Specialized Demand Contexts

Introduction and Context

Demand forecasting is most difficult when demand does not behave like a stable, continuously observed time series. Many operations decisions concern products purchased only occasionally, products affected by promotions or external events, newly launched offerings with little history, services whose demand is constrained by capacity, or items that become obsolete quickly. In such settings, a conventional seasonal ARIMA or exponential-smoothing model may produce precise-looking but operationally misleading forecasts.

Specialized demand contexts require a match between the demand-generating process and the forecasting method. Intermittent demand is characterized by many zero observations; lumpy demand combines intermittency with highly variable non-zero quantities; seasonal demand follows recurring calendar patterns; event-driven demand responds to identifiable interventions; censored demand is only partially observed because stockouts or capacity limits conceal true demand; and new-product demand must be inferred from analogies, market research, or diffusion theory.

The managerial objective is not merely to minimize forecast error. It is to support decisions about inventory, capacity, staffing, procurement, pricing, launch timing, service levels, and end-of-life liquidation. Consequently, forecast design should account for the forecast horizon, decision lead time, replenishment policy, cost asymmetry, information latency, and the consequences of error.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Distinguish intermittent, lumpy, seasonal, event-driven, censored, new-product, short-life-cycle, and service demand.
2. Select and implement Croston-type methods and understand their assumptions.
3. Apply diffusion models, analogical forecasting, and demand-sensing processes.
4. Recognize censoring and estimate lost demand rather than treating observed sales as demand.
5. Evaluate specialized forecasts using metrics appropriate to sparse and decision-dependent data.
6. Integrate statistical forecasts with managerial judgment, market intelligence, and operational constraints.
7. Govern specialized forecasting systems ethically and assess their limitations.

1. Demand Regimes and Their Operational Meaning

A useful starting point is to distinguish demand from sales. Demand is the quantity customers would purchase under specified conditions; sales are the transactions actually completed. If inventory, staffing, delivery capacity, or credit limits restrict transactions, sales are a censored observation of demand.

1.1 Intermittent and lumpy demand

For period t, let y_t denote demand. Intermittent demand contains many periods for which y_t = 0. Lumpy demand is intermittent demand in which the positive quantities vary substantially. Examples include aircraft components, emergency spare parts, specialist medical devices, and industrial maintenance items.

Two features should be separated:

• Demand occurrence: whether demand occurs in a period.
• Demand size: the quantity conditional on occurrence.

Let p be the probability of a non-zero demand in a period and μ_z the mean positive demand size. Under a simple independent model, the unconditional mean demand is:

Here, E(y_t) is measured in units per period, p is dimensionless, and μ_z is units per occurrence. The relationship assumes a stable occurrence probability and stable positive-size distribution. It is descriptive, not a guarantee that occurrence and size are independent.

A low occurrence probability creates difficulty for models based on recent consecutive observations. A single large order may dominate a moving average, while several zero periods may cause a forecast to collapse prematurely. Lumpy demand additionally requires attention to variance and tail risk, not only the mean.

1.2 Seasonal demand

Seasonal demand exhibits systematic variation associated with calendar position, weather, institutional schedules, or recurring commercial cycles. Seasonality may be additive or multiplicative. In an additive model, the seasonal effect is approximately constant in units; in a multiplicative model, its magnitude rises with the level.

A multiplicative decomposition can be written as:

where T_t is trend, S_t is the seasonal index, C_t represents cyclical movement, and I_t is irregular variation. The formulation assumes positive data and interpretable multiplicative effects. For a seasonal cycle of length m, normalized indices commonly satisfy:

The normalization prevents the seasonal indices from changing the overall level. Seasonal models should be adapted when holidays move between weeks, when seasons change length, or when promotions confound calendar effects.

1.3 Event-driven demand

Event-driven demand is caused by identifiable interventions: a product launch, sports final, election, severe storm, regulatory change, festival, media campaign, or competitor failure. The event may create a temporary pulse, a permanent level shift, or a changed growth rate.

A simple intervention model is:

where I_t is an indicator equal to 1 during the event window and 0 otherwise, β₂ estimates the event increment, and ε_t is unexplained error. A pulse intervention uses a temporary indicator; a step intervention remains 1 after the event. The model assumes that the intervention window is correctly defined and that other changes are either modeled or sufficiently controlled.

Event forecasts should include scenarios because event magnitude, timing, and customer response are uncertain. A point estimate without a range can be especially dangerous when capacity cannot be expanded quickly.

1.4 Censored demand

When a stockout occurs, recorded sales may be lower than customer demand. If [[MATH(s_t)]] is observed sales and [[MATH(d_t)]] is latent demand, a simple inventory-censoring relationship is:

where [[MATH(a_t)]] is available inventory or service capacity. This equation assumes that all available units are sold when demand exceeds availability and that no substitution or rationing effect changes behavior. In reality, customers may defer, substitute, abandon, or purchase from competitors.

Censoring creates a feedback problem: high demand causes stockouts, stockouts reduce observed sales, and a model trained on observed sales then under-forecasts future demand. Corrective methods include lost-sales estimation, censored regression, inventory-aware likelihood models, substitution analysis, and the use of unconstrained locations or comparable products as controls.

1.5 New, short-life-cycle, and service demand

A new product has limited or no direct historical data. A short-life-cycle product has a narrow selling window, so forecast learning and replenishment decisions occur simultaneously. Fashion apparel, entertainment releases, seasonal gifts, and technology products are examples.

Service demand has additional characteristics: it is often non-storable, arrival times matter, capacity can censor demand, and waiting time influences future arrivals. A call centre, hospital clinic, airline, hotel, or repair operation must forecast arrivals, workload, duration, abandonment, and sometimes no-shows. Forecasting units alone is inadequate when the operational requirement is labor hours or beds.

2. Croston-Type Forecasting for Intermittent Demand

2.1 Rationale and basic method

Croston's method separates the estimated size of a non-zero demand from the estimated interval between non-zero demands. Suppose a non-zero demand of size z occurs after an interval of τ periods. The original method updates the smoothed demand size and interval only when demand is non-zero:

Here, x_t is the positive demand size at an occurrence, q_t is the interval since the previous occurrence, α is a smoothing parameter between 0 and 1, z_t is the smoothed positive size, τ_t is the smoothed interval, and F_t+1 is forecast demand per period. Zero-demand periods do not update z_t or τ_t. The method assumes that occurrence intervals and positive sizes can be estimated separately and that the process is reasonably stable.

Initialization matters. A practitioner may initialize z₀ with the first positive demand and τ₀ with the interval to the first positive demand, or use several initial occurrences. Different initialization rules can produce materially different early forecasts.

2.2 Variants and bias correction

The original method can be biased because the ratio of two smoothed estimates is not generally an unbiased estimate of the long-run mean. Syntetos–Boylan Approximation (SBA) applies a correction commonly written as:

The correction is intended for intermittent demand under assumptions similar to those used in the theoretical analysis; it is not universally optimal. Teunter–Syntetos–Babai (TSB) models the probability of demand occurrence directly and updates it every period, including zero-demand periods:

Here, o_t equals 1 when demand occurs and 0 otherwise; β is the occurrence-probability smoothing parameter; and z_t is updated only for positive demand. TSB is particularly useful when demand becomes obsolete or the occurrence probability changes, because repeated zeros reduce p_t.

The following comparison is useful:

2.3 Worked example

Assume that positive demand sizes observed at periods 1, 4, 9, and 10 are 5, 7, 4, and 6 units. The intervals between occurrences are 3, 5, and 1 periods. Use α = 0.2, initialize z₁ = 5 and τ₁ = 3.

At the second occurrence:

z = 5 + 0.2(7 − 5) = 5.4

τ = 3 + 0.2(5 − 3) = 3.4

At the third occurrence:

z = 5.4 + 0.2(4 − 5.4) = 5.12

τ = 3.4 + 0.2(1 − 3.4) = 2.92

The Croston forecast after period 9 is therefore 5.12 ÷ 2.92 ≈ 1.75 units per period. At the fourth occurrence:

z = 5.12 + 0.2(6 − 5.12) = 5.296

τ = 2.92 + 0.2(1 − 2.92) = 2.536

Thus, the next forecast is 5.296 ÷ 2.536 ≈ 2.09 units per period. The calculation illustrates why a recent short interval increases the forecast even though the positive-size estimate changes only modestly. It should not be interpreted as a prediction that exactly 2.09 units will be sold in the next period; it is a rate forecast.

2.4 Evaluation and inventory decisions

Ordinary mean absolute percentage error is inappropriate when actual demand is zero, and it can become unstable when actual demand is very small. More suitable measures include mean absolute error, root mean squared error, scaled errors, cumulative forecast error, and service-oriented measures such as stockout frequency or fill rate. For intermittent demand, evaluation should be conducted over a replenishment cycle or lead-time-plus-review horizon, not only one period at a time.

A forecast rate must also be translated into a distribution. If demand occurrence is approximately Bernoulli with probability p and positive size has mean μ_z and variance σ_z², then, under independence, the one-period variance is:

This formula assumes a binary occurrence process and independent positive size. It shows that uncertainty arises both from the size of an order and from whether an order occurs. Inventory policy should use lead-time demand distributions, supplier reliability, ordering costs, and criticality rather than a point forecast alone.

3. Forecasting Seasonality and Events Together

Seasonal models should distinguish predictable calendar effects from event effects. A regression with seasonal indicators and event variables may be expressed as:

D_jt are seasonal dummy variables, γ_j are their effects, and δ is the event effect. One seasonal category is omitted to avoid perfect multicollinearity with the intercept. If demand variance grows with level, a logarithmic transformation may be considered, but zero values require care and the transformed forecast may need bias adjustment.

For events with uncertain magnitude, scenario forecasts are preferable:

Event sensing should begin before the event and continue during the event. Search activity, reservations, registrations, weather forecasts, traffic, social signals, and competitor availability may be leading indicators. Such variables should be validated out of sample; correlation during one event does not establish stable predictive value.

4. New-Product and Short-Life-Cycle Forecasting

4.1 Diffusion models

Diffusion models represent adoption over time. The Bass model describes the expected adoption rate using innovation and imitation effects:

where N(t) is cumulative adopters by time t, n(t) is the adoption rate, m is market potential, p is the coefficient of innovation, and q is the coefficient of imitation. The parameters p and q are dimensionless rates under the time scale used; m is customers or units. The model assumes a finite potential market and distinguishes externally influenced adoption from word-of-mouth adoption.

The Bass model is useful for strategic launch scenarios, but parameter estimates are weak when only a short pre-launch or early-launch history exists. Market potential is often uncertain, repeat purchases may violate the one-time adoption assumption, and advertising, price, distribution, and competition can change the process. A generalized model should include marketing and economic covariates when sufficient data exist.

4.2 Analogical forecasting

An analogy forecasts a new item using historical products, markets, regions, or customer segments judged similar. The analogy should be explicit rather than intuitive. Relevant dimensions include:

• customer problem and use case;
• price and positioning;
• channel and geographic reach;
• product novelty and switching cost;
• marketing support and distribution coverage;
• launch timing and competitive conditions;
• repeat-purchase and substitution behavior.

A weighted analogy forecast can be written:

where y_i,t is the aligned historical demand of analogue i and w_i is its weight. Alignment may require scaling for market size, distribution, or price. Weights should be justified and, where possible, estimated using historical back-testing rather than assigned solely by managerial preference.

4.3 Short-life-cycle control

For short-life-cycle products, a forecast is inseparable from a decision about initial commitment and later information acquisition. A large initial order may reduce stockouts but increase end-of-life markdowns. A small order may protect margin but lose sales and visibility. The newsvendor critical fractile provides a benchmark for a single-order decision:

where D is random demand, Q* is the recommended order quantity, C_u is underage cost per unit, and C_o is overage cost per unit. The formula assumes one replenishment opportunity, known cost parameters, and a specified demand distribution. It is not directly appropriate when multiple replenishments, capacity constraints, or strategic customer effects dominate.

5. Service Demand and Demand Sensing

Service forecasts should be expressed in operational drivers. A hospital may forecast arrivals by acuity and duration; a call centre may forecast contacts by interval and handling time; a hotel may forecast room nights, cancellations, and no-shows. If A_t is arrivals and h_t is average workload per arrival, estimated workload is:

where W_t is workload, measured for example in labor-hours, provided arrivals and workload per arrival are consistently defined. Variability in both terms must be considered; multiplying means does not fully describe the distribution of workload.

Demand sensing is a short-horizon process that updates a baseline forecast with rapidly arriving signals. A practical architecture has four layers:

1. Baseline: statistical forecast using historical demand.
2. Signal ingestion: orders, bookings, point-of-sale data, web traffic, weather, mobility, promotions, and operational status.
3. Signal qualification: latency, reliability, duplication, causality, and leakage checks.
4. Decision translation: inventory, labor, capacity, allocation, and exception actions.

Demand sensing is not synonymous with machine learning. A transparent regression, state-space model, or rules-based adjustment may outperform a complex algorithm when data are sparse or unstable. Machine learning is most useful when covariates are numerous, relationships are nonlinear, and sufficient back-testing data exist.

A crucial governance principle is that information unavailable at the forecast timestamp must not enter model training as if it were known. This problem, called data leakage, can make demand-sensing models appear highly accurate in development and fail in deployment.

6. Censored Demand: Detection and Correction

Censoring should be diagnosed using operational evidence. Indicators include zero sales while nearby locations sell out, persistent maximum-capacity utilization, unusually high substitution, abandoned carts, denied appointments, and abrupt sales declines immediately after stockouts. The analyst should preserve inventory-on-hand, availability, price, promotion, and substitution fields alongside sales.

Possible approaches include:

• Lost-sales imputation: estimate demand during unavailable periods using comparable periods or locations.
• Censored likelihood models: estimate parameters while recognizing that observations above capacity are censored.
• Survival or hazard models: useful when the event of purchase or arrival is time-dependent.
• Choice and substitution models: estimate where customers go when the preferred product is unavailable.
• Controlled experiments: vary availability or capacity where ethically and operationally feasible.

Imputation should produce uncertainty, not a falsely precise replacement value. A range of plausible lost demand can be propagated into inventory and capacity decisions.

7. Integrated Case: Illustrative Medical-Device Launch

Background

A medical-device manufacturer is launching a disposable catheter for a specialized procedure. The product has no direct sales history, hospitals purchase irregularly, demand is affected by surgeon training, and supply interruptions at competitors may produce sudden orders. The product has a twelve-month introductory window before a redesigned version is expected.

Problem

The commercial team proposes a conventional monthly moving average based on pilot orders. Operations expects intermittent demand and fears that early stockouts will conceal true market acceptance. Finance is concerned about obsolete inventory after the redesign.

Analysis

The team first separates demand into hospital adoption, procedure frequency, and units per procedure. Pilot orders are not treated as pure demand because some were samples and training allocations. Comparable devices are selected by clinical use, price, hospital type, channel, and training intensity. The team creates low, base, and high market-potential scenarios.

For established hospitals, TSB is selected because demand occurrence may increase during adoption and fall sharply when the replacement product is announced. For the launch curve, a Bass-style diffusion model is estimated using comparable products, with wide parameter ranges. A service-level model then converts expected procedures into device requirements.

Availability is recorded daily. When a hospital reaches zero inventory, the observation is flagged as censored. A control group of hospitals with uninterrupted availability is used to estimate the likely demand path. Forecast performance is evaluated over the supplier lead time rather than monthly MAPE.

Approach and solution

The governance team establishes a forecast timestamp and prevents future information from entering training data. Sales representatives may submit structured judgments, but every adjustment requires a reason code: training event, tender win, competitor shortage, or supply constraint. The system reports both baseline and adjusted forecasts.

Inventory is divided into launch stock, routine replenishment stock, and contingency stock. The contingency stock is released only when leading indicators and confirmed orders support the high scenario. Weekly reviews examine adoption, positive-demand intervals, stockout flags, cancellation rates, and forecast bias.

Outcome

In this illustrative case, the integrated approach avoids interpreting training samples as recurring demand, detects two periods of censored demand, and reduces the risk of both premature production expansion and excessive end-of-life inventory. The main benefit is not a single superior algorithm; it is alignment between demand definition, data governance, model selection, and operational decisions.

Learning points

1. New-product forecasting is a portfolio of evidence, not a single time-series exercise.
2. Availability data are essential when sales are constrained.
3. Intermittent demand requires separate treatment of occurrence and size.
4. Scenarios are more defensible than an apparently exact launch forecast.
5. Forecast overrides should be traceable, testable, and reversible.

8. Evaluation, Governance, and Best Practice

A specialized forecasting process should be evaluated at the level of the decision it supports. Important measures include bias, scaled absolute error, service level, fill rate, lost-sales rate, capacity utilization, markdowns, waste, and forecast value added. Forecast value added compares a process with and without a particular intervention, such as a sales override or demand-sensing signal.

Best practice includes:

• classify items by demand regime before selecting a model;
• use rolling-origin back-testing that preserves the information available at each historical forecast date;
• evaluate multiple horizons, especially lead time and review period;
• use probabilistic forecasts or scenarios where uncertainty is material;
• distinguish demand, sales, orders, shipments, and capacity;
• monitor parameter drift and structural breaks;
• document data lineage, transformations, overrides, and model ownership;
• use human judgment for contextual information, not for unrecorded optimism;
• retire models when products become obsolete or market conditions change.

Ethical and regulatory issues are significant in health care, employment services, credit-related operations, and public-sector allocation. A forecast can reproduce historical inequities if demand is under-recorded for underserved groups. Allocation decisions should not silently penalize customers whose previous demand was censored by poor access. Sensitive data must be minimized, protected, and used only for legitimate purposes. Automated recommendations should be auditable, and managers should retain responsibility for consequential decisions.

Chapter Summary

Specialized demand contexts require models that reflect how demand occurs, not merely how historical sales appear. Intermittent demand is decomposed into occurrence and positive size; Croston-type methods, SBA, and TSB provide useful but assumption-dependent tools. Seasonal and event-driven demand require explicit calendar and intervention variables, scenario analysis, and careful separation of recurring effects from exceptional events. Censored demand must be distinguished from genuinely low demand because stockouts and capacity limits conceal customer requirements. New-product and short-life-cycle forecasting combine diffusion models, analogies, market intelligence, and sequential learning. Service forecasting must translate arrivals into workload while accounting for capacity, duration, abandonment, and no-shows. Demand sensing can improve short-horizon responsiveness when signals are timely, causal, and governed against leakage. Across all contexts, evaluation should emphasize decision outcomes, uncertainty, bias, data quality, and operational consequences rather than a single accuracy statistic.

Key Terms

Intermittent demand: Demand containing many zero-observation periods.

Lumpy demand: Intermittent demand with highly variable positive quantities.

Demand occurrence: The event that demand is non-zero in a period.

Positive demand size: Quantity demanded conditional on an occurrence.

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

SBA: Syntetos–Boylan Approximation, a bias-adjusted Croston-type method.

TSB: Teunter–Syntetos–Babai method, which smooths demand occurrence probability.

Seasonality: Repeating variation associated with calendar or environmental cycles.

Intervention: An external event or change represented explicitly in a forecasting model.

Censored demand: Demand that is only partially observed because sales or service are constrained.

Lost sales: Potential demand not completed because the product or service was unavailable.

Diffusion model: A model describing adoption or market growth over time.

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

Analogy: A comparable historical product, market, or segment used to forecast a new offering.

Short-life-cycle product: An offering with a limited selling or decision window.

Newsvendor critical fractile: The target cumulative probability balancing underage and overage costs.

Demand sensing: Short-horizon updating using rapidly arriving operational and market signals.

Data leakage: Use of information in model development that would not be available at the forecast time.

Forecast censoring: Distortion created when observed sales or service reflect capacity rather than unconstrained demand.

Forecast value added: The measured contribution of a forecasting step or intervention relative to a benchmark.

Review Questions

Short Answer Questions

1. What distinguishes intermittent demand from lumpy demand?
2. Why can observed sales understate demand during a stockout?
3. What two quantities are separately estimated in Croston’s method?
4. What is the difference between a pulse and a step intervention?
5. Why is MAPE often unsuitable for intermittent demand?

Long Answer Questions

1. Explain Croston’s method, SBA, and TSB, including their assumptions and appropriate applications.
2. Discuss how seasonal and event-driven demand can be modeled together without confusing recurring and exceptional effects.
3. Compare diffusion models and analogical forecasting for a new product with no direct historical sales.
4. Explain the implications of censored demand for data collection, model estimation, and inventory decisions.
5. Design a demand-sensing process for a service operation and identify its principal governance risks.

Analytical / Application Questions

1. A spare-parts item has positive demand sizes of 4, 8, and 5 units at intervals of 2, 4, and 3 periods. Explain how you would initialize and apply a Croston-type forecast, and discuss how the result should be used in a replenishment policy.
2. A retailer observes falling sales during a promotion, but inventory records show repeated stockouts. Propose an analysis to distinguish weak demand from censored demand and identify the additional data required.
3. A technology product will be sold for six months before replacement. Construct a forecasting and decision framework combining analogies, diffusion scenarios, demand sensing, and end-of-life inventory controls.

Multiple Choice Questions with Answers

Question 1

Which feature most directly defines intermittent demand?

A. Demand varies only with weather

B. Many periods have zero demand

C. Demand is always capacity constrained

D. Demand has a fixed seasonal index

Correct Answer: B. Many periods have zero demand

Question 2

In Croston’s method, what is updated only when positive demand occurs?

A. The tax rate and price

B. The seasonal dummy variables

C. The positive demand size and inter-demand interval

D. The service capacity and lead time

Correct Answer: C. The positive demand size and inter-demand interval

Question 3

If p = 0.2 and mean positive demand is 10 units, the simple unconditional mean is:

A. 0.2 units per period

B. 2 units per period

C. 10.2 units per period

D. 50 units per period

Correct Answer: B. 2 units per period

Question 4

What does TSB model directly in addition to positive demand size?

A. Demand occurrence probability

B. Supplier purchase price

C. Seasonal cycle length only

D. Customer profit

Correct Answer: A. Demand occurrence probability

Question 5

Which situation is an example of censored demand?

A. Demand is zero because no customer wants the product

B. Sales are limited because inventory is exhausted

C. Demand follows a stable annual pattern

D. A product is sold in two colors

Correct Answer: B. Sales are limited because inventory is exhausted

Question 6

In the Bass model, the coefficient of imitation represents adoption influenced primarily by:

A. Word of mouth and existing adopters

B. Warehouse temperature

C. Accounting depreciation

D. Supplier lead time

Correct Answer: A. Word of mouth and existing adopters

Question 7

Why is MAPE problematic for intermittent demand?

A. It cannot be calculated when actual demand is zero

B. It always over-forecasts seasonal demand

C. It requires a diffusion model

D. It measures capacity instead of demand

Correct Answer: A. It cannot be calculated when actual demand is zero

Question 8

What is data leakage in demand sensing?

A. Deleting all historical observations

B. Using information unavailable at the forecast timestamp

C. Separating demand size from occurrence

D. Reducing the number of model variables

Correct Answer: B. Using information unavailable at the forecast timestamp

Question 9

The newsvendor critical fractile balances which two types of cost?

A. Fixed and variable labor cost

B. Underage and overage cost

C. Transport and tax cost

D. Marketing and research cost

Correct Answer: B. Underage and overage cost

Question 10

For service operations, multiplying arrivals by average workload per arrival estimates:

A. Market potential

B. Workload

C. Seasonal amplitude

D. Cumulative adoption

Correct Answer: B. Workload