A New Approach to Forecast Intermittent Demand and Stock-Keeping-Unit Level Optimization for Spare Parts Management

Abstract

The intermittent and lumpy demand of spare parts requires the choice of the right forecasting model among a variety of existing methods. Spare parts have an uneven lifecycle and mean time to failure for each individual item. As a result, they have a varied time of replacement, and consequently, a varied demand. This paper introduces a multi-cost function optimization approach that dynamically selects and adjusts forecasting models tailored to each spare part. The performance comparisons of the various demand forecasting methods led us to a new forecasting model, the Sfiris–Koulouriotis (SK) method, suited for highly lumpy and intermittent demand. A scaled version of the novel Stock-Keeping Unit-oriented Prediction Error Costs metric is also introduced. The composite negative-binomial–Bernoulli probability distribution for the stock control leveraged the replenishment policy. The best safety stock level is calculated for each individual item. Empirical validation in the automotive industry demonstrated that our approach significantly reduces safety stock while maintaining service levels, offering practical benefits for inventory management.

1. Introduction

1.1. Background and State of the Art in Spare Parts Demand Forecasting

Slow-moving spare parts often account for a large share of inventory value, with industry reports indicating up to 60% of total stock [1]. Their demand is intermittent and unpredictable, creating significant forecasting challenges and increasing the risk of both stock-outs and excess inventory under common stock control policies [2]. Effective management therefore requires accurate categorization by demand type and the selection of forecasting methods tailored to each stock-keeping unit (SKU) [3].

Intermittent-demand forecasting was formalized by Croston, who modeled separately (i) the sizes of non-zero demands and (ii) the intervals between them, applying simple exponential smoothing (SES) to each and taking their ratio for per-period forecasts [4]. Although widely implemented in ERP and forecasting software, Croston’s method was shown to be positively biased, leading to the Syntetos–Boylan Approximation (SBA), which mitigates this bias and often improves accuracy [5,6]. Later methods addressed issues such as obsolescence and declining demand; the Teunter–Syntetos–Babai (TSB) model replaces inter-demand intervals with demand-occurrence probabilities that decay exponentially, while hyperbolic exponential smoothing (HES) adopts hyperbolic decay. Both seek better performance as demand fades or disappears [7,8,9]. Reviews summarize these developments and the remaining challenges in handling extreme lumpiness, shifting occurrence processes, and industry-specific constraints [10,11].

Applied studies in spare-parts contexts reinforce the managerial importance of appropriate method choice and classification [12,13,14]. Despite several surveys, further work is needed to test alternative paradigms and validate methods on operational datasets [15].

Evaluating accuracy for intermittent series is non-trivial. Traditional metrics (MAE, MSE) can be scale-dependent and insensitive to long zero runs, while MAPE becomes unstable in the presence of many zeros [16,17]. Aggregation remedies may blur the timing of events and misrepresent service impact. Scaled metrics such as MASE aid cross-series comparison, and inventory-linked measures like periods in stock (PIS) and its scaled variant (sAPIS) connect errors to service performance, though difficulties persist for highly lumpy profiles [18,19]. More recently, the stock-keeping-unit-oriented prediction error costs (SPEC) metric [20] reframed evaluation in cost terms; here we adapt it into a scaled SPEC (sSPEC) tailored to automotive parts and use it alongside MASE and sAPIS for a multifaceted assessment. Recent surveys highlight movement toward such domain-aware metrics and hybrid evaluation frameworks [21].

Accurate demand categorization underpins these choices. Early schemes grouped series by variance into “smooth,” “sporadic,” and “slow-moving” categories [22,23]. Building on the concept of intermittence in Croston-type models, a widely used refinement distinguishes smooth, intermittent, lumpy, and erratic classes via demand frequency and size variability; this taxonomy is now standard in spare-parts research and practice [24,25,26].

1.2. Objectives and Structure of the Article

This study pursues the following six objectives: (i) to improve forecasting and inventory decisions for slow-moving automotive spare parts through a stock-keeping-unit-wise multi-method selection and optimization strategy; (ii) to introduce the Sfiris–Koulouriotis (SK) method for a subset of intermittent demand profiles, building on the Teunter–Syntetos–Babai framework by estimating demand occurrence non-parametrically as the empirical frequency of occurrence (non-zero counts over elapsed periods), enabling real-time adjustment with fewer parameters; (iii) to develop an SSA-aided variant (SK–SSA) aimed at stabilizing forecasts for highly lumpy and erratic series; (iv) to evaluate performance using a scaled stock-keeping-unit-oriented prediction error costs metric (sSPEC) alongside established measures such as MASE and sAPIS; (v) to examine demand classification and per-class selection policies, ensuring interpretability and practical deployability across heterogeneous items; and (vi) to connect forecast accuracy to operations via a continuous-review (R,Q) policy with negative-binomial–Bernoulli lead-time demand (LTD), reporting results in terms of service levels, safety stock, and backorders.

Briefly, the integrated framework improves accuracy over strong Croston-family baselines and yields tangible inventory gains. SK provides additional improvements for targeted intermittent profiles with low parameter burden; SK–SSA enhances stability on highly lumpy and erratic series. Using sSPEC to guide tuning leads to better inventory outcomes than traditional error metrics alone, and the (R,Q) + LTD linkage translates accuracy into lower safety stock and fewer backorders. The procedure remains computationally light and compatible with enterprise systems.

Recent advances in machine learning and probabilistic forecasting have also targeted intermittent demand. Transformer-based neural networks [27], modified k-Nearest-Neighbor frameworks [28], and feature-based or probabilistic combination models [29,30] have achieved competitive accuracy in richer data environments. However, these methods typically require long, dense training histories, extensive hyper-parameter tuning, and specialized hardware, which limits their deployability for sparse, SKU-level demand in enterprise systems. In contrast, the present study prioritizes parameter-light, interpretable models with closed-form updates that can operate efficiently within existing ERP infrastructures. The proposed SK and SK–SSA methods therefore complement rather than compete with data-hungry approaches, providing an implementable solution for organizations managing thousands of low-volume parts.

The remainder of the paper is organized as follows. Section 2 (Materials and Methods) presents the study context, data, and preprocessing (Section 2.1); the forecasting models, including comparators and proposed methods (Section 2.2); the error metrics, cost functions, and optimization objective (Section 2.3); the MCOST optimization framework, training protocol, and computing setup (Section 2.4); demand pattern classification and methods grouping (Section 2.5); the inventory control model and lead-time demand specification (Section 2.6); and the statistical testing, robustness analysis, and addressed research gaps (Section 2.7). Section 3 reports the empirical results. Section 4 provides the discussion. Section 5 presents the conclusions. A complete list of symbols and notation used throughout the paper is provided in Appendix B (

Table A2. Symbols and terms used in the study.

Symbol	Description
$y_t$	Observed demand in period $t$ (zero or non-zero)
$I_t$	Indicator of non-zero demand in period $t:$ $I_t=1\left\{y_t>0\right\}$
$n_z\left(t\right)$	Cumulative count of non-zero periods
$z_t$	Non-zero demand size observed at $t$ (only when $y_t>0$)
${\widehat{z}}_t$	Non-zero demand size forecast (smoothed estimate) at $t$
$x_t$	Inter-demand interval observed at $t$
${\widehat{x}}_t$	Inter-demand interval forecast (smoothed estimate) at $t$
${\widehat{y}}_{t+h|t}$	Forecast of demand $h$ steps ahead made at $t$
$r_t$	Demand rate in period $t$(in-sample values)
$n$	Number of in-sample periods
$h$	Forecast horizon (look-ahead)
$\alpha ,\beta$	Smoothing parameters (method-dependent)
$d_t$	Occurrence probability in period $t$
${\widehat{d}}_t$	Estimated occurrence probability at $t$
$W,W_t$	Window length and available window ($W_t=\min \left(W,t\right)$)
$\lambda$	Blend weight between windowed and cumulative estimators
${\alpha }_0,{\beta }_0$	Laplace correction parameters
ε	Small positive lower bound (size floor) applied to prevent numerical underflow in non-zero size updates (typical value ε = 10−6).
δ	Probability clipping constant to keep the occurrence estimate within (δ, 1 − δ) during cumulative/windowed/blended updates (typical value δ = 10−6).
$L_{SSA}$	SSA window (embedding) length used to form the trajectory matrix for size-series reconstruction in SK–SSA.
k	Number of leading SSA components retained in the reconstruction (controls smoothing strength) in SK–SSA.
${\tilde{y}}_{L_{SSA},k,t}$	SSA-smoothed demand series at $t$ using window $L_{SSA}$ and $k$ reconstructed components
${\tilde{z}}_{L_{SSA},k,t}$	SSA-smoothed non-zero demand size at $t$
ADI	Average inter-demand interval (Equation (31)).
$C{V}^2$	Squared coefficient of variation of non-zero demand sizes (Equation (32)).
KH-Select	Kourentzes–Hyndman classification
MCOST	Multi-cost optimization (per sku and model selection)
$M$	Sets of forecasting methods
$C$	Set of cost/accuracy objectives
$V_{ij}$	Out-of-sample loss for each method ($j\in M$) and cost objective $(i\in C)$.
$R_{ij}$	Adjusted MCOST score (Equation (30)) blending within-bounds success.
LTD	Lead-time demand random variable
$\mathrm{SS}$	Safety Stock
$\mathrm{R},\mathrm{Q}$	(R,Q) continuous-review policy parameters (reorder point, order quantity)
$L$	Supplier lead time (days)
NB, Bernoulli	Negative binomial size; Bernoulli occurrence
NBB	Negative-binomial-Bernoulli (composite NBB LTD)
sSPEC	Scaled stock-keeping-unit-oriented prediction error costs
sAPIS	Scaled absolute periods in stock
MASE	Mean absolute scaled error
MSR, MAR	Mean squared/absolute eate
PIS, APIS	Periods in stock/absolute PIS

Where symbols are overloaded (e.g., L for SSA window vs. lead time), the meaning is explicit from context or subscript.

).

2. Materials and Methods

2.1. Study Context, Data, and Preprocessing

The study focuses on slow-moving automotive spare parts with intermittent demand. For each stock-keeping unit (SKU), demand per period is recorded as a non-negative integer. Transactional records were aligned to a common observation window; missing entries were treated as zero demand; and invalid records (negative or non-integer values) were removed. Data were captured daily and aggregated to weekly series for forecasting. Each SKU contributes 104 consecutive weekly observations. Calendar effects such as public holidays and plant shutdowns were not modeled explicitly; any such effects are absorbed into the weekly totals. No seasonal adjustment was applied, consistent with the predominance of intermittent, low-signal series. All experiments use a univariate, one-step-ahead forecasting setup. Aggregate approaches from the literature (e.g., ADIDA, IMAPA) are acknowledged due to their relevance to intermittence via temporal aggregation [31,32], but they are not included in the empirical comparisons so as to maintain computational efficiency and a transparent link to inventory policy.

2.2. Forecasting Models: Scope, Comparators, and Proposed Methods

The model pool comprises established intermittent-demand methods together with newly developed variants. Closed-form update equations are provided in

Table 1. Definitions of intermittent demand forecasting methods used in this study.

Method	Formula	Equation Ref.
Croston	${\widehat{y}}_t={\widehat{z}}_t/{\widehat{x}}_t,$ $\left\{\begin{array}{l}{\widehat{z}}_t={\widehat{z}}_{t-1}+\alpha \left(z_t-{\widehat{z}}_{t-1}\right), {\widehat{x}}_t={\widehat{x}}_{t-1}+\alpha \left(x_t-{\widehat{x}}_{t-1}\right) if y_t>0 \\ {\widehat{z}}_t={\widehat{z}}_{t-1}, {\widehat{x}}_t={\widehat{x}}_{t-1} otherwise\end{array}\right.$	(1)
SBA	${\widehat{y}}_t=\left(1-\beta /2\right){\widehat{z}}_t/{\widehat{x}}_t,$ $\left\{\begin{array}{l}{\widehat{z}}_t={\widehat{z}}_{t-1}+\alpha \left(z_t-{\widehat{z}}_{t-1}\right), {\widehat{x}}_t={\widehat{x}}_{t-1}+\beta \left(x_t-{\widehat{x}}_{t-1}\right) if y_t>0 \\ {\widehat{z}}_t={\widehat{z}}_{t-1}, {\widehat{x}}_t={\widehat{x}}_{t-1} otherwise\end{array}\right.$ $\beta$ is a smoothing constant for demand intervals, further refining the original forecast.	(2)
TSB	${\widehat{y}}_t={\widehat{d}}_t{\widehat{z}}_t,$ $\left\{\begin{array}{l}{\widehat{z}}_t={\widehat{z}}_{t-1}+\alpha \left(z_t-{\widehat{z}}_{t-1}\right), {\widehat{d}}_t={\widehat{d}}_{t-1}+\beta \left(d_t-{\widehat{d}}_{t-1}\right) if y_t>0 \\ {\widehat{z}}_t={\widehat{z}}_{t-1}, {\widehat{d}}_t={\widehat{d}}_{t-1}+\beta \left(0-{\widehat{d}}_t\right) otherwise\end{array}\right.$	(3)
mSBA	${\widehat{y}}_t=\left(1-\beta /2\right){\widehat{z}}_t/{\widehat{x}}_t,$ $\left\{\begin{array}{l}{\widehat{z}}_t={\widehat{z}}_{t-1}+\alpha \left(z_t-{\widehat{z}}_{t-1}\right), {\widehat{x}}_t={\widehat{x}}_{t-1}+\beta \left(x_t-{\widehat{x}}_{t-1}\right) if y_t>0 \\ \left\{\begin{array}{l}{\widehat{z}}_t={\widehat{z}}_{t-1}\\ {\widehat{x}}_t=\left\{\begin{array}{l}{\widehat{x}}_{t-1} if x_t\le {\widehat{x}}_{t-1}\\ {\widehat{x}}_{t-1}+\beta \left(x_t-{\widehat{x}}_{t-1}\right) if x_t>{\widehat{x}}_{t-1} \end{array}\right.\end{array}\right.otherwise\end{array}\right.$	(4)
mTSB	${\widehat{y}}_t={\widehat{d}}_t{\widehat{z}}_t,$ $\left\{\begin{array}{l}{\widehat{z}}_t={\widehat{z}}_{t-1}+\alpha \left(z_t-{\widehat{z}}_{t-1}\right), {\widehat{d}}_t={\widehat{d}}_{t-1}+\beta \left(d_t-{\widehat{d}}_{t-1}\right) if y_t>0 \\ \left\{\begin{array}{l}{\widehat{z}}_t={\widehat{z}}_{t-1}\\ {\widehat{d}}_t=\left\{\begin{array}{l}{\widehat{d}}_{t-1}+\beta \left(0-{\widehat{d}}_t\right) if d_t\le {\widehat{d}}_{t-1}\\ {\widehat{d}}_{t-1}+\beta \left(1-{\widehat{d}}_t\right) if d_t>{\widehat{d}}_{t-1} \end{array}\right.\end{array}\right.otherwise\end{array}\right.$	(5)

, and a concise summary of each method’s purpose and key contribution is given in

Table 2. Method summary: purpose and key contribution.

Method	Purpose	Key Contribution
Croston	Forecasts intermittent demand by separating non-zero demand sizes and inter-demand intervals.	Foundational method but tends to introduce positive bias.
SBA	Improves Croston’s method by correcting the bias in demand estimation.	Reduces overestimation bias present in Croston’s method.
TSB	Addresses obsolescence issues by incorporating demand occurrence probabilities.	Better accounts for periods with zero demand, especially for end-of-lifecycle products.
mSBA	Further improves SBA by incorporating updates during zero-demand periods.	Updates inter-demand intervals during zero-demand periods.
mTSB	Enhances TSB by refining updates during zero-demand periods.	Combines benefits of mSBA and TSB.
SK	Simplifies probability-of-occurrence estimation using a non-parametric ratio, combined with exponentially smoothed non-zero size.	Parameter-light, interpretable updates that track intermittence without an extra smoothing parameter for occurrence.
SK–SSA	Applies SSA to stabilize non-zero size updates prior to the SK probability–size combination, targeting highly lumpy series.	Improved robustness on volatile, lumpy demand by reducing size-estimate variance while retaining SK’s simplicity.

. The comparator methods are as follows:

• Croston-type models: Croston decomposes demand into a non-zero demand size and an inter-demand interval, each updated by exponential smoothing; the per-period forecast is the ratio of these components [4]. Its positive bias motivated the Syntetos–Boylan Approximation (SBA), which reduces overestimation under intermittence [6];
• Probability-of-occurrence models: Teunter–Syntetos–Babai (TSB) replaces inter-demand intervals with a demand-occurrence probability updated every period, while the demand-size component is updated only when non-zero demand occurs, thereby accommodating obsolescence and declining usage [8];
• Modified Croston variants: mSBA updates the inter-demand interval during zero-demand runs by comparing observed and estimated spacing; mTSB analogously updates the occurrence probability during zero-demand periods [33].

The new methods introduced in this study are as follows:

• Sfiris–Koulouriotis (SK): Retains TSB’s probability-of-occurrence concept but estimates occurrence non-parametrically as the ratio of non-zero to zero occurrences within an online window, reducing free parameters and enabling responsive updates;
• SK–SSA: Applies a Singular Spectrum Analysis–inspired stabilization to highly lumpy sequences prior to the SK updating scheme to improve robustness.

The field of intermittent demand forecasting is evolving. Recent advancements include Transformer-based models, modified k-nearest-neighbor frameworks, and feature-based or probabilistic combinations [27,28,29,30]. The present study confines its empirical analysis to interpretable, non-aggregated Croston-family methods and the proposed SK/SK–SSA variants for three reasons: (i) alignment with an explicit inventory objective (sSPEC) and direct LTD linkage; (ii) item-wise deployment with closed-form, CPU-only updates suitable for ERP environments; and (iii) consistent evidence that sparse, short histories limit the practical advantage of data-hungry global models in spare-parts settings. A head-to-head comparison with recent machine-learning approaches is therefore outside the scope and left for future work.

We herein define the SK forecasting method for intermittent demand, extending the probability-of-occurrence idea of TSB while simplifying its updating and improving stability during extended zero-demand stretches.

The one-step-ahead forecast is

$${\widehat{y}}_{t+1}={\widehat{d}}_t{\widehat{z}}_t,$$

(6)

where ${\widehat{d}}_t$ denotes the estimated probability of non-zero demand (occurrence probability) and ${\widehat{z}}_t$ is exponentially smoothed size of non-zero demands (as in Croston-type/TSB schemes).

Let $y_t$ denote the observed demand in period $t$, and $I_s=1\left\{y_s>0\right\}$ be the indicator of a non-zero demand in period $s$. The cumulative count of non-zero periods is $n_z\left(t\right)={\displaystyle {\sum }_{s=1}^t{I}_s}$. The size component of the SK method follows an exponential-smoothing update (only when demand occurs) as follows:

$${\widehat{z}}_t=\left\{\begin{array}{ll}{\widehat{z}}_{t-1}={\widehat{z}}_t+\alpha \left(y_t-{\widehat{z}}_{t-1}\right),& y_t>0\\ {\widehat{z}}_{t-1}={\widehat{z}}_t,& y_t=0\end{array}\hspace{1em}\right.\alpha \in \left(0,1\right]$$

(7)

and the occurrence probability is estimated non-parametrically by

$${\widehat{d}}_t^{cum}={\displaystyle \frac{n_z\left(t\right)}{t}}={\displaystyle \frac{{\displaystyle {\sum }_{s=1}^t{I}_s}}{t}}$$

(8)

Equation (8) is the Bernoulli cumulative maximum-likelihood estimate under stationarity. A Laplace correction can be used at the start of a series as follows:

$${\widehat{d}}_t={\displaystyle \frac{n_z\left(t\right)+{\alpha }_0}{t+{\alpha }_0+{\beta }_0}}, {\alpha }_0,{\beta }_0 \ge 0$$

(9)

Optionally, for greater responsiveness, a trailing-window estimator may replace (or be blended with) Equation (8). Let $W_t=\min \left(W,t\right)$ be the available window length and define the windowed count of non-zero periods as

$$n_z^{\left(W\right)}\left(t\right)={\displaystyle \sum _{s=t-W_t+1}^t{I}_s}.$$

(10)

The corresponding occurrence estimate (with optional smoothing) is

$${\widehat{d}}_t^W={\displaystyle \frac{n_z^{\left(W\right)}\left(t\right)+{\alpha }_0}{W_t+{\alpha }_0+{\beta }_0}}$$

(11)

A convex blend of the windowed and cumulative estimators balances stability and adaptability as follows:

$${\widehat{d}}_t=\lambda {\widehat{d}}_t^{\left(W\right)}+\left(1-\lambda \right){\widehat{d}}_t^{cum}, \lambda \in \left[0,1\right]$$

(12)

To prevent numerical instability when early observations are zero or near-zero, initialization proceeds as

$${\widehat{z}}_0=\max \left(y_1,\epsilon \right), \epsilon > 0 \mathrm{small},$$

(13)

$${\widehat{d}}_0={\displaystyle \frac{{\alpha }_0}{{\alpha }_0+{\beta }_0}},$$

(14)

A lower bound is imposed to ensure numerical stability during long zero-demand runs as follows:

$${\widehat{d}}_t\in \left[\delta ,1-\delta \right],\hspace{1em}\delta ={\displaystyle \frac{1}{W_t+{\alpha }_0+{\beta }_0}}$$

The size estimate is similarly bounded by enforcing ${\widehat{z}}_t\ge \epsilon$ after each update of Equation (7).

During extended zero-demand stretches, ${\widehat{z}}_t$ remains fixed per Equation (7), while ${\widehat{d}}_t$ declines via Equations (8)–(11). Consequently, the forecast ${\widehat{y}}_{t+1}={\widehat{d}}_t{\widehat{z}}_t$ contracts appropriately for intermittent or lumpy demand. A concise pseudocode implementation is provided in Algorithm A1 (Appendix A) for reproducibility.

To improve robustness under erratic spikes and long zero runs, we introduce a stabilized variant, SK–SSA, which combines the SK structure with Singular Spectrum Analysis (SSA) smoothing. SSA decomposes the non-zero size series into additive components via lag embedding and singular-value decomposition [34,35], then reconstructs a denoised signal by retaining the leading $k$ components.

Let ${\tilde{z}}_t$ denote the SSA-reconstructed non-zero size at time $t$ obtained from the subset $\{z_s:y_s>0\}$. The SK–SSA update replaces $y_t$ in Equation (7) by ${\tilde{z}}_t$:

$${\widehat{z}}_t=\left\{\begin{array}{ll}{\widehat{z}}_{t-1}={\widehat{z}}_t+\alpha \left({\tilde{z}}_t-{\widehat{z}}_{t-1}\right),& y_t>0,\\ {\widehat{z}}_{t-1}={\widehat{z}}_t,& otherwise,\end{array}\right.$$

(15)

while keeping ${\widehat{d}}_t$ unchanged from SK. The one-step forecast remains ${\widehat{y}}_{t+1}={\widehat{d}}_t{\widehat{z}}_t$.

Given an embedding window length $L_{SSA}$ and rank $k\le L_{SSA}$, SSA forms an $L_{SSA}\times k$ trajectory matrix from the (non-zero) size series, computes its singular-value decomposition, and reconstructs the smoothed signal by diagonal averaging of the first $k$ components.

In practice, small values such as $L_{SSA}\in \left[6,24\right]$ and $k\in \left\{1,2,3\right\}$ avoid oversmoothing short or highly lumpy series. When zeros create gaps, the lagged autocovariance is computed from available pairs only (missing-pair omission), and the reconstructed signal is mapped back to the original time index. All reconstructions are bounded within $\left[\epsilon ,z_{\max }\right]$ to preserve the same numerical safeguards used for ${\widehat{z}}_0$. The smoothing parameter $\alpha$ is kept consistent with the SK update. SSA parameters $\left(L_{SSA},k\right)$ can be tuned on the training window alongside $\alpha$, or fixed at light defaults (e.g., $L_{SSA}=12,k=2$) for parsimony.

From a computational standpoint SK–SSA adds $O({L_{SSA}}^2)$ work per refresh, which reduces to $O(1)$ per period if updated periodically, remaining CPU-efficient for ERP-scale deployment. Empirically, SK–SSA reduces variance in ${\widehat{z}}_t$ without masking intermittence, yielding lower inventory-aligned costs under sSPEC.

Parameterization followed literature-informed bounds. SMA used a window length $W\in \left[10,30\right]$. SES used a single smoothing parameter $\alpha \in \left[0.10,0.40\right]$. CRO used $\alpha \in \left[0.10,0.40\right]$. SBA used $\alpha \in \left[0.05,0.40\right]$ with a second parameter $\beta \in \left[0.05,0.40\right]$. TSB used $\alpha \in \left[0.05,0.40\right]$ with a second parameter $\beta \in \left[0.10,0.40\right]$. SK used $\alpha \in \left[0.05,0.40\right]$. mSBA used $\alpha \in \left[0.05,0.40\right]$ with a second parameter $\beta \in \left[0.05,0.40\right]$. mTSB used $\alpha \in \left[0.05,0.40\right]$ and $\beta \in \left[0.05,0.40\right]$. SK–SSA used $\alpha \in \left[0.05,0.40\right]$ with SSA components $k\in \left\{1,2,3\right\}$. Final values were obtained via a rolling-origin search; for each SKU and chosen objective, the retained specification is the model–parameter combination with the lowest out-of-sample loss.

2.3. Error Metrics, Cost Functions, and Optimization Objective

To capture different operational priorities and the specific challenges of intermittent series, both conventional and intermittent-aware measures are considered. Formal definitions are given in

Table 3. Error metrics and cost functions: formulas and references.

Error Metric	Formula	Key Citation	Equation Ref
Mean Squared Error	$MS{E}_n=n^{-1}{\displaystyle \sum _{t=1}^n{\left(y_t-{\widehat{y}}_t\right)}^2}$	Standard	(16)
Mean Absolute Error	$MA{E}_n=n^{-1}{\displaystyle \sum _{t=1}^n\left|y_t-{\widehat{y}}_t\right|}$	Standard	(17)
Mean Squared Rate	$MS{R}_n={\displaystyle \sum _{t=1}^n{r_t}^2 },$ $r_t={\widehat{y}}_t-t^{-1}{\displaystyle \sum _{j=1}^t{y}_j}$	Kourentzes, 2014 [19]	(18)
Mean Absolute Rate	$MA{R}_n={\displaystyle \sum _{t=1}^n\left|r_t\right|}, r_t={\widehat{y}}_t-t^{-1}{\displaystyle \sum _{j=1}^t{y}_j}$	Kourentzes, 2014 [19]	(19)
Period In Stock	$PI{S}_n=-{\displaystyle \sum _{t=1}^n{\displaystyle \sum _{j=1}^t\left(y_j-{\widehat{y}}_j\right)}}$	Wallstrom and Segerstedt, 2010 [18]	(20)
Absolute Period In Stock	$API{S}_n=\left|PI{S}_n\right|$	Wallstrom and Segerstedt, 2010 [18]	(21)
Stock-Keeping-Oriented Prediction Error Costs		Martin et al., 2020 [20]
$SPE{C}_{{\alpha }_1,{\alpha }_2}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{t=1}^n{\displaystyle \sum _{i=1}^t\left(max\left[0;min\left[y_i;{\displaystyle \sum _{k=1}^i{y}_k-}{\displaystyle \sum _{j=1}^t{\widehat{y}}_j}\right]\cdot {\alpha }_1;min\left[{\widehat{y}}_i;{\displaystyle \sum _{k=1}^i{\widehat{y}}_k-}{\displaystyle \sum _{j=1}^t{y}_j}\right]\cdot {\alpha }_2\right]\cdot \left(t-i+1\right)\right)}}$			(22)
The constants ${\mathsf{\alpha }}_1\in \left[0,\mathrm{\infty }\right],{\mathsf{\alpha }}_2\in \left[0,\mathrm{\infty }\right]$ define the opportunity and stock-keeping costs, respectively
Absolute Scaled Error	$AS{E}_h=\left|y_h-{\widehat{y}}_h\right|/\left(n^{-1}{\displaystyle \sum _{k=2}^n\left|y_k-y_{k-1}\right|}\right)$	Hyndman and Koehler, 2006 [16]	(23)
Mean Absolute Scaled Error	$MAS{E}_h={\displaystyle \frac{1}{n}}{\displaystyle \sum _{t=1}^{n}AS{E}_h}$	Hyndman and Koehler, 2006 [16]	(24)
Scaled Absolute Period In Stock	$sAPI{S}_h=n\left|-{\displaystyle \sum _{t=1}^h{\displaystyle \sum _{j=1}^t\left(y_j-{\widehat{y}}_j\right)}}\right|/{\displaystyle \sum _{k=1}^n{y}_k}$	Kourentzes, 2014 [19]	(25)
Scaled Stock-Keeping-Oriented Prediction Error Costs		This Paper
${sSPEC}_{h,\alpha 1,\alpha 2}={\displaystyle \frac{n}{h}}{\displaystyle \sum _{t=1}^h{\displaystyle \sum _{i=1}^t\left(max\left[0;min\left[y_i;{\displaystyle \sum _{k=1}^i{y}_k-}{\displaystyle \sum _{j=1}^t{\widehat{y}}_j}\right]\cdot {\alpha }_1;min\left[{\widehat{y}}_i;{\displaystyle \sum _{k=1}^i{\widehat{y}}_k-}{\displaystyle \sum _{j=1}^t{y}_j}\right]\cdot {\alpha }_2\right]\cdot \left(t-i+1\right)\right)}}/{\displaystyle \sum _{k=1}^n{y}_k}$			(26)

. Mean squared error (MSE) and mean absolute error (MAE) provide conventional accuracy assessments, while mean squared rate (MSR) and mean absolute rate (MAR) operate on demand rates to mitigate the effect of long zero runs. Periods in stock (PIS) and absolute PIS (APIS) relate forecast deviation to inventory drift over time. Mean absolute scaled error (MASE) and absolute scaled error (ASE) support scale-independent benchmarking across series. The stock-keeping-unit–oriented prediction error costs (SPEC) metric integrates both stock-keeping and opportunity costs through two non-negative weighting parameters, ${\alpha }_1$ and ${\alpha }_2$, which balance under- and over-forecast penalties. Its scaled form (sSPEC) normalizes SPEC by each SKU’s in-sample mean demand, allowing performance to be compared across items with different magnitudes while retaining the same economic interpretation. Further details and illustrative examples of the SPEC cost structure can be found in [20], from which the present sSPEC formulation is adapted.

In line with the study’s inventory focus, sSPEC serves as the primary optimization criterion, complemented by MASE and sAPIS for robustness across demand types and scales. In this study, ${\alpha }_1$ and ${\alpha }_2$ were set to equal weights (${\alpha }_1$ = ${\alpha }_2$ = 1) to ensure balanced penalization of over- and under-forecasting and to maintain comparability across SKUs; in applied settings, these coefficients can be adjusted to reflect organization-specific cost structures.

The optimization objective is chosen prior to model tuning. For each SKU and chosen cost, parameters are searched by rolling-origin evaluation with expanding windows, generating one-step-ahead forecasts across multiple origins. Parameters are updated based on recent forecast performance to adapt to evolving demand characteristics, and final performance is assessed on a reserved test horizon. Because no single objective consistently reflects performance across smooth, intermittent, erratic, and lumpy profiles, six costs—MSE, MAE, APIS, MAR, MSR, and SPEC—are reviewed and used within the framework. This multi-cost setup enables context-dependent selection and supports robust comparisons across diverse demand scenarios.

2.4. Optimization Framework, Training Protocol, and Computing Setup

This study employs a multi-method, multi-cost (MCOST) optimization carried out per stock-keeping unit (SKU). Let $M=\{\mathrm{SES}, \mathrm{CRO}, \mathrm{SBA}, \mathrm{TSB}, \mathrm{SK}, \mathrm{mSBA},$ $\mathrm{mTSB}, \text{SK-SSA}\}$ denote the set of forecasting methods ($\left|M\right|=9$) and $C=\{\mathrm{MSE}, \mathrm{MAE},$ $\mathrm{PIS}, \mathrm{MSR}, \mathrm{MAR}, \mathrm{SPEC}\}$ denote the set of optimization objectives ($\left|C\right|=6$). This results in $9\times 6=54$ method–objective cases evaluated per SKU. Parameter selection follows a cost-first protocol: choose the objective, then tune model parameters via rolling windows with expanding origin; an approach advocated for intermittent series [19] and evaluated on a reserved test segment in line with recommended practice [36].

For each SKU and case $\left(m,c\right)\in M\times C$, parameters are searched within the bounds in Section 2.2 using a rolling-origin scheme that produces one-step-ahead forecasts at each origin. The in-optimization objective is the chosen cost $c$. This rolling procedure adaptively refines parameters based on recent performance, allowing the fitted specification to track changes in occurrences (zeros versus non-zero) and non-zero size processes over time, consistent with the dynamic tuning rationale of cost-function optimization in [19]. When the optimizer converges to a boundary or the objective is monotone over the admissible range, the solution is retained and flagged as a boundary solution; a within-bounds success indicator is recorded per case. The best parameter vector for $\left(m,c\right)$ is then refit on the full training segment and evaluated on a reserved test window; results are summarized by averaging over multiple forecast horizons.

Traditional level-based losses (MSE, MAE) can be dominated by long zero runs. Rate-based objectives (MSR, MAR) operate on occurrence-adjusted demand rates; inventory-linked measures (PIS) and cost-based SPEC align tuning with managerial outcomes. For reporting and cross-SKU comparison, we use scale-independent summaries (MASE, sAPIS) and the scaled inventory cost sSPEC; sSPEC is emphasized as the primary inventory-facing criterion in the main results.

To compare methods and costs fairly, we down-weight cases that frequently hit parameter bounds. Let $n_{sku}$ be the number of SKUs. For each $\left(i,j\right)\in C\times M$, let $F_{ij}$ be the number of SKUs whose optimized parameters are within bounds. The within-bounds success rate is

$$p_{ij}={\displaystyle \frac{\left(n_{sku}-F_{ij}\right)}{n_{sku}}}$$

(27)

Let $V_{ij}$ denote the out-of-sample error (e.g., sSPEC) for case $\left(i,j\right)$. The SKU-average error over all SKUs is

$$A_{ij}={\displaystyle \frac{1}{n_{sku}}}{\displaystyle \sum _{sku}{V}_{ij}}$$

(28)

Let $C_{ij}=n_{sku}-F_{ij}$ be the number of successful (within-bounds) optimizations and

$$O_{ij}={\displaystyle \frac{1}{C_{ij}}}{\displaystyle \sum _{sku}{V}_{ij}}$$

(29)

the average performance over successful optimizations. The adjusted ranking matrix $R\in {ℝ}^{\left|C\right|\times \left|M\right|}$ blends these via the success probability:

$$R_{ij}=O_{ij}\cdot p_{ij}+\left(A_{ij}+O_{ij}\cdot p_{ij}\right)\cdot \left(1-p_{ij}\right)$$

(30)

For each method $j\in M$, the preferred cost is the column $i^{*}\left(j\right)$ with the lowest $R_{ij}.$ Final per-SKU selection then proceeds as follows: for each SKU, among all methods evaluated under their preferred cost $i^{*}\left(j\right)$, retain the model–parameter combination with the lowest out-of-sample loss under the MCOST criterion. This implements the cost-first idea (choose objective, then tune via rolling windows) while correcting comparisons for infeasible/boundary-dominated cases. The overall selection process is summarized in the selection procedure of

Table A1. MCOST optimization and selection procedure.

Step	Operation	Input(s)	Output(s)	Purpose/Notes
1. Data Preparation	Construct per-SKU weekly demand series (length = 104 periods)	Raw demand transactions	Cleaned, zero-padded weekly series	Align time frames, remove invalid entries
2. Define Model Pool	Select forecasting methods: SMA, SES, CRO, SBA, TSB, SK, mSBA, mTSB, SK–SSA	–	Model set M = {9 methods}	Includes all Croston-family and new SK variants
3. Define Objective Pool	Select optimization criteria: MSE, MAE, PIS, MAR, MSR, SPEC	–	Objective set C = {6 costs}	Covers level-based, rate-based, and inventory-linked losses
4. Rolling-Origin Evaluation	For each (method, cost) pair, forecast one step ahead using expanding-window rolling origins	M, C, historical demand	Forecasts and in-sample loss trajectories	Captures evolving dynamics and parameter adaptation
5. Parameter Optimization	Minimize chosen cost function within method’s parameter bounds	Loss per (method, cost)	Optimal parameter vector θ*	Retain best parameters per SKU and cost
6. Boundary Check	Identify parameter solutions hitting limits of search grid	θ*, parameter bounds	Success indicator (1 = within bounds)	Used later for adjusted rankings
7. Compute Out-of-Sample Performance	Evaluate each (method, cost) on reserved test window	Optimized θ*	Out-of-sample loss (e.g., sSPEC, MASE)	Enables cross-method comparison
8. Adjusted Ranking Matrix (Equation (30))	Combine within-bounds average loss and overall loss weighted by success rate	All per-SKU results	Adjusted R ranking matrix	Prevents overranking boundary-dominated cases
9. Preferred Cost per Method	For each method, identify the cost objective yielding lowest adjusted average loss	R	Best objective per method	Implements “cost-first” optimization philosophy
10. Final Per-SKU Selection	Among all methods optimized under their preferred cost, select method with lowest test loss	Per-SKU results	Chosen model–cost–parameter triple	Final MCOST-optimized forecast per SKU
11. Inventory Evaluation	Map forecasts to continuous-review (R,Q) policy with NBB LTD	Forecasts, lead time	Scaled safety stock, backlog metrics	Quantifies operational outcomes

and detailed in pseudocode in Algorithm A3 (Appendix A).

For each SKU, a single temporal split was used—95% of observations for optimization and 5% held out for evaluation. This short holdout window (≈5 weeks) was selected to preserve sufficient data for model learning given the limited 104-week series, while still providing an independent performance check. Each candidate parameter combination on a discrete grid (covering smoothing factors, window lengths, and SSA components) was evaluated on these rolling forecasts, and the setting with the lowest in-sample loss under the chosen cost function was retained. This grid-based rolling-origin tuning yields adaptive yet reproducible estimates without stochastic effects associated with random search. Final performance was then computed on the reserved out-of-sample segment. SKU-level losses were averaged over the out-of-sample window, and dataset-level summaries report medians and interquartile ranges across SKUs. Where multi-horizon results are shown, metrics (e.g., sSPEC) are averaged over horizons h = 1, …, 5.

No GPU acceleration was used. All computations were performed on a desktop with an AMD Ryzen 5 1600 (six cores) and 64 GB RAM. This setup underscores the practicality and parsimony of the approach relative to heavier machine-learning alternatives, and supports ERP-scale deployment.

2.5. Demand Pattern Classification and Methods Grouping

Demand classification informs both model choice and interpretation. Each time series is categorized as smooth, intermittent, erratic, or lumpy using the average inter-demand interval ($\mathrm{ADI}$) and the squared coefficient of variation of non-zero demand ($C{V}^2$). Thresholds and formulas are given in Equations (31) and (32), and decision regions appear in Figure 1; notation follows Table A2. Following the KH-Select scheme (Kostenko and Hyndman’s refinement of Syntetos–Boylan–Croston classification) [25], we assign each stock-keeping unit (SKU) to one of the four classes and consider a tailored candidate set per class (SMA, SES, CRO, SBA, TSB, mSBA, mTSB, SK, SK–SSA).

Consistent with Kourentzes (2014) [19], KH-Select assignment improves prediction by aligning method families with observed intermittence and size variability, without relying on ad hoc cut-offs tied to a single model. In contrast to the original SBC classification, which primarily contrasted CRO and SBA [24], KH-Select expands the candidate pool to include probability-of-occurrence methods (e.g., TSB), modified Croston variants (mSBA, mTSB) [33], and the proposed SK and SK–SSA. Within each class, we document the frequency with which methods are ultimately selected, providing practitioners with transparent, class-specific preferences.

Importantly, classification is not the terminal decision. Because accuracy alone may not yield the best inventory outcomes, the final per-SKU choice is made by the multi-method, multi-cost selection (MCOST; Section 2.3 and Section 2.4), which evaluates competing model–cost pairs on out-of-sample loss and supports inventory-aligned objectives (e.g., sSPEC). This preserves the interpretability of KH-Select while allowing the data, and the chosen objective, to determine the best method for each SKU. Inventory implications are then quantified under the continuous-review (R,Q) policy with negative-binomial–Bernoulli (NBB) lead-time demand (Section 2.6).

$$ADI={\displaystyle \frac{\mathrm{Total} \mathrm{Number} \mathrm{of} \mathrm{Periods}}{\mathrm{Total} \mathrm{Number} \mathrm{of} \mathrm{Demand} \mathrm{Buckets}}}$$

(31)

$$CV={\displaystyle \frac{\mathrm{Standard} \mathrm{Deviation} \mathrm{of} \mathrm{Demand} \mathrm{Sizes}}{\mathrm{Demand} \mathrm{Mean}}}$$

(32)

The four classes are as follows:

• Smooth: $ADI\le 1.32$ and $C{V}^2\le 0.49$; typically fast-moving items.
• Intermittent: $ADI>1.32$ and $C{V}^2\le 0.49$; sparse arrivals with relatively stable non-zero sizes.
• Erratic: $ADI\le 1.32$ and $C{V}^2>0.49$; frequent demand with volatile size.
• Lumpy: $ADI>1.32$ and $C{V}^2>0.49$; long zero runs and highly variable non-zero sizes.

2.6. Inventory Control Model and Lead-Time Demand Specification

Forecast accuracy is linked to operational performance through a continuous-review $\left(R, Q\right)$ policy with backordering. When the inventory position reaches the reorder point $R$, a fixed quantity $Q$ is ordered. Lead-time demand (LTD) is the random demand accruing over the supplier lead time $L$; in the empirical study $L$ is treated as deterministic and constant at three days ($L=3/7$ week). Cycle-service targets (probability of no stockout during a replenishment cycle) determine $R$, and safety stock is $\mathrm{SS}=R-\mathbb{E}\left[\mathrm{LTD}\right]$.

To capture both over-dispersion in demand sizes and intermittence in arrivals, LTD is modeled with a composite negative-binomial–Bernoulli (NBB) distribution: a negative binomial (NB) for non-zero demand sizes and a Bernoulli for occurrence. This choice follows the literature recommending NB for spare-parts demand and Bernoulli for arrivals under intermittence. Formally, the NB size component is parameterized by $\left(r,p\right)$, and the arrival process by $b$; the construction and basic properties used here are given in Equations (33)–(35). In our implementation, per-period mean and dispersion implied by the selected forecast (Section 2.4) are mapped to $\left(r,p,d\right)$ by method-of-moments estimators (Equations (36)–(38)); edge cases are handled by capping $r\ge 1$ when the implied value is <1, corresponding to geometric limit.

$$P\left(k\right)=\left(\begin{array}{l}k-1\\ r-1\end{array}\right)p^r{\left(1-p\right)}^{k-r}, \forall \mathrm{k}\ge \mathrm{r}$$

(33)

The above equation is constructed based on the summation of geometric distributions, and it is applicable within the set of natural numbers {1, 2, 3, …}, where $0<p\le 1, \mathrm{r}\ge 1 \mathrm{and} \mathrm{integer},$ having the following:

$$\mu ={\displaystyle \frac{r}{p}},$$

(34)

$${\mathsf{\sigma }}^2={\displaystyle \frac{r\left(1-p\right)}{p^2}}.$$

(35)

LTD equals $L$ multiplied by, aka the total demand expected over the lead time. The parameters $p$ and $r$ are estimated as follows:

$$p={\displaystyle \frac{LTD}{{\left(\sigma \cdot LTD\right)}^2+LTD}}$$

(36)

$$r=INT\left(LTD\cdot p\right).$$

(37)

INT denotes the integer part function. In those cases where $r$ < 1, we set $r$ = 1; this corresponds to the limit distribution when $p$ is very small. The parameter $b$ of NBB is calculated using the following formula:

$$b={\displaystyle \frac{1}{ADI}}$$

(38)

where ADI is the average inter-demand interval (see Equation (31)).

Given $\left(r,p,d\right)$ and lead time $L$, closed-form LTD moments and quantiles are used to set policy parameters. The reorder point $R$ is obtained from the upper quantile of the fitted NBB LTD at the chosen cycle-service target—$\mathrm{SS}=R-\mathbb{E}\left[\mathrm{LTD}\right]$. Since Z-scores do not apply directly to NBB, safety stock is determined via the NBB quantile (or its fast numerical approximation) rather than a normal approximation; the relationship to the heuristic form $\mathrm{SS}=k{\sigma }_{LTD}$ (Equations (39)–(41)) is recovered by interpreting $k$ as the NBB-implied service factor for the target.

$$\mathrm{SS}=k\times \sqrt{\mu }\times \sqrt{\mu -r}$$

(39)

$$\mathrm{SS}=k\times \sqrt{{\displaystyle \frac{r\left(r-p\right)}{p}}}$$

(40)

$$\mathrm{SS}=k\times \sigma$$

(41)

where ${\sigma }^2={\mu }^2-\mu$ is the standard deviation of the NBB distribution.

To balance stock holding and service, we simulate LTD draws from the fitted NBB (100,000 replications per setting in the empirical study) and trace (i) backlog–safety-stock trade-off curves and (ii) realized versus target service levels. This directly links forecasting choices to inventory outcomes and guides the selection of $\mathrm{SS}$ for practical service targets (e.g., 0.80, 0.85, 0.90, 0.95, 0.97, 0.99) using the same NBB parameters that encode demand variability and arrival intermittence.

This procedure bases $R$ and $\mathrm{SS}$ on the full LTD distribution—sizes and arrivals—rather than on ad-hoc buffers, leading to more effective and transparent inventory decisions under intermittent demand. In this study, adequacy of the distribution fit was verified empirically by confirming that simulated cycle-service levels and backlog–safety-stock curves under the fitted parameters align with the policy targets. Formal goodness-of-fit tests were not pursued, as prior evidence supports modeling non-zero sizes with a NB distribution and arrivals with a Bernoulli process for discrete intermittent (see [37]). Degenerate cases with implied r < 1 were handled by setting r = 1 (geometric limit) as noted in Equation (37).

2.7. Statistical Testing, Robustness Analysis, and Addressed Research Gaps

Model performance is evaluated at the SKU level and then summarized across SKUs and forecast horizons. Results are reported under multiple accuracy and cost measures to reflect different managerial perspectives (for example, sSPEC as the primary inventory-facing criterion, with MASE, sAPIS, MAR, MSR, MSE, and MAE as complementary views). The reporting emphasizes per-SKU outcomes and aggregated summaries (means/medians and dispersion), avoiding assumptions that are unlikely to hold for intermittent series.

To assess the statistical significance of observed performance differences, pairwise Wilcoxon signed-rank tests were applied across all SKUs for each forecasting method under the primary (sSPEC) and secondary (MASE) metrics. Holm correction was used to control the family-wise error rate. Non-parametric bootstrap confidence intervals (CIs) were also computed for median sSPEC differences between SK-based methods and each baseline. Detailed results, including parameter-sensitivity and computational-efficiency analyses, are reported in Appendix C.

Robustness is examined along three practical dimensions. First, the optimization objective is varied; each model is re-optimized under alternative costs (sSPEC, MASE, sAPIS, MAR, MSR) and conclusions are compared to check that findings are not an artifact of a single metric. Second, sensitivity to parameter ranges is assessed by widening and narrowing the grids for exponential-smoothing and probability-update parameters; stability of the selected model–parameter combinations is inspected. Third, heterogeneity across demand types is explored by repeating summaries on the most challenging subset—items classified as highly intermittent or lumpy by the demand-categorization scheme—verifying that the proposed methods remain competitive where zero runs and size variability are most pronounced. Inventory implications are computed using the negative-binomial–Bernoulli specification within a continuous-review (R,Q) policy, ensuring that accuracy gains translate into service and stock-holding outcomes.

The analyses are designed to address the research gaps identified in the literature. Specifically, the study contributes the following: (i) the SK forecasting model and its SK–SSA analogue, targeting highly intermittent and lumpy demand while maintaining parsimony; (ii) the scaled stock-keeping-oriented prediction error costs metric (sSPEC), aligning model tuning and evaluation with inventory-relevant costs and enabling cross-SKU comparability; (iii) a multi-cost, multi-method optimization framework that selects the most suitable forecasting method per SKU rather than prescribing a single best model; (iv) an inventory linkage through the negative-binomial–Bernoulli representation of lead-time demand embedded in an (R,Q) policy; and (v) a refined use of demand categorization to guide method selection. Section 3 reports the empirical assessment of these components, and

Table 4. Summary of key related literature and research gap identification.

Study	Method	Application	Research Gap and Open Direction
Croston, 1972 [4]	Croston’s Method	Intermittent slow-moving items	Foundational approach; known positive bias and no explicit handling of zero-demand stretches.
Syntetos and Boylan, 2001 [5]	Bias analysis/early correction	Intermittent-demand items across sectors	Showed CRO’s systematic positive bias for intermittent series and derived an adjustment to reduce that bias (foundation of SBA).
Johnston et al., 2003 [1]	Traditional spare-parts forecasting	General spare parts demand	Highlighted challenges of intermittent demand in slow-moving spare parts; did not focus on optimization strategies tailored to distinct demand profiles.
Syntetos et al., 2005 [6]	SBA	Intermittent/lumpy demand	Applied a closed-form bias correction to CRO (multiplicative factor on size/interval ratio) to reduce overestimation under intermittence.
Syntetos et al., 2005 [6]	Demand categorization	Intermittent demand	Provided a practical taxonomy; mapping categories to broader method sets and policy impacts invited further work.
Wallström and Segerstedt, 2010 [18]	PIS/APIS metrics	Inventory-linked errors	Connected error to stock behavior; further handling of long zero runs and high lumpiness encouraged.
Boylan and Syntetos, 2010 [3]	Review	Spare parts management	Called for stronger forecasting–inventory integration, comparative benchmarks on real industrial datasets, better treatment of obsolescence and non-stationarity, and practical guidance on method selection and parameterization for short, sparse histories.
Louit et al., 2011 [2]	Horizontal stock control	Spare parts inventory	Underscored the value of classification; opened space for methods that link categorization to forecasting and policy.
Teunter et al., 2011 [8]	TSB	Obsolescence, slow-moving items	Reduced bias relative to Croston; scope to extend toward more complex patterns.
Bacchetti and Saccani, 2012 [12]	Reviews	Manufacturing spare parts	Synthesized practice–research gap; encouraged empirical testing of alternative models and metrics.
Romeijnders et al., 2012 [13]	Croston-based models	Intermittent spare-parts demand	Discussed bias and pattern sensitivity; further extensions to declining/obsolescent demand remained to be explored broadly.
Prestwich et al., 2014 [9]	HES	Slow-moving and obsolete parts	Addressed decay; applications beyond specific contexts suggested.
Kourentzes, 2014 [19]	MSR/MAR, sAPIS, global cost optimization	Intermittent demand accuracy	Advanced rate-based/objective-first tuning; integration with inventory policies and highly lumpy cases remained an open direction.
Hu et al., 2018 [10]	Review	Spare-parts inventory	Comprehensive overview; called for new empirical models for highly volatile demand.
Turrini and Meissner, 2019 [37]	NB/Bernoulli distributions	Spare-parts demand fitting	Supported NB and Bernoulli for sizes/arrivals; combining them operationally for LTD and policy tuning offered scope.
Martin et al., 2020 [20]	SPEC metric	Lumpy demand in inventory systems	Reframed evaluation in cost terms; scaling/normalization for cross-SKU comparability was a natural next step.
Zhang et al., 2023 [27]	Transformer-based ML	Sparse/irregular demand	Demonstrated potential; real-world deployment can face data and computational constraints.
Hasan et al., 2024 [28]	Modified k-NN	Intermittent demand	Showed ML gains; highlighted training-data and implementation considerations.
Wang et al., 2024 [30]	Probabilistic combinations	Dynamic intermittent demand	Advanced combinations; ERP integration and computational efficiency remain important considerations.
This study	MCOST with SK/SK–SSA; sSPEC; NBB LTD within (R,Q)	Automotive spare parts (intermittent/lumpy)	Contributes a per-SKU, objective-first selection aligned with inventory costs (sSPEC), introduces a non-parametric occurrence estimator (SK) and an SSA-stabilized variant (SK–SSA), and links forecasts to policy via NBB LTD, with CPU-only implementation suitable for ERP environments.

maps the targeted literature gaps to the corresponding elements of the framework.

3. Results

The detailed, comprehensive evaluation of the proposed framework was conducted on an industrial automotive dataset of 2050 stock-keeping units (SKUs), each with 104 weekly observations. This section provides a concise, precise account of the empirical results, their interpretation, and the experimental conclusions. The reporting mirrors the experimental design in Section 2 and proceeds in the same order: data characteristics (Section 2.1); model behavior and parameterization (Section 2.2); objective functions and optimization (Section 2.3 and Section 2.4); demand classification (Section 2.5); and inventory implications under a continuous-review (R,Q) policy with negative-binomial–Bernoulli (NBB) lead-time demand (Section 2.6).

Weekly aggregation of daily transactions produced 104 observations per stock-keeping unit (SKU), as specified in Section 2.1. Descriptive statistics confirm the setting is dominated by intermittence and lumpiness: the cross-SKU average inter-demand interval is about 2.2 weeks, the average demand per period is 18.9 units, and non-zero sizes cluster around 34.9 units. These statistics, summarized in

Table 5. Spare parts data—104 weeks of demand for real dataset of 2050 SKUs.

2050 SKUs	Demand Intervals		Demand Size		Demand Per Period
	Mean	St. Deviation	Mean	St. Deviation	Mean	St. Deviation
Min	1.1	0.2	1.1	0.3	0.4	0.1
25%ile	1.6	0.9	16.6	5.6	7.4	3.0
Mean	2.2	1.6	34.9	16.0	18.9	18.1
75%ile	2.8	2.1	49.7	22.5	25.8	22.2
Max	6.4	7.0	165.5	146.5	101.2	227.9

, support the use of Croston-family forecasting methods and intermittent-aware objectives.

The multi-cost selection protocol (MCOST; Section 2.4) was applied per SKU. Figure 2 depicts the workflow, weekly series construction, rolling-origin forecasting for all method–objective pairs, cost computation, ranking and parameter optimization, out-of-sample evaluation, and inventory analysis. This implementation mirrors the cost-first principle of Section 2.3 and operationalizes the training protocol and computing setup.

All candidate forecasters from Section 2.2 were tuned over literature-informed bounds (

Table 6. Optimization Setup.

Forecast Method	Parameters to Optimize	First Parameter Lower Bound	Upper Bound	Second Parameter Lower Bound	Upper Bound
SMA	1	10	30	-	-
SES	1	0.1	0.4	-	-
CRO	1	0.1	0.4	-	-
SBA	2	0.05	0.4	0.1	0.4
TSB	2	0.05	0.4	0.05	0.4
SK	1	0.05	0.4	-	-
mSBA	2	0.05	0.4	0.1	0.4
mTSB	2	0.05	0.4	0.05	0.4
SK–SSA	2	0.05	0.4	1	3

). Parameter optimization followed a deterministic grid-search procedure, independently applied per SKU and objective. For single-parameter methods (SMA, SES, CRO, SK), the smoothing constant α was evaluated on a uniform grid of 20 equally spaced points within the bounds shown in Table 6. For two-parameter methods (SBA, TSB, mSBA, mTSB, SK–SSA), a 20 × 20 grid was used across both smoothing parameters, resulting in 400 evaluations per SKU–objective pair. Each candidate combination was assessed via expanding rolling-origin evaluation, generating one-step-ahead forecasts across the training window. The configuration yielding the lowest in-sample loss under the chosen cost function was retained. Boundary optima were observed primarily in cases of monotone cost response under prolonged zero-demand runs, consistent with theoretical expectations.

To avoid overstating performance in cases with frequent boundary hits, the adjusted MCOST ranking matrix (Equation (30)) combines within-bounds averages with overall averages using the within-bounds success rate. This correction is applied throughout

Table 7. Cost functions ranking of real-world data set based on sSPEC averaged over five look-ahead horizons.

ALL SKUS AV. SSPEC	SMA	SES	CRO	SBA	TSB	SK	MSBA	MTSB	SK–SSA
1. MSE	0.797	0.767	0.779	0.831	0.779	0.799	0.928	0.803	0.889
2. MAE	0.794	0.757	0.783	0.870	0.793	0.828	0.961	0.838	1.084
3. PIS	0.795	0.786	0.777	0.818	0.730	0.794	0.926	0.811	0.821
4. MSR	0.797	0.767	0.780	0.813	0.776	0.796	0.928	0.801	0.802
5. MAR	0.797	0.769	0.781	0.808	0.780	0.797	0.927	0.802	0.799
6. SPEC	0.797	0.792	0.776	0.817	0.729	0.795	0.926	0.811	0.804
MCOST	0.797	0.767	0.750	0.742	0.672	0.748	0.869	0.732	0.711
%OPTIMIZED
1. MSE	64.15%	99.71%	96.49%	86.24%	65.80%	72.73%	81.95%	66.49%	41.32%
2. MAE	74.93%	86.15%	90.34%	86.29%	84.44%	86.10%	83.61%	83.95%	11.12%
3. PIS	84.77%	2.23%	83.62%	74.00%	24.92%	34.29%	43.79%	30.72%	42.13%
4. MSR	6.12%	100.00%	98.47%	94.58%	62.78%	96.75%	95.73%	63.61%	96.75%
5. MAR	5.61%	100.00%	99.87%	97.07%	72.21%	98.79%	95.60%	73.61%	98.79%
6. SPEC	93.44%	0.00%	81.58%	52.39%	15.11%	33.78%	37.79%	21.67%	36.65%
MCOST	97.45%	100.00%	98.47%	98.77%	91.46%	99.64%	98.85%	96.65%	99.69%
ADJUSTED SSPEC OF OPTIMIZED
1. MSE	0.980	0.769	0.799	0.901	0.883	0.924	1.047	0.901	1.104
2. MAE	0.943	0.848	0.844	0.967	0.893	0.923	1.091	0.944	1.186
3. PIS	0.896	0.794	0.877	0.959	0.851	0.952	1.154	0.947	1.021
4. MSR	0.838	0.767	0.786	0.849	0.893	0.769	0.954	0.907	0.777
5. MAR	0.833	0.769	0.782	0.825	0.878	0.779	0.953	0.896	0.781
6. SPEC	0.843	0.792	0.890	1.020	0.822	0.963	1.138	0.932	0.989
MCOST	0.814	0.767	0.759	0.737	0.696	0.734	0.855	0.709	0.711
RANK
1	MAR	MSR	MAR	MAR	SPEC	MSR	MAR	MAR	MSR
2	MSR	MAR	MSR	MSR	PIS	MAR	MSR	MSE	MAR
3	SPEC	MSE	MSE	MSE	MAR	MAE	MSE	MSR	SPEC
4	PIS	SPEC	MAE	PIS	MSE	MSE	MAE	SPEC	PIS
5	MAE	PIS	PIS	MAE	MSR	PIS	SPEC	MAE	MSE
6	MSE	MAE	SPEC	SPEC	MAE	SPEC	PIS	PIS	MAE

Bold values show optimal results; bold in headings is for clarity.

and underpins the comparative statements below. Rate-based objectives (mean squared rate, MSR; mean absolute rate, MAR) tended to favor occurrence-aware methods (SK, SK–SSA), while the inventory-aligned costs (SPEC, sSPEC) more often selected methods that temper costly inventory drift (e.g., TSB when decay or obsolescence is present). Scaled stock-keeping-oriented prediction error costs (sSPEC) served as the primary criterion; mean absolute scaled error (MASE) and scaled absolute periods in stock (sAPIS) are reported as complementary views.

Optimized parameters concentrate away from extremes for most methods. Figure 3 and Figure 4 show bar charts and box plots of the tuned parameters across SKUs. Dispersion is wider in items with very long zero runs, reflecting the greater difficulty of stabilizing occurrence estimates; SK–SSA mitigates this by smoothing the size component prior to SK updating (Section 2.2).

Table 8. Setup example of spare-part weekly demand over 104 periods, showing sequences of zero and non-zero demand observations. Each cell represents the demand quantity per week, with zeros indicating no demand events.

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
13	13	0	5	6	0	0	1	0	5	6	0	3	1	11	0	4
18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34
14	0	7	1	0	0	0	0	8	0	0	14	0	0	0	0	0
35	36	37	38	39	40	41	42	43	44	45	46	47	48	49	50	51
5	10	4	13	0	8	7	0	7	9	0	6	2	12	0	4	2
52	53	54	55	56	57	58	59	60	61	62	63	64	65	66	67	68
2	15	16	1	0	0	0	0	7	0	5	0	5	0	7	0	4
69	70	71	72	73	74	75	76	77	72	78	79	80	81	82	83	84
6	7	5	0	0	0	0	4	12	0	11	3	5	2	5	2	6
85	86	87	88	89	90	91	92	93	94	95	96	97	98	99	100	101
7	3	3	0	7	11	9	8	7	0	2	3	4	0	0	2	5
102	103	104
0	0	0

and

Table 9. Optimized values of the parameters of each forecast method for the spare part of Table 8.

Forecast Method	Number of Parameters Optimized	First Parameter Name/Best Value	Second Parameter Name/Best Value	Parameter Meaning/Notes
SMA	1	Window|11	-	MA window length
SES	1	α (level)|0.15	-	Level smoothing
CRO	1	α|0.25	-	Croston: smooths size and interval
SBA	2	α (size)|0.30	β (interval)|0.40	Bias-corrected Croston
TSB	2	α (size)|0.30	β (interval)|0.30	Size and probability of occurrence
SK	2	α (size)|0.40	-	Size smoothing; occurrence is non-parametric
mSBA	2	α (size)|0.05	β (interval)|0.40	Interval updates during zeros
mTSB	2	α (size)|0.40	β (interval)|0.40	Occurrence updates during zeros
SK–SSA	2	α (size)|0.40	k|1	SSA-stabilized size and reconstructed components

present a 104-week example with average inter-demand interval 1.60 weeks and squared coefficient of variation 0.37. Optimized parameters are listed for all methods;

Table 10. Scaled metric comparisons on the example time series.

Metric	SMA (11)	SES (0.15)	CRO (0.25)	SBA (0.30, 0.40)	TSB (0.30, 0.30)	SK (0.40)	mSBA (0.05, 0.4)	mTSB (0.40, 0.40)	SK–SSA (0.40, 1, 3)
MASE	0.556	0.544	0.626	0.518	0.434	0.508	0.461	0.568	0.400
sAPIS	3.194	1.466	1.886	0.510	0.478	0.504	0.467	0.545	0.460
sSPEC	0.329	0.181	0.191	0.169	0.166	0.167	0.167	0.186	0.163

compares performance using MASE, sAPIS, and sSPEC over five horizons. Results match known properties: CRO exhibits positive bias; SBA corrects that bias; TSB performs well when occurrence decays; mSBA improves SBA via interval updates during zero runs; mTSB is not uniformly dominant; SK sits between SBA and TSB; SK–SSA delivers the best overall performance for this SKU.

The KH-Select demand classification from Section 2.5 (smooth, intermittent, erratic, lumpy) was used to define scenario subsets and to report selection frequencies. Scenario-level averages across three horizons (t + 1, t + 3, t + 5) show MCOST ranking first overall, followed by MSR and MAR (

Table 11. Performance comparison of error metrics under various model selection scenarios across three different horizons.

Error	MASE			sSPEC			sAPIS
	t + 1	t + 3	t + 5	t + 1	t + 3	t + 5	t + 1	t + 3	t + 5
Scenario 1: SES, CRO, SBA
MSE	0.951	0.940	0.929	0.616	0.850	1.158	1.187	2.661	4.448
MAE	0.966	0.952	0.943	0.644	0.906	1.284	1.208	2.730	4.611
MSR	0.910	0.900	0.892	0.589	0.814	1.114	1.138	2.553	4.276
MAR	0.896	0.887	0.878	0.579	0.802	1.092	1.122	2.519	4.203
PIS	1.022	1.010	0.998	0.659	0.897	1.198	1.273	2.840	4.641
SPEC	1.055	1.048	1.034	0.682	0.950	1.254	1.320	2.966	4.808
MCOST	0.851	0.875	0.873	0.569	0.742	1.068	1.070	2.345	3.784
Scenario 2: SES, CRO, SBA
MSE	0.951	0.940	0.929	0.616	0.850	1.158	1.187	2.661	4.448
MAE	0.966	0.952	0.943	0.644	0.906	1.284	1.208	2.730	4.611
MSR	0.910	0.900	0.892	0.589	0.814	1.114	1.138	2.553	4.276
MAR	0.896	0.887	0.878	0.579	0.802	1.092	1.122	2.519	4.203
PIS	1.022	1.010	0.998	0.659	0.897	1.198	1.273	2.840	4.641
SPEC	1.055	1.048	1.034	0.682	0.950	1.254	1.320	2.966	4.808
MCOST	0.851	0.875	0.873	0.569	0.742	1.068	1.070	2.345	3.784
Scenario 3: SES, CRO, SBA, TSB, SK
MSE	1.018	1.003	0.988	0.649	0.880	1.200	1.249	2.769	4.606
MAE	0.971	0.961	0.952	0.649	0.915	1.281	1.216	2.743	4.590
MSR	0.957	0.951	0.940	0.604	0.829	1.144	1.172	2.627	4.376
MAR	0.940	0.934	0.924	0.596	0.824	1.122	1.157	2.599	4.308
PIS	1.051	1.049	1.036	0.667	0.912	1.196	1.297	2.881	4.619
SPEC	1.058	1.053	1.041	0.682	0.934	1.209	1.319	2.918	4.631
MCOST	0.833	0.857	0.857	0.559	0.699	1.055	1.042	2.155	3.325
Scenario 4: SES, CRO, SBA, TSB, SK, mSBA, mTSB, SK–SSA
MSE	1.051	1.035	1.022	0.696	0.969	1.342	1.309	2.939	4.908
MAE	0.950	0.937	0.930	0.673	1.017	1.530	1.186	2.802	4.789
MSR	0.940	0.935	0.925	0.594	0.827	1.157	1.145	2.587	4.329
MAR	0.930	0.923	0.913	0.593	0.828	1.140	1.141	2.575	4.283
PIS	1.067	1.060	1.049	0.690	0.974	1.313	1.311	2.947	4.809
SPEC	1.059	1.051	1.042	0.691	0.964	1.293	1.314	2.944	4.757
MCOST	0.761	0.803	0.811	0.539	0.686	1.048	0.949	2.007	3.169

). Selection frequencies by class (

Table 12. Performance comparison of per-item demand categorization for various model scenarios.

Pool of Models	MASE			sSPEC			sAPIS
	t + 1	t + 3	t + 5	t + 1	t + 3	t + 5	t + 1	t + 3	t + 5
CRO (single model)	0.848	0.860	0.856	0.542	0.743	0.901	1.065	2.355	3.879
SBA (single model)	0.782	0.816	0.817	0.528	0.728	0.885	0.978	2.152	3.524
CRO, SBA	0.758	0.805	0.811	0.507	0.686	0.839	0.950	2.057	3.318
SES, CRO, SBA	0.741	0.796	0.803	0.497	0.667	0.805	0.927	1.987	3.171
TSB (single model)	0.805	0.832	0.841	0.516	0.712	0.863	0.965	2.175	3.533
CRO, SBA, TSB	0.703	0.777	0.789	0.477	0.639	0.758	0.879	1.875	2.927
SES, CRO, SBA, TSB	0.699	0.776	0.787	0.474	0.631	0.747	0.875	1.856	2.890
mSBA (single model)	0.779	0.800	0.807	0.569	0.841	1.106	0.968	2.206	3.724
CRO, mSBA	0.693	0.765	0.777	0.491	0.673	0.823	0.864	1.848	2.943
CRO, mSBA, mTSB	0.655	0.753	0.768	0.474	0.637	0.781	0.835	1.783	2.825
SES, CRO, mSBA, mTSB	0.651	0.747	0.763	0.468	0.635	0.769	0.827	1.758	2.766
mTSB (single model)	0.801	0.827	0.834	0.539	0.757	0.938	1.009	2.166	3.691
SK (single model)	0.793	0.826	0.830	0.521	0.722	0.859	0.951	2.139	3.528
TSB, SK	0.725	0.791	0.799	0.489	0.654	0.775	0.908	1.937	3.026
SES, CRO, SBA, TSB, SK	0.635	0.782	0.783	0.460	0.612	0.719	0.845	1.785	2.756
SMA, SES, CRO, SBA, TSB, SK	0.630	0.740	0.834	0.456	0.606	0.711	0.838	1.767	2.721
mSBA, mTSB, SK–SSA	0.589	0.710	0.732	0.433	0.628	0.746	0.752	1.627	2.557
CRO, SK, SK–SSA	0.611	0.722	0.740	0.453	0.623	0.756	0.760	1.644	2.585
SK, mSBA, mTSB	0.649	0.745	0.761	0.499	0.647	0.786	0.832	1.781	2.809
SMA, SES, CRO, SBA, TSB, SK, mSBA, mTSB, SK–SSA	0.539	0.689	0.715	0.395	0.529	0.612	0.702	1.481	2.259

and Figure 5) reveal that SK and SK–SSA are often preferred for intermittent and lumpy items, while SBA is frequently favored in lumpy cases, consistent with its interval-update logic.

Repeating the analysis on isolated intermittent, lumpy, and erratic subsets confirms two patterns (Figure 6 and the corresponding tables): (i) SK often outperforms TSB even without SSA on very sparse sequences, and (ii) SK–SSA is typically top-ranked across subsets, supporting the value of light stabilization before probability updates when zero runs are long and sizes are volatile. The 75th percentile scores, as indicated in

Table 13. Isolated demand categorized datasets.

176 Lumpy SKUs
	Demand Intervals		Demand Size		Demand per Period
	Mean	St. Dev	Mean	St. Dev	Mean	St. Dev
Min	1.3	0.6	2.5	1.8	1.3	1.0
25%ile	1.5	0.9	7.3	5.9	4.0	4.8
Mean	1.7	1.1	38.9	33.2	24.3	36.2
75%ile	1.8	1.2	63.5	49.6	37.6	49.1
Max	3.2	3.1	165.5	146.5	101.2	227.9
1572 Intermittent SKUs
	Mean	St. Dev	Mean	St. Dev	Mean	St. Dev
Min	1.3	0.6	1.1	0.3	0.4	0.1
25%ile	1.8	1.2	17.2	4.9	6.8	2.4
Mean	2.5	1.8	33.5	12.6	15.4	9.6
75%ile	3.0	2.3	44.5	17.2	18.7	11.7
Max	6.4	7.0	99.3	50.4	69.9	72.7
74 Erratic SKUs
	Mean	St. Dev	Mean	St. Dev	Mean	St. Dev
Min	1.1	0.3	5.5	4.4	4.1	6.4
25%ile	1.1	0.4	14.2	11.4	12.0	24.4
Mean	1.2	0.5	35.6	27.0	30.5	64.3
75%ile	1.2	0.5	52.8	41.2	44.1	90.7
Max	1.3	0.7	80.4	58.2	69.6	195.3

, confirmed the notably strong forecasting performance for the majority of cases. Particularly noteworthy was the observation that our proposed SK model often outperformed TSB in several instances, establishing itself as a compelling candidate model, even without SSA.

A light sensitivity analysis was performed by varying the exponential-smoothing parameter α (0.05–0.4) and, for SK–SSA, the SSA window length $L_{SSA}\in \left\{6,8,10,12\right\}$ and number of reconstructed components $k\in \left\{1,2,3\right\}$. Forecast performance and inventory outcomes remained stable within these ranges, indicating robustness to moderate parameter shifts. Computational efficiency was also verified: on a standard desktop CPU (AMD Ryzen 5 1600 (six cores) with 64 GB RAM), full MCOST optimization across 2050 SKUs completed in under 15 min without GPU acceleration (

Table 14. Performance comparison of forecast methods for the isolated data sets.

Intermittent SKUs sSPEC	SMA	SES	CRO	SBA	TSB	SK	mSBA	mTSB	SK–SSA	Optimal
Average	0.615	0.618	0.586	0.573	0.581	0.577	0.618	0.583	0.526	0.448
Median	0.286	0.297	0.277	0.245	0.252	0.243	0.199	0.248	0.141	0.130
75th Percentile	0.768	0.775	0.729	0.761	0.700	0.773	0.909	0.739	0.780	0.572
25th Percentile	0.219	0.207	0.219	0.167	0.175	0.203	0.135	0.175	0.081	0.070
Maximum	2.089	2.171	2.044	2.115	2.120	2.037	2.267	2.161	2.067	2.007
Minimum	0.093	0.092	0.086	0.065	0.034	0.078	0.060	0.047	0.036	0.023
Lumpy SKUs sSPEC	SMA	SES	CRO	SBA	TSB	SK	mSBA	mTSB	SK–SSA	Optimal
Average	0.525	0.525	0.489	0.463	0.483	0.476	0.485	0.480	0.434	0.392
Median	0.229	0.240	0.223	0.190	0.220	0.208	0.162	0.200	0.127	0.112
75th Percentile	0.329	0.383	0.324	0.287	0.314	0.303	0.378	0.300	0.271	0.204
25th Percentile	0.185	0.157	0.149	0.103	0.128	0.131	0.098	0.124	0.070	0.039
Maximum	2.173	2.147	2.081	2.189	2.138	2.049	2.250	2.066	2.074	2.062
Minimum	0.027	0.042	0.033	0.010	0.015	0.025	0.010	0.026	0.022	0.009
Erratic SKUs sSPEC	SMA	SES	CRO	SBA	TSB	SK	mSBA	mTSB	SK–SSA	Optimal
Average	0.470	0.475	0.441	0.424	0.445	0.416	0.468	0.434	0.411	0.370
Median	0.231	0.210	0.193	0.174	0.204	0.211	0.194	0.189	0.169	0.147
75th Percentile	0.634	0.695	0.613	0.643	0.641	0.539	0.746	0.592	0.625	0.483
25th Percentile	0.096	0.107	0.092	0.056	0.081	0.069	0.051	0.067	0.049	0.022
Maximum	1.593	1.690	1.701	1.692	1.677	1.656	1.826	1.689	1.676	1.537
Minimum	0.019	0.026	0.014	0.007	0.020	0.013	0.005	0.018	0.007	0.001

), supporting ERP-scale deployability.

Furthermore, both the mSBA and mTSB methods exhibited superior performance compared to their counterparts, SBA and TSB, respectively, while the SK–SSA method consistently outperformed all others on average. The final column of Table 14 presents the optimal results obtained through demand categorization based on cost function minimization. The last column is a benchmark case representing the best achieved per-SKU cost (not a theoretical bound). Across all datasets, per-item optimization yielded substantial improvements in demand forecasting. This multi-method approach efficiently captured micro-trends and time decay by selecting the most appropriate forecast method for each item.

The bar chart comparisons in Figure 6 reaffirmed the superiority of the SK–SSA method, which emerged as the preferred choice across all three critical datasets. Additionally, the TSB method received high preference for the intermittent dataset, while the mSBA method excelled for the lumpy dataset. In the case of the erratic dataset, the SK–SSA, mSBA, SK, and TSB methods demonstrated comparable performance.

Per-SKU forecasts were embedded in the continuous-review (R,Q) model with backordering (Section 2.6). Reorder points R were set from NBB LTD quantiles at target cycle-service levels, with safety stock augmenting R. For each setting we simulated 100,000 NBB draws to estimate achieved service levels and backlog–safety-stock trade-offs. This simulation design directly implements the LTD specification and quantile-based R setting described in Section 2.6.

Backlog–safety-stock curves (Figure 7a) and realized service versus safety stock (Figure 7b) show that methods capturing micro-trends and decay deliver more efficient frontiers (curves closer to the axes). Figure 7c reports realized service levels relative to targets; Figure 7d compares scaled safety stock at a 97% target across methods and demand classes.

Table 15. Scaled safety stock and backlogs (in parenthesis) for various service levels.

Model	Service Level
	80%	85%	90%	95%	97%	99%
SMA	6.1 (10,980)	8.7 (7670)	12.4 (4546)	18.6 (1851)	23.2 (925)	32.9 (192)
SES	6.9 (9730)	9.5 (6811)	13.3 (4001)	19.6 (1644)	24.2 (834)	34.0 (185)
CRO	6.8 (9679)	9.5 (6592)	13.2 (3917)	19.4 (1628)	23.9 (837)	33.8 (190)
SBA	8.6 (8486)	11.4 (5823)	15.5 (3381)	22.2 (1335)	27.2 (667)	37.8 (148)
TSB	7.2 (9887)	10.0 (6803)	13.8 (4085)	20.2 (1674)	24.9 (870)	35.1 (193)
SK	7.1 (9052)	9.76 (6232)	13.5 (3656)	19.8 (1478)	24.5 (728)	34.3 (152)
mSBA	9.6 (5423)	12.6 (3719)	16.8 (2241)	24.1 (922)	29.3 (501)	40.6 (138)
mTSB	6.9 (6453)	9.6 (4514)	13.4 (2748)	19.7 (1204)	24.3 (679)	34.2 (189)
SK–SSA	9.3 (5381)	12.2 (3758)	16.4 (2267)	23.4 (1005)	28.6 (547)	39.6 (150)

summarizes scaled safety stock and backlogs across service levels for the full dataset. On average, SK and especially SK–SSA require lower safety stock at a given service level, while SBA and TSB also limit backorders effectively. Differences are more pronounced at the item level than in aggregate, reinforcing the value of per-SKU selection rather than a single, horizontal policy.

Figure 8 shows a positive association between improvements in forecasting metrics and reductions in backlog counts. Against the organization’s existing non-optimized SBA policy, the proposed MCOST-based framework achieved approximately 15–20% lower safety stock at matched service levels, without increasing backorders.

Three conclusions follow from the results: (i) per-SKU MCOST outperforms single-objective optimization on average; (ii) enlarging a small, interpretable Croston-based pool with SK and SK–SSA increases the chance of matching heterogeneous, intermittent patterns; and (iii) inventory outcomes improve materially when forecasts are operationalized via (R,Q) with NBB LTD, yielding stable reorder points and consistently lower safety stocks for a given target service level.

Non-parametric tests confirmed that SK–SSA significantly outperformed all other Croston-family models on the primary cost metric (sSPEC, p < 0.01 after Holm correction). SK also achieved significant gains (p < 0.05) over SBA and TSB in the intermittent and lumpy subsets. Bootstrap 95% confidence intervals for the median sSPEC differences further indicated stable positive effects of SK and SK–SSA, validating that the observed improvements are statistically robust.

4. Discussion

This study set out to improve forecasting and inventory decisions for intermittent and lumpy spare-parts demand by combining the following three elements: (i) a pool of interpretable Croston-family methods [Croston (CRO), Syntetos–Boylan Approximation (SBA), Teunter–Syntetos–Babai (TSB), and modified variants], complemented by a new Sfiris–Koulouriotis (SK) forecasting method and its Singular Spectrum Analysis variant (SK–SSA); (ii) an inventory-aligned evaluation metric, the scaled Stock-keeping-oriented Prediction Error Costs (sSPEC); and (iii) a per-SKU multi-cost optimization (MCOST) procedure that selects both the objective and the method based on empirical evidence rather than prescription. Across 2050 SKUs with 104 weekly observations each, the empirical results indicate consistent gains in forecast accuracy and, crucially, in inventory performance, while remaining computationally light and compatible with enterprise resource planning systems (Table 5, Table 6, Table 7, Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, Table 14 and Table 15; Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8).

For intermittent series, Croston-type forecasts are a natural baseline but are positively biased. SBA reduces that bias, and probability-of-occurrence updates as in TSB perform well when obsolescence or demand decay is present (Table 10). Building on these ideas, SK retains the TSB occurrence-probability rationale with simpler parameterization, and SK–SSA further stabilizes highly lumpy sequences. Overall, SK and SK–SSA frequently rank among the top choices under rate-based costs, that is mean squared rate (MSR) and mean absolute rate (MAR), and under sSPEC (Table 7, Table 9, Table 10, Table 11 and Table 12; Figure 5 and Figure 6). This supports the hypothesis that enlarging a small, interpretable pool increases the chance of matching heterogeneous item-level patterns. The strong performance of SK–SSA on the most challenging subsets (intermittent and lumpy) highlights the value of light smoothing of non-zero sizes before probability updates when zero runs are long and size variability is high.

In practice, SK is well suited to intermittent demand because it targets the operational driver for intermittence, namely the per-period demand-occurrence probability, while keeping the model simple and transparent. Particularly, SK (a) estimates the occurrence probability directly and pairs it with a single-parameter exponential smoother for non-zero size; (b) adapts quickly to regime shifts because it avoids a second smoothing layer on occurrence; (c) remains stable through long zero runs with straightforward start-up safeguards; (d) preserves interpretability as “probability of occurrence × typical size”; and (e) links naturally to the Bernoulli component of the negative-binomial–Bernoulli lead-time demand model used for inventory decisions. These properties help explain why SK and SK–SSA translate forecast gains into lower safety stocks at a given service level in our experiments.

Selecting a single accuracy metric can be misleading for intermittent series because traditional level-based losses [Mean Squared Error (MSE), Mean Absolute Error (MAE)] overweight magnitude and ignore spacing. Consistent with prior work [19], rate-based objectives (MSR, MAR) and inventory-linked costs [periods in stock (PIS), SPEC] capture different operational trade-offs. The MCOST protocol (Section 2.4; Figure 2) formalizes this by choosing the objective first, tuning parameters via rolling windows, and adjusting comparisons for boundary solutions through an explicit success-rate weighting. Empirically, MCOST dominates single-objective optimization on average (Table 7), and sSPEC is more discriminative than sAPIS for SK and TSB tuning, which supports the idea that a cost function that incorporates stock-keeping and opportunity costs is better aligned with managerial priorities. These results substantiate the working hypothesis that per-SKU, objective-aware optimization outperforms uniform horizontal tuning. Extended numerical summaries and sensitivity tests confirming these findings appear in Appendix C.

Using KH-Select for demand classification preserves interpretability and connects to established practice (Section 2.5). The framework supports multiple scenarios, while the per-SKU multi-cost optimization (MCOST) procedure selects the best method based on out-of-sample loss (Section 2.4). In the results, the frequency maps (Figure 5) and scenario comparisons (Table 12) show that SK and SK–SSA are frequently preferred in intermittent and lumpy regions, while mSBA is often favored in lumpy cases, which is consistent with its interval-update logic during zero-demand runs. This two-stage process balances parsimony and adaptivity: a human-interpretable starting point followed by data-driven final selection.

At the policy layer, the study adopts a continuous-review (R,Q) model with backordering and a negative-binomial–Bernoulli (NBB) lead-time demand (LTD) specification (Section 2.6). Negative binomial handles over-dispersion in spare-parts demand, and Bernoulli captures occurrence. Simulations using 100,000 NBB draws per setting (Figure 7) show that forecast improvements translate into more favorable backlog–safety-stock trade-offs, with SK and especially SK–SSA requiring lower safety stock at a given service level on average. These findings corroborate inventory–literature recommendations on the suitability of the negative binomial for spare-parts [37] and demonstrate policy robustness—recommended (R,Q) settings, method rankings, and trade-offs remain consistent under the tested scenarios (Section 2.6). The observed 15–20% reduction in safety stock relative to a non-optimized SBA policy underscores the practical value of aligning the forecasting objective (sSPEC) with inventory outcomes.

This study does not employ aggregate models or heavy machine learning. Aggregate approaches such as ADIDA and IMAPA mitigate irregularity via temporal aggregation and reconciliation, and recent machine-learning proposals (for example, transformers, modified k-Nearest Neighbors, feature-based combinations) target sparse sequences. These directions are acknowledged; however, the experiments focus on non-aggregated, Croston-family methods to preserve computational efficiency, interpretability, and a transparent link to inventory policy. The small hardware footprint (desktop CPU, no GPU) and closed-form updates facilitate integration at item scale, which is a practical constraint that can limit adoption of global, data-hungry models in spare-parts environments.

While modern machine-learning techniques such as deep sequential or probabilistic models represent valuable future directions, they were intentionally excluded here to maintain interpretability, computational lightness, and ERP compatibility. The proposed methods thus address a complementary need: practical, explainable forecasting for large SKU portfolios where long training histories are unavailable.

The results suggest three practical implications for inventory managers:

• Objective selection matters. Choosing an inventory-aligned cost (sSPEC) for parameter tuning leads to materially different and better inventory outcomes than traditional errors alone;
• Method diversity pays. Maintaining a small, interpretable pool (CRO, SBA, TSB, modified variants, SK, SK–SSA) and selecting per SKU via MCOST improves both accuracy and stockholding/backorder trade-offs;
• Policy robustness. Using NBB for LTD within an (R,Q) policy yields stable reorder points and safety stock across service-level targets, with clear trade-offs (Figure 7) that support policy communication.

The proposed framework performs strongly across demand categories but is not uniformly superior for every SKU. This pattern is expected given the diversity of item-level demand processes and the intentionally compact, interpretable model pool we adopt. Our aim was not to enforce single-method dominance; rather, it was to deliver a reproducible, explainable, and computationally efficient benchmark that translates forecasting gains into inventory improvements within ERP environments.

Several extensions follow naturally. First, the evaluation centers on Croston-family methods; broadening the pool to include hyperbolic exponential smoothing and selected probabilistic or hybrid approaches could increase coverage where data richness permits. Second, we treated lead time as deterministic; embedding time-varying or stochastic lead times within the LTD specification, or imposing explicit service-level constraints, would provide a more general policy layer. Third, calendar and event effects were not modelled explicitly; in environments with pronounced seasonality or shutdowns, simple indicators or hierarchical reconciliation may add value. Finally, while the dataset is large and industrial, it reflects a single sector (automotive) and weekly aggregation; replication across industries (e.g., electronics, machinery) and sampling frequencies would further test generalizability. This focus was intentional, chosen for data completeness, operational relevance, and ERP alignment, ensuring methodological consistency and practical applicability. Extending the framework beyond automotive is a natural next step given its parameter-light, ERP-ready design.

We view these as avenues for refinement rather than limitations of the core framework, whose contribution is to offer a transparent, SKU-wise selection that is practical to deploy and demonstrably linked to inventory outcomes.

Other possible extensions include the following:

• Forecasting

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  Enlarge the candidate pool with additional intermittent-aware models.

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  Develop hybrid SK variants that combine probability updates with regime detection.

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  Investigate joint tuning with SSA windowing.

• Objectives

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  Compare sSPEC with alternative cost-to-serve surrogates.

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  Explore dynamic weighting between opportunity and stock-keeping costs.

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  Extend MCOST to multi-horizon settings or service-level-constrained tuning.

• Inventory

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  Generalize LTD to composite models with stochastic lead time.

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  Integrate maintenance signals or condition-based triggers (e.g., advance demand information, semi-Markov maintenance coupling) for anticipatory planning.

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  Evaluate deterioration-risk-aware replenishment policies for items subject to obsolescence or shelf-life constraints.

These extensions would complement the current framework’s per-item, interpretable selection and reinforce the link between forecasting and decision-relevant costs. Overall, the evidence indicates that a transparent, per-SKU MCOST selection over a Croston-based pool, augmented by SK and SK–SSA and evaluated with sSPEC, is an effective and practical route for organizations facing intermittent spare-parts demand. It improves forecasts, reduces safety stock and backorders, and integrates cleanly into existing enterprise systems.

5. Conclusions

This study proposes a practical framework for intermittent-demand forecasting of spare parts and for optimizing inventory decisions. The framework combines a Croston-based pool of interpretable forecasting methods with a new SK forecasting method and its SK–SSA variant, an inventory-aligned evaluation metric (sSPEC), and a per-SKU MCOST selection scheme. Forecasts are operationalized through a continuous-review (R,Q) policy using negative-binomial–Bernoulli (NBB) lead-time demand (LTD), which links statistical accuracy directly to service levels, safety stock, and backorders. Favoring Croston-family models preserves transparency for implementation in practice, while enlarging the method pool increases the chance of matching diverse irregular and lumpy patterns at the item level. Across a large automotive dataset, the framework improves forecast accuracy and reduces safety stock and backorders relative to strong baselines. In the empirical study, the SK–SSA method achieved, on average, the lowest safety stock at a given service level. The procedure is computationally light and compatible with enterprise systems, which enables item-wise deployment without specialized hardware.