Inventory Planning Models and Replenishment

Introduction and Context

Inventory planning determines when and how much stock an organisation should obtain. The central managerial problem is a trade-off: excessive inventory ties up capital, occupies space, increases insurance and obsolescence costs, and can conceal process problems; insufficient inventory causes stockouts, lost sales, production disruption and poor customer service.

Replenishment decisions must therefore consider demand, supplier lead time, ordering and holding costs, desired service levels, and uncertainty. No single model is best for every item. A stable, high-value item may require continuous monitoring, while a low-value consumable may be managed through a simple periodic review rule.

This chapter develops the principal inventory planning models used in procurement and supply chain operations: economic order quantity (EOQ), reorder point systems, safety stock, service-level decisions, and periodic review policies.

Learning Objectives

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

1. Explain the cost trade-offs underlying inventory replenishment.
2. Calculate and interpret EOQ.
3. Calculate reorder points under constant and variable demand and lead time.
4. Estimate safety stock using demand and lead-time variability.
5. Relate safety stock to cycle-service level.
6. Compare continuous-review and periodic-review systems.
7. Select an appropriate replenishment rule and recognise its limitations.
8. Apply inventory models responsibly using accurate data and professional judgement.

1. Foundations of Replenishment Planning

1.1 Important inventory quantities

Several quantities must be distinguished:

• On-hand inventory is physically present stock.
• On-order inventory has been purchased but not yet received.
• Inventory position is on-hand inventory plus on-order inventory minus backorders or committed demand.
• Lead time is the elapsed time between placing an order and receiving usable stock.
• Cycle stock is inventory held because replenishment arrives in batches.
• Safety stock is additional inventory held to protect against uncertainty.
• Pipeline inventory is stock in transit during lead time.
• Stockout occurs when demand cannot be fulfilled from available inventory.

A replenishment decision should normally use inventory position rather than on-hand inventory alone. If 100 units are on hand and 200 units are already on order, treating the position as only 100 units may create unnecessary purchasing.

1.2 Relevant costs

The main costs are:

A sound policy evaluates total relevant cost rather than focusing only on unit price. A discount that creates excess, obsolete inventory may increase total cost.

2. The Economic Order Quantity Model

2.1 Logic of EOQ

The basic EOQ model identifies the order quantity that minimises annual ordering cost plus annual holding cost. If orders are very small, ordering cost is high. If orders are very large, average inventory and holding cost are high.

The classical assumptions are:

• annual demand is known and constant;
• demand occurs at a uniform rate;
• replenishment is instantaneous or effectively instantaneous;
• lead time is constant;
• shortages are not permitted;
• unit price is constant, with no quantity discount;
• ordering cost per order and holding cost per unit per year are constant;
• inventory is received in a complete batch.

These assumptions make EOQ a benchmark, not a complete representation of real operations.

Let:

• D = annual demand in units per year;
• S = ordering or setup cost per order in currency/order;
• H = annual holding cost per unit in currency/unit/year;
• Q = order quantity in units.

The average cycle stock is Q/2. Annual ordering cost is (D/Q)S, and annual holding cost is (Q/2)H. Therefore:

where TC(Q) excludes purchase cost when unit price is constant.

The EOQ is:

The units are consistent: DS/H has units of units², so its square root has units of units.

At EOQ, annual ordering cost equals annual holding cost:

The number of orders per year is:

The average time between orders is approximately:

where the operating-time unit must match the desired result, such as days per year.

2.2 Worked example: EOQ

A distributor sells 12,000 units per year. Ordering cost is $40 per order and annual holding cost is $2.50 per unit.

The recommended order quantity is therefore approximately 620 units.

Annual ordering cost:

Annual holding cost:

The small difference is caused by rounding. The organisation would place approximately:

EOQ does not determine the reorder point. It determines the size of the replenishment order; timing requires demand and lead-time information.

2.3 Extensions and limitations

If a supplier offers quantity discounts, each price bracket should be evaluated using a total-cost comparison. The organisation must first determine whether the EOQ is feasible within each bracket, then compare purchase, ordering and holding costs. A discount should not be accepted automatically.

For gradual production replenishment, the economic production quantity model adjusts for the fact that inventory is consumed while it is being produced. For items with highly variable demand, significant shortages, capacity constraints or strong seasonality, a basic EOQ may be misleading.

3. Reorder Points and Continuous Review

A continuous-review, or Q,r, system places a fixed quantity Q when inventory position reaches the reorder point r. The order quantity may be EOQ, a supplier minimum, a pallet quantity or another operationally suitable amount.

3.1 Constant demand and lead time

If demand rate and lead time are constant:

where d is demand per time unit and L is lead time in the same time unit. The expression dL is lead-time demand.

For example, if daily demand is 80 units and lead time is 6 working days, the reorder point is 480 units when no safety stock is required.

3.2 Variable demand with constant lead time

If daily demand varies but lead time is constant, a common model is:

where μ_d is mean demand per day and SS is safety stock.

Assuming independent, approximately normally distributed daily demand, demand during lead time has standard deviation:

where σ_d is the standard deviation of daily demand. If the desired cycle-service level is associated with standard-normal factor z, then:

and:

The square-root term is important: uncertainty grows with the square root of time, not in direct proportion to time, under the stated independence assumptions.

3.3 Variable demand and variable lead time

When both demand and lead time vary, a commonly used approximation for the standard deviation of lead-time demand is:

where L̄ is mean lead time, σ_L is lead-time standard deviation, and demand and lead time are assumed independent. The reorder point is:

A more general expression includes the interaction term when demand and lead time are correlated:

The covariance term should not be ignored when late deliveries systematically coincide with high demand. In practice, many organisations use the simpler formula because covariance is difficult to estimate; this should be documented as an approximation.

3.4 Worked example: reorder point and safety stock

A retailer has mean daily demand of 100 units and daily demand standard deviation of 20 units. Lead time is constant at 5 days. Management wants a 95% cycle-service level. For a one-sided normal approximation, use z = 1.645.

Lead-time demand:

Standard deviation during lead time:

Safety stock:

Reorder point:

Thus, a continuous-review policy would trigger replenishment when inventory position falls to approximately 574 units. The order quantity remains a separate decision.

4. Service Levels and Safety Stock

4.1 Meaning of service level

“Service level” must be defined precisely. Two common measures are:

• Cycle-service level (CSL): probability that no stockout occurs during a replenishment cycle or lead-time protection period.
• Fill rate: percentage of total demand satisfied immediately from available stock.

A 95% CSL does not mean that 95% of units demanded are filled. A cycle can experience a stockout involving only a few units, or a stockout can be severe. Fill rate is therefore often more useful where shortage quantities matter.

Under a normal lead-time-demand model, the CSL is approximated by:

where Φ is the standard normal cumulative distribution function. Conversely:

Typical one-sided values include approximately 1.282 for 90%, 1.645 for 95%, and 2.326 for 99% CSL.

Higher service levels increase safety stock disproportionately because the upper tail of the demand distribution becomes more expensive to protect. The appropriate service target should reflect item criticality, shortage consequences, customer commitments and financial cost—not an arbitrary universal percentage.

4.2 Data and distribution cautions

The normal approximation is useful for high-volume items with reasonably stable demand. It may be inappropriate for intermittent demand, short product life cycles, promotion-driven demand, or very small quantities. For such items, empirical percentiles, Poisson-type models, Croston-style methods, or simulation may be more appropriate.

Historical data must be cleaned for stockout censoring. Observed sales during a stockout may understate true demand because customers could not purchase what was unavailable. Promotions, product launches, calendar effects and supplier disruptions should also be identified rather than blindly included in averages.

5. Periodic Review Policies

In a periodic-review, or P,S, system, inventory is checked every P time units and raised to an order-up-to level S. The order quantity varies:

The protection period is the review interval plus lead time. Stock must cover demand until the next review opportunity and the subsequent replenishment arrives.

If demand per period has mean μ_d and standard deviation σ_d, review period is P, lead time is L, and both are expressed in the same units, then:

This formula assumes independent demand across periods, constant lead time, and a normal approximation. If the periodic demand standard deviation is already measured for the whole protection period, it must not be multiplied by √(P+L) again.

Worked example: periodic review

A wholesaler reviews an item every 7 days. Lead time is 4 days. Average daily demand is 60 units and daily standard deviation is 12 units. The target CSL is 95%, so z = 1.645.

Protection period:

Expected protection-period demand:

Protection-period standard deviation:

Safety stock:

Order-up-to level:

If inventory position at review is 310 units, the order is approximately 416 units. The actual order may need rounding to a case pack or minimum order quantity.

Continuous versus periodic review

A periodic system generally needs more protection stock for the same service objective because uncertainty accumulates over a longer protection period. However, grouping orders can reduce transport and administrative costs.

6. Selecting a Replenishment Rule

A practical selection process is:

1. Classify the item. Use value, criticality, demand pattern, perishability and supply risk—not value alone.
2. Measure demand and lead time. Use consistent units and distinguish working days from calendar days.
3. Estimate uncertainty. Calculate appropriate averages, standard deviations and relevant percentiles.
4. Choose review logic. Select continuous review where rapid response is important; select periodic review where review economies are valuable.
5. Set the order quantity. Consider EOQ, minimum order quantity, case pack, truck utilisation, shelf life and supplier constraints.
6. Set the protection level. Use a justified service target and include safety stock where uncertainty exists.
7. Validate operational feasibility. Check storage, cash, supplier capacity, receiving capacity and expiry restrictions.
8. Monitor performance. Review stockouts, fill rate, excess stock, forecast error, supplier lead-time reliability and parameter drift.

ABC analysis can prioritise managerial attention by annual consumption value, while a demand-variability classification can distinguish smooth, seasonal and intermittent items. Combining value and variability is more informative than applying one rule to every SKU.

7. Applied Case: Illustrative Case of Meridian Medical Supplies

Background

Meridian Medical Supplies distributes sterile examination kits to clinics. Annual demand is 24,000 kits. The supplier charges a constant unit price. Ordering and receiving cost is $75 per order, and annual holding cost is $3 per kit. Average daily demand is 100 kits, with a daily standard deviation of 25 kits. Supplier lead time averages 6 working days and has a standard deviation of 1 working day. Demand and lead time are treated as independent. The organisation targets a 95% cycle-service level.

Problem

The purchasing team currently orders 2,000 kits whenever stock appears low. This practice creates high average inventory and does not formally account for supplier lead-time variability. Management wants a defensible replenishment policy.

Analysis

EOQ is:

A practical order quantity might be 1,100 kits if compatible with packaging and supplier constraints.

Mean lead-time demand is:

Using the variable-demand and variable-lead-time approximation:

With z = 1.645, safety stock is:

The reorder point is:

Approach and solution

Meridian adopts a continuous-review policy with an order quantity of approximately 1,100 kits and a reorder point of approximately 793 kits. The inventory position—not merely physical stock—is monitored. Because kits are sterile and have expiry dates, the organisation also applies first-expiry, first-out issuing and reviews whether the order quantity creates expiry risk.

Outcome

The policy reduces the batch size from 2,000 to about 1,100 kits while explicitly protecting against both demand and lead-time variability. The organisation should measure the result over several replenishment cycles rather than assuming that the calculated service level will occur automatically.

Learning points

• EOQ determines order size, not reorder timing.
• Lead-time variability materially increases safety stock.
• Inventory position prevents duplicate ordering when purchase orders are outstanding.
• A mathematically valid policy can still be operationally unsuitable if expiry, packaging or supplier minimums are ignored.
• Service performance must be measured against actual stockout and fill-rate data.

8. Technology, Governance and Professional Practice

Enterprise resource planning and warehouse-management systems can calculate inventory position, issue reorder alerts, and integrate purchase orders with receipts. Barcode scanning, RFID and supplier portals improve transaction accuracy and lead-time visibility. Forecasting and machine-learning tools can identify seasonality or anomalous demand, but they do not remove the need for sound assumptions, human review and master-data governance.

Important controls include approved parameter ownership, version-controlled supplier lead times, review dates for safety-stock settings, segregation of purchasing authority, and audit trails for overrides. AI-generated recommendations should be explainable enough for a buyer to understand the data, assumptions and service-cost trade-off. Automated ordering must not bypass approval limits, product-quality controls or regulatory requirements, especially for medical, food, chemical and safety-critical products.

9. Challenges and Limitations

Inventory models are decision aids, not guarantees. Common limitations include:

• inaccurate or incomplete demand history;
• demand lost during stockouts;
• non-stationary demand caused by promotions or market change;
• correlated demand and lead-time shocks;
• supplier minimums and transport constraints;
• quantity discounts and price changes;
• obsolescence, shelf-life and product substitution;
• normal-distribution assumptions applied to intermittent demand;
• service-level definitions that confuse CSL with fill rate;
• parameters that are not updated after operational changes.

Managers should use sensitivity analysis. For example, calculate the effect of a longer lead time, a higher service target or a 20% increase in demand variability. The purpose is not false precision; it is to reveal which assumptions drive the decision.

Best-Practice Checklist

• Use inventory position for replenishment decisions.
• Separate order quantity from reorder point.
• Match units for demand, lead time and review period.
• Include both demand and lead-time variability where relevant.
• State the service-level definition.
• Use total cost, not unit price alone.
• Respect minimum quantities, pack sizes, capacity and expiry constraints.
• Validate parameters with procurement, operations and finance.
• Monitor actual service, excess stock and forecast error.
• Review policies when suppliers, products or demand patterns change.

Chapter Summary

Inventory replenishment balances ordering, holding and shortage costs while protecting customer or production service. EOQ provides a benchmark order size under relatively stable assumptions, using the trade-off between ordering cost and holding cost. The reorder point determines when replenishment should be triggered and equals expected demand during the relevant protection period plus safety stock. Safety stock reflects uncertainty and is linked to a clearly defined service measure, such as cycle-service level or fill rate.

Continuous-review systems respond promptly through a fixed-quantity reorder rule, while periodic-review systems inspect inventory at intervals and order up to a target level. Periodic review requires protection over the review interval plus lead time. Variable demand and lead time require careful estimation, and simple normal approximations should not be applied blindly to intermittent or highly seasonal demand. Effective professional practice combines quantitative models with data governance, operational constraints, supplier knowledge, technology and ongoing performance review.

Key Terms

EOQ: Order quantity that minimises relevant annual ordering and holding costs under stated assumptions.

Ordering cost: Cost incurred each time an order is placed and processed.

Holding cost: Annual cost of carrying one unit in inventory.

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

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

Reorder point: Inventory-position level that triggers replenishment.

Lead-time demand: Demand expected during replenishment lead time.

Safety stock: Additional stock held to absorb uncertainty.

Cycle stock: Inventory resulting from ordering in batches.

Cycle-service level: Probability of avoiding a stockout during a defined replenishment cycle.

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

Continuous review: Inventory system monitored continuously or near continuously.

Periodic review: Inventory system checked at predetermined intervals.

Protection period: Time demand must be covered by available or incoming stock.

Order-up-to level: Target inventory position in a periodic-review system.

Demand variability: Variation in demand around its average.

Lead-time variability: Variation in supplier replenishment time.

ABC analysis: Classification of items by annual consumption value or similar importance.

Inventory position: The replenishment-control quantity including outstanding orders and deductions for commitments.

Review Questions

Short Answer Questions

1. What is the difference between inventory position and on-hand inventory?
2. State the assumptions of the basic EOQ model.
3. Distinguish cycle-service level from fill rate.
4. Why does a periodic-review system require protection over review period plus lead time?
5. What is the operational purpose of safety stock?

Long Answer Questions

1. Explain the cost trade-off that produces EOQ and discuss situations in which basic EOQ may be inappropriate.
2. Describe how demand variability and lead-time variability affect reorder-point calculations.
3. Compare continuous-review and periodic-review systems, including their advantages and limitations.
4. Explain why inventory models should use inventory position rather than on-hand stock alone.
5. Discuss how data quality, governance and technology influence replenishment performance.

Analytical / Application Questions

1. Annual demand is 18,000 units, ordering cost is $50 per order, and annual holding cost is $4 per unit. Calculate EOQ and explain how rounding to a case pack could affect the decision.
2. Daily demand averages 80 units with a standard deviation of 15 units. Lead time is constant at 9 days and the target CSL is 95% using z = 1.645. Calculate safety stock and reorder point.
3. An organisation reviews stock every 5 days. Average daily demand is 120 units, daily standard deviation is 20 units, lead time is 3 days, and z = 1.645. Calculate the periodic-review order-up-to level and explain two practical factors that may require adjustment.

Multiple Choice Questions with Answers

Question 1

Which quantity is normally used to trigger replenishment in a continuous-review system?

A. On-hand inventory only

B. Inventory position

C. Annual purchase value

D. Average supplier price

Correct Answer: B. Inventory position

Question 2

Annual demand is 10,000 units, ordering cost is $25 per order, and annual holding cost is $2 per unit. What is the approximate EOQ?

A. 250 units

B. 354 units

C. 500 units

D. 707 units

Correct Answer: B. 354 units

Question 3

If daily demand is 50 units and constant lead time is 4 days, with no safety stock, what is the reorder point?

A. 12.5 units

B. 54 units

C. 200 units

D. 250 units

Correct Answer: C. 200 units

Question 4

Under the normal approximation, which factor is used to convert lead-time-demand standard deviation into safety stock?

A. The unit purchase price

B. The z-value associated with the target service level

C. The number of suppliers

D. The annual demand divided by order quantity

Correct Answer: B. The z-value associated with the target service level

Question 5

A 95% cycle-service level means approximately that:

A. 95% of demand units are always filled

B. 95% of replenishment cycles avoid a stockout, under the model

C. Inventory is held for 95 days

D. 95% of inventory is safety stock

Correct Answer: B. 95% of replenishment cycles avoid a stockout, under the model

Question 6

If daily demand standard deviation is 10 units and constant lead time is 9 days, what is the lead-time-demand standard deviation under the independent-demand model?

A. 10 units

B. 19 units

C. 30 units

D. 90 units

Correct Answer: C. 30 units

Question 7

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

A. Ordering cost plus holding cost

B. Review period minus lead time

C. Review period plus lead time

D. Annual demand divided by EOQ

Correct Answer: C. Review period plus lead time

Question 8

A periodic-review system has an order-up-to level of 900 units and an inventory position of 350 units. Ignoring rounding and constraints, what order is placed?

A. 350 units

B. 550 units

C. 900 units

D. 1,250 units

Correct Answer: B. 550 units

Question 9

Which statement about EOQ is correct?

A. It determines the exact reorder point in all situations

B. It assumes shortages and stockouts are always optimal

C. It balances relevant ordering and holding costs under stated assumptions

D. It automatically accounts for demand seasonality

Correct Answer: C. It balances relevant ordering and holding costs under stated assumptions

Question 10

When both demand and lead time vary, why may safety stock need to increase?

A. Because uncertainty exists during the replenishment protection period

B. Because unit price must always increase

C. Because ordering cost becomes zero

D. Because periodic review eliminates demand

Correct Answer: A. Because uncertainty exists during the replenishment protection period