Inventory Planning and Replenishment Models

Introduction

Inventory planning converts uncertain customer demand into decisions about how much to order, when to order, and how much protection to hold against uncertainty. In retail, these decisions affect availability, working capital, markdowns, storage capacity, supplier relationships, and customer experience. A policy that minimizes purchase cost may create stock-outs; a policy that maximizes availability may generate excessive working capital and obsolescence.

Inventory planning therefore involves balancing competing objectives:

• high product availability and customer service;
• low ordering, purchasing, holding and shortage costs;
• efficient use of warehouse, shelf and distribution capacity;
• responsiveness to seasonality, promotions and assortment changes; and
• compliance with governance, audit and sustainability requirements.

The models in this chapter are decision aids rather than universal answers. Their quality depends on demand data, lead-time measurement, cost estimation, parameter review and managerial judgment.

Learning Objectives

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

1. Explain the logic and assumptions of major inventory replenishment models.
2. Calculate economic order quantity, reorder point and safety stock.
3. Select appropriate service-level and uncertainty assumptions.
4. Apply periodic-review, min–max, base-stock and newsvendor models.
5. Evaluate quantity discounts, capacity restrictions and lead-time variability.
6. Compare continuous-review and periodic-review policies under uncertainty.
7. Design an auditable replenishment process for retail operations.

1. Foundations of Replenishment Decisions

1.1 Inventory position and inventory status

On-hand inventory is stock physically available at a location. On-order inventory has been purchased but has not yet been received. Backorders are confirmed requirements not yet fulfilled. A useful control measure is inventory position:

where IP is inventory position, OH is on-hand inventory, OO is on-order inventory and BO is backorders. All quantities must be expressed in the same units.

Inventory position differs from available stock. A retailer may have 100 units on hand but an inventory position of 140 if 40 units are on order. Replenishment decisions should normally use inventory position; otherwise, outstanding orders may be ignored and duplicate orders created.

1.2 Demand, lead time and cycle time

Demand may be measured per day, week or review period. The planning interval must match the unit used in the model. Lead time is the elapsed time between placing an order and receiving usable stock. It may include order transmission, supplier processing, transportation, receiving and quality inspection.

The order cycle is the interval between replenishment orders. In a continuous-review system, inventory is monitored continuously or near-continuously. In a periodic-review system, inventory is reviewed at fixed intervals.

1.3 Cost categories

Relevant inventory costs generally include:

• ordering or setup cost per order;
• purchase cost per unit;
• holding cost per unit per period;
• shortage, lost-margin or substitution cost;
• markdown and obsolescence cost;
• transportation and receiving cost; and
• capacity and financing cost.

Holding cost is often estimated as a percentage of unit value. If annual holding-rate percentage is i and unit purchase price is C, annual holding cost per unit is:

For example, a 25% annual rate on a $40 item gives H = 0.25 × 40 = $10 per unit-year. The rate should include only costs relevant to the decision and should not be treated as a universal accounting convention.

2. Economic Order Quantity

2.1 Basic EOQ model

The economic order quantity balances ordering cost against cycle-stock holding cost. Under the basic model, annual demand and lead time are known, replenishment is instantaneous, shortages are not permitted, unit price is constant, and demand is stable.

where:

• Q* = economic order quantity, in units per order;
• D = annual demand, in units per year;
• S = ordering or setup cost, in currency per order; and
• H = annual holding cost, in currency per unit-year.

The units are consistent because DS/H has units of units², whose square root is units.

Annual relevant cost, excluding purchase cost when unit price is constant, is:

The first term is annual ordering cost and the second is annual cycle-stock holding cost. Differentiating with respect to Q and setting the derivative to zero gives:

which produces the EOQ formula. At EOQ, annual ordering cost equals annual cycle-stock holding cost:

This equality is a diagnostic property, not a separate ordering rule.

2.2 Worked example: EOQ

A retailer sells 24,000 units annually. Ordering cost is $75 per order and annual holding cost is $3 per unit.

The practical order quantity is approximately 1,095 units, subject to supplier pack sizes. Orders per year are:

Average cycle stock is approximately 547.7 units. Annual ordering and holding costs are each about $1,643, so total relevant annual cost is approximately $3,286. Purchase cost is excluded because it does not change when price is constant and every feasible policy buys the same annual volume.

2.3 Limitations of EOQ

EOQ is less reliable when demand is intermittent, products are seasonal, supplier minimums are binding, prices change, lead times are unreliable, or capacity is scarce. It also ignores safety stock unless the holding-cost expression is expanded. EOQ should therefore be treated as a starting point for cycle-stock sizing, not as a complete replenishment policy.

3. Reorder Point and Safety Stock

3.1 Deterministic reorder point

If demand during lead time is known and shortages are unacceptable, the reorder point is lead-time demand:

where d is average demand per unit time and L is lead time in the same time units. If daily demand is 40 units and lead time is 6 days, ROP = 40 × 6 = 240 units.

A reorder point is not necessarily the quantity ordered. The reorder point triggers a replenishment; the order quantity may be EOQ, a supplier minimum, or a target-level order quantity.

3.2 Safety stock under demand variability

When demand varies but lead time is fixed, a common approximation for safety stock is:

where σ_d is the standard deviation of demand per time unit, L is fixed lead time, and z is the standard-normal factor associated with the desired cycle service level. The resulting reorder point is:

This assumes independent demand observations and approximately normal aggregate demand. It is inappropriate for very low-volume, highly intermittent or strongly seasonal demand without modification.

3.3 Safety stock under lead-time variability

If daily demand and lead time are independent random variables, the variance of demand during lead time can be approximated by:

Thus:

where σ_L is the standard deviation of lead time. The reorder point is:

The formula captures two distinct risks: demand fluctuation during an average lead time and additional uncertainty caused by fluctuating lead time. It assumes independence. If demand and lead time are correlated, a covariance term is required:

where σ_dL represents the relevant demand–lead-time covariance approximation. In practice, empirical simulation is often safer when the data are non-normal or correlated.

3.4 Worked example: variable demand and lead time

A store sells an average of 100 units per day. Daily demand standard deviation is 20 units, average lead time is 5 days, and lead-time standard deviation is 1 day. The desired cycle service level is 95%, for which z ≈ 1.645.

Therefore:

and:

The operational reorder point is approximately 681 units if whole units are required. This calculation does not mean that 95% of all demand is guaranteed; it approximates a 95% probability of avoiding a stock-out during a replenishment cycle under the stated assumptions.

3.5 Service-level measures

A cycle service level is the probability of experiencing no stock-out during a replenishment cycle. A fill rate is the proportion of demand satisfied immediately from stock. These measures are not interchangeable. A system can avoid stock-outs in most cycles but still have a low fill rate if occasional shortages are large. Managers must specify the service measure before selecting a safety-stock factor.

For a normally distributed lead-time demand with mean μ_DL and standard deviation σ_DL, a target cycle service level CSL uses:

where Φ⁻¹ is the inverse standard-normal cumulative distribution function. Typical values are approximately 1.282 for 90%, 1.645 for 95% and 2.326 for 99%.

4. Continuous-Review Policies

4.1 The (Q, R) policy

In a continuous-review (Q,R) policy, an order of quantity Q is placed whenever inventory position reaches or falls below reorder point R. A practical implementation is:

EOQ can supply Q, while the probabilistic reorder-point formula supplies R. This separates cycle-stock and protection-stock decisions.

4.2 The (s, S) policy

A min–max or (s,S) policy orders enough to raise inventory position to an order-up-to level S whenever position is at or below the reorder level s:

This is useful when demand occurs in batches, when orders are reviewed frequently but not continuously, or when variable order quantities are operationally preferable. It prevents an unnecessary fixed order when inventory position has changed materially since the previous review.

4.3 Min–max interpretation

The lower bound is usually linked to expected lead-time demand plus protection stock. The upper bound covers lead-time demand and the intended replenishment cycle. The values should be calculated using the same demand horizon and service objective; arbitrary minimum and maximum values often create either chronic excess or repeated shortages.

5. Periodic-Review and Base-Stock Models

5.1 Periodic-review order-up-to policy

In a periodic-review system, inventory is checked every P time units. Because the retailer cannot order again until the next review, protection is required over the protection period:

For fixed lead time and variable demand, the order-up-to level is:

The order at a review is:

The formula assumes independent demand, fixed review and lead-time intervals, and approximately normal aggregate demand. With variable lead time, use the standard deviation of demand over the protection period rather than simply substituting an average lead time without assessing variance.

5.2 Base-stock policy

A base-stock policy attempts to maintain inventory position at a target level. At every review or demand event, the replenishment quantity offsets the reduction in inventory position. It is conceptually simple and provides high responsiveness, but it may generate frequent small orders and can be expensive when ordering or transportation costs are significant.

Base-stock logic is particularly suitable for critical items, short lead times, stable supplier integration and automated electronic ordering. It is less attractive for slow-moving products with high fixed order costs.

5.3 Comparing policy types

6. The Newsvendor Model for Single-Period Inventory

The newsvendor model applies when inventory has one principal selling opportunity or when leftover stock has a materially different value, as in fashion, fresh food, event merchandise and seasonal promotions.

Let c be unit purchase cost, p selling price, v recovery or salvage value, and g be shortage or lost-margin consequence. The standard critical fractile is:

where under the basic profit formulation:

Thus:

Here Cu is underage cost and Co is overage cost. The target quantity is the demand quantile corresponding to this fractile. The formulation assumes one ordering decision, known price and costs, and no replenishment opportunity during the selling period.

For example, if p = $50, c = $30 and v = $10, then:

The retailer should order the median of the demand distribution. If shortage has additional reputational or contractual cost, that cost should be included in Cu, but double-counting lost margin must be avoided.

7. Quantity Discounts and Capacity Constraints

7.1 All-units quantity discounts

When unit price falls for the entire order after a quantity breakpoint, the basic EOQ is not sufficient. For each price bracket:

1. calculate EOQ using that bracket’s holding cost, often H = iC;
2. adjust the EOQ upward to the smallest quantity permitted in that bracket if the EOQ is below its range;
3. calculate total annual cost for each feasible candidate; and
4. select the candidate with the lowest total cost.

A general annual cost expression is:

where C is unit price. If safety stock is included and constant across candidates, its holding cost may be added separately; if quantity affects lead time or storage, that assumption must be reconsidered.

7.2 Capacity constraints

An order quantity can be economically attractive yet operationally infeasible. Constraints may include shelf space, warehouse cube, vehicle capacity, cash budget, supplier allocation and maximum inventory days. A simple storage constraint is:

where v_i is space per unit of item i and K is available capacity. If several products compete for scarce space, the decision should compare contribution, service importance, handling cost and shortage consequences per unit of constrained resource. A blanket EOQ for every SKU is rarely optimal under shared capacity.

8. Selecting Policies Under Uncertainty

A sound policy-selection process follows these steps:

1. Classify the item. Consider sales value, margin, criticality, perishability, demand variability, substitution and lifecycle stage.
2. Diagnose demand. Separate baseline demand from promotion, seasonality, trend and one-time events.
3. Measure lead time. Use actual receipt dates, not only supplier promises. Examine mean, dispersion and late-tail behavior.
4. Choose the service measure. Decide whether the objective is cycle service level, fill rate, revenue protection or another metric.
5. Select review logic. Use continuous review where stock-outs are costly and transactions are automated; use periodic review where routes, labor or supplier schedules favor batching.
6. Set parameters. Estimate demand, variability, cost and capacity using a stated historical window and documented exclusions.
7. Stress-test. Simulate demand spikes, supplier delays, promotions, returns and capacity loss.
8. Monitor and revise. Track forecast error, stock-outs, excess stock, inventory turns and parameter drift.

The apparent precision of a model should not exceed the quality of its data. A complex stochastic model with biased demand and unreliable lead-time records can perform worse than a transparent min–max rule supported by disciplined review.

9. Illustrative Case: UrbanStyle’s Seasonal Sneaker Replenishment

Background

UrbanStyle, a fictional mid-sized fashion retailer, sells a limited-edition sneaker through 20 stores and an online channel. The product has a 12-week selling season, supplier lead time averages 3 weeks, and the supplier permits only one initial order and one expedited replenishment. Unsold units can be marked down heavily at the end of the season.

Problem

The buying team proposes EOQ because annual demand is estimated at 18,000 units. However, the item is seasonal, demand is uncertain, and the replenishment opportunity is limited. Applying annual EOQ would ignore the product’s short selling window and markdown risk.

Analysis

The merchandise team estimates seasonal demand from comparable launches, adjusts for planned digital advertising, and constructs a probability distribution. The unit purchase cost is $42, planned selling price is $80, and expected end-season recovery value is $20. Under the basic newsvendor logic:

Therefore:

The initial order should therefore approximate the 63.33rd percentile of seasonal demand, not the mean and not an annual EOQ. The team also reserves limited expedited capacity for a second order, using real-time sales and store-level availability data.

Approach and solution

UrbanStyle uses a hybrid policy:

• a newsvendor quantity for the initial seasonal commitment;
• a weekly periodic review during the season;
• a base-stock target adjusted for remaining weeks and observed sell-through;
• higher service targets for large stores with strong demand and lower targets for stores with substitute products; and
• an explicit markdown trigger when residual demand falls below available stock.

The audit trail records the demand distribution, price and salvage assumptions, approval of the initial order, changes to the forecast and authorization for expedited replenishment.

Outcome

The hybrid approach reduces reliance on a false annual-demand assumption. It preserves the value of a second order while limiting end-season exposure. The case does not imply that the 63.33rd percentile is universally correct; it is valid only for the stated one-period assumptions and cost interpretation.

Learning points

1. Product lifecycle determines model choice.
2. Salvage value and markdown risk are central to seasonal decisions.
3. Replenishment flexibility changes the appropriate model.
4. A service target must reflect customer substitution and store role.
5. Documentation is essential when judgments alter model parameters.

10. Technology, Governance and Auditability

Modern retail systems can calculate inventory position continuously, integrate point-of-sale data, estimate lead-time distributions and recommend replenishment quantities. Machine-learning forecasts may improve demand prediction when sufficient clean data exist, but an algorithm does not eliminate inventory-policy risk. Forecast accuracy and replenishment performance are different: a good forecast can be undermined by incorrect pack sizes, stale lead times or unavailable capacity.

Governance controls should include:

• approved ownership of demand, lead-time and cost parameters;
• version-controlled replenishment rules;
• segregation between parameter maintenance and order approval where risk warrants it;
• exception reports for manual overrides, unusual order quantities and stock-outs;
• reconciliation of purchase orders, receipts, inventory records and invoices;
• periodic back-testing of service and cost outcomes; and
• documented treatment of returns, shrinkage, damaged goods and substitutions.

AI-assisted recommendations should be explainable at the level required for managerial approval. Bias can arise when historical sales reflect stock-outs: observed sales may understate true demand. Promotional data can also contaminate baseline demand. Ethical inventory management includes avoiding systematic under-allocation to communities or stores merely because historical availability was poor.

11. Common Errors and Best Practices

Common errors

• mixing daily demand with annual lead time;
• using on-hand stock instead of inventory position;
• treating forecast error as demand standard deviation without checking definitions;
• using a cycle-service z value to claim a fill-rate result;
• ignoring lead-time variability;
• applying EOQ to single-season or perishable items;
• accepting quantity-discount breakpoints without total-cost comparison;
• counting safety stock twice; and
• rounding quantities without respecting pack, pallet or capacity constraints.

Best practices

• define units and time bases before calculating;
• maintain separate parameters for cycle stock and safety stock;
• estimate lead-time variability from actual receipt history;
• segment SKUs rather than imposing one policy on all products;
• use simulation for non-normal, intermittent or highly seasonal demand;
• include capacity and cash constraints in the final decision;
• review parameters after promotions, assortment changes and supplier disruptions; and
• connect replenishment outcomes to financial and service measures.

Chapter Summary

Inventory replenishment balances ordering, holding, shortage and purchase costs while protecting customer service. EOQ determines an economical cycle-stock quantity under stable-demand assumptions; it does not by itself determine when to order. The reorder point combines expected lead-time demand with safety stock. Safety-stock calculations must distinguish demand variability from lead-time variability and must state the selected service measure.

Continuous-review (Q,R) and (s,S) policies are appropriate when inventory position can be monitored frequently. Periodic-review policies protect demand over both the review interval and lead time, while base-stock systems continually restore a target inventory position. The newsvendor model is appropriate for single-period or highly seasonal decisions in which overage and underage costs differ materially.

Quantity discounts require comparison of feasible total-cost candidates, and capacity constraints may invalidate an otherwise attractive order quantity. Effective policy selection requires item segmentation, reliable data, stress testing, governance and ongoing parameter review. Technology can improve visibility and forecasting, but transparent assumptions and auditable controls remain essential.

Key Terms

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

Lead time: Time between placing an order and receiving usable inventory.

Cycle stock: Inventory held to cover expected demand between replenishments.

Safety stock: Additional inventory held to protect against uncertainty.

Economic order quantity (EOQ): Order quantity minimizing basic ordering plus cycle-stock holding cost.

Reorder point (ROP): Inventory-position level that triggers replenishment.

Service level: Probability or proportion used to express inventory availability performance.

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

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

Continuous review: Inventory monitoring in which replenishment can be triggered whenever a threshold is reached.

Periodic review: Inventory evaluation at predetermined time intervals.

Min–max policy: Rule that orders up to a maximum when inventory reaches a minimum.

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

Newsvendor model: Single-period model balancing overage and underage costs.

Underage cost: Economic consequence of ordering too little.

Overage cost: Economic consequence of ordering too much.

Quantity discount: Lower unit price available when an order reaches a specified quantity.

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

Inventory audit trail: Documented evidence of data, assumptions, approvals and replenishment actions.

Review Questions

Short Answer Questions

1. Distinguish on-hand inventory from inventory position.
2. State the assumptions of the basic EOQ model.
3. Why does lead-time variability increase safety stock?
4. Distinguish cycle service level from fill rate.
5. What is the protection period in a periodic-review system?

Long Answer Questions

1. Explain the EOQ model, derive its cost-minimization logic, and evaluate its limitations in retail.
2. Compare continuous-review, periodic-review, min–max and base-stock policies.
3. Explain how demand and lead-time variability affect reorder-point calculations.
4. Describe how quantity discounts should be evaluated when capacity is limited.
5. Discuss governance and audit controls for technology-enabled replenishment.

Analytical / Application Questions

1. Daily demand averages 80 units with a standard deviation of 15 units. Lead time averages 4 days with a standard deviation of 1.5 days. Assuming independence and a 95% cycle service level, calculate safety stock and reorder point using the variable-lead-time approximation.
2. A seasonal item sells for $70, costs $40 and has a salvage value of $15. Calculate the critical fractile and explain what demand percentile should determine the order quantity.
3. A retailer is choosing between continuous review and weekly periodic review for a high-value item. Develop a recommendation framework using service risk, transaction cost, lead-time variability, capacity and data availability.

Multiple Choice Questions with Answers

Question 1

Which expression correctly defines inventory position?

A. On-hand inventory plus backorders

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

C. On-order inventory minus on-hand inventory

D. Annual demand divided by order quantity

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

Question 2

Under the basic EOQ assumptions, what two costs are balanced at the optimum?

A. Purchase cost and shortage cost

B. Transportation cost and markdown cost

C. Annual ordering cost and annual cycle-stock holding cost

D. Salvage cost and backorder cost

Correct Answer: C. Annual ordering cost and annual cycle-stock holding cost

Question 3

A product has average demand of 40 units per day and a fixed lead time of 6 days. With no safety stock, what is the reorder point?

A. 46 units

B. 240 units

C. 400 units

D. 1,440 units

Correct Answer: B. 240 units

Question 4

Which factor is directly included in the approximate variance of demand during variable lead time?

A. Only unit purchase price

B. Only annual ordering cost

C. Demand variability and lead-time variability

D. Only salvage value

Correct Answer: C. Demand variability and lead-time variability

Question 5

A periodic-review system reviews inventory every P days and has lead time L. What period must safety stock protect?

A. P − L

B. P only

C. L only

D. P + L

Correct Answer: D. P + L

Question 6

If selling price is $50, purchase cost is $30 and salvage value is $10, what is the newsvendor critical fractile?

A. 0.25

B. 0.40

C. 0.50

D. 0.75

Correct Answer: C. 0.50

Question 7

In an (s,S) policy, what normally happens when inventory position reaches or falls below s?

A. The retailer permanently stops ordering

B. The retailer orders enough to raise inventory position to S

C. The retailer orders exactly s units

D. The retailer writes off all on-order inventory

Correct Answer: B. The retailer orders enough to raise inventory position to S

Question 8

Why can EOQ alone be unsuitable for a limited-season fashion item?

A. EOQ always includes markdown cost automatically

B. EOQ assumes every item is perishable

C. The item may have a single selling opportunity and substantial leftover risk

D. EOQ cannot use any demand information

Correct Answer: C. The item may have a single selling opportunity and substantial leftover risk

Question 9

In an all-units quantity-discount problem, which candidate should be compared with other feasible candidates?

A. Only the unconstrained EOQ at the lowest price

B. The feasible EOQ or breakpoint quantity for each price bracket

C. Only the smallest supplier quantity

D. Only the quantity used in the previous year

Correct Answer: B. The feasible EOQ or breakpoint quantity for each price bracket

Question 10

A 95% cycle service level under a normal approximation uses a z-value of approximately:

A. 0.645

B. 1.000

C. 1.645

D. 2.576

Correct Answer: C. 1.645