Specialized Demand Forecasting Problems

Introduction

Standard forecasting methods assume that demand is observed frequently, is reasonably continuous, and has a stable relationship with its history. Logistics systems often violate all three assumptions. Spare parts may be demanded only occasionally; a fast-moving fashion item may have a life of eight weeks; a new product has no sales history; and a port closure or epidemic can abruptly invalidate historical patterns. These situations require specialized forecasting rather than simply applying a more sophisticated version of a conventional moving average.

This chapter examines five important problem classes: intermittent and lumpy demand, new products, short-life-cycle items, and demand under disruption. It develops Croston-family methods, analogy and diffusion approaches, demand sensing, and scenario-based forecasting. The emphasis is not only on point forecasts, but also on uncertainty, decision consequences, data quality, model governance, and the distinction between forecasting demand and planning supply.

Learning objectives

After studying this chapter, a learner should be able to:

1. Distinguish intermittent, lumpy, new-product, short-life-cycle, and disruption-related demand.
2. Select and apply Croston-family methods while recognizing their assumptions and biases.
3. Use analogies, diffusion models, and structured judgment for products without adequate history.
4. Design forecasts for short life cycles using launch, peak, decline, and end-of-life information.
5. Combine demand sensing with statistical forecasts without confusing signals with causal evidence.
6. Construct scenarios and translate forecast uncertainty into inventory, capacity, and service decisions.
7. Evaluate specialized forecasts using metrics appropriate to the decision and demand pattern.

1. Why specialized demand forecasting is difficult

Demand is called intermittent when zero-demand periods occur frequently. It is lumpy when both the timing and the size of non-zero demand vary substantially. A spare-parts series such as 0, 0, 4, 0, 0, 12, 0, 2 illustrates both features. Aggregating such data to monthly totals may hide operational shortages, while forecasting every zero as ordinary low demand may produce systematic understocking.

A product is a new product when its own sales history is absent or too short to estimate stable parameters. A short-life-cycle item has a limited selling window, after which replenishment has little value. Seasonal fashion, promotional products, event merchandise, and many electronics fall into this category. Disruption demand occurs when normal demand or its fulfilment environment is altered by a shock such as a natural disaster, industrial action, cyberattack, regulatory change, pandemic, or sudden competitor exit.

Four distinctions are especially important:

• Demand versus sales: observed sales may be censored by stockouts, allocation, or lost orders.
• Demand occurrence versus demand size: intermittent demand has two processes—whether demand occurs and how much is requested.
• Forecast uncertainty versus supply uncertainty: an accurate demand forecast does not guarantee supply availability.
• Prediction versus decision: the best statistical forecast may not be the best inventory decision when shortage and excess costs are asymmetric.

2. Intermittent and lumpy demand

Let y_t denote demand in period t, with y_t = 0 in many periods. Define the occurrence indicator

and, when I_t = 1, define the non-zero demand size z_t = y_t. A useful conceptual decomposition is

The equation is an identity under the stated definitions: expected demand equals the probability of occurrence multiplied by the conditional mean size. It is useful because occurrence and size may be driven by different causes. For example, failures determine whether a spare part is needed, whereas the number of units requested may depend on the failure severity.

Two descriptive measures are commonly used. The average demand interval (ADI) is the average number of periods between non-zero demands. If the time between successive non-zero observations is q_j, for j = 1, ..., n, then

where n is the number of observed demand events. The squared coefficient of variation of non-zero demand is

where z̄ is the mean non-zero demand and s_z is its sample standard deviation. High ADI indicates infrequent demand; high CV² indicates variable demand size. A common classification is:

The boundaries used to label these categories are heuristics, not universal laws. They should be calibrated to the planning frequency, lead time, and cost of error.

Croston’s method

Croston’s method separately smooths non-zero demand size and the interval between demands. At a demand occurrence, let [[MATH{z_t}]] be the observed non-zero size and [[MATH{q_t}]] the interval since the previous occurrence. With smoothing parameter [[MATH{α}]], where [[MATH{0 < α < 1}]], the updates are

The estimated demand per period is

Between demand occurrences, the estimates are normally left unchanged. The method is applicable when zero observations mainly represent genuine absence of demand and when the process is sufficiently stable for exponential smoothing to be meaningful. It does not automatically correct for stockout-censored demand, changing installed base, planned promotions, or a structural break.

Croston’s original estimator is historically important but has been shown to possess bias under some stochastic assumptions, particularly because the ratio of two estimates is not generally an unbiased estimate of the ratio of their expectations. Consequently, implementations should not treat it as universally optimal.

Croston-family improvements

Several variants address timing, bias, or implementation concerns.

• Syntetos–Boylan Approximation (SBA): applies a correction to the Croston estimate, commonly represented as

The correction is derived for particular intermittent-demand assumptions and is not a guarantee of superiority in every dataset.

• Teunter–Syntetos–Babai (TSB): smooths the probability of demand occurrence rather than the interval. Let p_t be the estimated probability of a non-zero demand and ẑ_t the smoothed non-zero size. At every period,

Then

where [[MATH{0 < α, β < 1}]]. TSB can adapt more naturally when the probability of demand is declining, such as at product obsolescence, because the occurrence probability is updated even during zero-demand periods.

• Intermittent-demand state-space and count models: zero-inflated Poisson, negative binomial, hurdle, and dynamic generalized linear models can explicitly model occurrence and size. They require more data, careful estimation, and diagnostic testing. A zero-inflated model is not automatically appropriate merely because zeros are present; structural zeros and sampling zeros must be distinguished.

Worked example: Croston estimation

Suppose a spare-part series has non-zero demand sizes 4, 8, and 5, with intervals 2, 3, and 2 periods. Let [[MATH{α = 0.2}]]. Initial estimates are [[MATH{ẑ_0 = 4}]] and [[MATH{q̂_0 = 2}]]. After the second event,

[[MATH{ẑ_1 = 0.2 × 8 + 0.8 × 4 = 4.8}]]

[[MATH{q̂_1 = 0.2 × 3 + 0.8 × 2 = 2.2}]]

After the third event,

[[MATH{ẑ_2 = 0.2 × 5 + 0.8 × 4.8 = 4.84}]]

[[MATH{q̂_2 = 0.2 × 2 + 0.8 × 2.2 = 2.16}]]

Thus the Croston rate is [[MATH{4.84 ÷ 2.16 ≈ 2.24}]] units per period. The forecast is a rate, not a claim that exactly 2.24 units will be demanded in every period. For replenishment over a lead time, the rate must be combined with lead-time uncertainty and service objectives.

3. Forecast evaluation and inventory implications

Ordinary percentage errors are problematic when actual demand is zero. Prefer scale-dependent or decision-oriented measures such as mean absolute scaled error (MASE), mean absolute error on non-zero events, interval-based measures, and cost-weighted service outcomes. For a forecast [[MATH{f_t}]] and actual demand [[MATH{y_t}]], MAE is

where [[MATH{T}]] is the number of evaluation periods. MAE is interpretable in units but can favour forecasts that perform well on frequent zeros. Therefore, evaluation should report several views: all-period accuracy, non-zero-size accuracy, demand-occurrence accuracy, bias, fill rate, stockout frequency, and inventory cost.

Time-series cross-validation must respect chronology. Rolling-origin evaluation trains on an earlier window, forecasts a future horizon, records the error, and then advances the origin. Randomly shuffling intermittent observations can leak future information and produce misleading performance.

For a simple continuous-review approximation, expected demand during lead time [[MATH{L}]] is [[MATH{Lμ}]], where [[MATH{μ}]] is average demand per period. If lead-time demand has standard deviation [[MATH{σ_L}]], a reorder point may be written as

where [[MATH{z_γ}]] is the standard-normal quantile associated with target cycle-service probability [[MATH{γ}]]. This formula assumes a suitable distributional approximation, known lead time, and a service definition consistent with the quantile. For intermittent demand, the distribution of lead-time demand is often skewed and discrete; simulation or empirical quantiles may be safer than imposing normality.

4. New products: forecasting without own history

New-product forecasting is fundamentally an information and identification problem. The forecaster must distinguish four sources of evidence:

1. Analogy: sales of comparable products, markets, or previous launches.
2. Judgment: structured estimates from sales, marketing, engineering, and channel experts.
3. Market evidence: concept tests, preorder data, pilot sales, search activity, and channel commitments.
4. Theory: adoption and diffusion models that describe how awareness and adoption may evolve.

Analogy-based forecasting

A defensible analogy is not simply the product with the highest superficial similarity. Select analogues using attributes that affect demand: price, customer segment, functionality, channel, geography, launch support, installed base, and competitive context. A weighted analogue forecast may be written as

where [[MATH{A_{i,t}}]] is the adjusted demand profile of analogue [[MATH{i}]], [[MATH{w_i}]] is its weight, and [[MATH{m}]] is the number of analogues. Weights should be justified by similarity and validated against historical launches. Adjustments may account for market size, distribution coverage, price, and launch timing.

A useful practice is to forecast a range rather than a single number. Experts provide a base case, an optimistic case, and a conservative case, with explicit assumptions. The Delphi method can reduce dominance by senior individuals by collecting and iterating anonymous expert judgments, although consensus is not proof of accuracy.

Bass diffusion model

The Bass model represents adoption by innovators and imitation. Let [[MATH{N}]] be market potential, [[MATH{Y_t}]] cumulative adopters by time [[MATH{t}]], [[MATH{p}]] the coefficient of innovation, and [[MATH{q}]] the coefficient of imitation. The continuous-time adoption rate is

Here [[MATH{0 ≤ Y(t) ≤ N}]], and [[MATH{p, q ≥ 0}]]. The first term captures external influence; the second captures social imitation. The model assumes a defined market potential and relatively stable innovation and imitation parameters. It may perform poorly when price changes, supply constraints, competition, regulation, or multiple product generations materially alter adoption.

In practice, diffusion models should be treated as structured hypotheses. Early sales data can update [[MATH{N, p, q}]], but parameter estimates are highly uncertain when only a small portion of the adoption curve has been observed.

5. Short-life-cycle and end-of-life items

For a short-life-cycle product, forecast value declines rapidly as the selling window closes. The process should therefore be managed as a sequence of decisions:

• pre-launch: estimate market size, launch timing, and channel allocation;
• launch: update using orders, traffic, conversion, and early sell-through;
• growth and peak: distinguish genuine demand acceleration from pipeline filling;
• decline: reduce replenishment and protect availability for profitable locations;
• end of life: decide markdown, substitution, transfer, return, recycling, or disposal.

The classic newsvendor logic is useful for a single ordering decision before uncertain demand is realized. Let [[MATH{D}]] be demand, [[MATH{Q}]] the order quantity, [[MATH{C_u}]] the underage cost per unit, and [[MATH{C_o}]] the overage cost per unit. Under a continuous demand distribution and risk-neutral expected-cost minimization, the critical fractile is

Thus [[MATH{Q*}]] is the corresponding demand quantile. The result assumes a single decision, known cost parameters, no replenishment after ordering, and a suitable demand distribution. In real logistics, salvage value, markdown timing, capacity, substitution, and fairness across channels may alter the decision.

For multiple locations and several review points, dynamic programming, stochastic programming, or simulation may be more appropriate. A common practical error is to continue using a stable seasonal forecast after the product has entered decline. Lifecycle stage should be a model input, not merely a descriptive label.

6. Demand sensing

Demand sensing uses high-frequency, near-real-time signals to update a baseline forecast. Signals can include point-of-sale data, orders, shipment confirmations, web traffic, search intensity, promotions, weather, competitor prices, mobility, and channel inventory. A conceptual decomposition is

The decomposition is a modelling assumption, not an accounting identity. Signals must be tested for lead time, stability, causal relevance, and data latency. A high correlation between searches and sales may reflect common seasonality rather than predictive causation.

A robust architecture commonly contains:

1. a statistical baseline at the SKU-location-time level;
2. a feature layer that cleans, aligns, and time-stamps external signals;
3. a short-horizon adjustment model;
4. human review for exceptional events;
5. forecast versioning, monitoring, and override governance.

Machine-learning models such as gradient boosting or temporal neural networks can capture nonlinear interactions, but they do not remove the need for leakage controls, recalibration, explainability, and hierarchy reconciliation. A forecast that uses a signal unavailable at the forecast issue time is operationally invalid, even if its historical accuracy is impressive.

7. Demand under disruption

Disruption forecasting begins by identifying what has changed: demand level, demand mix, demand timing, supply availability, lead time, or the relationship between variables. Historical data from a stable regime may be uninformative or actively misleading. The forecaster should create an event timeline, separate observed sales from constrained sales, identify affected nodes and lanes, and define plausible recovery paths.

Scenario-based forecasting

A scenario is a coherent set of assumptions, not a random best-case or worst-case number. A useful set may include:

For scenario [[MATH{s}]], a weighted expected demand can be expressed as

where [[MATH{D_{s,t}}]] is scenario demand, [[MATH{π_s}]] is its probability, [[MATH{π_s ≥ 0}]], and [[MATH{∑π_s = 1}]]. Probabilities should be used only when they represent a defensible belief; otherwise, scenario ranges and robust decisions are preferable. A weighted average can conceal severe downside risk and should not replace stress testing.

Scenario planning can be linked to decisions through a loss function. If action [[MATH{a}]] produces cost [[MATH{C(a,D_s)}]] under scenario [[MATH{s}]], expected cost is

A robust policy may instead minimize the maximum scenario cost or impose service constraints in every critical scenario. The appropriate criterion depends on risk tolerance, criticality, reversibility, and the consequences of failure.

8. Integrated applied case: aircraft spare parts during a disruption

The following is an illustrative case. An airline distributor manages a hydraulic-pump seal used across a fleet. The part has 24 months of weekly data. Demand is zero in 72% of weeks, and non-zero demand sizes range from 1 to 9. A supplier fire increases replenishment lead time from four to ten weeks. Simultaneously, aircraft utilization falls initially but is expected to recover unevenly.

Problem

A conventional seasonal model predicts a low average because it treats most zero weeks as ordinary observations. The distributor faces two risks: stockouts during aircraft recovery and excess inventory if utilization remains depressed. Sales are also censored because some historical orders were backordered.

Analysis

The team first reconstructs latent demand from backorders and distinguishes genuine zero demand from periods in which stock was unavailable. It calculates ADI and non-zero [[MATH{CV²}]], confirming a lumpy pattern. A TSB model is selected for the baseline because demand occurrence may decline during the utilization shock and then recover. A Croston-family forecast is not used alone: failure rates, fleet utilization, maintenance schedules, and known aircraft groundings provide causal information.

Three scenarios are built for the next 26 weeks:

• slow recovery: utilization remains 20% below plan and supply lead time is 12 weeks;
• base recovery: utilization returns gradually and lead time falls to eight weeks;
• rapid recovery: utilization returns to normal by week 10 and lead time falls to six weeks.

For each scenario, demand occurrence and size are simulated from historical non-zero observations adjusted by utilization. Inventory is evaluated using empirical lead-time demand quantiles rather than a normal approximation. Critical aircraft-on-ground orders receive a higher service priority than routine replenishment.

Approach and solution

The distributor adopts a two-level policy. A protected emergency stock is reserved for safety-critical failures, while routine demand is supplied from a separate allocation. Weekly demand sensing updates the utilization adjustment, but updates are capped to prevent a single noisy order from causing excessive replenishment. The forecast is reviewed at a weekly control meeting, with all overrides recorded and expiry dates assigned.

The policy also includes supplier diversification, repair-and-return of serviceable seals, and substitution rules approved by engineering. Scenario triggers are defined: if utilization exceeds the base path for three consecutive weeks, the rapid-recovery policy is activated; if supplier lead time exceeds ten weeks, emergency procurement is escalated.

Outcome and learning points

The case illustrates that specialized forecasting is not a contest between one algorithm and another. The useful result comes from combining occurrence-size modelling, censored-demand correction, external drivers, scenarios, service differentiation, and explicit governance. It also shows why a forecast must be connected to an inventory policy: a point estimate alone does not specify which customers receive scarce stock or how much risk the organization accepts.

9. Challenges, governance, and best practices

Specialized forecasting has several limitations. Sparse data make parameters unstable; analogues may not be truly comparable; diffusion models can be overinterpreted; external signals may be endogenous; and scenario probabilities may be politically rather than analytically assigned. Forecast combination can help, but combining weakly justified models does not guarantee improvement.

Governance should address:

• Data integrity: preserve timestamps, stockout flags, order cancellations, substitutions, and returns.
• Fair allocation: during scarcity, define transparent priorities rather than allowing powerful customers to dominate informally.
• Privacy: use customer or mobility data only with lawful purpose, proportionality, access control, and retention limits.
• Explainability: document features, overrides, model versions, and reason codes.
• Bias and feedback: a model trained on constrained sales may perpetuate under-allocation to smaller markets.
• Cybersecurity and resilience: protect demand data and maintain a usable fallback process.
• Human accountability: assign ownership for approving overrides and acting on scenario triggers.

Best practice is to begin with a strong data audit, establish a baseline, compare specialized methods through rolling-origin evaluation, measure both statistical and operational outcomes, and maintain a champion–challenger process. Forecasts should be reconciled across product, location, and channel hierarchies where decisions require consistency. Intermittent forecasts should be evaluated over a horizon relevant to replenishment, not only one period ahead.

Contemporary developments

Probabilistic forecasting, hierarchical reconciliation, Bayesian updating, and probabilistic machine learning are increasingly useful for specialized demand. Digital twins can test inventory policies under alternative disruption paths. Foundation models and generative AI may assist with document extraction, event detection, and scenario narration, but generated explanations are not evidence of predictive validity. Human review, data lineage, controlled experiments, and out-of-time validation remain essential.

A promising research direction is hybrid modelling: combining structural knowledge—such as failure rates, market potential, and lead-time mechanisms—with flexible statistical or machine-learning components. The central methodological challenge is identifying when a regime has changed enough that old data should be down-weighted or discarded.

Chapter Summary

Specialized demand forecasting is required when demand is sparse, volatile, rapidly evolving, or affected by structural shocks. Intermittent demand should be understood through separate occurrence and size processes. Croston’s method and its SBA and TSB variants are useful baselines, but their assumptions and possible bias require empirical validation. New products require carefully selected analogies, structured judgment, market evidence, and, where appropriate, diffusion theory. Short-life-cycle items require lifecycle-aware updating and decisions that recognize overage and underage costs. Demand sensing can improve short-horizon responsiveness only when signals are timely, causal or predictive, and free from leakage. Disruptions call for scenario-based forecasts, stress testing, empirical lead-time distributions, and explicit decision rules. Across all applications, forecast quality includes data governance, fairness, explainability, and the operational consequences of error.

Key Terms

Intermittent demand: Demand containing many zero-demand periods.

Lumpy demand: Demand with both infrequent occurrence and highly variable non-zero size.

Demand occurrence: Whether demand is positive in a specified period.

ADI: Average demand interval between non-zero demand events.

CV²: Squared coefficient of variation of non-zero demand size.

Croston’s method: A method that separately smooths non-zero demand size and inter-demand interval.

SBA: Syntetos–Boylan Approximation, a bias-corrected Croston-family method.

TSB: Teunter–Syntetos–Babai method, which updates the probability of demand occurrence.

New product forecasting: Estimation of demand before sufficient product-specific history exists.

Analogue: A comparable historical product, market, or launch used as forecasting evidence.

Diffusion model: A model describing adoption over time through external and social influences.

Bass model: A diffusion model using innovation and imitation coefficients.

Short-life-cycle item: A product with a limited period in which replenishment is economically valuable.

Newsvendor model: A single-period inventory model balancing underage and overage costs.

Demand sensing: Use of recent high-frequency signals to update a baseline forecast.

Disruption: An event that materially changes demand, supply, lead time, or their relationships.

Scenario forecast: A set of coherent demand paths based on explicit assumptions.

Forecast reconciliation: Adjustment of forecasts so that related hierarchical forecasts are consistent.

Censored demand: Demand not fully observed because of stockouts, allocation, or other constraints.

Forecast governance: Controls covering data, models, overrides, accountability, and monitoring.

Review Questions

Short Answer Questions

1. What distinguishes intermittent demand from lumpy demand?
2. Why does Croston’s method estimate demand size and interval separately?
3. State two conditions under which a diffusion model may be inappropriate.
4. What is the purpose of demand sensing?
5. Why are percentage errors problematic when actual demand is zero?

Long Answer Questions

1. Explain the assumptions, mechanics, and limitations of Croston’s method, SBA, and TSB.
2. Discuss how analogies and structured expert judgment can be combined for new-product forecasting.
3. Explain how short-life-cycle forecasting differs from forecasting a stable replenishment item.
4. Describe a complete scenario-based forecasting process for a logistics disruption.
5. Evaluate the governance risks associated with machine-learning demand sensing.

Analytical / Application Questions

1. A spare part has non-zero demand sizes 3 and 7, with intervals 2 and 4. Using [[MATH{α = 0.25}]], initial estimates [[MATH{ẑ_0 = 3}]] and [[MATH{q̂_0 = 2}]], calculate the updated estimates after the second event and the Croston rate.
2. A seasonal item has underage cost 12 currency units and overage cost 3 currency units. Determine the newsvendor critical fractile and explain what demand quantile it requires.
3. A disruption produces three demand scenarios of 800, 1,200, and 2,000 units with probabilities 0.3, 0.5, and 0.2. Calculate expected demand and explain why the result should not replace stress testing.

Multiple Choice Questions with Answers

Question 1

Which feature most directly characterizes intermittent demand?

A. High demand in every period

B. Frequent zero-demand periods

C. A constant upward trend

D. A fixed seasonal cycle

Correct Answer: B. Frequent zero-demand periods

Question 2

In Croston’s method, the point forecast is calculated as:

A. Smoothed interval divided by smoothed size

B. Smoothed size multiplied by smoothed interval

C. Smoothed non-zero size divided by smoothed interval

D. The median of all observations

Correct Answer: C. Smoothed non-zero size divided by smoothed interval

Question 3

What does TSB update explicitly during every period, including zero-demand periods?

A. The probability of demand occurrence

B. The supplier’s purchase price

C. The product’s market potential only

D. The normality of lead-time demand

Correct Answer: A. The probability of demand occurrence

Question 4

The identity [[MATH{E(y_t) = P(I_t = 1) × E(z_t | I_t = 1)}]] separates demand into:

A. Trend and seasonality

B. Price and promotion

C. Occurrence and non-zero size

D. Supply and transportation

Correct Answer: C. Occurrence and non-zero size

Question 5

Which metric is generally unsuitable without modification when many actual demands are zero?

A. MAE

B. MASE

C. A percentage error based on division by actual demand

D. Non-zero demand-size error

Correct Answer: C. A percentage error based on division by actual demand

Question 6

In the Bass model, the coefficient of imitation represents influence associated mainly with:

A. Social contagion and existing adopters

B. Supplier lead time

C. Inventory carrying cost

D. Random measurement error

Correct Answer: A. Social contagion and existing adopters

Question 7

The newsvendor critical fractile is:

A. [[MATH{C_o/(C_u + C_o)}]]

B. [[MATH{C_u/(C_u + C_o)}]]

C. [[MATH{C_u − C_o}]]

D. [[MATH{C_u × C_o}]]

Correct Answer: B. [[MATH{C_u/(C_u + C_o)}]]

Question 8

A major danger in using observed sales during a stockout is that they may:

A. Overstate unconstrained demand

B. Understate unconstrained demand

C. Always remove seasonality

D. Guarantee unbiased parameters

Correct Answer: B. Understate unconstrained demand

Question 9

A scenario forecast should primarily be understood as:

A. A random selection of optimistic numbers

B. A coherent path based on explicit assumptions

C. A replacement for all statistical models

D. A guarantee of the most likely outcome

Correct Answer: B. A coherent path based on explicit assumptions

Question 10

Why is random shuffling inappropriate for ordinary time-series cross-validation?

A. It increases the number of observations

B. It can leak future information into model training

C. It eliminates all zero demand

D. It forces the use of Croston’s method

Correct Answer: B. It can leak future information into model training