The Gamma Distribution and Inventory Control: Disruptive Lead Times Under Conventional and Nonclassical Conditions

Abstract

Background: Foundational research on the gamma distribution and inventory control highlighted its flexibility and practicality for managing fast-moving finished goods. Nonetheless, concerns remain about conventional statistical approximations of lead-time demand (LTD) distributions. Real-world lead times often result in nonstandard LTD forms, and traditional methods may introduce parameter mismatches under nonclassical conditions. Despite these challenges, this research demonstrates that a gamma LTD approximation is an effective method for managing these goods. Methods: This study employs numerical experiments to assess accuracy at high service levels, focusing on errors in system cost and product availability. Three propositions are validated: (1) a standard distribution generally characterizes the demands of fast-moving items; (2) demand variability systematically modifies the form of nonstandard LTD distributions, enhancing accuracy; (3) nonclassical conditions generally improve the accuracy of properly parameterized gamma approximations. A purposive sample of disruptive lead-time distributions found in global maritime supply chains drives numerical experiments. Results: Externally validated evidence provides the following findings within our study context: (1) a nonstandard lead-time distribution does not necessarily result in a similar LTD distribution, as it also depends on demand variability; (2) demand variability positively affects the form of a nonstandard LTD distribution under conventional conditions, with nonclassical conditions enhancing this effect; (3) the shape transformations almost always improve the accuracy of a gamma approximation. Conclusions: A gamma LTD approximation can manage inventory for fast-moving finished goods effectively, even with disruptive lead times under both conventional and nonclassical conditions.

1. Introduction

In a pioneering study on the gamma distribution in inventory control, Burgin [1] argued that its versatility, tractability, and ease of application should give it a prominent role in the management of fast-moving finished goods. Although the extant supply chain literature includes numerous studies with gamma applications, many writers have reservations about the use of a standard statistical form to approximate realistic lead-time demand (LTD) distributions [2,3,4,5,6,7,8,9]. The principal concern is that lead-time distributions often have nonstandard attributes, such as multiple peaks, gaps, and outliers, which give rise to LTD distributions with similar features. The differences between the actual LTD distribution and a standard statistical approximation can lead to inaccurate inventory decisions, resulting in significant cost and product availability errors. Additionally, these errors grow as service targets near unity.

Another matter involves parameter mismatches. Conventional estimates of the LTD mean and variance rely on two assumptions. First, the demands per period (D) and the lead times (L) are independent and identically distributed (i.i.d.), and mutually independent random variables. Under these conditions, the convolution of D and L generates an i.i.d. LTD distribution (X) with mean and variance estimates accurately formulated from the parameters of D and L [10] (p. 51), [11] (p. 310). Dependent demands and correlated LTD forecast errors, however, can seriously inflate the true LTD variance [5].

Second, each replenishment order arrives in the issue sequence [12] (pp. 201–202). The assumption is valid when the probability of more than one open per replenishment cycle is negligible. A fixed order cost that is high enough to drive the order quantity above the expected LTD typically satisfies this condition. Otherwise, multiple open orders can randomly arrive out of sequence. It is well-established that this phenomenon generates an “effective” lead-time (ELT) distribution with a smaller variance than the original lead-time distribution [12,13]. In other words, random order crossover introduces another potentially significant parameter mismatch.

This study demonstrates the promise of a gamma LTD approximation for managing fast-moving inventory items with nonstandard lead times under conventional and nonclassical conditions. The enterprise examines accuracy as a function of service, defined by the probability of meeting all demand in one replenishment cycle (P₁). Accuracy metrics reflect cost and product availability through the absolute percent error in annual total cost (ATC) and the fraction of demand met from available inventory (P₂), or item fill rate. The research includes three propositions. First, fast-moving item demand typically follows a normal, gamma, lognormal, or exponential distribution. Second, demand variability, as measured by the coefficient of variation (CV), systematically alters the form of the LTD distribution in ways that enhance accuracy. Third, nonclassical conditions generally improve the accuracy of approximations based on reasonably accurate parameter estimates.

A purposive sample of documented nonstandard cases of L in global maritime supply chains drives the numerical experiments. The selected cases possess attributes that are expected to challenge the accuracy of any standard LTD approximation. The results validate the propositions and yield original findings that will be of interest to practitioners, educators, and researchers.

The paper proceeds as follows. Section 2 reviews the pertinent literature that motivates the debate as to the accuracy of a standard LTD approximation. Section 3 describes the business setting, model formulations, and numerical experiments. Section 4 reviews the results. Section 5 discusses the validity of experimental values and the evidence supporting the propositions. Section 6 summarizes the investigation and presents the conclusions.

2. Literature Review

The literature review concentrates on studies examining the accuracy of standard LTD approximations under conventional conditions. The principal issues involve the shape statistics, the accuracy constructs, and the effects of nonstandard lead times on the LTD distribution. These studies are reviewed in chronological order to track the development of these issues and to record the competing findings. For more comprehensive reviews, see [4,14,15]. Section 3 discusses studies relevant to nonclassical conditions.

2.1. Higher Moments

Four connected studies launched a debate as to the importance of commonly used statistical moments in modeling the LTD distribution. Naddor [16] calls for systematic LTD shape research by investigating eight theoretical distributions. The results support the finding that optimal policies often “depend on means and standard deviations of demand but not on the specific forms of distributions” [16] (p. 1770). This finding implies that inventory decisions tend to be more sensitive to the relative variation of the distribution (σ/µ) than to the higher moments—namely, skewness (ν) and kurtosis (κ).

Fortuin [17] answers the call by investigating five popular statistical distributions, each with a low CV_X (0.167). The numerical experiments calculate P₂ values based on an experimental safety stock multiplier (k) that ranges from 0 to 2, where R = ${\mu }_X$ + k${\sigma }_X$, Q = 1.25${\mu }_X$, and ${\sigma }_X={\mu }_X/6$. A narrow P₂ bandwidth across the five distributions supports Naddor’s [16] findings.

Tadikamalla [18] then uses the same distributions to evaluate the minimum ATC as a function of a CV_X in the 0.33-to-3.0 range and P₁ in the 0.67-to-0.97 range. Only the results for a CV_X ≤ 0.5 corroborate Naddor’s findings.

Heuts et al. [19] present two sets of experiments that challenge the findings reported by Naddor [16] and, indirectly, those in Fortuin [17]. They first replicated Naddor’s experiments with Schmeiser-Deutsch (S-D) distributions. The results show that “—optimal solutions may significantly depend on the shape of the distribution even if the mean and the variance are the same in two situations” [19] (p. 7). The second set of experiments, which uses other S-D distributions under different conditions, corroborates that finding.

Two subsequent studies contribute to the foregoing debate. Bartezzaghi et al. [20] investigate the impact of six theoretical distributions of X on inventory levels as a function of a P₂ target. The distributions include basic symmetric forms (uniform and normal) and more complex (unimodal and bimodal) shapes. Among other things, the findings reveal that “—not only multi-modality and asymmetry can be considered important, but also the kind of asymmetry (right rather than left) plays a key role” [20] (p. 400).

Rossetti and Ünlü [21] revisit popular unimodal distributions. They develop rules based on the first two moments for selecting the most accurate approximation of the true form in terms of ready rates, backlogs, and on-hand inventory levels. The results indicate that selection rules have considerable potential.

2.2. Accuracy Constructs

Three linked studies present competing findings based on different accuracy constructs and experimental service levels. Each study considers a normal approximation of various nonstandard LTD distributions in an R, Q inventory system.

Eppen and Martin [5] use a mixture of marginal LTD distributions to evaluate the correct reorder points for given P₁ targets. They show that a normal demand distribution with a CV less than 0.14, along with two examples of discrete nonstandard lead times, leads to similar nonstandard LTD distributions. When P₁ = 0.95, a normal LTD approximation based on the first lead-time example results in errors of 10% in R and 32% in safety stock. The second example produces comparable errors when the P₁ target is 0.917. Among other things, these results imply that a normal LTD approximation “is unwarranted in general” [5] (p. 1381).

Tyworth and O’Neill [22] investigate the sensitivity of ATC to errors in R and safety stock. This study, too, uses a mixture of marginal LTD distributions approach based on normally distributed D, combined with industry-based examples of nonstandard L. Numerical experiments examine errors when realistic freight-rate structures influence lot sizes. The results show an absolute ATC error < 5% in most of the cases studied when the CV_D = 0.40 and the items have a target P₂ in the 0.90–to–0.99 range. These findings indicate that a normal approximation is robust in the setting studied.

Lau and Lau [6] challenge the robustness of the normal LTD approximation. They use a beta demand distribution combined with a lead time of one period to represent the true LTD distribution in an R, Q system that minimizes the ATC. To help ensure the fairness of the test, the writers use two constraints to enhance the likelihood of small ATC errors—namely, a Q > 2${\mu }_X$ and a CV_X < 0.3. They also consider the role of higher statistical moments by first specifying a μ = 1 and a σ < 0.3 and then manipulating the parameters of the beta distribution to create unimodal, bimodal, and nonmodal (uniform) LTD distributions with different v and k values. Finally, they conduct experiments to evaluate ATC errors as a function of a unit shortage cost (b) that ranges from ≈0.5 to ≈447. The findings show that a normal LTD approximation generates ATC errors >15% when the P₁ ≥ 0.999 and the derived P₂ ≥ 0.9995. By contrast, the corresponding conditions in Tyworth and O’Neill [22] are a P₁ ≥ 0.93 and a P₂ ≤ 0.99. Two additional findings contribute to the discussion of accuracy. The first is that ATC errors do not necessarily increase monotonically with b (or the corresponding P₁ level). The second finding is that ν and κ are not reliable indicators of ATC error levels.

Cobb et al. [23] evaluate, among other things, the accuracy of a normal approximation of an LTD distribution with lognormal demands and nonstandard lead times. Testing 100 randomly selected parameter values, they find that ATC errors range from 0.01% to 206%, with an average of around 35%.

Several writers have explored the gamma LTD approximation for fast-moving items in nonstandard lead-time situations. Saldanha [7] uses normal and gamma LTD approximations of R to benchmark estimates developed from a proposed bootstrap approximation. This study investigates hypothetical LTD distributions with nonmodal, unimodal, and bimodal shapes in a continuous review inventory system with a P₂ target and a pre-specified Q > ${\mu }_X$. Experimental controls include a bootstrap sample size (n) ≤ 24, a ${CV}_X$ in the 0.20-to-2 range, and a P₂ target in the 0.80-to-0.99 range. The results show that the bootstrap LTD approximation is generally more accurate than the normal and gamma approximations.

In a related case study, Saldahna et al. [8] examine the proposed bootstrap approximation in two sets of numerical experiments. The first approach employs bootstrap and standard approximation methods, as well as safety stock holding and shortage costs, to determine errors in the safety stock. Experiments assume the true compound LTD distribution uses a gamma-distributed D with a CV of 0.2 and lognormal and bimodal distributions of L. The second set includes nine SKUs in an R, Q system, where R meets a P₁ target of 0.95, and Q is the supplier’s minimum order quantity. The results show that the bootstrap method is as reliable as the gamma approximation for unimodal LTD distributions but more accurate for bimodal distributions. The authors warn that small LTD datasets (n ≤ 24) may make the bootstrap method unpredictable, particularly when P₁ ≥ 0.99 [8] (p. 15).

In summary, variations in cost components and service levels can lead to fragmented results even under comparable demand and lead-time conditions. Additionally, two findings require further investigation. First, skewness and kurtosis might not serve as reliable indicators of ATC error levels. Second, ATC errors do not necessarily increase when P₁ is in the 0.90–0.999 range.

3. Methods

Fast-moving items suited for multi-period inventory control systems often exhibit demand processes that align with a mixture of marginal distributions (MMD) method for developing accurate LTD distribution, density, and loss functions. This well-established parametric approach involves significant numerical complexity under conventional conditions [2,5,22,24,25]. Previous work in this area is advanced by (1) introducing and evaluating a demand-variability theory showing why a gamma LTD approximation can be effective when lead times follow a nonstandard distribution and (2) expanding the scope of investigation to include nonclassical conditions and simultaneous considerations of P₁ and P₂ service metrics.

3.1. Business Setting

The business setting includes functional products with stable demand and low obsolescence risk, as identified by Fisher [26] and Harris and Farrington [27], where items with “A” or “B” inventory classifications require continuous review and fill rates in the 0.95-to-0.99 range (see, e.g., [2,28,29]). Modern enterprise resource planning (ERP) systems provide suitable demand data, while ERP modules and third parties supply lead-time data.

3.1.1. Demand

Researchers commonly use unimodal distributions to model fast-moving item demand. Vernimmen et al. [30] reviewed 66 articles, finding that 14 used a normal distribution and eight used a gamma distribution. Hayya et al. [31] reported similar findings. More recently, Chatfield and Prichard [32] and Saldanha et al. [8] have employed gamma distributions in numerical experiments concerning fast-moving demands.

Table 1. Grounded evidence of normal and gamma distributions for fast-moving items.

Normal		Gamma
Source	Data	Source	Data
Das et al. [33]	Retail merchandise	Bischak et al. [34]	Experience with industrial datasets
Disney et al. [4]	Industrial equipment	Silver and Robb [35]	Building products
Fang et al. [36]	Truck/auto components	Teunter et al. [37]	Bike and auto parts
Kapuscinski et al. [38]	Office equipment
Leachman [16]	Retail merchandise
Yang et al. [39]	Household appliance parts

presents grounded studies that utilize normal and gamma demand distributions.

3.1.2. Lead Time

The lead-time setting focuses on the backend of a global maritime supply chain, where the supply node includes factories and distribution centers, and the demand node includes import distribution centers and stores for retailers, wholesalers, or resellers. Intermodal shipping itineraries often experience disruptions that generate nonstandard lead-time distributions [4,33]. Additionally, transportation cost indivisibilities encourage shippers to fully load containers by pre-specifying order quantities or consolidating orders across items and suppliers [40,41]. These actions may result in substantial fixed order costs. Additionally, ancillary order planning and control activities often entail significant fixed costs per order [9,42,43]. For example, Kropf and Sauré [44] uses a unique dataset for Swiss export trades to infer a fixed cost per shipment that “—comprises the monetary equivalent of the time spent filling in customs forms, organizing trade credit and monitoring and coordinating the actual transportation to the receiver” [44] (p. 166). Their most conservative specification of this cost had an average net present value of 5723 Swiss francs (CHF) in 2014. They indicate that this level is consistent with the direct measures found in the Doing Business survey of the World Bank in 2014. The CHF 5723 figure is equivalent to ≈$8400 in U.S. dollars (USD) in 2024 based on the January Producer Price Index for Total Retail Trade Industries and a 2024 exchange rate.

3.2. Conventional LTD Scenario (X)

The following notation describes the independent and identically distributed (i.i.d.) random variables and statistical functions that inform the (R, Q) inventory system under conventional conditions.

L: ${\mu }_L$,${\sigma }_L$—lead time

D: ${\mu }_D$,${\sigma }_D$—demand per period

X: ${\mu }_X$,${\sigma }_X$—lead-time demand (LTD)

$\widehat{X}$: ${\mu }_X$,${\sigma }_X$—gamma approximation of X

F(·)—cumulative distribution function (CDF)

F⁻¹(·)—inverse CDF

G(·)—expected loss function (units per replenishment cycle)

f(·)—probability density function (PDF)

P(·)—probability mass function (PMF)

Table 2. Continuous review (R, Q) model elements.

Notation	Description	Metric	Formulation	Equation
Decisions
R	actual reorder point|P1 target	units	$=F_X^{-1}$ (P1)	(1)
$\widehat{R}$	approximate reorder point|P1 target		$=F_{\widehat{X}}^{-1}$ (P1)	(2)
Q	pre-specified order quantity	units	=EOQ = [(2AS)/h]0.5	(3)
Service Measures
P1	probability of satisfying all demand in one cycle, or cycle service level (given)	0 ≤ P1 ≤ 1	=Pr[X ≤ x]	(4)
P2	actual fraction of annual demand satisfied from stock, or item fill rate (derived)	0 ≤ P2 ≤ 1	$=1-G_X$(R)/Q	(5)
$\widehat{\mathrm{P}}$2	derived fraction of annual demand satisfied from stock, or item fill rate	0 ≤ P2 ≤ 1	$=1-G_X$($\widehat{R}$)/Q	(6)
Systems Variables and Cost Calculations
b	shortage cost	$/unit	input
h	holding cost factor	$/unit/year	input
A	fixed order cost	$/order	input
S	annual demand	units/year	input
ATC	annual total cost	$/year	=AOC + ASC + AHC	(7)
AOC	annual order cost	$/year	=A·S/Q
ASC	annual shortage cost	$/year	$=G_X\left(R\right)$·b·S/Q
AHC	annual holding cost	$/year	=SSC + CSC
SSC	safety stock holding cost	$/year	=(R − µx)·h
CSC	cycle stock holding cost	$/year	=Q/2·h

Notes: X represents lead-time demand under conventional conditions; $\widehat{R}$ replaces R in the ASC and SSC elements of Equation (7) for the gamma approximation of X.

shows the notation, description, metrics, and formulation of the inventory system elements. The reorder point satisfies a pre-specified P₁ target as shown in Equations (1) and (2), while the system uses the classic economic order quantity (EOQ) calculation in Equation (3) for Q. Equations (4)–(6) define the service measures, where the P₁ target is an experimental input, and the actual and approximate P₂ levels are derived. As shown in Equation (7), the ATC comprises ordering, shortage, and holding costs. These cost formulations are reasonably accurate when the derived P₂ ≥ 0.90 [7,12,35,45]. Additionally, the choice between a complete backorder model and a complete lost sales model has a negligible effect on the safety stock cost (SSC) component at high P₂ levels, say, above 0.95 [46]. Finally, it is useful to note that by definition, the mismatch between ${\widehat{F}}_X$ and $F_X$, and thereby $\widehat{R}$ and R, affects the ATC through the annual shortage cost (ASC) and SSC components.

The LTD mean ${\mu }_X$ and standard deviation ${\sigma }_X$ are calculated from the known or estimated D and L parameters as shown in conventional Equations (8) and (9). Given values for ${\mu }_X$ and ${\sigma }_X$, the gamma LTD approximation $\widehat{X}$ uses the method of matching moments to calibrate its shape (α) and scale (β) parameters as follows: α = ${\mu }_X^2$/${\sigma }_X^2$ and β = ${\sigma }_X^2$/${\mu }_X.$

$${\mu }_X={\mu }_D{\mu }_L$$

(8)

$${\sigma }_x=\sqrt{{\mu }_L{\sigma }_D^2+{\mu }_D^2{\sigma }_L^2}$$

(9)

When the distribution of D has a standard form, such as the normal, gamma, lognormal, or exponential, one can accurately model the LTD distribution as mixture of marginal distributions X|{L = l} with marginal distributions defined by D(l${\mu }_D$, l^0.5${\sigma }_D$) for each instance of discrete L [5,12,24,25,47,48]. The cumulative distribution function is evaluated by summing each F_X|{L = l}(x) weighted by the probability that L = l, as shown in Equation (10)

$$F_X(x)=\sum _{l=1}^{l_{max}}{F}_{X|L=l}(x;l{\mu }_D,l^{0.5}{\sigma }_D)·p\left(l\right)$$

(10)

Equation (11) presents a comparable formulation for the first-order loss function, which is utilized to determine the expected units short per replenishment cycle.

$$G_X(x)=\sum _{l=1}^{l_{max}}{G}_{X|L=l}(x;l{\mu }_D,l^{0.5}{\sigma }_D)·p(l)$$

(11)

Finally, Equation (12) models the actual probability mass function $P_X(·)$ over a suitable range of x values—namely, where j = 1, 2, …, $x_{max}$ and $x_{max}$ is large enough to satisfy $\sum p{(x}_j)=1$ at the ${10}^{-4}$ level of precision.

$$P_X\left(X=x_j\right)=F_x\left(x_j+0.5\right)-F_x\left(x_j-0.5\right)$$

(12)

3.3. Numerical Experiments

The experimental design includes accuracy measures, demand and lead-time conditions, treatments, and system inputs.

Table 3. Experimental design elements for base case LTD scenario X.

Notation	Description	Metric	Formulation	Equation
Dependent Variables
Accuracy
Cost	absolute percent error in ATC	%	$=\left|\widehat{\mathrm{A}\mathrm{T}\mathrm{C}}-\mathrm{A}\mathrm{T}\mathrm{C}\right|/\mathrm{A}\mathrm{T}\mathrm{C}$∙100	(13)
Service	absolute error in P2	decimal fraction	=|$\widehat{\mathrm{P}}$2 − P2|	(14)
Independent Variables
Setting
D	random variable for demand	units/period	$\mathrm{gamma} ({\mathsf{\mu }}_{\mathrm{D}}=10,{\mathsf{\sigma }}_{\mathrm{D}}={\mathsf{\mu }}_{\mathrm{D}}·\mathrm{C}\mathrm{V})$
L	random variable for lead time	periods	empirical nonstandard, discrete
X	${\mathrm{F}}_{\mathrm{X}}\left(\mathrm{x}\right)={\displaystyle \sum _{\mathrm{i}=1}^{{\mathrm{l}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}}{\mathrm{F}}_{\mathrm{D}|\mathrm{L}={\mathrm{l}}_{\mathrm{i}}}(\mathrm{x};\mathrm{l}\mathsf{\alpha },\mathsf{\beta })·\mathrm{p}\left({\mathrm{l}}_{\mathrm{i}}\right)$			(15)
	${\mathrm{G}}_{\mathrm{X}}\left(\mathrm{x}\right)={\displaystyle \sum _{\mathrm{i}=1}^{{\mathrm{l}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}}\left\{{\mathrm{l}}_{\mathrm{i}}\mathsf{\alpha }\mathrm{B}\left[1-{\mathrm{F}}_{\mathrm{D}|\mathrm{L}={\mathrm{l}}_{\mathrm{i}}}\left(\mathrm{x};{\mathrm{l}}_{\mathrm{i}}\mathsf{\alpha }+1,\mathsf{\beta }\right)\right]-\mathrm{x}\left[1-{\mathrm{F}}_{\mathrm{D}|\mathrm{L}={\mathrm{l}}_{\mathrm{i}}}\left(\mathrm{x};{\mathrm{l}}_{\mathrm{i}}\mathsf{\alpha },\mathsf{\beta }\right)\right]\right\}\mathrm{p}\left({\mathrm{l}}_{\mathrm{i}}\right)$			(16)
Experiments
P1	service target	0 ≤ P1 ≤ 1	0.90-to-0.999 0.01 increments ≤ 0.90, 001 increments > 0.90
${\mathrm{C}\mathrm{V}}_D$	$\mathrm{demand} \mathrm{variability} ({\mathsf{\sigma }}_{\mathrm{D}}$$/{\mathsf{\mu }}_{\mathrm{D}}$)	decimal fraction	0.1-to-1 0.1 increments ≤ 0.4 0.3 increments > 0.4
System Parameters
h	holding cost	$/unit/year	$1
S	annual volume	units	$={\mathsf{\mu }}_{\mathrm{D}}$∙52 weeks/yr	(17)
A	fixed order cost|Q = EOQ	$/order	$=(h{·Q}^2$)/2∙S	(18)
b	shortage cost	$/unit	=(h∙Q/S∙P1)/(1 − P1)	(19)

classifies the elements as dependent and independent variables and includes Equations (13)–(19) that define accuracy and identify system inputs.

3.3.1. Dependent Variables

Two dependent variables capture accuracy. The first is the absolute percent error in the ATC as defined by Equation (13). The second is the absolute deviation in the fill rate (P₂) as defined by Equation (14). For clarification, ${\mathrm{P}}_2$ and ${\widehat{\mathrm{P}}}_2$ are fill rates derived, respectively, from the actual (R, Q) and approximate ($\widehat{R}$, Q) solutions for the same P₁ target.

3.3.2. Independent Variables

Independent variables characterize the setting, experimental, and inventory-system elements.

Setting Variables

The setting variables describe demand and lead time conditions. As previously documented, studies show that normal and gamma distributions often characterize fast-moving item demands with appropriate accuracy. The gamma is chosen for its reasonably accurate approximations of non-negative, bell-shaped forms when the CV level is less than 0.30 and positive-skewed shapes at higher levels [1]. Equations (15) and (16) supply the specific formulations of the distribution and loss functions. The experiments assume that ${\mu }_D$ = 10, with pre-specified CV levels determining the standard deviation (${\sigma }_D={\mu }_D\times {CV}_D)$. As with Bischak et al. [34] (p. 767), the demand level minimally affected the results in this study.

Several studies and reports document cases of non-standard lead time distributions based on a common dataset comprising more than 125,000 liner-carriage shipments of finished goods for retailers, manufacturers, and freight forwarders [33,49,50,51,52]. Each distribution corresponds to a specific lane (origin-destination pair), where shipments over each lane follow an intermodal landside-ocean-landside itinerary that includes travel through ports in Africa, Asia, the Americas, and the European Union. These lanes account for about 80% of the total traffic volume [52] (p. 43). About 40 shippers, carriers, and third-party logistics providers confirmed the distributions in a global ocean transportation roundtable sponsored by the MIT Center for Transportation and Logistics [33] (p. 63). Additionally, Disney et al. [4] document distributions of L developed from more than 9000 container shipments across 13 lanes. One instance involves office products transported from North America to China, while the remaining cases involve retailers and forwarders shipping from Chinese ports to multiple U.S. ports.

The investigation selected five out of 20 cases for analysis. Many studies use hypothetical distributions of L to demonstrate that multimodality and symmetry present significant challenges for accurate standard LTD approximations [3,5,22,53]. Consequently, the selection process initially excluded unimodal and symmetric distributions and then chose cases within categories jointly defined by skewness (low, moderate, and high) and kurtosis (leptokurtic and platykurtic) as outlined in Figure 1. The chosen cases have the highest absolute kurtosis within each category to rigorously evaluate the gamma LTD approximation.

Simulation techniques are employed as needed to standardize distribution periods, measured in days to weeks. This ensures consistency across all instances and alignment with the sailing schedules of liner containerships. Supplementary File S1 includes charts of the 20 distributions along with their shape attributes and sources.

Figure 2 displays each lead-time case in the first panel, along with its directly corresponding LTD density and distribution functions that materialize when the CV_D = 0.1 in the other two panels.

The charts in the second panel show multiple peaks, characterizing multimodality. A peak is a probability loading with smaller nearby loadings. The “p” markers show that this method applies to both large mounds and small bumps. The third panel highlights discrepancies between the actual and gamma distribution functions, which lead to reorder-point errors. Multimodality produces distinct step-like F(X) curves, in contrast to smooth gamma curves. Wide, flat steps pose significant challenges for gamma approximations because they commonly introduce substantial absolute deviations and oscillating error patterns. Meanwhile, elevated levels of ν and κ progressively stretch the actual curve beyond the gamma curve, leading to substantial errors as the F(X) approaches unity.

Experimental Variables

Experimental levels of P₁ and CV_D drive the inventory-system calculations for each lead-time case. The P₁ values range from 0.90 to 0.99 in increments of 0.01 and from 0.991 to 0.999 in increments of 0.001. This approach supports a detailed investigation of concurrent P₂ levels encompassing the 0.97-to-0.99 sweet spot for most firms in the setting of interest. The ${CV}_D$ ranges from 0.1 to 1.0 to capture both smooth and erratic demand patterns, with a smooth time series benchmarked at CV ≤ 0.49 [55]. The 0.1 to 0.4 range is divided into 0.1 increments, with 0.3 increments after that, as nonstandard lead times most affect the LTD distribution at lower levels.

The CV_D experiments are based on the idea that increased demand variability enhances the interaction among the marginal LTD densities, or the probability of joint demands in D_L=l ≡ X_L=l. The effect is akin to rolling multiple six-sided dice, wherein certain sums have a higher probability due to the greater number of possible combinations leading to those outcomes. Both situations promote a unimodal tendency in the distribution. The probability loadings of L, however, have a distinctive effect on the form of X.

Nevertheless, increased interactions among marginal densities can systematically change multimodal distributions of X at low CV_D levels into unimodal forms at higher levels. Figure 3 graphically illustrates this concept, drawing inspiration from a notable example in Eppen and Martin [5] (Example 1). The charts show the impact of CV_D on the conditional LTD distributions, along with the associated density and distribution functions when demand follows a gamma distribution with μ = 10 and σ = CV∙0.1. The first column presents the interaction between the conditional density curves. The second column shows density functions transitioning from bimodal to unimodal distributions, which display higher values of CV, ν, and κ. The third column illustrates the convergence of the actual and approximate distribution functions.

When the CV_D = 0.1, the gap between the peaks of the P_X curve creates an F_X curve with step function qualities. A step creates oscillating mismatches as the actual FX curve intersects the approximate curve from both below and above. The width of these steps can pose significant challenges to accuracy. Although not shown, a continuous univariate mixture of two normal distributions with means of 5 and 10, both having a standard deviation of 1, coupled with a mixture rate of 0.5, yields comparable results.

System Variables

The experimental process begins with h = 1 and S = m × ${\mu }_D$, where m = 52 periods per year. Consistent values for A and b are then formulated without loss of generalization. The formulation of A in Equation (18) uses inputs from h, S, and Q, where Q reflects a prescribed order quantity large enough to support the conventional one open-order assumption. If this quantity represents the EOQ, then Equation (18) aligns the input for A with known values for Q, h, and S. Other LTD scenarios relax the EOQ condition to account for multiple open orders in a replenishment cycle.

Known values also enable the calculation of b in Table 3, Equation (19). The P₁ target reflects the critical ratio (CR) as defined by b/(b + $\overline{h}$), where $\overline{h}$ = h∙Q/S, and Q/S represents the fraction of the year that one unit is held. The CR assumption allows R and Q decisions to achieve almost optimal ATC levels. Looking at the notation and system cost formulations in Table 2, the EOQ optimally balances the cycle stock cost (CSC) with the annual order cost (AOC), while the R that satisfies the P₁ target optimally balances the SSC with the ASC.

3.4. Nonclassical LTD Scenarios

The investigation addresses two nonclassical LTD scenarios. The first considers the distribution of cumulative LTD demand forecast errors. The second scenario explores the effective LTD distribution that emerges in situations involving multiple open orders. Established heuristics supply reasonably accurate estimates of the correct LTD mean and variance in these scenarios.

3.4.1. Serial Correlation (U)

Eppen and Martin [5] supply procedures for evaluating reorder points in the presence of dependent demands and correlated forecast errors. As background information, the authors observed that firms often use a time-series model to forecast the demand per period based on past observations and have sufficient historical data to characterize the distribution of L accurately. The distribution of U guides inventory decisions in this setting, where autocorrelated forecasts and dependent demands can seriously inflate the variance ${\sigma }_U^2$. They develop a method for estimating the true variance based on a first-order moving average process with differencing degree of one and a smoothing factor (α).

Saoud et al. [11] supply detailed theoretical and practical rationales for this approach, along with a comprehensive review of the pertinent literature. Other writers supply specific applications. Babai et al. [56] use the heuristics when exploring the viability of three strategies to estimate the ${\sigma }_U^2$ in the presence of stochastic lead times. Prak et al. [57] present an intuitive reformulation of an Eppen and Martin [5] heuristic for situations in which the demand process is i.i.d., the lead time is deterministic, and the forecast errors are correlated.

The procedures of interest assume that the forecast error (${ϵ}_t)$ in period t has a translated demand distribution, ${ϵ}_t={\mu }_{ϵ}$− θ, where θ is a random variable defined by a standard statistical distribution with mean ${\mu }_{ϵ}$ and variance ${\sigma }_{ϵ}^2$, that generates an E(${ϵ}_t$) = 0. The heuristic estimates the variance of the total forecast era ror through L = l forecast periods as the product of a markup factor B_L=l and a known or estimated ${\sigma }_{ϵ}^2$, where Equation (20) defines the markup.

$$\begin{gathered} B_{L=l}=[2/6-2/3·{(1-\alpha )}^2]·l^3 \\ +[1/2-1/2·{(1-\alpha )}^2]·l^2 \\ +[1/6+2/3·(1-\mathsf{\alpha })+1/6·{(1-\alpha )}^2]·l \end{gathered}$$

(20)

The assumption here is that a gamma demand process characterizes θ(${\mu }_{ϵ}, {\sigma }_{ϵ})$ with ${\mu }_{ϵ}$ ≅ ${\mu }_D$ and ${\sigma }_{ϵ}$ = ${\mu }_{ϵ}$ × ${CV}_{ϵ}$ and experimental inputs supplying ${CV}_{ϵ}$ values. Equations (21) and (22) define the mixture distribution functions of U. They are analogous to the corresponding functions of X in Equations (15) and (16), except for the markup factor on the conditional standard deviation. The markup in Equations (15) and (16) is $l^{.5}$, whereas the markup in Equations (20) and (21) is $B_l^{.5}$.

$$F_U(u)=\sum _{l=1}^{l_{max}}{F}_{U|L=l}\left(u;l{\mu }_U, B_l^{.5}{\sigma }_{U|L=l}\right)·p\left(l\right)$$

(21)

$$G_U(u)=\sum _{l=1}^{l_{max}}{G}_{U|L=l}\left(u;l{\mu }_U, B_l^{.5}{\sigma }_{U|L=l}\right)·p\left(l\right)$$

(22)

3.4.2. Random Order Crossover (Y, W)

When lead times are random, issuing multiple orders in the same replenishment cycle can cause orders to arrive out of sequence [12,15]. This means a later order can randomly arrive before an earlier one. The random order crossover (ROC) phenomenon leads to an effective lead-time (ELT) distribution with less variance (${\mathsf{\sigma }}_{\mathrm{ELT}}^2$) and variability (CV_ELT) than the original lead-time distribution [31].

Firms can create an ELT distribution using a database with historical records of order issues and arrival periods. Sort each field in ascending order and calculate the ELT by subtracting the issue period from the arrival period for each record. Wensing and Kuhn [58] (p. 271) observe, “Since the effective lead time is not influenced systematically by the corresponding order’s position in the issue sequence”, one can use a “—common random variable to describe the effective lead time that an arbitrary delivery observes”. A popular approach, therefore, is to replace the distribution of L with the distribution of ELT [32,34,59]. In the absence of appropriate historical records, one can develop an ELT distribution from a given discrete, or discretized continuous, distribution of L in one of two ways. The first is to apply the numerical procedures found in [58]. The second method is to take random draws of L for orders issued over t = 1, 2, …, n periods, where the corresponding arrival period is t + L = l [27] (Table 3). This study follows the second approach to calculate 10,000 ELT observations for each of the five nonstandard cases of L.

The last step is to model the effective LTD distribution. Writers have relied on simulation methods to convolute D and ELT to accomplish this task (see, e.g., [32,34,59]). The procedure here is to replace the five parent nonstandard distributions with their simulated ELT distributions. In other words, the effective LTD distributions are Y = X|ELT and W = U|ELT. For clarification, Figure 4 diagrams the LTD scenarios in relation to the open-order and serial correlation assumptions.

The investigation of Y and W requires an order-interval (ρ) assumption. Shorter intervals naturally create more open orders and thereby opportunities for ROC to decrease the ELT’s variance. More importantly, perhaps, Wensing and Kuhn [58] show that ρ has a remarkable effect on the form of the ELT distribution. They show that progressively shorter intervals can transform an intermittent, bipolar ELT distribution into a unimodal distribution with contiguous positive probability loadings over a smaller range (see also [59,60]).

Hadley and Whitin [12] (pp. 201–202) inspire a theory for such findings. They use arrival probability densities for two consecutive orders to describe the region in which ROC is theoretically possible. A decrease in ρ enlarges this region and, thereby, increases the incidence of ROC and multiple arrivals in the same period. These effects supply more opportunities to create instances of P(L = l) > 0, as well as augment existing probability loadings. The experiments set the expected ρ = 1 to emphasize shape effects and then match the values of EOQ and A as follows: EOQ = ρ∙${\mu }_D$ and A = (h∙EOQ²)/(2∙S).

4. Results

4.1. One Open Order (X, U)

Table 4. Demand variability effects on attributes of X and U by the lead-time case.

	D	L				X				U
Case	CV	CV	ν	κ	Peaks	CV	ν	κ	Peaks	CV	ν	κ	Peaks
1		0.14	−0.56	5.81	2
	0.1					0.14	−0.44	5.26	6	0.15	−0.37	5.05	5
	0.2					0.16	−0.19	4.30	2	0.17	−0.04	4.02	2
	0.3					0.19	0.04	3.69	2	0.21	0.22	3.56	2
	0.4					0.23	0.22	3.43	1	0.26	0.40	3.48	1
	0.7					0.34	0.59	3.58	1	0.40	0.79	3.98	1
	1.0					0.47	0.88	4.18	1	0.56	1.12	4.91	1
2		0.33	0.75	2.45	2
	0.1					0.33	0.75	2.50	8	0.33	0.76	2.55	6
	0.2					0.34	0.74	2.64	3	0.34	0.78	2.82	3
	0.3					0.35	0.74	2.83	2	0.37	0.83	3.18	2
	0.4					0.38	0.74	3.04	1	0.40	0.88	3.57	1
	0.7					0.46	0.84	3.65	1	0.51	1.11	4.69	1
	1.0					0.57	1.02	4.30	1	0.65	1.38	5.91	1
3		0.18	0.96	6.97	2
	0.1					0.19	0.89	6.47	5	0.19	0.89	6.38	6
	0.2					0.21	0.77	5.48	3	0.22	0.79	5.39	1
	0.3					0.24	0.68	4.68	1	0.26	0.74	4.70	1
	0.4					0.28	0.65	4.21	1	0.30	0.75	4.38	1
	0.7					0.41	0.80	4.07	1	0.45	0.96	4.58	1
	1.0					0.55	1.07	4.71	1	0.62	1.27	5.55	1
4		0.16	1.27	7.75	2
	0.1					0.17	1.18	7.22	7	0.17	1.16	7.07	4
	0.2					0.18	0.99	6.11	2	0.19	0.96	5.86	1
	0.3					0.21	0.81	5.12	1	0.23	0.83	5.00	1
	0.4					0.23	0.72	4.48	1	0.27	0.79	4.55	1
	0.7					0.34	0.72	3.94	1	0.40	0.91	4.50	1
	1.0					0.46	0.90	4.26	1	0.55	1.17	5.22	1
5		0.30	2.59	15.64	3
	0.1					0.31	2.47	14.84	7	0.31	2.46	14.81	7
	0.2					0.33	2.15	12.60	4	0.33	2.18	12.95	4
	0.3					0.35	1.87	10.74	2	0.36	1.89	10.97	1
	0.4					0.39	1.61	8.93	1	0.40	1.67	9.41	1
	0.7					0.52	1.31	6.36	1	0.56	1.48	7.50	1
	1.0					0.68	1.43	6.32	1	0.74	1.66	7.90	1

Notes. A shaded value with red font signifies a reduction in absolute value compared to the prior CV level, while a bold font indicates a U value that is lower than the corresponding X value.

outlines the impact of the demand CV on the shape attributes of X and U by lead-time case. The CV levels of X and U consistently increase, with higher levels for U from the markup factor. By contrast, the number of peaks for X and U decreases non-monotonically to one. This transformation occurs with CV_D levels between 0.2 and 0.4 for both X and U. Additionally, both ν and κ show inconsistent responses to demand variability within and between the LTD scenarios. Key findings are that (1) the form of the LTD is sensitive to changes in demand variability at low levels; (2) the markup factor accelerates the unimodal LTD transition for U; and (3) the higher moments can react irregularly to the demand CV.

Figure 5 presents the maximum absolute errors in ATC and P₂ among all cases for P₁ targets by CV_D level. The results are for the LTD scenario X, as the outcomes for U are comparable, albeit typically displaying marginally lower levels. The top chart shows ATC error curves, where a 5% limit serves as a guideline for practical applications. This threshold accounts for the challenges in discerning the stochastic nature of LTD distributions as noted by Lau and Lau [6] (p. 50). Additionally, it acknowledges other practical difficulties encountered in acquiring precise cost parameter estimates, as discussed by Hadley and Whitin [12] (Chs. 4 and 9), and Silver et al. [46] (p. 275). Recall that a CV_D < 0.70 distinguishes between smooth and erratic demand time series, which characterize smooth items and erratic items.

The upper chart shows that demand variability generally lowers the ATC error curves. For smooth items, the curves first decline from 3.5% to below 1% as P₁ increases from 0.90 to 0.98 and then rise sharply. The corresponding P₂ values range from 0.987 to 0.997. In the case of erratic items, the maximum absolute errors remain nonsignificant until P₁ exceeds 0.99. At 0.999, these errors reach 5.2% for CV_D = 0.7 and 0.66% for CV_D = 1.0, with P₂ values of 0.998 and 0.9998. The bottom chart shows that maximum absolute errors for P₂ decrease non-monotonically with P₁. The grand maximum error is about a point (0.01). For smooth items, the error is below 0.005 when P₁ ≥ 0.94, while for erratic items, it stays below 0.002 along all P₁ values.

4.2. Multiple Open Orders (U, W)

To ensure clarity and brevity, the ensuing discussion focuses on results for U vs. W under worst-case conditions, as the findings for X vs. Y show similar patterns. Case 5 is significant as it primarily identifies the highest absolute ATC error levels, with CV_D = 0.1 resulting in the greatest errors. Figure 6 shows charts comparing its effective lead-time (ELT) and effective lead-time demand (ELTD) density and distribution functions. Panel A displays an ELT distribution with a lower CV than the L distribution and no gaps between probability loadings. The charts in Panel B reveal the effect of these differences on the respective density functions. Panel C shows that the gamma approximation matches W better than U when P₁ > 0.90.

Figure 7 expands the case analysis in Figure 6 to include ATC errors, where the ATC errors in Panel D are lower for W than U over an expanded P₁ range. The maximum and mean ATC errors for all cases in Panel E confirm the Case 5 findings and indicate that the highest P₂ level increases from 0.997 to 0.9996 in the scenario for W. The maximum absolute errors in P₂ also decrease, though not shown in Figure 7.

5. Discussion

The following discussion aims to validate the results and to review our study’s propositions.

5.1. External Validation

The level of A plays a critical role in defining LTD scenarios, order sizes, and frequencies, and P₂ levels. Among other things, the level of b influences a P₁ target based on the CR. Industry benchmarks validate experimental values. As indicated in Section 3.1.2, Kropf and Sauré [44] projected that the fixed cost associated solely with ancillary shipment planning and control activities would be approximately $8400 in 2024 US dollars. For comparison, the extent of experimental A values can be developed from a representative range of unit costs (c) for given holding-cost factors (w) and experimental A/h ratios. With values of c ranging from $50 to $5000 U.S. dollars, a w of 15%, and the minimum A/h level across all cases, the imputed A ranges from $10 to $1041. At the highest A/h level, the results range from $72 to $7212. Increasing w from 15% to 30% naturally doubles these ranges. Thus, by the $8400 benchmark, which is conservative in the sense that it does not consider fixed shipping-cost elements, the experimental values of A cover a wide range of realistic item costs in the one open-order LTD scenarios. In the multiple open-order scenarios, however, the levels of A do not exceed $150. Determining the realism of such levels is beyond the scope of this study.

Empirical studies report b values ranging from 3 to 20 in retail [61,62].

Table 5. Experimental values of b and industry benchmarks.

Critical Ratio Service Target	Mean Fill Rate	Mean Shortage Cost per Unit	Mean Absolute Percent Error in Total System Cost
P1	P2	b	ATC
0.900	0.9851	1.15	0.4%
0.910	0.9865	1.29	0.4%
0.920	0.9878	1.47	0.3%
0.930	0.9893	1.69	0.2%
0.940	0.9909	2.00	0.2%
0.950	0.9923	2.42	0.2%
0.960	0.9937	3.06	0.1%
0.970	0.9950	4.12	0.1%
0.980	0.9966	6.25	0.1%
0.990	0.9983	12.62	0.9%
0.991	0.9986	14.04	1.2%
0.992	0.9988	15.81	1.5%
0.993	0.9989	18.09	2.0%
0.994	0.9992	21.12	2.6%
0.995	0.9994	25.37	3.6%
0.996	0.9995	31.75	5.0%
0.997	0.9997	42.37	7.2%
0.998	0.9998	63.62	11.1%
0.999	0.9999	127.37	19.9%

Notes: shaded values are within industry benchmarks, which range from 2 to 20; mean values among all cases and CV_D levels; the mean unit cost of overage ($\overline{h}$) = 0.128.

shows B levels, P₂ values, and ATC errors for each P₁ target. Values highlighted meet the B benchmarks, where the P₁ target ranges from 0.94 to 0.994, the mean P₂ ranges from 0.99 to 0.999, and the mean absolute ATC errors are between 0.2% and 2.6%. Caplice and Kalkanci [49] report a 0.92 CR upper limit, approximately matching a 0.99 fill rate in Table 5.

5.2. Propositions

The first proposition is that fast-moving item demands typically follow a standard distribution. Section 3.1.1 documented a significant body of empirical evidence that supports the use of standard statistical distributions, such as normal and gamma, to model demand. The evidence, however, does not imply that the lead-time demand (LTD) distribution invariably follows the demand models, particularly with nonstandard lead times. This study employed a gamma distribution for efficiency, given its ability to model an exponential form when the CV = 1, and a normal form when CV < 0.30. Different standard distributions over specific CV ranges should not materially affect the results found in this study.

The second proposition is that demand variability systematically alters the form of the LTD distribution in ways that enhance accuracy. Several original findings support this proposition. First, a nonstandard distribution of L does not necessarily lead to a similar LTD distribution, as it is also contingent upon demand variability. Second, even at low levels (CV_D ≤ 0.4), demand variability significantly impacts the form of the LTD. Third, the changes to the form almost always improve the accuracy of a gamma approximation. More specifically, the maximum absolute errors in ATC remain below 5% under conventional conditions. For items with a smooth demand process (CV < 0.7), the errors decrease from approximately 3.5% at CV = 0.1 to below 1% at CV = 0.4 across a range of P₁ targets capable of achieving a P₂ as high as 0.997. For items with erratic demand, errors remain below 1% for P₁ targets from 0.90 to 0.999, where the maximum P₂ exceeds 0.999. Meanwhile, the maximum absolute P₂ error decreases non-monotonically across the entire P₁ range, from about 0.01 to under 0.005 for smooth demand items, and stays below 0.002 for erratic demand items.

The third proposition is that nonclassical conditions generally improve the accuracy of approximations when they are based on reasonably accurate parameter estimates. Under the one open-order condition, established heuristics supply suitable estimates of the variance of U in the presence of dependent demands and correlated forecast errors. These heuristics incorporate a markup factor on the variance that produces ATC and P₂ errors with marginally lower levels for U compared to X. Under the multiple open-order condition, heuristic methods also generate reasonably accurate parameter estimates of the ELT distribution that emerges from random order crossovers. It has been shown that increasing the order frequency converts a nonstandard distribution of L into a well-behaved one regarding modality, symmetry, and intermittency. This paper presented an explanation for this phenomenon and demonstrated its benefit to accuracy. The findings show that a gamma LTD approximation with proper parameter estimates improves accuracy compared to conventional methods.

Other results corroborate findings in Lau and Lau [53], indicating that higher moments may not effectively predict ATC error levels at increasing service levels. The findings indicate that decreasing returns to service from incremental increases in reorder points and safety stock as P₁ approaches unity do not necessarily result in higher ATC errors.

6. Conclusions

This paper systematically examined the impact of demand variability on the shape of the LTD distribution and the accuracy of the gamma approximation. Additionally, it expanded the usual inquiry to include nonclassical conditions. The business setting encompasses functional items that usually have demand in each period, with quantities exhibiting either smooth or erratic time series patterns. Fill rate targets in the 0.97–0.99 range commonly satisfy a high product availability priority. Modern ERP systems provide demand data, and system modules or external sources collect appropriate lead-time data. Significantly, this setting supports the use of standard statistical forms to represent demand processes accurately.

This study introduced a theory predicting that elevated levels of demand variability systematically change the LTD distribution in ways that improve the accuracy of a gamma LTD approximation. The investigation examined documented nonstandard lead-time distributions selected to challenge gamma LTD approximations. Thresholds of 5% for absolute ATC errors and 0.01 for fill rate errors guide practical applications.

In context, the gamma LTD approximation in an R, Q inventory system can effectively manage fast-moving goods with disruptive lead times. At the lowest experimental CV_D level, smooth inventory items can achieve P₂ targets up to 0.997, with maximum absolute errors ≤ 0.01 and maximum ATC errors ≤ 3.5%. These benchmarks drop to 0.002 and ≈1% for erratic items. The error curves decrease non-monotonically at higher CV_D levels and generally decrease further under nonclassical conditions. Thus, the gamma LTD approximation can improve the reliability of ERP systems that use normal LTD approximations. It also offers a convenient, flexible way to double-check the values of b, P₂, and A that align with a P₁ target based on the critical ratio.

Certain multimodal lead-time forms can challenge gamma LTD approximations in practice. These forms often exhibit wide gaps between peaks and distant outliers (see, e.g., [3,5,58]). In such cases, it is advisable to identify and manage the sources of outliers, if possible, before utilizing sophisticated alternatives to a gamma approximation. The management of risks associated with suppliers and geopolitical factors falls outside the scope of this study. Future research could use AI, digital technologies, and big data to reduce risks and improve flawed supply chain processes. This approach is supported by various studies (see, e.g., [63,64]). These technologies can also clarify the potential for gamma LTD approximations in other business sectors where lead-time variability is important, and demand is likely to follow a standard statistical distribution.

The paper closes with the following remarks. Considering in-transit inventory holding costs and landed costs, which include the item cost, in the ATC calculation can significantly reduce the absolute percent errors. While these costs do not influence reorder point decisions, they offer a total logistics cost perspective on the potential advantages of a more accurate, yet numerically complex, method. Finally, as Burgin [1] (p. 509) cautions, “versatile as the gamma distribution is, it is not a universal distribution to be employed in all circumstances”. However, as a generalization, it has great potential for controlling the inventory of fast-moving goods under both conventional and non-classical conditions.