Periodic Review and Constrained Replenishment

Introduction

Periodic-review inventory systems evaluate stock at predetermined intervals rather than continuously. At each review, the decision maker determines an order quantity that must cover expected demand until the next review and, usually, the supplier lead time. These systems are especially common when ordering is administratively expensive, deliveries are scheduled, suppliers impose order cycles, or several products are consolidated into one shipment.

The central managerial challenge is to balance customer-service objectives against working capital, storage capacity, transport availability, supplier restrictions, and purchasing budgets. A policy that is optimal for one item in isolation may be infeasible or inefficient for the portfolio. Consequently, periodic review is both an inventory-control problem and a constrained resource-allocation problem.

Learning Objectives

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

1. Explain periodic-review, min–max, base-stock, and order-up-to policies.
2. Calculate order-up-to levels under deterministic and uncertain demand.
3. Incorporate review periods, lead times, safety stock, and service targets.
4. Analyze minimum-order, budget, storage, transport, and supplier constraints.
5. Design joint-replenishment and multi-item policies.
6. Evaluate implementation risks, digital-support requirements, and managerial trade-offs.

1. Inventory Position and the Logic of Periodic Review

A replenishment decision should be based on inventory position rather than physical stock alone. Define inventory position as

where OH_t is on-hand inventory at time t, OO_t is open or on-order inventory, and BO_t is backorders. All terms must use the same units. On-hand stock includes usable inventory; damaged, quarantined, or obsolete material should be excluded unless it can realistically satisfy demand.

In a periodic-review system, inspections occur every R time units. If review occurs at time t, the next opportunity to order is t+R. With replenishment lead time L, an order placed at t must protect against demand over the protection period

assuming that lead time and review intervals are measured in the same units and that replenishment is not available before L. If the review schedule is irregular, P should be replaced by the time from the current review until the next possible replenishment arrival, including the relevant lead time.

A practical periodic-review decision is therefore:

where S_t is the target, or order-up-to, level. If Q_t > 0, the system orders Q_t; if Q_t ≤ 0, it orders nothing, subject to minimum-order and contractual rules.

A periodic system differs from a continuous-review system in an important way: demand occurring immediately after a review cannot trigger another order. The review interval therefore creates additional exposure to stockout. Longer review intervals generally reduce ordering and administrative costs but require more cycle stock and safety stock.

2. Core Policy Families

2.1 Order-up-to or base-stock policy

An order-up-to policy raises inventory position to a specified level at every review:

The level S is selected to cover expected demand during the protection period plus a service cushion. In a pure base-stock policy, the target is continuously restored after every demand event. In practice, the terms are often used interchangeably, although “base-stock” more strictly describes continuous replenishment while “order-up-to” commonly describes periodic replenishment.

2.2 Min–max policy

A min–max policy defines a reorder minimum [[MATH,s]] and maximum [[MATH,S]]. At review, if inventory position is at or below s, the system orders enough to reach S:

The minimum is usually related to expected demand during lead time plus safety stock, whereas the maximum covers the review cycle and lead time. Unlike an order-up-to rule applied at every review, min–max may deliberately defer replenishment when stock remains above the trigger.

2.3 Fixed-cycle ordering

In a fixed-cycle system, orders are placed every R periods, but the quantity may be constrained by supplier calendars, truck schedules, or a target shipment size. This is often the most practical arrangement for distributors and manufacturers purchasing many stock-keeping units (SKUs) from the same supplier.

2.4 Comparison of policies

3. Setting the Order-Up-To Level

Let average demand per period be d, demand standard deviation per period be σ_d, and assume independent demand across periods. Expected demand during the protection period is

If demand is independent and identically distributed, its standard deviation over P periods is

The independence assumption is important. With serial correlation, seasonality, or forecast errors, simply multiplying the one-period standard deviation by √P can understate or overstate uncertainty.

A common order-up-to level is

where z is the standard-normal quantile associated with the desired cycle-service probability. For example, z ≈ 1.645 corresponds to approximately 95% protection against demand exceeding the target under a normal-demand approximation.

The formula is appropriate when demand uncertainty can reasonably be represented by a normal approximation, the protection period is known or sufficiently stable, and replenishment is available as assumed. For intermittent, highly skewed, seasonal, or short-history demand, empirical quantiles, Poisson or negative-binomial models, simulation, or forecast-error distributions may be more defensible.

If lead time is also random and independent of demand, with mean L̄ and variance σ_L², a useful approximation for demand during random lead time is

For a periodic system, replace lead time with the protection period and use the corresponding variance. This expression assumes stationary demand and independence between demand and lead time. If demand and lead time are correlated, a covariance term is required and simulation may be preferable.

Worked example: order-up-to calculation

A retailer reviews a product weekly. Supplier lead time is two weeks, average weekly demand is 120 units, and weekly demand standard deviation is 30 units. The desired cycle-service level is 95%, using z = 1.645.

The protection period is

Expected protection-period demand is 120 × 3 = 360 units. Demand standard deviation is 30√3 ≈ 51.96 units. Safety stock is 1.645 × 51.96 ≈ 85.47 units. Thus,

The operational target is normally rounded upward to 446 units. If current on-hand inventory is 180 units, open orders are 220 units, and backorders are 20 units, then

The unconstrained order is 446 − 380 = 66 units.

4. Service Measures and Safety Stock

A cycle-service level measures the probability of avoiding at least one stockout during a replenishment cycle. It differs from fill rate, which measures the proportion of demand supplied immediately. A policy can have a high cycle-service level but a lower fill rate when occasional shortages are large; conversely, frequent small shortages may produce a different relationship.

Safety stock should therefore be linked to the selected service measure, shortage cost, and demand structure. It should not be treated as an arbitrary percentage of average demand. Excessive safety stock can conceal poor forecasting, unreliable suppliers, obsolete inventory, and weak parameter governance.

For a normal-demand model, safety stock is often written as

The units of SS are inventory units, σ_P is inventory units, and z is dimensionless. If the organization uses a fill-rate target, a loss-function or simulation approach is generally more appropriate than directly equating fill rate with a normal quantile.

5. Constraints on Replenishment

The unconstrained order quantity may be operationally impossible. Constraints should be represented explicitly rather than handled informally by buyers.

5.1 Minimum order quantities and lot sizes

Let MOQ be a minimum order quantity and m a required order multiple. A simple rule is

where q_t = S − IP_t is the unconstrained requirement and ⌈x⌉ denotes rounding upward to the nearest integer. The first expression is best understood operationally: if positive demand exists but is below the MOQ, the firm must either order the MOQ, defer the order, or use an approved exception. Ordering the MOQ can create excess stock and holding cost; deferring can create shortage risk.

5.2 Budget constraints

For items i = 1,…,n, with unit purchase cost c_i, a purchasing budget B imposes

The problem is not solved correctly by proportionally scaling all orders unless all items have identical marginal service value, costs, and constraints. A better approach ranks or optimizes orders according to shortage cost, criticality, supplier commitments, and expected service improvement.

5.3 Storage constraints

If v_i is the storage volume per unit and available capacity is [[MATH,V]], then

where I_i is the relevant post-receipt inventory measure. The organization must specify whether capacity is measured in pallet positions, cubic metres, weight, or a mixed capacity index. Volume alone may be inadequate where hazardous-material segregation, temperature control, or compatibility rules apply.

5.4 Transport and supplier constraints

Truck capacity, dock appointments, supplier production limits, and delivery calendars can be expressed as constraints such as

where w_i is shipment weight per unit and W is available transport capacity. Supplier capacity may be represented by Q_i ≤ U_i, where U_i is the maximum allocable quantity. Delivery-day restrictions may require binary decision variables indicating whether a supplier or route is activated.

5.5 Multi-constraint formulation

A simplified multi-item decision model is:

Here h_i is a holding-cost rate, p_i is a shortage penalty, and expectations reflect uncertainty. This is a conceptual formulation; exact solution may require mixed-integer programming, stochastic programming, robust optimization, or simulation-based search. The objective should include relevant costs without double-counting purchase cost, transfer prices, or already sunk costs.

6. Joint Replenishment and Multi-Item Policies

Joint replenishment coordinates several items so that orders share a supplier visit, purchase order, truck, or setup. It can reduce ordering and transportation cost, but synchronization may force some items to order earlier or later than their independent optimum.

A common structure is a common review interval R for a family of items. Each item then has an order-up-to level based on its own demand, uncertainty, and service requirement. The review interval should reflect the combined cost of ordering and holding inventory. If the supplier offers a discount or fixed delivery cost, joint replenishment can be evaluated using total landed cost rather than unit price alone.

A useful practical procedure is:

1. Group items by supplier, route, temperature class, or replenishment calendar.
2. Estimate each item’s protection-period demand and uncertainty.
3. Set differentiated service targets according to criticality and shortage cost.
4. Calculate unconstrained order quantities.
5. Apply MOQ, pack-size, budget, capacity, and supplier rules.
6. Resolve conflicts using marginal service value and operational feasibility.
7. Simulate performance before deployment.

Items should not automatically receive identical service levels. A low-cost, easily substituted item may justify a lower target than a safety-critical component whose stockout stops production.

7. Advanced Decision Methods

7.1 Marginal allocation under a binding budget

When budget is binding, the next unit should generally be assigned where it produces the greatest relevant marginal benefit. That benefit may be avoided shortage cost, increased contribution margin, production continuity, or customer-priority protection. A simple ratio is

This is a heuristic, not a universal optimality theorem. It is valid only when marginal effects can be estimated consistently and interactions among items are limited. Shared components, substitution, and production-bill-of-materials relationships require a broader model.

7.2 Robust and stochastic approaches

Stochastic optimization uses probability distributions for demand, lead time, and supply. Robust optimization protects against specified uncertainty sets rather than relying entirely on estimated probabilities. Robust policies are useful when data are sparse or forecasts are structurally unreliable, but they may be conservative and increase inventory.

Simulation is often the most transparent validation method. A discrete-event simulation can reproduce daily demand, review timing, lead-time variation, order constraints, stockouts, substitutions, and capacity. Performance should be reported through distributions, not only average inventory: fill rate, cycle service, backorders, inventory turns, budget utilization, expedited freight, and percentile outcomes are relevant.

8. Illustrative Case: Northstar Components

The following is a fictional illustrative case.

Background

Northstar Components assembles industrial pumps and purchases three items from one regional supplier. Orders are reviewed every two weeks, supplier lead time is two weeks, and one truck is available at each review. The truck can carry 1,000 kg. Management has a replenishment budget of 10,000 monetary units per review.

Analysis

The protection period is P = 2 + 2 = 4 weeks. For each item, S_i = d_iP + z_iσ_i√P.

For A: expected demand is 320 and standard deviation is 40, so S_A = 320 + 1.65×40 = 386. The unconstrained order is 136 units.

For B: expected demand is 140 and standard deviation is 20, so S_B = 140 + 1.28×20 = 165.6, rounded to 166. The unconstrained order is 36 units.

For C: expected demand is 48 and standard deviation is 8, so S_C = 48 + 2.05×8 = 64.4, rounded to 65. The unconstrained order is 20 units.

The unconstrained purchase cost is 136×18 + 36×42 + 20×95 = 6,648. Weight is 136×2 + 36×5 + 20×8 = 632 kg. Both constraints are satisfied. Thus the recommended initial order is 136 A, 36 B, and 20 C, subject to pack sizes and supplier limits.

Suppose the supplier changes the truck limit to 500 kg. The initial plan is infeasible. Reducing item A by 66 units frees 132 kg, but it creates a substantial service loss. A better managerial response is to examine marginal shortage costs: C may be critical to production despite its weight, while A may be substitutable. The team should test alternative allocations through simulation, discuss supplier split deliveries, and consider whether an expedited shipment or a temporary production substitution is cheaper than accepting stockout risk.

Outcome and learning points

The case demonstrates that independent order-up-to calculations are only a starting point. Feasibility, item criticality, service differentiation, and cross-functional decisions determine the final policy. Northstar should also monitor whether the assumed demand variability remains valid, whether inventory position data are accurate, and whether the two-week review schedule creates avoidable exposure.

9. Implementation, Governance, and Technology

Effective periodic review depends on master-data quality. Each SKU should have an approved review interval, lead-time estimate, demand model, service target, unit cost, pack size, MOQ, storage classification, and supplier calendar. Changes should be version-controlled and attributable.

An enterprise resource planning or advanced planning system can calculate inventory position, generate proposals, consolidate supplier orders, and enforce constraints. AI and machine-learning methods can improve forecasts or detect unusual demand, but they do not remove the need for policy logic, explainability, and human oversight. A forecast that improves mean accuracy may still worsen service if it systematically misses demand spikes or produces unstable parameters.

Governance issues include:

• preventing deliberate inflation of forecasts to obtain scarce budget;
• separating service-critical exceptions from routine overrides;
• maintaining audit trails for manual changes;
• protecting supplier and customer data;
• avoiding discriminatory allocation of scarce products among customers;
• documenting emergency-order and allocation rules.

Periodic parameters should be reviewed after major assortment, supplier, demand, or capacity changes. A useful control cycle is: measure performance, diagnose causes, update parameters, test the proposed policy, approve changes, and monitor the result.

10. Limitations and Best Practices

Periodic review is less responsive than continuous review and can perform poorly during rapid demand changes, highly unreliable lead times, or severe supply disruption. Normal-demand formulas may be misleading for intermittent demand. Safety stock can become excessive when forecast error, lead-time variability, and review-period risk are counted twice. Minimum orders can create “lumpy” inventory and hidden obsolescence.

Best practices include:

1. Use inventory position, not on-hand stock alone.
2. Calculate protection periods explicitly.
3. Distinguish cycle service from fill rate.
4. Estimate uncertainty from forecast errors and lead-time data.
5. Segment SKUs by criticality, value, variability, and substitutability.
6. Apply lot-size and capacity constraints before releasing orders.
7. Use simulation for nonlinear or highly constrained systems.
8. Measure total cost and service, not purchase price alone.
9. Review exceptions and overrides for bias and root causes.
10. Coordinate inventory, purchasing, transport, finance, and operations decisions.

Chapter Summary

Periodic-review systems replenish at scheduled inspections and must protect demand over the review interval plus lead time. The inventory position equation provides the correct basis for decisions because it includes on-hand stock, open orders, and backorders. Order-up-to and min–max policies translate service objectives into replenishment rules, while safety stock reflects demand and lead-time uncertainty.

In real organizations, unconstrained quantities must be reconciled with minimum order quantities, pack sizes, budgets, storage, transport, supplier capacity, and delivery calendars. Joint replenishment can reduce ordering and logistics cost but may sacrifice item-level responsiveness. Multi-item allocation is therefore a constrained optimization problem involving service value, cost, capacity, and operational risk. Reliable implementation requires accurate master data, transparent governance, simulation or analytical validation, and continuous performance review.

Key Terms

Inventory position: On-hand inventory plus open orders minus backorders.

Periodic review: Inventory evaluation at predetermined time intervals.

Protection period: Review interval plus replenishment lead time.

Order-up-to level: Target inventory position after replenishment.

Base-stock policy: A policy that restores inventory position to a target level.

Min–max policy: A rule that orders to a maximum when inventory position reaches a minimum.

Review interval: Time between scheduled inventory reviews.

Safety stock: Inventory held to protect against uncertainty.

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

Fill rate: Proportion of demand supplied immediately from stock.

Lead time: Time between placing and receiving an order.

Minimum order quantity: Smallest permitted supplier order.

Lot multiple: Required order increment or pack-size multiple.

Joint replenishment: Coordinated ordering of multiple items.

Inventory constraint: A limit on budget, space, weight, or another resource.

Shortage cost: Economic consequence of unavailable inventory.

Robust optimization: Optimization designed to perform acceptably under specified uncertainty.

Inventory governance: Rules controlling parameters, overrides, accountability, and auditability.

Review Questions

Short Answer Questions

1. Why is inventory position more useful than on-hand inventory for periodic-review decisions?
2. Define the protection period and explain its significance.
3. Distinguish cycle-service level from fill rate.
4. What is the purpose of a min–max policy?
5. State two risks associated with minimum order quantities.

Long Answer Questions

1. Explain how an order-up-to level is calculated under uncertain demand and discuss the assumptions behind the normal-demand approximation.
2. Compare periodic order-up-to, min–max, and joint-replenishment policies.
3. Discuss how budget, storage, transport, and supplier constraints change multi-item replenishment decisions.
4. Explain why safety stock should not be set solely as a percentage of average demand.
5. Evaluate the role of ERP, analytics, simulation, and AI in periodic-review inventory management.

Analytical / Application Questions

1. A product is reviewed every week, has a three-week lead time, average demand of 200 units per week, weekly standard deviation of 40 units, and a 98% cycle-service target with z = 2.05. Calculate the order-up-to level, assuming independent normal demand.
2. Three SKUs meet their individual order-up-to targets but exceed a shared truck-weight limit. Design a defensible allocation procedure and identify the data required.
3. A supplier imposes a minimum order of 500 units on an item whose calculated requirement is 120 units. Analyze the alternatives of ordering 500, deferring, negotiating, or changing the review interval, including the relevant costs and risks.

Multiple Choice Questions with Answers

Question 1

In a periodic-review system, the protection period normally equals:

A. Demand lead time minus the review interval

B. Review interval plus replenishment lead time

C. Replenishment lead time divided by demand

D. Safety stock plus cycle stock

Correct Answer: B. Review interval plus replenishment lead time

Question 2

Inventory position is correctly defined as:

A. On-hand inventory plus backorders

B. On-hand inventory minus open orders

C. On-hand inventory plus open orders minus backorders

D. Open orders minus on-hand inventory

Correct Answer: C. On-hand inventory plus open orders minus backorders

Question 3

Average weekly demand is 120 units, the review interval is one week, and lead time is two weeks. Expected demand during the protection period is:

A. 120 units

B. 240 units

C. 360 units

D. 480 units

Correct Answer: C. 360 units

Question 4

Under independent demand, the standard deviation over four periods is:

A. Four times the one-period standard deviation

B. Two times the one-period standard deviation

C. The one-period standard deviation divided by four

D. The one-period standard deviation divided by two

Correct Answer: B. Two times the one-period standard deviation

Question 5

A min–max policy normally places an order when:

A. Inventory position is at or below the minimum level

B. On-hand inventory is above the maximum level

C. Demand is exactly equal to average demand

D. The supplier has excess capacity

Correct Answer: A. Inventory position is at or below the minimum level

Question 6

If average demand is 100 units per period, the protection period is 3 periods, and safety stock is 40 units, the order-up-to level is:

A. 140 units

B. 240 units

C. 300 units

D. 340 units

Correct Answer: D. 340 units

Question 7

Which constraint directly limits total purchase expenditure?

A. ∑v_i(I_i + Q_i) ≤ V

B. ∑c_iQ_i ≤ B

C. ∑w_iQ_i ≤ W

D. Q_i ≤ U_i

Correct Answer: B. ∑c_iQ_i ≤ B

Question 8

The principal purpose of joint replenishment is to:

A. Eliminate all safety stock

B. Coordinate items to share ordering or delivery resources

C. Guarantee identical service levels for all items

D. Remove supplier lead time

Correct Answer: B. Coordinate items to share ordering or delivery resources

Question 9

If current on-hand inventory is 180 units, open orders are 220 units, and backorders are 20 units, inventory position is:

A. 140 units

B. 380 units

C. 420 units

D. 460 units

Correct Answer: B. 380 units

Question 10

Which statement best describes a fill rate?

A. The probability of no stockout in a cycle

B. The percentage of demand supplied immediately from stock

C. The percentage of orders placed on schedule

D. The proportion of warehouse capacity used

Correct Answer: B. The percentage of demand supplied immediately from stock