Periodic-Review and Service-Level Models

Introduction and context

Inventory decisions differ fundamentally according to how often stock status is observed. In a continuous-review system, inventory is monitored continuously and an order is released when a trigger is reached. In a periodic-review system, inventory is examined at fixed intervals and replenishment decisions are made only at those review times. Periodic review is common when many stock-keeping units are managed together, when suppliers accept scheduled orders, when physical counting is conducted in batches, or when transport and administrative costs favour consolidated orders.

The central managerial problem is not simply how much to order. It is how to coordinate review frequency, replenishment quantity, protection-period stock, demand uncertainty, supplier lead time, and customer-service objectives. This chapter develops the principal periodic policies—(P,S), order-up-to, and min–max policies—and explains how service levels should be defined, calculated, monitored, and traded against cost.

Learning objectives

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

• distinguish periodic-review from continuous-review inventory systems;
• formulate and interpret (P,S), order-up-to, and min–max policies;
• calculate protection-period demand and safety stock under appropriate assumptions;
• connect cycle-service and fill-rate targets with inventory decisions;
• analyse the effects of review period, lead time, variability, and aggregation;
• compare periodic and continuous review using cost, risk, and operational criteria;
• identify implementation, governance, data, and technology requirements.

1. Foundations of periodic review

Let inventory position be the stock measure used for replenishment decisions:

Here, IP_t is inventory position at time t, OH_t is on-hand inventory, OO_t is outstanding on-order inventory, and BO_t is backorders. All quantities must use the same unit, such as units or kilograms. Allocated stock may be subtracted from on-hand inventory where the organisation defines inventory position as available-to-promise stock.

At a review epoch, the decision-maker observes inventory position and releases replenishment. Between review epochs, no routine order is placed under a pure periodic-review policy. Consequently, the system must cover demand during both the supplier lead time and the time until the next opportunity to order.

If reviews occur every P time units and replenishment lead time is L, the protection period is:

The protection period is the interval during which demand must be protected after a review decision. This is the defining difference from a continuous-review reorder-point model, which generally protects only lead-time demand.

A periodic system does not necessarily imply that every item receives the same review period. High-value or critical items may be reviewed daily, while low-value, stable-demand items may be reviewed weekly or monthly. The choice should reflect demand risk, item criticality, ordering costs, supplier constraints, and the cost of monitoring.

2. The (P,S) and order-up-to policy

The standard periodic policy is written (P,S). Every P time units, the inventory position is reviewed and raised to order-up-to level S. The replenishment quantity at review time t is:

The maximum operator prevents a negative order. If inventory position already exceeds S, the usual policy places no order, although exception rules may be needed for excess or obsolete stock.

The policy is called an order-up-to policy because each review attempts to restore inventory position to a specified level. The order-up-to level should cover expected demand over the protection period plus a risk allowance:

where μ_T is expected demand during protection period T, and SS is safety stock. If average demand per time unit is d, and demand is stationary, then:

The units are consistent: units/time multiplied by time gives units.

Safety stock under normally distributed demand

Suppose protection-period demand is approximately normal with standard deviation σ_T. For a one-sided cycle-service target CSL, the safety stock is:

where z_CSL is the standard-normal quantile satisfying Pr(D_T ≤ S) = CSL, and D_T is protection-period demand. Thus:

This formulation assumes that the relevant demand distribution is adequately represented by a normal approximation, the parameters are stable over the planning horizon, replenishment is available at the stated lead time, and shortages are measured as failure to meet demand during a replenishment cycle. It should not be applied mechanically to intermittent, highly skewed, bounded, or strongly seasonal demand.

If independent demand has standard deviation σ_d per time unit and demand is measured in discrete periods, a common approximation is:

This follows from variances adding for independent periods. If demand observations are autocorrelated, the covariance terms must be included; the square-root expression may then understate or overstate risk.

If lead time is also random and independent of demand, with mean E(L) and variance Var(L), a commonly used random-lead-time approximation is:

This expression assumes demand rate and lead time are suitably represented by their means and that demand and lead time are independent. When both demand and lead time vary substantially, simulation or an empirical quantile is preferable.

Worked example 1: order-up-to calculation

A distributor reviews a component every two weeks. Supplier lead time is three weeks. Weekly demand has mean 400 units and standard deviation 120 units. Assume independent weekly demand, constant lead time, and a 95% cycle-service target, for which z = 1.645.

The protection period is:

T = 2 + 3 = 5 weeks.

Expected protection-period demand is:

μ_T = 400 × 5 = 2,000 units.

Protection-period standard deviation is:

σ_T = 120√5 ≈ 268.3 units.

Safety stock is:

SS = 1.645 × 268.3 ≈ 441.4 units.

Therefore:

S ≈ 2,000 + 441.4 = 2,441.4 units.

If the inventory position at review is 1,760 units, the order is approximately 681 units, subject to the organisation’s rounding, pack-size, and minimum-order rules. The calculation protects the five-week interval, not merely the three-week supplier lead time.

3. Min–max policies

A min–max policy specifies a lower threshold [[MATH,s]] and an upper level [[MATH,S]]. At each review, an order is placed when inventory position is at or below the lower level, usually raising inventory position to S:

The notation (s,S) distinguishes this policy from (P,S). In a strict periodic min–max system, the review frequency is fixed, but an order is conditional on the threshold. An order-up-to system orders at every review whenever inventory position is below S; a min–max system can suppress small orders while inventory position remains above s.

A useful design approach is to set the maximum level for protection-period coverage and choose the minimum level to balance order frequency against shortage risk. In practice, s may be based on expected lead-time demand plus lead-time safety stock, while S covers lead time plus the review interval. However, the exact relationship depends on whether the threshold is checked continuously or only at review epochs.

For a periodic threshold system, the risk between reviews means that the lower level cannot be interpreted as a complete protection point unless the policy’s timing convention is explicit. Organisations should document whether inventory position includes open orders, whether an order is triggered at equality, and whether an order can arrive before the next review.

Min–max policies are attractive for operational simplicity and for environments with minimum order quantities or batching. They can, however, generate irregular order timing and may be less transparent for service-level optimisation than a direct order-up-to rule.

4. Service-level models

Service level is not a single universal measure. Two definitions are especially important.

Cycle service level

Cycle service level is the probability of experiencing no stockout during a replenishment cycle or protection period:

It is a probability, not the percentage of demand supplied. A 95% cycle service level means that approximately 95% of protection periods have no stockout under the model assumptions. It does not mean that 95% of units are immediately supplied.

For normal protection-period demand:

where Φ is the standard-normal cumulative distribution function.

Fill rate

The item fill rate is the proportion of demand supplied immediately from available stock. Let D_T be protection-period demand and (D_T − S)^+ denote unmet demand, where x^+ = max(0,x). The expected fill rate is:

This measure is volume-oriented. A system can have a high cycle service level but a lower fill rate if the occasional shortages are large; conversely, a low cycle service level can coexist with a high fill rate when stockout events are frequent but very small.

For normally distributed demand, expected shortage per protection period can be written:

where z = (S − μ_T)/σ_T, and φ and Φ are the standard-normal density and cumulative distribution functions. The normal model permits analytical evaluation, but empirical demand or simulation is usually safer for intermittent or highly non-normal items.

Choosing a target

The target should reflect the economic and operational consequences of shortage. Critical spare parts, safety equipment, and items supporting contractual service commitments may require a high cycle-service or fill-rate target. Low-value items with easily substituted demand may justify a lower target. A target should never be selected solely because it is a familiar number such as 95%; it should be connected to shortage cost, customer promise, production disruption, and working-capital constraints.

Service differentiation is generally superior to applying one target to every SKU. A segmentation may combine annual usage value, demand variability, lead-time exposure, substitutability, and criticality. Governance should prevent commercial pressure from producing undocumented target changes.

5. Protection-period stock and demand aggregation

Periodic review increases exposure because demand accumulates over P + L. If demand is independent, standard deviation rises with the square root of time while mean demand rises linearly. Consequently, increasing the review period raises average cycle stock and usually raises safety stock as well.

Demand aggregation can reduce relative uncertainty. If n independent locations have identical mean demand μ and standard deviation σ, aggregate mean is nμ and aggregate standard deviation is σ√n. The coefficient of variation therefore falls from σ/μ at one location to σ/(μ√n) for the aggregate. This is a statistical benefit, not a guarantee of lower total cost: transportation, response time, allocation, and risk pooling constraints may offset it.

Seasonality requires time-phased forecasts rather than a constant d. If expected demand in each future period is d_i, then protection-period mean is:

Forecast error, not merely demand variation, must be included in σ_T. Promotional events, new products, obsolescence, and intermittent demand can make historical standard deviations misleading.

6. Periodic versus continuous review

The comparison should include more than inventory quantity. It should consider service responsiveness, information requirements, ordering cost, transport economics, labour, systems capability, and resilience.

A continuous-review system can often achieve comparable service with less cycle stock because it does not wait for the next review. A periodic system may nevertheless be cheaper because it consolidates purchase orders, deliveries, inspections, and administrative transactions. Continuous review is not automatically superior: poor data, unreliable scanning, supplier constraints, or excessive transaction costs can reduce its value.

The choice can be framed as a total-cost comparison:

Here, TC is total relevant cost per planning period; C_h is holding cost per unit-period; I is average inventory; C_o is ordering or setup cost per order; N_o is number of orders; C_t is transport or delivery cost per shipment; N_t is number of shipments; C_s is shortage cost per relevant unit or event; and C_e represents other costs such as expediting, review, systems, and obsolescence. The equation is a decision framework rather than a universal accounting identity. Cost components must be defined consistently and not double counted.

7. Implementation architecture and control

A reliable periodic-review process requires five controls.

Policy data. Each SKU should have a review period, lead time, target service measure, demand forecast, uncertainty estimate, order constraints, and unit-cost data. Lead times should be based on actual receipt history, including variability and supplier performance.

Inventory accuracy. A mathematically sound policy fails when on-hand, allocated, in-transit, or backordered quantities are inaccurate. Cycle counting, transaction discipline, and reconciliation are essential.

Exception management. Planners need alerts for projected stockout, abnormal demand, late supply, excess stock, forecast changes, and parameter drift. Exceptions should not silently overwrite policy parameters.

Parameter governance. Changes to P, S, s, or service targets should be traceable, authorised, and periodically reviewed. The organisation should distinguish a genuine demand change from a temporary promotion or data error.

Performance measurement. Monitor fill rate, cycle service, backorders, inventory turns, inventory value, forecast bias, supplier lead-time adherence, order frequency, and obsolete stock. A single service metric can encourage dysfunctional behaviour; balanced measures are preferable.

Advanced planning systems can calculate empirical demand quantiles, simulate alternative review periods, and optimise targets across a portfolio. Machine-learning forecasts may improve the estimate of future demand, but they do not remove the need for uncertainty calibration, explainability, override controls, and monitoring for bias. AI-generated replenishment recommendations should remain subject to human accountability, especially for critical materials and contractual service commitments.

8. Illustrative case: periodic review in a hospital consumables network

Background. A regional hospital network manages 1,200 routine consumable items through weekly purchasing. Individual wards previously placed ad hoc orders, causing duplicated administrative work and uneven stock availability. Critical emergency items were already monitored continuously.

Problem. Routine items experienced both shortages and excess. The existing practice used a nominal two-week maximum without incorporating supplier lead-time variation or distinguishing fill rate from cycle service.

Analysis. The supply team classified items by criticality and demand pattern. Stable, non-critical items were assigned weekly periodic review. High-criticality items retained continuous review. For each periodic item, the team calculated protection-period demand using the seven-day review interval plus measured supplier lead time. Items with seasonal demand used time-phased forecasts. Service targets were specified as fill-rate targets for routine consumables and cycle-service targets for selected clinical-risk items.

Approach. The hospital implemented an order-up-to policy with inventory position including on-hand, open purchase orders, and ward backorders. Pack sizes were incorporated by rounding order quantities upward, while a maximum-capacity check prevented unsafe accumulation. Late receipts generated an exception rather than an automatic increase in every target. Monthly reviews examined forecast error, supplier reliability, stockouts, and expiry.

Outcome. The network reduced routine order transactions and improved visibility of stock risk. The most important improvement was not a universal increase in stock; it was the alignment of review frequency and service definitions with clinical criticality. Items with short shelf life were managed using expiry-aware allocation and smaller replenishment quantities. Emergency items remained subject to a tighter control regime because the cost of failure dominated administrative savings.

Learning points. Periodic review is a portfolio design choice, not a rule that must apply to every SKU. Inventory position must include pipeline stock correctly. Service targets should reflect consequences, and safety stock should be recalculated when demand, lead time, or review frequency changes. Finally, operational adoption depends on clear ownership and exception processes as much as on formulas.

9. Limitations and critical evaluation

Periodic-review formulas are useful approximations, not substitutes for judgement. Normality may be unsuitable for intermittent demand, zero-inflated demand, promotions, and product launches. Independence assumptions fail when demand is correlated across periods or when supplier delays coincide with demand surges. Backorders, lost sales, substitutions, rationing, and priority customers complicate the meaning of service level.

The order-up-to calculation also presumes that replenishment orders are feasible. Minimum order quantities, fixed order multiples, supplier capacity, shipment calendars, and budget limits can cause actual inventory position to differ from the ideal target. Capacity constraints may require a constrained allocation model rather than independent SKU calculations.

High service targets can produce sharply increasing safety stock because the normal quantile rises rapidly in the upper tail. A target of 99% is not simply 4% more demanding than 95%; it may require substantially more stock, particularly when demand uncertainty is large. Managers should therefore evaluate marginal service benefit against marginal working capital and shortage avoidance.

Ethically, service differentiation must not conceal unacceptable discrimination between customers or patients. Critical goods require transparent risk criteria, escalation paths, and continuity plans. In regulated environments, records of parameter changes, traceability, lot control, expiry management, and auditability may be mandatory. Inventory decisions should support safety and fairness rather than merely optimise financial cost.

Best-practice checklist

1. Define inventory position and shortage treatment explicitly.
2. Calculate protection-period demand over review interval plus lead time.
3. Separate cycle service from fill rate in reports and targets.
4. Use empirical or simulation methods when distributions are irregular.
5. Segment items by value, variability, criticality, and substitutability.
6. Include lead-time uncertainty and supplier performance in risk estimates.
7. Apply pack-size, minimum-order, capacity, and expiry constraints.
8. Compare periodic and continuous review using total relevant cost.
9. Govern parameter changes and preserve an auditable decision trail.
10. Revalidate policies after structural demand or supply changes.

Chapter Summary

Periodic-review inventory systems inspect inventory at fixed intervals and must protect demand over the review period plus replenishment lead time. The (P,S) or order-up-to policy raises inventory position to S at each review, with quantity Q_t = max(0,S−IP_t). Under stable, independent, approximately normal demand, the target is expected protection-period demand plus safety stock, S = μ_T + zσ_T.

Min–max policies use a lower threshold and an upper replenishment level, making them useful where order batching or minimum quantities matter. Cycle service level measures the probability of no stockout during a protection period, whereas fill rate measures the proportion of demand supplied immediately; they are not interchangeable. Review frequency, demand uncertainty, lead-time variability, seasonality, and aggregation all affect protection stock.

Periodic review can reduce transaction and transport costs but introduces delayed response and typically greater exposure than continuous review. The appropriate choice depends on total cost, criticality, data quality, supplier conditions, and operational capability. Effective implementation requires accurate inventory position, governed parameters, exception management, balanced performance measures, and responsible use of analytics and AI.

Key Terms

Periodic review: Inventory is checked and replenishment decisions are made at scheduled intervals.

Continuous review: Inventory status is monitored continuously or near continuously for replenishment triggers.

Review period (P): Time between successive inventory reviews.

Lead time (L): Time from placing a replenishment order to its usable receipt.

Protection period: Review period plus lead time in a periodic system.

Inventory position: On-hand plus on-order inventory minus backorders or equivalent commitments.

Order-up-to level (S): Target inventory position to which a periodic system replenishes.

(P,S) policy: A policy reviewing every P periods and ordering up to S.

Min–max policy: A policy that orders up to S when inventory position is at or below s.

Safety stock: Stock held to protect against demand or supply uncertainty.

Cycle service level: Probability of no stockout during a replenishment cycle or protection period.

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

Protection-period demand: Demand occurring during review period plus lead time.

Demand aggregation: Combining demand streams to exploit risk pooling.

Forecast bias: Systematic tendency of forecasts to overstate or understate demand.

Backorder: Customer demand recorded for later fulfilment rather than immediately lost.

Stockout: Condition in which available inventory cannot satisfy current demand.

Parameter governance: Controlled authorisation, documentation, and review of inventory-policy settings.

Review Questions

Short Answer Questions

1. Why does a periodic-review system protect demand over P+L rather than only over lead time?
2. Define inventory position and identify its principal components.
3. Distinguish cycle service level from fill rate.
4. State two assumptions behind the square-root-of-time standard-deviation formula.
5. Why might a hospital retain continuous review for emergency items while using periodic review for routine consumables?

Long Answer Questions

1. Explain the operation of a (P,S) policy and derive its order quantity expression. Discuss the role of safety stock.
2. Compare periodic-review and continuous-review systems in terms of cost, responsiveness, information requirements, and risk.
3. Explain how demand variability, lead-time variability, seasonality, and review frequency influence protection-period stock.
4. Critically evaluate the use of normal-distribution service-level models for intermittent and highly seasonal demand.
5. Design a governance framework for setting, monitoring, and changing service targets across an inventory portfolio.

Analytical / Application Questions

1. A firm reviews an item every three days, has a lead time of five days, mean daily demand of 80 units, and independent daily standard deviation of 20 units. Using a 95% cycle-service target with z=1.645, calculate protection-period demand, protection-period standard deviation, safety stock, and order-up-to level.
2. An organisation is considering changing review frequency from weekly to monthly. Analyse the likely effects on cycle stock, safety stock, order consolidation, responsiveness, and supplier coordination. State what additional data you would require before deciding.
3. A product has frequent small shortages but very few completely failed replenishment cycles. Recommend whether management should focus on cycle service or fill rate, and explain how the choice would affect policy evaluation and customer communication.

Multiple Choice Questions with Answers

Question 1

In a periodic-review system with review period P and lead time L, the protection period is:

A. L − P

B. P ÷ L

C. P + L

D. P × L

Correct Answer: C. P + L

Question 2

If inventory position is 1,760 units and the order-up-to level is 2,441 units, the order quantity is approximately:

A. 681 units

B. 1,760 units

C. 2,441 units

D. 4,201 units

Correct Answer: A. 681 units

Question 3

Which expression defines inventory position in the chapter?

A. On-hand − on-order + backorders

B. On-hand + on-order − backorders

C. On-hand + backorders − on-order

D. On-order − on-hand − backorders

Correct Answer: B. On-hand + on-order − backorders

Question 4

A 95% cycle service level means that:

A. 95% of all units are always supplied immediately

B. 95% of protection periods are expected to have no stockout under the model

C. Shortage quantity is limited to 5 units

D. Inventory turnover must equal 95%

Correct Answer: B. 95% of protection periods are expected to have no stockout under the model

Question 5

Under independent demand with standard deviation σ per period, protection-period standard deviation over T periods is commonly approximated by:

A. σT

B. σ ÷ T

C. σ√T

D. σ + T

Correct Answer: C. σ√T

Question 6

Which policy orders up to S only when inventory position is at or below s?

A. (P,S) policy

B. Periodic min–max policy

C. Economic production quantity policy

D. First-in-first-out policy

Correct Answer: B. Periodic min–max policy

Question 7

A fill rate primarily measures:

A. The proportion of demand supplied immediately

B. The probability that every cycle has zero demand

C. The percentage of suppliers meeting lead-time promises

D. The fraction of inventory that is obsolete

Correct Answer: A. The proportion of demand supplied immediately

Question 8

In the worked distributor example, the expected demand during the five-week protection period was:

A. 400 units

B. 600 units

C. 1,200 units

D. 2,000 units

Correct Answer: D. 2,000 units

Question 9

Which condition most directly supports use of the square-root-of-time variability approximation?

A. Strong positive autocorrelation

B. Independent demand across periods

C. Unlimited supplier capacity only

D. Zero demand variance

Correct Answer: B. Independent demand across periods

Question 10

Compared with periodic review, continuous review generally provides:

A. No need for inventory records

B. Less responsiveness to sudden demand

C. The ability to trigger replenishment without waiting for the next review

D. Guaranteed lower total cost

Correct Answer: C. The ability to trigger replenishment without waiting for the next review