Ann Oper Res DOI 10.1007/s10479-016-2157-9 S.I. : AVI-ITZHAK-SOBEL:PROBABILITY

Inventory turns and finite-horizon Little’s Laws Benjamin Melamed1 · Rudolf Leuschner2 · Weiwei Chen2 · Dale S. Rogers3 · Min Cao4

© Springer Science+Business Media New York 2016

Abstract Over the past 30 years managers have increasingly focused on improving inventory management both within their own firms and across the supply chain. To this end, inventory turns metrics have been adopted as a popular tool for measuring flow velocity through the inventory and the efficiency of inventory-related asset utilization. There are multiple computational methods thereof currently used by practitioners, but each has some flaws. In particular, consensus is lacking as to (1) how to measure inventory flow and (2) how to measure the inventory level. Clearly, high-quality accounting information is essential for an accurate assessment of all inventory performance metrics. Unfortunately, when comparing efficiencies across firms or diagnosing and correcting intra-firm inefficiencies, choices of specific accounting rules and distortions between fair market values and corresponding accounting book values can lead to potentially misleading results. This paper presents finite-horizon

B

Benjamin Melamed [email protected] Rudolf Leuschner [email protected] Weiwei Chen [email protected] Dale S. Rogers [email protected] Min Cao [email protected]

1

Department of Supply Chain Management, Rutgers Business School – Newark and New Brunswick, Rutgers University, 100 Rockafeller Rd., Piscataway, NJ 08854, USA

2

Department of Supply Chain Management, Rutgers Business School – Newark and New Brunswick, Rutgers University, 1 Washington Park, Newark, NJ 07102, USA

3

Department of Supply Chain Management, Carey School of Business, Arizona State University, Main Campus, P.O. Box 874706, Tempe, AZ 85287, USA

4

Department of Accounting and Information Systems, Rutgers Business School – Newark and New Brunswick, Rutgers University, 100 Rockafeller Rd., Piscataway, NJ 08854, USA

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versions of Little’s Law and elucidates their connection to inventory turns and restricted sojourn times in inventory, defined as the portion of the full sojourn time that falls within a prescribed time period. In particular, it explains when the reciprocal of sample inventory turns coincides with the sample average of restricted sojourn times. As such, the paper provides a unified prescriptive model for correctly relating inventory turns to sojourn times through an inventory system, thereby facilitating more accurate intra-firm and cross-firm comparisons. Keywords Finite-horizon Little’s Law · Infinite-horizon Little’s Law · Inventory · Inventory turns · Little’s Law · Supply chain · Supply chain financial management

1 Introduction The setting of this paper is a large class of flow systems with a well-defined boundary separating the system from its exogenous environment. A flow of mobile entities (entities, for short) traverses the system, where entities are identifiable parcels of the flow. Thus, entities enter the system from the environment, spend some time in the system and then exit back into the environment. In its full generality, the model admits both discrete entities which arrive and depart “in lockstep” (to be referred to as synchronous entities), or entities that arrive and depart “gradually” (to be referred to as asynchronous entities), or mixtures thereof; formal definitions are deferred until Sect. 4. Furthermore, entity flows into and out of the system (arrivals and departures) can be in discrete mode or continuous mode. In the limiting case where entities become “infinitesimally small”, the system becomes a continuous-flow system, where entities are akin to “molecules” in a fluid flow and lose their identities in the sense of being uncountably infinite. In all cases, entities are never trapped in the system. At this point we need not specify the exact nature of the entities. In fact, they can correspond to physical products, their monetary values or some other numerical attribute, or commodity flows in a pipeline or chemical plant. The abstracted description above encompasses many systems, including inventories, where entities arrive as replenishment, spend some time “on the shelf” until shipped to customers. Furthermore, the generality of such systems stems from the fact that we leave unspecified the nature of the arrival stream, the order of entity departure, and what the entities do in the system. For example, entities can arrive singly or in batches, discretely or continuously, and can depart first-in-first-out (FIFO), last-in-first-out (LIFO) or in any permutation of the arrival order. We merely observe the system over time and record the entity level over time. We now focus on inventory systems as a specialized setting, though the treatment to follow applies to general flow systems. The inventory turnover ratio metric (abbreviated as inventory turns) is a member of a class of metrics that aim to measure an inventory’s operational and financial efficiency over a given period. It is loosely defined as the ratio of some notion of “flow in the inventory system” to some notion of “average inventory level”, both over the same period. As such it aims to capture flow velocity through the inventory and the efficiency of inventory-related asset utilization. A companion metric closely related to inventory turns is a notion of “average sojourn time in inventory” (say, a Days Inventory Outstanding metric, abbreviated as DIO), which captures the notion of “average time spent by entities in the inventory system”. In this paper we use the terms “DIO” and “average sojourn times” interchangeably, where the former is commonly used by practitioners and the latter by mathematicians. A similar comment applies to the time units used to assess inventory turns. Thus, a low inventory turns metric (or high DIO metric) can be evidence of

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slow-moving inventory and tends to be an indication of poor inventory management resulting in high inventory carrying costs. Conversely, high inventory turns (and low DIO) translate into low inventory carrying costs, but may yield poor customer service. For most executives, inventory turnover is a popular key performance indicator (KPI), and managers are incentivized to improve this metric. Analysts can estimate and benchmark inventory turns based on publicly available data. Although it is a widely used and accepted metric, there is no standardized computation method for it both inside and outside the firm. This state of affairs gives rise to a number of risks: 1. Different measurement and computational procedures can easily render the resultant metrics incomparable. In other words, this may give rise to an “apples and oranges” comparison. For example, most managers compute this metric as the ratio of cost of goods sold (COGS) to average inventory, but some use the ratio of sales to average inventory, and others may use the ratio of purchases to average inventory. Each of these versions yields a different result which, when used without a deep understanding of the underlying intricacies, can cause managers to misgauge inventory management efficiency. Furthermore, the reciprocal of the inventory turn metric is often used to compute the average sojourn times of inventory items (Weygandt et al. 2012, p. 289), even though, as we shall see, this is not an identity, but merely an approximation and often of questionable quality. 2. Too much leeway in computing these metrics may give rise to biased and perhaps selfserving outcomes. In particular, financial reports that cite these metrics as KPIs may use supply chain and accounting shenanigans to yield outcomes which are unduly favorable to the firm or its managers. The risks above motivate the need to base the inventory turns concept on firmer foundations than the aforementioned vague operational definition. We propose to do that by connecting inventory turns to some theoretical concepts and results from the domain of Queueing Theory. These theoretical concepts are not only mathematically crisp, but can also serve to clarify the components of inventory turns and suggest robust ways of using empirical data to compute DIO metrics from inventory turns. In some flow systems, inventory turns metrics and DIO metrics are related by a wellknown identity, called Little’s Law. Little’s Law is very general in scope, including the classic discrete entity setting (cf. Little and Graves 2008 for a survey), as well as fluid flow variants where entities become infinitesimally small (Konstantopoulos et al. 1996; Wiendahl and Breithaupt 2000). This paper addresses the concept of inventory turns and its relation to sojourn times in the framework of Little’s Law adapted to finite time intervals. Such a law will be referred to as a Finite-Horizon Little’s Law (FHLL), in contrast to a traditional variant of Little’s Law, to be referred to as Infinite-Horizon Little’s Law (IHLL). An IHLL is generally formulated in terms of long-run averages or steady-state settings, the exception being finite intervals that start and end with an empty system (cf. Little and Graves 2008 and references therein). However, business decisions are perforce based on finite time horizons and short-term statistics and metrics. A typical example is an accounting reporting period that can be annual or quarterly, or even as short as a month or less for internal purposes. Such time intervals may be too short to support adequate modeling via an IHLL. It is then of interest to identify FHLL versions which are guaranteed exact over finite intervals for a broad class of flow systems with any starting and ending inventory levels, and to elucidate their proper interpretation and usage. This paper makes several contributions. First, we exhibit several FHLL versions, and describe business reasons for their usage. The results include discrete and continuous flows of entities or attributes thereof. Second, we use the developed formulas to point out account-

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ing practices that lead to potential distortions due to grossly approximate computations of inventory turns and incorrect computation of average sojourn times as reciprocals of inventory turns. The impact of inexact computations on metric values is then illustrated through a simple example. The remainder of this paper is divided into the following sections. Section 2 reviews the literature on inventory turns. Section 3 reviews the classic IHLL where entities arrive as single units, and presents an FHLL counterpart. Section 4 generalizes Sect. 3 by replacing a single entity with a numerical attribute thereof, and then extends the results to admit both discrete and continuous flows. Section 5 addresses FHLL for the important special case of monetary flows. Finally, Sect. 6 concludes the paper.

2 Literature review of inventory turns The management of inventory has received attention in the literature since the 1950s. The two primary roles of inventory are (1) to provide a buffer for demand (e.g., production and sales); and (2) to connect various value-adding production and distribution units (e.g., raw material storage, manufacturing, and finished goods storage) of the firm (cf. Chikán 2009, 2011). In addition, inventory permits the decoupling of nodes in a network of customers and suppliers. From an operational perspective, management needs the quantity and timing of replenishment (production or purchase), while from a financial perspective, the goal is to minimize costs and maximize net profit by balancing setup, holding, and stockout costs (Simchi-Levi et al. 2008; Katehakis and Smit 2012; Shi et al. 2013, 2014). Inventories have a significant impact on the financial performance of the firm. Because a significant part of the firm’s assets is often invested in inventory, analysts pay close attention to the efficiency of inventory management. As management continuously seeks to reduce inventory levels, the markets reward it on productivity gains (Eroglu and Hofer 2011; Gaur et al. 2005; Mishra et al. 2013). There has been considerable interest in evaluating how levels of inventory turns have changed over time, their impact on operational improvements and on operational and financial performance (Balakrishnan et al. 1996; Billesbach and Hayen 1994; Chang and Lee 1995; Hendricks and Singhal 1997; Huson and Nanda 1995). Four studies track inventory turns at the firm and industry levels. In one study, the researchers determine whether inventory turns have decreased over time, using aggregate industry-level data for 20 industrial sectors over a 33-year period (1961–1994) (Rajagopalan and Malhotra 2001). Another study tracks inventory turns in the retail industry by using financial data for 311 publicly listed retail firms for the years 1987–2000 to estimate the correlation of inventory turns with gross margin, capital intensity, and other financial metrics (Gaur et al. 2005). Kolias et al. (2011) later replicates this study with data from Greece. Finally, higher inventory turns seem to correlate with better financial performance in retail firms over the period 1985–2010 (Alan et al. 2014). A number of operations management studies have used inventory turns as a proxy for management effectiveness (Demeter and Matyusz 2011; Eroglu and Hofer 2011; Zeng and Hayya 1999). In another study, Schonberger (2003) calls inventory turns an “indication of effort” when evaluating firm performance. An alternative to standard inventory turns are turnover curves that identify economies of scale in inventory holding by drawing on the relationship between sales and inventories at multiple stocking locations of a firm (Ballou 1981, 2000, 2005).

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The aforementioned studies assumed that the traditional method of computing inventory turns via publicly available information is valid. However, the thrust of this paper is to caution against uncritical computation methodologies.

3 Inventory turns and simple variants of Little’s Laws In this section we consider flow systems with discrete (countable) entities, which traverse the system as distinct units. Thus, their arrival and departure times can be measured, as can their number (or some associated magnitude) in the system at any given time. The total time an entity spends in the system (called sojourn time) is computed as the difference between the entity’s departure time from the system and arrival time in it, while for a given time interval, the corresponding restricted sojourn time is the portion of the sojourn time that intersects that interval. We next present the simplest variants of IHLL and FHLL.

3.1 Simple IHLL Let l(t) be an integrable function over an interval [a, b], finite or infinite. Then its associated b 1 time average is given by b−a a l(t) dt, provided it exists. In this section, l(t) is a realization (sample path) of the number of entities in the system at time t, and the time average is obtained as the limit with a = 0 and b tending to infinity. A traditional IHLL assumes that the underlying stochastic process (random number of entities in the system over time) approaches (or is in) steady state. Under mild regularity conditions (always assumed to hold in practice), Little’s Law in traditional notation asserts the long-run (equilibrium) relation L = λ × W. (1) Here, L is the equilibrium mean number of entities in the system (also the long-run time average of the number of entities in the system over an infinite time horizon), λ is the equilibrium mean arrival rate of entities to the system, and due to steady state, it equals the equilibrium mean departure rate of entities from the system (mean number of arriving entities or departing entities per unit time), and W is the equilibrium mean sojourn time of entities flowing through the system (also long-run average of sojourn times). The nice feature of Eq. (1) is that if we know the value of any two of its symbols, we can easily compute the value of the remaining one. This is particularly handy when only two of the symbols can be practically measured, and the remaining one cannot. To connect this IHLL to inventory turns, rewrite Eq. (1) as 1 λ = W L

(2)

Thus, in inventory context, the right-hand side of Eq. (2) can be interpreted as an inventory turns metric, since the numerator is the expected flow rate through the inventory and the denominator is the equilibrium mean inventory level. Intuitively, this metric provides the multiple of “mean inventory size” which “moves through the inventory per time unit” (a reporting period can serve as a “time unit”). Equation (2) states that W1 can be interpreted as inventory turns, so that the long-run mean sojourn time of inventory items through the system is the reciprocal of the inventory turns metric. This suggests that the term “inventory turns” is a bit of a misnomer as it would be more accurate to call it “inventory turns rate”.

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3.2 Simple FHLL counterpart The FHLL counterpart of the IHLL of Sect. 3.1 is formulated in terms of realizations of the  corresponding random variables underlying Eq. (1) over a finite time horizon, t1 , t2 with t1 < t2 . In particular, recall that such a finite time horizon may be an accounting reporting   period. In the remainder of this paper and unless otherwise specified, the interval t1 ,t2 is fixed,  and  all quantities are deterministic (typically, realizations of random variables). For t∈ t1 ,t2 , let lP(t) be the total number of entities present in  thesystem at time t, and let wP(i) be the sojourn time of entity i, restricted to the interval t1 ,t2 . Here, lP(t) is assumed to be a bounded integer-valued step function with unity jumps, where up and down jumps signal entity arrivals and departures, respectively. Define the following sets of entities.   • E A,D : set of all entities that arrived at the system and departed from it during t1 ,t2 .   • E A,ND : set of all  entities that arrived at the system during t1 ,t2 and did not depart from it during t1 ,t2 , but excluding entities that arrived at t2 .   • ENA,D : setof all entities that did not arrive at the system during t1 ,t2 and departed from it during t1 ,t2 , but excluding entities that departed at t1 . • ENA,ND : set of all entities that arrived at the system before t1 and departed  it after t2 .  from • EA = EA,D ∪ EA,ND : set of all entities that arrived at the system during t1 ,t2 .  • ED = EA,D ∪ ENA,D : set of all entities that departed from the system during t1 ,t2 . • EP = E A,D ∪ EA,ND ∪ ENA,D ∪ ENA,ND : set of all entities that were present in the system during t1 ,t2 . In a similar vein, we say that an entity is present in the system during the interval t1 ,t2 , if it is a member of EP . We assume that each of the sets above is finite and let the cardinality of set S be denoted by |S|. Since EP is a union of disjoint sets, one has           E  = E  + E      P A,D A,ND + ENA,D + ENA,ND       E  = E  + E   A   A,D   A,ND  E  = E  + E  D

A,D

NA,D

    Denoting kP = |EP | and noting that |ENA,D NA,ND = lP t1 , one has         kP = lP t1 + |EA,D+|EA,ND  = lP t1 + EA .  | + E

(3)

To avoid trivialities stemming from null sets of present entities, we shall henceforth adopt the convention that ratios of zeroes evaluate to zero. The following proposition states an FHLL. 1 Consider a flow system with discrete entities. Then, for any time interval Proposition  t1 ,t2 and associated entity set EP , 1 t2 − t1



t2

t1

kP l P(t) dt = × t2 − t1

k P

i=1 w P(i)

Proof We first establish the equality  t

kP 2 l P(t) dt = w P(i). t1

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i=1

kP

(4)

(5)

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Fig. 1 Graphical illustration of Eq. (5)

The equality holds because each side of Eq. (5) is just the area under the curve lP(t) over the   interval t1 ,t2 , but each side represents the same area in a different way. Intuitively, the lefthand side expresses the integral via Riemann sums of “vertical strips”, while the right-hand side does the same via “horizontal strips”, as illustrated graphically in Fig. 1. Equation (4) follows by dividing both sides of Eq. (5) by t2 − t1 and multiplying and dividing its right-hand side by kP .   ˆ (on left) to the product of a sample Note that Eq. (4) relates a sample time average, L, ˆ rate, λ (“rate of entity presence”), and arithmetic average, Wˆ , of restricted  a sample   sojourn  times (on right), all over the interval t1 ,t2 . In other words, for any time interval t1 ,t2 , one has the FHLL identity Lˆ = λˆ × Wˆ , (6) which is analogous to the IHLL identity of Eq. (1). All quantities in Eq. (6) can be exactly computed in principle from empirical observations. Note, however, that while Eqs. (1) and (6) are analogous and have the same algebraic structure and related semantics, their settings are fundamentally different. The former (IHLL) is formulated in terms of theoretical expectations, which are rarely known in practice, and therefore, must be estimated, whereas the latter (FHLL) is formulated in terms of realizations, which can in principle be precisely measured in the field. Another important difference is that an IHLL assumes an equilibrium setting, so Eq. (1) is typically a long-run identity, often with an unbounded time horizon. In contrast, an  FHLL holds for any interval t1 ,t2 , regardless of the underlying probability law. This is useful from an accounting viewpoint, since if a sequence of such intervals represents reporting periods, an FHLL readily allows inventory turns and times in inventory to be associated with distinct reporting periods. In particular, the inventory turns version of Eq. (6) per period is 1 λˆ = . Wˆ Lˆ

(7)

  Let ES be any subset of entities present in the system during the interval t1 ,t2 . In the sequel, we shall systematically carry over a corresponding subscript to all its associated quantities, such as lS(t), kS , wS(i), etc. A useful extension of the simple FHLL above is provided by the following. Corollary   1 Let ES ⊆ EP be any subset of entities present in the system during the interval t1 ,t2 . Then, Eq. (4) holds for ES , provided lP(t), kP , and wP(i) are replaced by their counterparts associated with ES .

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Proof Follows from the observation that Eq. (5) still holds after the stated replacements.   Note carefully that an FHLL generally has separate formulas for arriving and departing entity streams (corresponding to the entity sets EA and ED , respectively). In contrast, its IHLL counterpart formula is unique, since the arrival rate and the departure rate are equal in equilibrium, which is not generally the case in the FHLL version. However, a single FHLL obtains, for example, in the special case where lP(t1 ) = lP(t2 ) = 0, since then EP = EA = ED and all restricted sojourn times are equal to their unrestricted counterparts.

4 Attribute-oriented FHLLs A simple IHLL deals with discrete entities where each entity arrival increments the system’s entity counts by one and each departure decrements it by one, so the system level is integervalued. However, discrete flows can be readily generalized from entity counts to attribute magnitudes associated with each entity. For example, corporate finance and accounting are more concerned with the monetary value of entities flowing through inventories than the underlying entities per se. In other cases, we may be interested in the flow of product unit footprints or volumes rather than the underlying product units. Moreover, in many cases we need to model entities which do not move in lockstep, but rather arrive or depart gradually over time rather than all at once. For example, consider arrivals of palette entities comprised of batches of bottles and demand sizes consisting of bottle lots possibly smaller than a full palette. In a similar vein, consider entities consisting of bulk material which arrives by tanker truck or pipeline, is transferred continuously to inventory on a conveyor belt or through a pipe, and is dispensed in any quantity to arriving demands. Thus, it is of interest to extend the FHLL of Sect. 3.2 to include such cases. Formally, let each entity i be mapped by a positive attribute function, a, to a real-valued magnitude a(i)>0. We interpret a(i) as the magnitude of a parcel of the total inflow injected into the system by entity i over its total sojourn in the system. Thus, in an attribute-oriented flow system, we have entity flows which are always discrete, and attribute flows which may be discrete or continuous. Let a(i,t) ≥ 0 be the attribute magnitude associated with entity i at time t. Thus, a(i,t) > 0 is the contribution of entity i to the total system level of this attribute at time t, thereby signaling that entity i is then present in the system. The definition of a(i,t) should capture the behavior of an entity traversing the system; in particular, its total of (positive) increments cannot exceed a(i) and similarly for the total of its absolute decrements. Next, we the definition  generalize    of a present entity as follows: we say that entity i is present in t1 ,t2 if there is t∈ t1 ,t2 , such that a(i,t)>0. As before, we let EP as the set of all   entities present during t1 ,t2 and denote kP = EP . However, we adapt the level function, lP , to attribute-oriented settings by defining lP(t) =

k P i=1

a(i,t)

(8)

  to be the total of attribute levels of entities present in the system at time t∈ t1 ,t2 . In the sequel, we distinguish between discrete entities whose attributes arrive and depart in lockstep and those that do so gradually. Thus, an entity is synchronous if a(i,t)>0 implies a(i,t) = a(i) independently of t; otherwise, the entity is asynchronous. Note that Sect. 3.1 just treats the special case of synchronous entities with a(i) = 1.

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We now proceed to exhibit the forms assumed by FHLL variants in attribute-oriented settings. To this end, we generalize the model setting of Sect. 3.2 in three directions. 1. Synchronous attribute-oriented setting. We keep the notion of entities flowing through a system, but with each entity we associate an attribute of a positive magnitude (referred to as a parcel in contradistinction to the underlying entity), such as monetary value (e.g., in dollars) or physical characteristics (e.g., footprint or volume). Here, parcels move synchronously. 2. Asynchronous attribute-oriented setting. We keep the notion of entities flowing through a system as before except that parcels move asynchronously. Here, parcels can arrive and depart gradually, either discretely or continuously. 3. Continuous flow setting. The system is a fluid-flow system where entities lose their identity and the notion of attribute is obviated. Here all flows are continuous.

4.1 Synchronous attribute-oriented setting In this subsection we consider flow systems with synchronous entities. Here, lP(t) is assumed to be a bounded step function with arbitrary jumps, where up and down jumps are of magnitudes a(i), signaling entity arrivals and departures, respectively. For each i = 1, 2, . . . kP , define wP(i) to be the restricted sojourn time of present entity i, in the interval t1 ,t2 . The following proposition establishes two generalized FHLL variants, dubbed the EntityAveraged FHLL and Attribute-Averaged FHLL, respectively. Proposition 2 Consider a flow system with synchronous entities. (a) An Entity-Averaged FHLL is given by k P  t 2 w (i) a(i) 1 kP lP(t) dt = × i=1 P t2 − t1 t1 t2 − t1 kP

(9)

(b) An Attribute-Averaged FHLL is given by k P k P  t 2 wP(i) a(i) 1 j=1 a( j) lP(t) dt = × i=1 k P t2 − t1 t1 t2 − t1 a( j)

(10)

j=1

Proof Using an argument similar to that in Proposition 1, we establish the equality  t

kP 2 lP(t) dt = wP(i) a(i). t1

i=1

(11)

The results follow by dividing both sides of Eq. (11) by t2 − t1 , and then multiplying and kP a( j) to obtain dividing the resultant right-hand sides by kP to obtain Eq. (9), and by j=1 Eq. (10).   2 Let ES ⊆EP be any subset of entities present in the system during the interval  Corollary t1 ,t2 . Then, Eqs. (9) and (10) hold for ES , provided lP(t), kP , a( j) and wP(i) are replaced by their counterparts associated with ES . Proof Follows from the observation that Eq. (11) still holds after the stated replacements.   Note that Proposition 2 is a proper generalization of Proposition 1, since setting a(i) = 1 k P for all i = 1, 2, . . . , kP implies j=1 a( j) = kP , and consequently, both Eqs. (9) and (10)

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reduce to Eq. (4). Note further that Eqs. (9) and (10) have the respective structures Lˆ = λˆ e ×Wˆ e and Lˆ = λˆ a ×Wˆ a ; both relate a sample time average, Lˆ (on left) to a product (on right) of a sample rate (λˆ e and λˆ a , respectively) and a sample average of restricted sojourn times (Wˆ e and Wˆ a , respectively). In both cases, the term wP(i)a(i) is the sum of the common restricted sojourn time, wP(i), experienced by the corresponding a(i) attribute units, and as such their sum represents the total restricted sojourn times experienced by all present entities over t1 ,t2 . However, the sums wP(i)a(i) are averaged differently: in Eq. (9), via entity averaging over the total number of present entities, kP , and in Eq. (10), via attribute averaging over the k P a( j). Since all entities are synchronous, corresponding total magnitude of attributes, j=1 the former case can be validly interpreted as average restricted sojourn time per entity, while the latter can be validly interpreted as average restricted sojourn time per present attribute unit. Furthermore, defining the total present attributes, αP , by αP =

|EP | j=1

  |EA | a( j) = lP t1 + a( j) j=1

(12)

constitutes a proper generalization of kP in Eq. (3). The inventory turns versions of Eqs. (9) and (10) are, respectively, λˆ e 1 = Wˆ e Lˆ λˆ a 1 = ˆ Wa Lˆ

(13) (14)

Clearly, the inventory turns variants on the right-hand sides of Eqs. (13) and (14) are generally different.

4.2 Asynchronous attribute-oriented setting In this subsection we consider flow systems with the more complex case of asynchronous entities. Since synchronous entities are just a special case of asynchronous ones, the treatment of the latter will be a proper generalization of the former. To begin with, we note that distinct portions of an asynchronous entity’s attribute may experience different sojourn times. Therefore, the definition of an entity sojourn time is not clear cut, since the natural definition of the time elapsed from the moment the inflow of a(i) began until its outflow ended is not useful in many key flow systems (for example when we have monetary flows). Consequently, we only define it via attribute averaging as the ratio of the weighted sum of attribute parcels that experience the same sojourn time (weighted by such parcel magnitudes) divided by the total magnitude of attribute parcels that experienced some sojourn time (present parcels). Before proving the main result, we introduce some preliminaries that generalize the synchronous case to the asynchronous one. Let l(t) be a nonnegative integrable bounded real-valued function (say, of time) over t1 ,t2 . Define the function  t 2 (15) wl(x) = 1[x,∞)(l(t)) dt, x ≥ 0, t1

where 1S(y) is the indicator function of set S. In attribute-oriented inventory context, l(t) plays the role of a general level function lP(t), x plays the role of an inventory level, and wl(x) stands for the sojourn time “experienced” by an infinitesimal attribute  parcel at level x. Thus, wl(x) maps each argument x to the time duration in the interval t1 ,t2 during which

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the function l(t) was at or exceeded level x. Following this reasoning, we conclude that  ∞ (16) wl(x) d x 0

stands for the total restricted sojourn times in the system, weighted by the corresponding attribute parcels that experienced it. Accordingly, setting l = lP in Eq. (16) gives rise to a proper kP asynchronous-case generalization of their synchronous-case counterparts, i=1 wP(i) a(i). Next, for i = 1, . . . , kP , let α(i) denote the total magnitude of attribute parcels of entity i   that arrived at the system (discretely or continuously) during the interval t1 ,t2 . Generalizing the reasoning of Eq. (12), we define the total present attributes in the asynchronous case by   |EA | α( j). (17) αP = lP t1 + j=1

  Note that lP t1 is the total magnitude of attribute parcels of all entities present in the system |EA | α( j) is the total magnitude of attribute parcels that arrived at the at time t1 , whereas j=1 system after t1 up until time t2 ; however, recall that entities that arrived at t2 are excluded from Eq. (17). We are now in a position to state a generalized FHLL for asynchronous entities. The following proposition provides only the Attribute-Averaged Little’s Law, since as argued before, the Entity-Averaged counterpart does not support adequate semantics. Proposition 3 Consider a flow system with asynchronous entities. Then, an AttributeAveraged FHLL is given by  t2   |EA |  t lP t1 + j=1 α( j) 2 t wlP (x) d x 1 lP(t) dt = ×  1 (18) |EA | t2 − t1 t1 t2 − t1 l t + α( j) P

Proof We first establish the equality   t 2 lP(t) dt = t1

0

∞

wl (x) d x. P

1

j=1

(19)

The proof of Eq. (19) follows by observing that the integral on the left expresses the integral via Riemann sums of “vertical strips”, and the right side via “horizontal strips”. A graphical proof demonstrates that the graph of the function wl can be obtained from the graph of the P function lP by rotating the latter of 90◦ counterclockwise and eliminating “blank space”; see Fig. 2 for an illustration. Furthermore, this transformation is area preserving. Equation (18) follows by dividing both sides of Eq. (19) by t2 − t1 , and then multiplying   |EA | and dividing the resultant right-hand side by lP t1 + j=1 α( j).   We mention that Proposition 3 is a very general FHLL, which subsumes all previous results. However, the presentation as a progression of increasingly general results is easier to follow. Finally, we observe the following corollary. 3 Let ES ⊆EP be any subset of entities present in the system during the interval Corollary  t1 ,t2 . Then, Eq. (18) holds for ES , provided EA , lP(t), and wl (x) are replaced by their P counterparts associated with ES . Proof Follows from the observation that Eq. (19) still holds after the stated replacements.  

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Fig. 2 Graphical illustration of Eq. (19)

4.3 Continuous flow setting In this case, all flows are continuous, and we can distinguish among flows. Entities can be defined as parcels of fluid, but they would necessarily be asynchronous. Since an EntityAveraged FHLL would not have adequate semantics for the reasons explained in Sect. 4.2, we can dispense with entities altogether. Let flow  P be governed by an arrival rate function λP(t) and departure rate function μP(t), where t∈ t1 ,t2 for both. Assume that φP(t) = λP(t)−μP(t) is the net flow rate in the system such that  t     (20) φP (τ ) dτ ≥ 0, t ∈ t1 ,t2 , lP(t) = lP t1 + t1

  and define the present flow in the interval t1 ,t2 by  t   2 λP(t) dt. αP = lP t1 +

(21)

t1

The next proposition presents a fluid-flow FHLL. Proposition 4 Consider a fluid flow system with integrable lP(t). Then, an AttributeAveraged FHLL is given by ∞   t  t lP t1 + t 2 λP(t) dt 2 1 0 wl (x) d x 1 (22) lP(t) dt = ×   Pt t2 − t1 t1 t2 − t1 l t + 2 λ (t) dt P

1

t1

P

Proof Similar to that of Proposition 3 with the present flow of Eq. (21) replacing its counterpart in Eq. (17).     Corollary 4 Let S be any sub-flow of P during the interval t1 , t2 . Then, Eq. (22) holds, provided lP(t), λP(t), and wl (x) are replaced by their counterparts associated with sub-flow S. P

Proof Follows from the observation that Eq. (19) still holds after the stated replacements.  

5 Monetary flows The foregoing discussion shows that inventory turns can come in various flavors and serve different purposes. For example, entity-oriented FHLLs may typically be used to measure

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the operational efficiency of inventory management, while attribute-oriented counterparts can also be used to measure the corresponding economic efficiency. In practice, companies may be content to use just economic efficiency by computing some estimate of inventory turns of the form (Gaur et al. 2005) Total monetary flow through the inventory during a reporting period Time average of monetary value of inventory over same reporting period

(23)

Monetary flows of interest can assume several forms: accountants use the cost of goods sold (COGS) to assign monetary values to inventory contents and flows, while managers may also use internally purchase prices, revenues or net profits for the same. Some of these metrics can often be computed by outsiders from the company’s financial statements, while others may be computed from data available only internally and thus accessible only to company managers. We now proceed to analyze financial inventory turns of the form of Eq. (23). The first observation is that the notion of monetary flow in the numerator of Eq. (23) is not crisp in the sense that different companies use different proxies for it as pointed out above. Thus, in the absence of systematic proxies, the resulting inventory turns metrics may not be entirely comparable. In a similar vein, companies may use inconsistent or poor approximations of the denominator of Eq. (23). For example, companies that measure inventory value only at the end of each reporting period (as these values must appear on their Balance Sheet) often do a poor job of computing inventory turns. To wit, a common practice is to average the reporting period’s beginning and ending inventory values. Not only is such an average a crude approximation of the exact time average of inventory level over the reporting period, the tendency to lower inventories at the end of reporting periods can introduce a systematic and serious bias into resultant inventory turns. More precisely, understatements of inventory time averages in the denominator of Eq. (23) could substantially overstate the inventory turns metric. Moreover, managers, whose compensation is tied to high inventory turns, may become a source of agency (Kolias et al. 2011). It is best to use the full-fledged time average of the inventory’s monetary values to avoid such distortions. Another issue is the relation of the inventory turns metric of Eq. (23) to average sojourn times, and consequently, to variants of generalized FHLLs. The metric in Eq. (23) is a bona fide inventory turns metric with clear semantics, that is, “how many times average inventory flowed out of the system during the reporting period”. However, not every inventory turns metric satisfies an FHLL, and consequently, its reciprocal may not have a meaningful interpretation in terms of average sojourn times. Eq. (23) is a case in point. Note that the numerator and denominator are computed here over generally differing entity sets (respectively, ED and EP ), which are related by ED ⊆EP . However, the propositions in this paper require the two sets to be the same for the corresponding FHLL to hold. It follows that no FHLL in this paper generally holds for Eq. (23). The second observation is that if ED is properly contained in EP , then the reciprocal of Eq. (23) is generally not the average restricted sojourn time, but an approximation    at best, and a poor one at worst (as mentioned before, an exception is the case lP t1 = lP t2 = 0 which implies EP = EA = ED ). In all other cases, Eq. (23) will understate the inventory turns of entity set EP and overstate that of entity set ED , while the reciprocal of Eq. (23) will overstate the average restricted sojourn time of entity set EP and understate that of entity set ED . It may be argued that the former is a better approximation of the average unrestricted sojourn times, and that the goodness of the approximation would improve as the length of the reporting period increases, since then the “relative discrepancy” between sets EP and ED narrows. However, the modeler should tread here carefully since the goodness of the approximation is generally unknown.

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Fig. 3 Inventory level realization of a FIFO inventory system subject to lost sales

The following simple example illustrates the points made above by comparing the inventory turns metric of Eq. (23) generated by entity sets EP and ED , respectively, in a simulation of synchronous entities flowing through an MTS inventory system with base stock level s = 100, and subject to the lost sales rule. For simplicity we set a(i) = 1 as the monetary value of each entity i, so entities are product units and we can identify entities with their monetary values. The demand arrival process was Poisson with a time varying rate parameter: 0.5 in the time interval [0,100] and 1.0 in the time interval (100,200]. Demand sizes were deterministic at 1 unit per arrival, and inventory units were dispensed to arriving demands in FIFO order. Replenishments arrived at the inventory as singletons and inter-replenishment times were iid (independent and identically distributed) exponentially distributed with a   time-dependent and level-dependent rate parameter, s − lP(t) × 0.01. A Monte Carlo simulation of the inventory system was run starting at time 0 with empty inventory. Figure 3 depicts a realization of the resulting inventory level process over a complete cycle starting and ending with inventory level 0. Here, the time interval of the displayed cycle is denoted by [ts ,te ], where ts = 4.2 and te = 171.3. Over this interval one has EP = ED , so unrestricted sojourn times as well as restricted ones can be computed for every present entity over any subinterval of [ts ,te ]. To compare various inventory turns computations based on Eq. (23), we defined a set of   reporting periods, t1 , t2 , within the shown cycle, satisfying ts < t1 < t2 < te . For any such reporting period, the numerator of Eq. (23) is just the total number, |ED |, of inventory units sold (i.e., departing entities). The corresponding denominator of Eq. (23) was computed in three ways as follows:   D0: time average of inventory level for ED over t1 ,t2   D1: time average of inventory level for EP over t1 ,t2 D2: inventory level averaged over its values at t1 and t2 only Using the common numerator, |ED |, and dividing it in turn by each of the three denominators above yielded three corresponding inventory turns, T0, T1 and T2, respectively, with corresponding reciprocals, W0, W1 and W2, respectively. Note that only W0 satisfies a simple FHLL, thereby providing the correct average restricted sojourn times of sold inventory

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Fig. 4 Inventory turns (left) and average restricted sojourn times (right) of sold inventory units and their approximations

Fig. 5 Percentage relative absolute deviations of approximated average restricted sojourn times of sold inventory

units in ED ; the others are approximations, where W1 is larger than W0 and W2 could be larger or smaller. Next, we fixed t1 = 20 and varied t2 between 60 and 160 by increments of 1. Figure 4 consists of two graphs, each depicting three curves as functions of t2 : the first depicts inventory turns metrics corresponding to the values of T0, T1, T2, respectively, while the second depicts the corresponding values of W0, W1, and W2. Figure 4 confirms that T1 underestimates T0 and W1 overestimates W0. In contrast, T2 and W2 are quite inaccurate, where the inaccuracy depends strongly on the span of the reporting period. Figure 5 depicts the percentage relative absolute deviations of W1 and W2, respectively, from W0 as functions of t2 . It shows that the estimation error of W1 and W2 can be quite large. Furthermore, the estimation error of W1 diminishes as the reporting period increases (in our case, as t2 approaches te ), while that of W2 is more erratic and ill-behaved. Finally, we observe that for a version or variant of Little’s Law to hold meaningfully, a flow conservation principle should be in effect in the sense that all increases and decreases in

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system levels should correspond to actual flow arrivals and flow departures. In our case that means that the incoming monetary flow should be valuated at its ultimate outgoing sale value and kept at that value thereafter. If the levels of attribute parcels in the system are allowed to fluctuate in time not due to “real” arrivals and departures but due to intermediate valuations pursuant to market conditions, write-downs, etc., then value fluctuation would be tantamount to creating virtual arrivals and departures that violate flow conservation and muddle the interpretation of the resulting metrics. We shall revisit this point in the next section.

6 Conclusion The major contribution of this paper is the theoretical development of finite-horizon versions and variants of Little’s Law. A nice feature of all such versions and variants is that they hold for multiple streams of entities, as well as any sub-system with a well-defined boundary. They also hold for any hybrid combination of synchronous and asynchronous flows, including discrete and continuous flows. Consequently, both inventory turns and restricted average sojourn time metrics can be computed, in principle, for individual SKUs or groups thereof with any related numerical attribute or attribute set. In particular, they can also be applied to groups of inventories that form a sub-system, and the financial burden of carrying inventory across multiple reporting periods can be readily broken down into sequential components, each attributable to a distinct reporting period. Another useful feature of finite-horizon versions and variants of Little’s Law is that there are two methods of computing the same quantity (inventory turns and the reciprocal of average restricted sojourn times). This fact provides data processing departments the option of selecting the method that has the lower computational complexity. Modelers and analysts, however, must carefully understand the semantics of inventory turns and DIO metric computations, as uncritical application of FHLL formulas can easily produce significantly distorted results. As an example, consider the case of an inventory with monetary flows, and suppose that the value of an inventory item can fluctuate over time as function of supply and demand (e.g., precious metals and petrochemicals), or lose value due to impairment (e.g., breakage, spoilage of perishables, or obsolescence of electronic parts). From a mathematical point of view, such value fluctuations look like arrivals at the inventory and departures from it. However, these semantics do not always comport with business logic when inventory turns and DIO metrics are computed from fluctuating monetary values. For example, consider an inventory turns metric where the writing down of some impaired inventory results in a re-computation and restatement of the inventory time average at the end of the reporting period. Focusing on the inventory turns metric of Eq. (23), we see that such restatement would decrease the denominator but increase the numerator since COGS includes write-downs. The result would be a nominally higher inventory turns metric! Considering that write-downs typically occur in conjunction with poor inventory management, this could unduly reward ineffective inventory managers. The anomaly is a consequence of conflating “departures” of monetary value due to sales with those due to write-downs. An opposite anomaly could occur if inventory turns are internally based on selling price (say, from precious commodity sales), the selling price inflates in value due to market conditions, and inventory values are marked to market periodically. Thus, while the theory developed here allows an entity attribute a(i,t) to fluctuate over time, such fluctuations should be “well behaved” and not lead to anomalies. This is a modeling problem whose resolution is the responsibility of the modeler. It should also be recognized that economic efficiency is not perfectly correlated with operational efficiency.

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The findings of this paper have relevance to both supply chain operations managers and financial managers. Many strategic and tactical supply chain decisions are taken based on inventory performance. In particular, inventory turns and DIO metrics are key measures used by supply chain managers of all kinds to gauge the success of inventory management within the firm and throughout the supply chain. Consequently, careful attention should be paid to how these important metrics are measured and used in decision making.

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