Continuous-Review Stochastic Inventory Models

Introduction

Inventory decisions become substantially more difficult when demand and replenishment lead time are uncertain. In a deterministic model, a firm can calculate the exact time at which stock will reach zero and schedule replenishment accordingly. In practice, demand varies from day to day, suppliers may deliver early or late, and shortages may be backordered or permanently lost. A continuous-review stochastic model addresses these conditions by monitoring inventory continuously and triggering an order when the inventory position reaches a specified reorder point.

The central policy is the (Q, r) policy:

• Q is the fixed replenishment quantity.
• r is the reorder point.

Whenever the inventory position is at or below r, an order of Q units is placed. The reorder point must protect against demand occurring during the replenishment lead time. Consequently, its design requires a probability model for lead-time demand and an explicit service objective.

These models are important in spare-parts management, retail distribution, healthcare, manufacturing maintenance, and e-commerce fulfilment. They are particularly appropriate for high-value or critical items for which inventory is reviewed continuously through an enterprise resource planning (ERP), warehouse management, or point-of-sale system.

Learning objectives

After studying this chapter, the reader should be able to:

1. Explain the logic and assumptions of continuous-review stochastic inventory systems.
2. Design a (Q, r) policy under uncertain demand and lead time.
3. Calculate safety stock using appropriate demand and lead-time distributions.
4. Distinguish cycle-service level from fill rate.
5. Model backorders, lost sales, and variable replenishment conditions.
6. Evaluate the assumptions, limitations, and implementation risks of stochastic inventory models.
7. Interpret inventory-policy results in managerial and operational terms.

1. The inventory-position logic

On-hand inventory is the physical stock currently available. It does not, by itself, indicate whether a replenishment decision is required. The relevant control variable is inventory position:

where:

• IP = inventory position, in units;
• OH = on-hand inventory;
• OO = inventory on order;
• BO = outstanding backorders.

Inventory position increases when an order is placed and decreases when demand occurs. A continuous-review (Q, r) policy places an order of Q units whenever IP ≤ r. If the order quantity is fixed, the policy is often called a fixed-order-quantity system.

The distinction between on-hand inventory and inventory position is essential. Suppose on-hand inventory is 40 units, 100 units are already on order, and 15 units are backordered. Then:

A reorder decision based only on on-hand stock would incorrectly suggest an immediate order, whereas the inventory position shows that substantial replenishment is already committed.

2. Components of a stochastic (Q, r) policy

A practical policy contains at least four decisions:

1. Order quantity Q: commonly based on the economic order quantity, a supplier minimum, a production batch, or a transportation constraint.
2. Reorder point r: the inventory position at which replenishment is triggered.
3. Service target: such as a cycle-service level or fill rate.
4. Shortage treatment: whether unmet demand is backordered, lost, substituted, or expedited.

2.1 Order quantity

Under the basic EOQ assumptions, the economic order quantity is:

where D is annual demand in units per year, S is ordering cost per order, and H is annual holding cost per unit. The units are consistent because DS has currency per year and H has currency per unit per year, giving units squared under the square root.

The EOQ is only a starting point in a stochastic system. A manager may choose a different Q because of supplier batch sizes, truckload economies, production setup constraints, quantity discounts, shelf-life restrictions, or a desired ordering frequency. Increasing Q generally reduces order frequency but increases average cycle stock, approximately Q/2 when replenishment is instantaneous and shortages are excluded.

2.2 Reorder point

The reorder point consists of expected demand during lead time plus safety stock:

where μ_LT is mean lead-time demand and SS is safety stock. This expression assumes that the inventory position is measured correctly and that the selected safety stock corresponds to the stated service objective.

2.3 Lead-time demand

Let demand per period be the random variable D_t, and let lead time be L. Lead-time demand is:

If lead time is fixed at L periods, demand is independent across periods, with mean μ_D and standard deviation σ_D, then:

The second expression assumes independent demand periods and constant lead time. If demand is autocorrelated, the variance must include covariance terms. If both demand and lead time vary, the simple formula σ_D√L is not generally valid.

3. Safety stock under uncertain demand

Assume lead-time demand is approximately normally distributed:

For a target cycle-service level CSL, the reorder point is:

and safety stock is:

Here, z is the standard-normal quantile satisfying P(D_L ≤ r) = CSL. For example, a 95% cycle-service level uses approximately z = 1.645, while a 99% target uses approximately z = 2.326.

The formula is appropriate when the normal approximation is reasonable, demand observations are sufficiently stable, and the policy is intended to control the probability of no stockout during a replenishment cycle. It is not automatically appropriate for intermittent demand, very small volumes, severe skewness, or highly variable lead times.

Worked example 1: fixed lead time and variable demand

A distributor sells 80 units per day. Daily demand has a standard deviation of 20 units. Lead time is constant at 5 days. The desired cycle-service level is 95%.

Mean lead-time demand is:

The standard deviation of lead-time demand is:

Using z = 1.645:

Therefore:

The policy should place a replenishment order whenever inventory position reaches approximately 474 units. The result does not mean that 74 units will always remain physically available. It means that, under the stated assumptions, the probability of avoiding a stockout during a replenishment cycle is approximately 95%.

4. Variable lead time and variable demand

When both demand and lead time vary, lead-time demand is a random sum. A widely used approximation assumes that daily demand and lead time are independent and that demand is identically distributed. Then:

where μ_L and σ_L are the mean and standard deviation of lead time. Thus:

and, under the normal approximation:

The first variance component, μ_Lσ_D², reflects demand uncertainty. The second, μ_D²σ_L², reflects lead-time uncertainty. This decomposition is useful managerial information: it indicates whether process improvement should focus on forecasting demand or improving supplier reliability.

The formula assumes independence. If demand and lead time are correlated, an additional covariance contribution is required. In general, for a random lead time and correlated demand, simulation or a direct empirical distribution of observed lead-time demand is safer than applying the independent approximation.

Worked example 2: variable demand and lead time

Daily demand has mean 100 units and standard deviation 25 units. Lead time has mean 4 days and standard deviation 1.5 days. Demand and lead time are assumed independent. The target cycle-service level is 95%.

Mean lead-time demand:

Variance of lead-time demand:

Therefore:

Safety stock is:

The reorder point is:

Lead-time variation is the dominant source of uncertainty in this example. Eliminating or reducing supplier lead-time variability could therefore lower safety stock more effectively than a modest improvement in demand forecasting.

5. Cycle-service level and fill rate

Two service measures are frequently confused.

5.1 Cycle-service level

The cycle-service level (CSL) is the probability that no stockout occurs during a replenishment cycle:

A cycle is usually defined as the interval between successive replenishment orders, or as the protection period associated with one order cycle. CSL treats a cycle with one unit short and a cycle with 100 units short in the same way: both are counted as stockout cycles.

5.2 Fill rate

The fill rate is the proportion of demand satisfied immediately from available stock. In a long-run demand-based form:

For a continuous-review (Q, r) system with complete backordering and one replenishment order of Q units per cycle, a common approximation is:

where (x)^+ = max(x,0). This expression assumes that shortages are measured during the protection period and that the order quantity is the relevant demand denominator. It must be adapted when demand is lost, multiple outstanding orders interact, or the replenishment process is not a simple one-order cycle.

A high CSL does not necessarily imply a high fill rate. For example, if 95% of cycles have no shortage but the remaining 5% have very large shortages, the fill rate may be considerably below 95%. Conversely, a cycle can experience a small shortage while satisfying most demand, producing a lower CSL but a high fill rate.

6. Backorders and lost sales

6.1 Backordering

Under a backorder policy, unmet demand remains outstanding and is fulfilled when replenishment arrives. Backorders are appropriate when customers accept delay, when demand is contractually recorded, or when the item is essential and substitution is difficult.

Backorders create two consequences. First, on-hand inventory can be zero or negative in a records-based representation while inventory position remains meaningful. Second, shortages generate explicit delay costs, expediting costs, customer dissatisfaction, or production stoppage. A shortage-cost model may include a per-unit backorder cost p_b and, where relevant, a per-unit per-period delay cost.

For a normally distributed lead-time demand variable, expected shortage in one protection period is:

where L(z) = φ(z) − z[1 − Φ(z)] is the standard normal loss function, φ(z) is the standard normal density, and Φ(z) is its cumulative distribution function. This relationship assumes normal lead-time demand and a reorder point expressed as r = μ_LT + zσ_LT.

6.2 Lost sales

Under a lost-sales policy, demand that cannot be met immediately disappears rather than becoming a backorder. This is common in retail, hospitality, and some online markets where customers switch suppliers or abandon purchases.

Lost sales generally produce a lower effective demand history because observed sales are censored: recorded sales equal available stock rather than true customer demand. Forecasting systems must therefore distinguish sales from demand. If stockouts are treated as zero demand, future forecasts may be biased downward, causing inadequate replenishment.

Backorder and lost-sales models also differ in inventory dynamics. In a backorder system, shortage demand remains part of the inventory-position calculation. In a lost-sales system, it does not. Consequently, the same reorder point and order quantity can generate different service outcomes under the two shortage assumptions.

7. Selecting Q and r jointly

A common practical sequence is:

1. Estimate demand and lead-time distributions.
2. Select Q using EOQ, supplier constraints, or an optimization model.
3. Select a service target based on customer and operational consequences.
4. Calculate r from the distribution of lead-time demand.
5. Simulate or test the policy under realistic conditions.
6. Review performance using fill rate, CSL, inventory, shortages, and cost.

A cost-based formulation may minimize annual ordering, holding, and shortage costs. A simplified objective is:

where D C is purchasing cost when unit purchase cost is C, S is order cost, h is the relevant holding-cost rate, and the shortage-cost term depends on the backorder or lost-sales mechanism. The exact expected-cost expression depends on the replenishment assumptions and should not be substituted by a universal formula.

The economic trade-off is clear: increasing r raises safety stock and holding cost but normally reduces shortages; increasing Q raises cycle stock but may reduce ordering and transportation cost. A service constraint can be imposed instead of a shortage-cost objective, for example requiring FR ≥ 98%.

8. Replenishment conditions beyond the basic model

8.1 Order lead-time uncertainty

Supplier lead times may be affected by production queues, transport congestion, customs, quality holds, and supplier capacity. Managers should use actual receipt dates, not only promised dates, to estimate the lead-time distribution. Separating administrative lead time, production lead time, and transport lead time can reveal improvement opportunities.

8.2 Review and transaction delays

A nominal continuous-review system may not be truly continuous. If transactions are processed in batches or data are updated every few hours, the system has an effective review interval. The protection period then includes lead time plus the time until the next review. Ignoring this delay understates required safety stock.

8.3 Minimum and maximum constraints

Suppliers may impose minimum order quantities, case-pack quantities, or order multiples. If the calculated Q is not feasible, it must be rounded upward or replaced by a constrained quantity. A maximum inventory or shelf-life constraint may require a smaller order quantity and more frequent orders.

8.4 Multiple outstanding orders

When Q is large relative to demand or lead time is long, several orders may be outstanding simultaneously. The inventory position remains the correct trigger variable, but simple cycle-based fill-rate approximations may become inaccurate. Discrete-event simulation can represent order timing, partial deliveries, cancellations, and allocation rules more realistically.

8.5 Partial deliveries and expediting

If suppliers deliver in installments, the replenishment process is not a single random lead time. The model should represent the distribution of first receipt, completion receipt, and usable quantity. Expediting can be incorporated as an alternative supply mode with its own cost, capacity, and lead-time distribution.

9. A detailed applied case: illustrative medical-device distributor

Background

The following is a fictional illustrative case. A distributor supplies a sterile replacement cartridge used by hospitals. Stockouts can delay treatment, but excessive inventory creates expiry risk. Demand and supplier lead time are both variable.

Daily demand has mean 30 units and standard deviation 8 units. Supplier lead time has mean 6 days and standard deviation 2 days. Demand and lead time are initially treated as independent. Management wants a 97.5% cycle-service level. The proposed order quantity is 240 units, based on supplier packaging and transport economics.

Problem

The distributor must determine the reorder point and assess whether a cycle-service target is sufficient. Because the item is date-sensitive, the company also needs to monitor expiry, emergency shipments, and actual customer fill rate.

Analysis

For a 97.5% target, z ≈ 1.96. Mean lead-time demand is:

Variance of lead-time demand is:

Thus:

Safety stock is:

The reorder point is:

The proposed policy is therefore approximately (240, 304), subject to validation.

The calculation shows that lead-time uncertainty contributes most of the variance: 30² × 2² = 3,600, compared with demand uncertainty of 384. A supplier-development project that reduces lead-time standard deviation from 2 days to 1 day would materially reduce safety stock.

Approach and solution

The distributor implements the policy in its ERP system using inventory position rather than on-hand stock. It also records actual demand during stockouts through hospital order logs, rather than treating unavailable units as zero demand. A three-month pilot uses daily data and compares:

• actual cycle-service level;
• unit fill rate;
• emergency orders;
• expiry losses;
• average inventory;
• supplier lead-time distribution.

Because a normal approximation may be imperfect for hospital demand, the company bootstraps historical daily demand and samples observed lead times in a Monte Carlo simulation. The simulation includes partial deliveries and expiry dates.

Outcome

The analytical policy provides a transparent initial setting, while simulation identifies the consequences of skewed demand and late deliveries. Management discovers that the nominal 97.5% CSL produces a lower unit fill rate during several unusually large hospital orders. The firm therefore retains the reorder point as a baseline but adds allocation rules, emergency-supply agreements, and a separate service target for critical hospitals.

Learning points

1. A mathematically calculated reorder point is not a substitute for data governance.
2. The dominant uncertainty source should guide improvement investment.
3. Cycle-service and fill rate measure different customer outcomes.
4. Stockout observations must be corrected for censored demand.
5. Critical products may require segmentation rather than one universal service target.

10. Technology, analytics, and governance

Modern inventory systems can calculate inventory position continuously from sales, receipts, returns, reservations, transfers, and backorders. Statistical forecasting platforms estimate demand distributions rather than only point forecasts. Machine-learning methods can improve forecasts when demand depends on promotions, weather, customer characteristics, or network conditions. However, a more complex forecast does not automatically produce a better inventory policy.

AI-supported systems should be governed through:

• traceable data sources and versioned forecasts;
• monitoring for forecast bias and stockout-censored demand;
• human review of abnormal recommendations;
• documented service and cost objectives;
• scenario testing before automatic parameter changes;
• controls against manipulation of supplier or customer data.

For regulated products, such as pharmaceuticals and medical devices, inventory policies must also respect expiry, traceability, quality release, recall procedures, and applicable regulatory requirements. Ethical considerations arise when allocation rules determine which customers receive scarce stock. Priority rules should be explicit, defensible, and consistent with contractual and clinical obligations.

11. Limitations and best practices

Important limitations

The normal approximation can be poor for intermittent or highly skewed demand. Demand observations may not be independent, lead times may be correlated with demand surges, and supplier performance may change over time. EOQ assumptions may fail where prices, capacity, shelf life, or setup constraints dominate. A target service level also has no economic meaning unless the consequences of shortage and excess inventory are understood.

Best practices

1. Use inventory position, not only on-hand stock.
2. Estimate demand during actual lead time, including transaction and review delays.
3. Separate forecast uncertainty from supplier lead-time uncertainty.
4. Match the service measure to the customer promise.
5. Correct demand histories for lost sales and stockout censoring.
6. Segment items by criticality, value, intermittency, and substitutability.
7. Validate analytical formulas with historical back-testing or simulation.
8. Review parameters when demand, suppliers, product design, or service requirements change.
9. Monitor both average performance and tail events.
10. Document assumptions so that policy parameters remain auditable.

Chapter Summary

Continuous-review stochastic inventory models use inventory position to trigger replenishment under uncertain demand and lead time. In a (Q, r) policy, Q determines the replenishment quantity and r protects against demand during the replenishment lead time. With fixed lead time and independent demand, lead-time demand has mean Lμ_D and standard deviation σ_D√L. With independent variable demand and lead time, a useful approximation is Var(D_L) = μ_Lσ_D² + μ_D²σ_L². Safety stock is normally calculated as a service-factor multiple of lead-time-demand standard deviation, but the selected distribution and assumptions must be justified.

Cycle-service level measures the probability of avoiding any stockout during a cycle, whereas fill rate measures the proportion of demand supplied immediately. Backorders retain unmet demand for later fulfilment; lost sales remove it from the current order stream and can censor demand observations. Variable replenishment, partial deliveries, review delays, minimum order quantities, expiry, and multiple outstanding orders may require simulation or richer stochastic models. Effective implementation combines quantitative analysis, data governance, supplier management, service segmentation, and continuous performance review.

Key Terms

Inventory position: On-hand inventory plus on-order inventory minus outstanding backorders.

(Q, r) policy: A continuous-review policy ordering Q units when inventory position reaches r.

Reorder point: Inventory-position level that triggers replenishment.

Safety stock: Inventory held to protect against demand or lead-time uncertainty.

Lead-time demand: Total demand occurring during the replenishment lead time.

Cycle-service level: Probability of no stockout during a replenishment cycle.

Fill rate: Proportion of demand fulfilled immediately from available stock.

Backorder: Unmet demand recorded and fulfilled later.

Lost sale: Demand that disappears because immediate supply is unavailable.

Cycle stock: Inventory arising from ordering in batches, commonly approximately Q/2.

EOQ: Economic order quantity balancing ordering and holding costs under specified assumptions.

Service factor: Standard-normal quantile z associated with a target service probability.

Normal approximation: Representation of lead-time demand by a normal distribution.

Demand censoring: Distortion of observed demand because stockouts prevent sales from being recorded.

Shortage function: Expected amount by which demand exceeds available inventory.

Lead-time variability: Statistical variation in replenishment duration.

Inventory on order: Replenishment quantities ordered but not yet received.

Continuous review: Monitoring inventory position whenever relevant transactions occur.

Monte Carlo simulation: Repeated random sampling used to estimate policy performance under uncertainty.

Review Questions

5 Short Answer Questions

1. Define inventory position and explain why it differs from on-hand inventory.
2. What are the two parameters in a (Q, r) policy?
3. State the assumptions behind σ_LT = σ_D√L.
4. Distinguish cycle-service level from fill rate.
5. Explain one important difference between backorders and lost sales.

5 Long Answer Questions

1. Explain the structure of a continuous-review stochastic inventory system and discuss the managerial meaning of Q and r.
2. Derive the reorder-point expression for fixed lead time and normally distributed lead-time demand.
3. Discuss how variable lead time changes safety-stock calculations and identify the assumptions behind the common variance approximation.
4. Compare backorder and lost-sales systems in terms of inventory records, customer consequences, and demand-data quality.
5. Evaluate the limitations of using EOQ and normal-distribution assumptions in a modern supply chain.

3 Analytical / Application Questions

1. Daily demand has mean 120 units and standard deviation 30 units. Lead time is constant at 4 days, and the required cycle-service level is 99%. Using z = 2.326, calculate lead-time demand mean, safety stock, and reorder point.
2. A supplier has mean lead time 5 days and standard deviation 1 day. Daily demand has mean 50 units and standard deviation 10 units. Assuming independence and a 95% cycle-service level, calculate the approximate reorder point using z = 1.645. Identify which uncertainty component is larger.
3. A retailer reports a 98% cycle-service level but only a 94% fill rate. Explain how both results can occur and recommend two additional measures that should be monitored before changing the reorder point.

Multiple Choice Questions with Answers

Question 1

In a continuous-review system, which expression defines inventory position?

A. On-hand inventory minus annual demand

B. On-hand inventory plus on-order inventory minus backorders

C. On-order inventory minus safety stock

D. On-hand inventory plus cycle stock plus annual demand

Correct Answer: B. On-hand inventory plus on-order inventory minus backorders

Question 2

Daily demand has a standard deviation of 10 units and fixed lead time is 4 days. Assuming independent daily demand, what is the standard deviation of lead-time demand?

A. 2.5 units

B. 14 units

C. 20 units

D. 40 units

Correct Answer: C. 20 units

Question 3

A reorder point under a normal lead-time-demand approximation is generally calculated as:

A. Mean lead-time demand minus safety stock

B. Order quantity plus annual demand

C. Mean lead-time demand plus safety stock

D. Safety stock divided by lead time

Correct Answer: C. Mean lead-time demand plus safety stock

Question 4

Which statement best describes cycle-service level?

A. The proportion of demand units filled immediately

B. The probability of no stockout during a replenishment cycle

C. The percentage reduction in ordering cost

D. The average number of units ordered per cycle

Correct Answer: B. The probability of no stockout during a replenishment cycle

Question 5

Which service measure is directly based on the proportion of demand units supplied immediately?

A. Fill rate

B. Cycle-service level

C. Lead-time reliability

D. Order frequency

Correct Answer: A. Fill rate

Question 6

Daily demand has mean 100 units and lead time has mean 3 days. What is the mean lead-time demand under the standard independent-demand assumption?

A. 33.3 units

B. 97 units

C. 103 units

D. 300 units

Correct Answer: D. 300 units

Question 7

If demand and lead time are independent, which term in the common lead-time-demand variance approximation represents lead-time uncertainty?

A. μ_Lσ_D²

B. μ_D²σ_L²

C. μ_Dσ_D

D. σ_L/μ_L

Correct Answer: B. μ_D²σ_L²

Question 8

What is the principal effect of a lost-sales assumption?

A. Unmet demand remains recorded as a future obligation

B. Unmet demand disappears rather than becoming a backorder

C. Lead time becomes deterministic

D. Safety stock is always zero

Correct Answer: B. Unmet demand disappears rather than becoming a backorder

Question 9

A system has on-hand inventory of 40 units, 70 units on order, and 10 units backordered. What is its inventory position?

A. 20 units

B. 40 units

C. 100 units

D. 120 units

Correct Answer: C. 100 units

Question 10

Which action is most appropriate when supplier lead-time variability is the dominant contributor to lead-time-demand variance?

A. Ignore supplier performance and increase forecast frequency only

B. Reduce supplier lead-time variability through supplier-development measures

C. Treat all stockouts as zero demand

D. Lower the reorder point without changing service objectives

Correct Answer: B. Reduce supplier lead-time variability through supplier-development measures