective in practice? We must posit assumptions about the
demand process that gives rise to the historic data. in detail, if not concept,
as to how to calculate the expected service level associated with not ordering
now. PFErr is implemented in today’s supply-chain software systems as follows. ?
First, stock-status level (or some analogous measure of inventory) is converted
into an ordinate of a standardized normal distribution by subtracting off the
point forecast for demand over lead time plus a review period and dividing by
the variability of historic forecast error (usually, the standard deviation of
past forecast errors, properly scaled using a square root calculation to provide
an interval of lead time plus a review period). A serious complication arises
when lead time is uncertain. We discuss this issue later. ? Second, a
standardized unit normal loss function is referenced to get the expected lost
demand associated with the ordinate. ? Third, this standardized expected loss is
scaled up to original demand units to provide expected lost demand over lead
time plus a review period. ? Finally, expected service is measured as the ratio
of expected lost demand to the comparable point forecast of demand. An
alternative articulation of this process works backward from a target service
level to an implied critical ordinate of the standardized unit normal loss
function, and finally to a reorder level. Looked at this way, all SKUs that have
the same target service level have the same critical ordinate. That common
ordinate value is a consequence of assuming that forecast errors for a SKU are
independently and normally distributed, so that a single inverse function
applies. Missing from this algorithm is a correction that may be required due to
forecast bias. Although mean forecast error need not be close to zero, mean
square error is rarely used in practice. If lead time is uncertain, the above
calculations for variability become more complicated (and often more dif?cult to
implement because historic data for estimating lead-time variation is limited).
If expected lead time is two weeks, but it turns out that an order does not
arrive until four weeks, some customer demand in the third and fourth weeks may
not be satisfied from inventory on hand. Whereas the demand distribution for a
given lead time may be viewed as concentrated around a single modal value,
adding uncertainty in lead time can deform the one period demand distribution
into a composite distribution with several modal values corresponding to a
discrete number of weeks of lead time. If targeted service level is high, then
safety stock has to cover the rightmost possible values for demand. The
composite distribution need not be well approximated by a normal distribution.
You and in practice two different approaches to accommodating uncertain lead
time. One assumes that lead time is fixed at a value safely above its mean (such
as the 90th percentile). This approach impacts the forecast demand over lead
time, but does not affect the value expressing demand forecast uncertainty. The
other uses a standard formula. We seem to have a chicken-or-the-egg
conundrum. This logic puzzle is a familiar one to statisticians, who examine
analogous issues in data-driven analyses. When is it prudent to assume an
underlying normal distribution when making a decision based on limited data?
Determining the appropriate value for safety stock is a serious challenge for
practitioners because in today’s business environment, management wants its
supply chain to perform at an extremely high level of customer service. At such
high service levels, this determination is the hardest to make—the accuracy of
data-driven estimates is poor.
5. SIMULATE TO CALIBRATE There are enlightened practitioners who do have a way
of dealing with this issue. I designate this process as model calibration. These
practitioners apply what is called retrospective simulation to a collection
(possibly a stratified random sample) of SKUs. They utilize historic data rather
than estimated distributions in a simulation of SKU replenishment, where the
target service level is made identical for all SKUs in the collection. This
yields a simulated measure of service, and is compared to the target service
level that is used in the PFErr or implied by the OREst formula. As an
illustration, a data-driven simulation that targets .98 service overall may
result in obtaining .93 service overall from the retrospective simulations. In a
subsequent step, practitioners adjust the targeted service amount to get the
desired actual service. It is now feasible to perform this calibration process
using spreadsheet software on a fast PC. The consequence of avoiding an assumed
probability distribution for demand, such as a normal distribution, becomes
clear once a retrospective simulation is performed. A mathematical distribution
like the normal has a positive probability of exceeding any given value for the
ordinate. Hence, no matter what value is used for safety stock, there is a
chance that some demand goes unfilled. In contrast, in a retrospective
simulation using finite historic data, there is a value of safety stock for each
particular SKU that gives 100% service. As a consequence, as the target service
level is raised in the calibration process, more and more SKUs reach 100%
service. But other SKUs may not be close to the desired service level, and
therefore the overall service level may not be sufficiently high. Raising the
target level further provides no improvement for those SKUs already at 100%
service, but adds to their inventory. The irony is that all unfilled demand in a
retrospective simulation was actually filled historically, since these data
represent actual sales, not potential demand. Consider what happens when
calibration becomes an integral part of the systems design process. The
calibration mechanics enable a practitioner to ascertain trade-offs between
simulated service overall, inventory investment overall, as well as other
ancillary measures. Hence, using calibration, it is possible to make direct
comparisons of alternative replenishment approaches, such as PFErr and / 223
OREst, by looking at an aggregation of results from simulating a multitude of
SKUs. The calibration process is a cross-sectional variation on what
statisticians call bootstrap analysis. The same replenishment architecture (for
example, PFErr) and target service level are used for a sample of SKUs, where
each SKU has its own historic demand. Calibration is best practice in system
design, but it is not standard practice by any means, and not easily
accomplished with today’s supply-chain software. For that reason, the typical
PFErr implementation of an automatic inventory replenishment system is fraught
with worse-than expected outcomes, and consequent overrides of the underlying
replenishment logic.
6. RESEARCH OPPORTUNITIES Unlike areas of OR where current research is
technically (mathematically) far ahead of practice, current inventory research
investigates issues of central importance to practice. Therefore, to the extent
that such research reveals important insights about supply-chain architecture,
the research expands knowledge in a way that deepens understanding of how these
processes behave. Further, the enhanced understanding builds on the established
base of prior knowledge. Recognizing this, it also must be said that incremental
mathematical research is not likely to enhance practice. The reason is that
mathematical inventory research is blind to all the data issues discussed above
and far removed from the entrenched software that now drives supply-chain
systems. The references at the end of this article contain a few examples of
research efforts that have dealt with statistical issues. For the most part,
these studies assume that demand occurs in a stationary environment, and that
lead time is a known value. I am aware of other relevant articles with less
restrictive assumptions, but I have not included them in the references. It
seems promising to explore a new research avenue that takes as its starting
point a finite history of demand data for a family of SKUs. Unlike the
publications of 40 to 50 years ago, the new research should assume plentiful,
recognizably dirty, demand data. Lead-time uncertainty needs to be examined from
a data perspective as well. The objective should be stocking and replenishment
logic that is driven by such data, and operating in the context of a supply
chain. The overall research objective is to and sufficiently robust approaches
that produce reliable trade-off assessments between service and other aspects of
inventory management. The theory should rest on the analysis of a family of
SKUs, although the replenishment strategy itself may be single-item focused. The
family of SKUs should be defined in a way that its members are recognizable by
reference, at least, to historic demand over a finite history. The theory should
encompass data-oriented tests as to when the actual replenishment rules may be
reliably applied. The theory should provide measures of accuracy statistics
problem growing out of the research program of RAND’s Logistics Department. The
central theme of the thesis was the design of top-level managerial systems for
controlling lower echelons in a supply chain. The research applied statistical
principles in fashioning a control system comprised of s S inventory policies.
Both Ken and Bob gave generously of their time in challenging my thinking about
these issues. In 1962 the thesis was published in the Operations Research
Society’s research monograph series. I never had the opportunity to collaborate
with William Cooper at Carnegie. But Bill’s dedication to doctoral students was
already legendary before the 1960s, and Bill provided a role model for me early
on: I am forever grateful. Robert Fetter, who was on the Sloan School faculty
when I was studying at MIT, was interested in practical applications of
operations research, and we had many lengthy and wide-ranging conversations
about advancements in operations research. It was Bob who introduced me to
McKinsey & Co. while I was on the Stanford faculty. Then in 1967, Bob became a
close colleague again when I joined him on the faculty at Yale. My working
relationship with McKinsey & Company began in 1960 in San Francisco. Shortly
thereafter, David Hertz became a partner at McKinsey in New York. I first met
David when he served as editor of the Operations Research Society research
monograph series. David kindly tutored me on the skills that are required to
take university research and make it relevant to business. Over four decades, I
have had innumerable opportunities to participate with McKinsey client service
teams in designing and implementing what is now called supply chain logistics
systems. In the early 1960s it already was evident that software packages
developed by leading computer companies such as IBM were going to be critically
important in getting businesses to adopt operations research inventory models.
Whereas commercial applications of mathematical programming algorithms seemed,
from the 1960s onward, to develop in concert with advances in computing
technology, inventory modeling did not fare so well. This article takes a
retrospective look as to why. ACKNOWLEDGMENTS I am grateful for suggestions made
by my colleagues Jack Evans, Geraldo Ferrer, Kyle Catanni, Ann Marucheck,
Jayashanker Swaminathan, and Clay Whybark, and the editors Lawrence Wein and
Frederic Murphy. with respect to estimates of future system wide behavior
comparable to measures of estimation error commonplace in standard statistics.
The theory should recognize that most demand environments change over time,
usually gradually but sometimes abruptly. It should assume that on occasion
human intervention is called for, and the theory should assist in signaling when
such occasions are warranted. Finally, it should take notice of rules of thumb
that may, at ?rst glance, seem too simple (such as replenishment rules based on
weeks of supply, or kth largest prior week of demand), but that may function
well in a nonstationary environment that contains both demand and lead-time
uncertainties.
PERSONAL RECOLLECTIONS My career interest in inventory and statistical processes
was fostered in the beginning by eight scholars, some being mentors, some
colleagues, a few both, and all good friends. The pivotal circumstances occurred
in four places, Stanford University, RAND Corporation, Massachusetts Institute
of Technology, and McKinsey & Co. Kenneth Arrow and Gerald Lieberman at Stanford
wisely counseled me as an undergraduate during the early 1950s about exciting
career opportunities in the emerging field of operations research. They opened
my eyes about studying economics and mathematical statistics. Until then I never
realized that applied mathematics could provide a livelihood. Jerry guided me in
learning about important business applications of statistical analyses. Ken
assisted me in spending summers at RAND, starting in 1953, and thereby first
sparked my interest in inventory modeling. In 1957, I became a junior colleague
of Jerry’s in the Department of Industrial Engineering at Stanford. I had an
office in Serra House, in proximity to Ken and his preeminent co-authors. They
were actively collaborating on inventory theory, and writing papers that
comprised several Stanford University Press research monographs. In the early
l960s, Arthur (Pete) Veinott, Jr. also joined the faculty in Stanford’s
Department of Industrial Engineering. Pete and I shared a deep interest in
inventory and production algorithms, resulting in a dialog that lasted for
years. Along the way, we developed a practical algorithm to compute s S policies
for discrete probability distributions. Pete’s broad perspective on the issues
central to inventory and production modeling was inspiring. In 1955–1957, while
doing doctoral studies in mathematical economics at MIT, I was fortunate to be
assigned as a research assistant to Thomson Whitin, then on the faculty of the
Sloan School of Management. Tom was preparing the second edition of his
monograph on inventory processes, and posed the core problem that led to our
collaboration on the dynamic economic lot-size model. Robert Solow, in guiding
my doctoral studies, counseled me as to the importance of broadly examining
alternative analytic perspectives in researching a scholarly area. I selected
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