Inventory Planning and Replenishment Models

Introduction and context

Inventory planning determines how much stock to hold, when to replenish it, and which service and cost trade-offs to accept. In a supply chain, inventory protects operations from uncertainty and enables customer service, but it also consumes working capital, warehouse space, insurance, labour and management attention. The central planning problem is therefore not simply to minimize inventory. It is to provide the required service at an economically and operationally acceptable total cost.

Replenishment models convert information about demand, lead time, ordering cost, holding cost and supply uncertainty into practical decisions. Some models assume stable demand and are useful for establishing a benchmark; others explicitly address uncertainty, quantity discounts, production constraints or short product life. A competent inventory professional must understand both the mathematics and the conditions under which a model should not be trusted without adjustment.

Learning objectives

After studying this chapter, you should be able to:

• explain the cost and service logic of inventory replenishment;
• calculate and interpret the economic order quantity (EOQ);
• evaluate quantity-discount and economic production quantity decisions;
• distinguish continuous-review and periodic-review systems;
• apply min–max, two-bin and newsvendor approaches;
• identify assumptions, limitations and implementation requirements; and
• select a replenishment policy appropriate to the item and operating environment.

1. Inventory-planning foundations

1.1 Relevant inventory costs

The main costs affected by replenishment decisions are:

The relevant objective is usually to minimize expected total relevant cost while satisfying a specified service requirement. A simplified annual cost expression is:

where TC is annual total relevant cost, D is annual demand in units, C is unit purchase cost, Q is order quantity, S is ordering cost per order, and H is annual holding cost per unit. The expression assumes a constant unit price, instantaneous replenishment, no shortages and average cycle stock of Q/2.

If holding cost is expressed as a fraction i of unit value, then H = iC, with H measured in currency per unit per year. All annual quantities and time units must be consistent.

1.2 Cycle stock, safety stock and pipeline stock

Cycle stock is the inventory used between replenishments. Safety stock protects against uncertainty in demand or lead time. Pipeline stock is inventory that has been ordered but has not yet arrived. These categories are conceptually distinct, although an inventory record may combine them.

The inventory position used by many systems is:

where IP is inventory position, OH is on-hand inventory, OO is on-order inventory and BO is backorders or committed shortages. Replenishment decisions should generally use inventory position rather than on-hand stock alone; otherwise, multiple outstanding orders can cause over-ordering.

1.3 Service measures

The cycle service level (CSL) is the probability of not experiencing a stockout during a replenishment cycle. The fill rate is the proportion of demand supplied immediately from stock. They are not identical: a single stockout involving one unit and a stockout involving a large quantity may both count as one failed cycle, but their fill-rate effects differ. Planning should specify which service measure matters for the customer and the business.

2. Economic order quantity

2.1 Logic and derivation

EOQ balances ordering cost and holding cost. As Q increases, annual ordering cost (D/Q)S falls, while annual cycle-stock holding cost (Q/2)H rises. Ignoring purchase cost because it is constant when price does not depend on quantity, the relevant cost is:

Differentiating with respect to Q and setting the derivative to zero gives:

Therefore:

Q* is the EOQ in units per order. The formula applies when demand is reasonably stable, replenishment is effectively instantaneous, shortages are not intentionally planned, unit price is constant, ordering cost is fixed per order, and holding cost is proportional to average inventory. The solution is a theoretical benchmark, not an automatic purchasing instruction.

At EOQ, annual ordering cost equals annual cycle-stock holding cost:

The number of orders per year is D/Q*, and the time between orders is approximately Q*/D years. If there are W working days per year, the order interval in working days is WQ*/D.

2.2 Worked example: EOQ

A distributor has annual demand of 12,000 units. Each order costs $50 to process and receive. Annual holding cost is $2 per unit. Demand is constant and replenishment is immediate.

The recommended order quantity is approximately 775 units. Annual ordering cost is:

Annual cycle-stock holding cost is:

Thus, the combined relevant cost is approximately $1,549.20 per year, excluding purchase cost and safety stock. If the supplier requires orders in cartons of 100, the planner must compare feasible quantities such as 700 or 800 rather than blindly using 774.6.

3. Quantity-discount models

A supplier may reduce the unit price when an order reaches a quantity breakpoint. In this situation, purchase cost changes with Q, so it cannot be ignored. For each price bracket, calculate the candidate order quantity and compare total annual cost only at feasible quantities.

The total cost for a price level C_j is:

If holding cost is a percentage of price, H_j = iC_j. The procedure is:

1. Calculate EOQ using the holding cost associated with each price level.
2. Check whether that EOQ falls within the bracket’s feasible quantity range.
3. If it does not, use the nearest feasible breakpoint in that bracket.
4. Calculate total annual cost for every feasible candidate, including the lowest quantity that qualifies for each discount.
5. Select the candidate with the lowest total cost.

Worked example: all-unit discount

Annual demand is 10,000 units, ordering cost is $40, and holding cost is 20% of unit price. The supplier offers:

For the first bracket, H = 0.20 × 10 = $2.00, so:

This is not feasible in the first bracket; the best candidate there is 499. For the second bracket, H = $1.96, producing an EOQ of approximately 639, which is feasible. For the third bracket, H = $1.90, producing an EOQ of approximately 649, which is below the bracket minimum; the candidate is therefore 1,000.

Approximate total costs are:

• At 499: 10,000(10) + (10,000/499)(40) + (499/2)(2) ≈ $103,602.
• At 639: 10,000(9.80) + (10,000/639)(40) + (639/2)(1.96) ≈ $101,253.
• At 1,000: 10,000(9.50) + (10,000/1,000)(40) + (1,000/2)(1.90) = $98,400.

The 1,000-unit order is economically preferred under the stated assumptions. In practice, the decision should also test cash-flow capacity, shelf life, storage capacity and obsolescence risk.

4. Economic production quantity

EOQ assumes that the entire order arrives at once. When an organisation manufactures an item internally, replenishment occurs gradually while demand continues. The economic production quantity (EPQ) accounts for this overlapping production and consumption.

Let p be the production rate and d the demand rate, expressed in the same units of time, with p > d. The maximum inventory during a production run is Q(1 − d/p). Average inventory is half of that amount. The EPQ is:

As p becomes very large relative to d, the factor (1 − d/p) approaches 1 and the formula approaches EOQ. The model assumes a stable production rate, stable demand, reliable quality, fixed setup cost and no disruptive breakdowns. Capacity, campaign scheduling, minimum run lengths and changeover losses may make a finite-production model more realistic than EOQ, but they may also require a broader production-planning model.

5. Reorder points and safety stock

5.1 Deterministic reorder point

With known demand and lead time, the reorder point is demand during lead time:

where d is demand per time unit and L is lead time in the same time unit. If daily demand is 40 units and lead time is 5 days, ROP = 40 × 5 = 200 units.

5.2 Safety stock under demand uncertainty

If demand during lead time is approximately normally distributed and lead time is constant, a common policy is:

where μ_LT is mean lead-time demand, σ_LT is its standard deviation, and z is the standard-normal safety factor associated with the target cycle service level. Safety stock is zσ_LT. This relationship is appropriate only when the distributional approximation, data quality and service interpretation are reasonable.

If daily demand has mean μ_d and standard deviation σ_d, and lead time is fixed at L days:

If both daily demand and lead time are variable, and they are assumed independent, a commonly used approximation is:

where σ²_LT is the variance of lead-time demand, E[L] is mean lead time and σ²_L is lead-time variance. If demand and lead time are correlated, an additional covariance term is required; blindly using the independent form can understate risk.

6. Continuous-review and periodic-review systems

6.1 Continuous review: the (Q, r) policy

Under continuous review, inventory position is monitored whenever transactions occur. When IP ≤ r, an order of Q units is placed. The policy is called (Q,r), where Q is order quantity and r is reorder point.

Advantages include rapid response to demand changes and suitability for important, expensive or critical items. Disadvantages include the need for accurate transaction recording, system connectivity and disciplined exception management. Barcode scanning, RFID and warehouse-management-system integration make continuous review more feasible, but technology cannot correct inaccurate stock counts or unrecorded losses.

6.2 Periodic review: the (P, S) policy

Under periodic review, inventory position is checked at fixed intervals P. The order raises the position to an order-up-to level S:

A periodic system must protect against demand during the protection period, which is review interval plus lead time:

For constant lead time and independent daily demand:

Periodic review is convenient when suppliers visit on a schedule, when many items are ordered together, or when administrative simplicity is important. It exposes stock to risk for the time until the next review, so its safety stock is generally higher than that of an equivalent continuous-review system.

7. Min–max and two-bin systems

A min–max policy establishes a lower trigger and an upper target. When the inventory position reaches or falls below the minimum, the order quantity raises it toward the maximum:

The minimum commonly reflects lead-time demand plus safety stock, while the maximum also covers expected demand until the next replenishment opportunity. The policy is simple, transparent and useful where demand is intermittent or review occurs at practical intervals. However, the meaning of “minimum” and “maximum” must be documented; otherwise different users may interpret them as on-hand stock, inventory position or physical shelf levels.

A two-bin system divides stock into two containers. The first bin is used in normal operations; when it is emptied, the second bin supplies demand while the first bin triggers replenishment. The quantity in the second bin is intended to cover lead-time demand and protection. Two-bin systems are inexpensive and visually clear for low-value, stable-demand consumables. They are weak for expensive, highly variable, serialized or rapidly changing items, and they depend on correct bin sizing and prompt replenishment.

8. The newsvendor model

The newsvendor model addresses a single ordering decision for a short-life, seasonal or otherwise time-sensitive product. The decision balances the cost of ordering too much against the cost of ordering too little.

Let C_u be underage cost per unit, C_o be overage cost per unit, and F(Q) be the cumulative distribution function of demand. The critical fractile is:

If selling price is p, purchase cost is c, and salvage value is v, then, in the basic case:

The model assumes one order opportunity, uncertain demand, no replenishment after the decision and a known demand distribution. It is particularly relevant to fresh food, fashion, newspapers, event merchandise and promotional products. It should not be used for ordinary replenishable items merely because demand is uncertain.

Worked example: seasonal product

A retailer sells a seasonal item for $30, buys it for $18 and can salvage unsold units for $6. Therefore, C_u = 30 − 18 = $12 and C_o = 18 − 6 = $12.

The retailer should order the 50th percentile, or median, of the demand distribution. If demand is normally distributed with mean 1,000 and standard deviation 200, the 50th percentile is 1,000 units. This result is not a guarantee of profit; it is the quantity that balances expected marginal underage and overage costs under the model.

9. Applied case: replenishment redesign at Meridian Facilities Supply

This is an illustrative case. Meridian supplies maintenance materials to factories. Fasteners, lubricants and electrical components are purchased from multiple vendors. The company has experienced both emergency purchases and excess stock. A review found that buyers were ordering by intuition, on-hand balances excluded open orders, and all products were managed with the same monthly review cycle.

Problem

A critical bearing has annual demand of 3,600 units, ordering cost of $75, unit cost of $40 and annual holding-rate estimate of 25%. Supplier lead time averages 10 working days, with demand averaging 15 units per working day and a standard deviation of 5 units. The target cycle service level is approximately 95%, for which z ≈ 1.645.

Analysis

Annual holding cost is:

EOQ is:

A practical order quantity might be 230 or 240 units, subject to packaging and supplier constraints.

Mean lead-time demand is 10 × 15 = 150 units. Its standard deviation, assuming fixed lead time, is:

Safety stock is:

Thus:

Approach and solution

Meridian introduced a continuous-review (Q,r) policy for the critical bearing: order approximately 230 units when inventory position reaches 176 units. The inventory system was amended to include open purchase orders and reservations. Demand and lead-time statistics were reviewed quarterly, while buyers retained authority to override the model when a supplier announced a disruption or a major maintenance shutdown was scheduled.

Lower-value consumables were moved to two-bin control, and seasonal promotional items were planned with a newsvendor-style analysis. Supplier minimums, shelf space and cash-flow effects were included in approval decisions.

Outcome and learning points

The redesign reduced purely intuition-based ordering and made accountability clearer. Its main lesson is that a mathematically sound parameter is ineffective if transaction records are inaccurate. A second lesson is segmentation: critical bearings, routine consumables and seasonal goods require different policies. Finally, service targets should be explicit. A high service target for every item can create excessive working capital, while a low target for a production-critical component can create disproportionate operational damage.

10. Model selection, governance and contemporary practice

A practical selection framework asks five questions:

1. Is demand replenishable, seasonal or one-time?
2. Is review continuous or periodic in the available system?
3. Are demand and lead time stable enough for deterministic calculations?
4. Are price, production capacity, shelf life or minimum order constraints material?
5. What service, financial and operational consequences follow from shortage?

ABC analysis may be combined with demand-variability or criticality classifications. For example, an A-critical item may receive continuous review and carefully estimated safety stock, while a C-routine item may use two bins. Classification should be revisited because demand value and operational criticality are not the same dimension.

Modern systems use enterprise-resource-planning, warehouse-management and supplier portals to automate reorder signals. Forecasting and machine-learning tools can improve demand estimates when sufficient clean history exists, but they do not remove the need to specify lead-time distributions, service objectives, constraints and exception rules. Automated recommendations should be explainable, auditable and subject to human review for unusual events.

Governance issues include segregation of purchasing authority, approval of parameter changes, protection against fraudulent supplier or quantity data, audit trails and periodic stock reconciliation. Ethical practice requires that service decisions do not unfairly prioritize customers without an agreed commercial or public-interest basis. In regulated or safety-critical environments, traceability, expiry control and documented approval may be mandatory.

Challenges and limitations

All replenishment models simplify reality. Important limitations include:

• nonstationary demand caused by promotions, substitution or market change;
• correlated demand across products and common-cause supply disruption;
• inaccurate forecasts, lead-time records or inventory balances;
• capacity, transport, cash-flow and storage constraints;
• supplier minimums, price changes and unreliable delivery;
• obsolescence, deterioration and quality holds; and
• the risk that a calculated optimum creates operationally impractical orders.

Sensitivity analysis is therefore essential. Planners should vary demand, lead time, holding cost and service assumptions and observe whether the decision changes. If a small parameter change produces a very different policy, managerial judgment and scenario planning deserve more attention than a false impression of numerical precision.

Best-practice checklist

• Define the service measure before calculating safety stock.
• Use inventory position, including open orders and commitments.
• Match time units and verify physical order constraints.
• Segment items by value, variability, criticality and life cycle.
• Recalculate parameters after structural demand or supply changes.
• Compare model recommendations with capacity, cash and shelf-life limits.
• Record assumptions, overrides and parameter ownership.
• Measure outcomes using fill rate, stockouts, inventory turns, excess stock and emergency freight—not only purchase price.

Chapter Summary

Inventory replenishment models balance ordering, holding, purchasing and shortage consequences. EOQ provides a benchmark when demand, price and replenishment conditions are relatively stable. Quantity-discount analysis adds changing purchase prices, while EPQ recognizes gradual internal production. Reorder points combine lead-time demand with safety stock when uncertainty exists. Continuous-review systems react whenever inventory position reaches a trigger; periodic-review systems replenish at scheduled intervals and must protect demand over the review-plus-lead-time period. Min–max and two-bin systems provide practical visual or rule-based control, whereas the newsvendor model is designed for one-time or short-life decisions. Effective implementation requires accurate data, explicit service objectives, item segmentation, governance and sensitivity analysis. No model replaces professional judgment about capacity, criticality, cash, shelf life or disruption risk.

Key Terms

EOQ: Economic order quantity that balances ordering and cycle-stock holding costs.

EPQ: Economic production quantity for gradual replenishment during internal production.

Cycle stock: Inventory expected to be consumed between replenishments.

Safety stock: Additional inventory held against uncertainty.

Pipeline stock: Ordered inventory not yet received.

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

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

Holding cost: Cost of carrying inventory over time.

Reorder point: Inventory-position level that triggers replenishment.

Lead-time demand: Demand expected while replenishment is in transit or being produced.

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

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

Continuous review: Inventory monitoring after transactions or in near real time.

Periodic review: Inventory assessment at fixed time intervals.

Min–max policy: Rule that replenishes inventory from a minimum trigger toward a maximum level.

Two-bin system: Visual system in which depletion of one bin signals replenishment.

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

Critical fractile: Target demand percentile determined by underage and overage costs.

Inventory position: The control quantity used to reflect stock, open orders and commitments.

Review Questions

Short Answer Questions

1. Why does EOQ ignore purchase cost when unit price is constant?
2. Distinguish cycle stock from safety stock.
3. What is the difference between cycle service level and fill rate?
4. Why is inventory position more useful than on-hand stock for replenishment decisions?
5. State two situations in which a two-bin system is appropriate.

Long Answer Questions

1. Derive the EOQ formula and explain each assumption on which it depends.
2. Explain the procedure for solving an all-unit quantity-discount problem.
3. Compare continuous-review and periodic-review systems in terms of control, data requirements and safety stock.
4. Explain the EPQ model and show how it differs from EOQ.
5. Discuss how governance, data quality and item segmentation affect replenishment performance.

Analytical / Application Questions

1. Annual demand is 8,000 units, ordering cost is $60 and annual holding cost is $3 per unit. Calculate EOQ, annual ordering cost and annual holding cost at EOQ. State the assumptions required.
2. Daily demand has mean 50 units and standard deviation 8 units. Lead time is fixed at 6 days, and the target cycle service level is 95% with z = 1.645. Calculate mean lead-time demand, safety stock and reorder point.
3. A seasonal product sells for $45, costs $25 and has a salvage value of $10. Calculate the critical fractile and explain what demand percentile should guide the order quantity. Discuss one limitation of applying the model in practice.

Multiple Choice Questions with Answers

Question 1

Which expression represents inventory position?

A. On-hand inventory + backorders − on-order inventory

B. On-hand inventory + on-order inventory − backorders

C. On-order inventory − on-hand inventory − backorders

D. On-hand inventory ÷ on-order inventory

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

Question 2

In the basic EOQ model, increasing order quantity causes annual ordering cost to:

A. Increase linearly

B. Remain constant

C. Decrease

D. Become equal to purchase cost

Correct Answer: C. Decrease

Question 3

Annual demand is 10,000 units, ordering cost is $50 and annual holding cost is $2. What is the approximate EOQ?

A. 224 units

B. 500 units

C. 707 units

D. 1,000 units

Correct Answer: C. 707 units

Question 4

Which condition is required for the standard EPQ formula?

A. Production rate is less than demand rate

B. Production rate is greater than demand rate

C. Demand must be zero during production

D. Unit price must increase with quantity

Correct Answer: B. Production rate is greater than demand rate

Question 5

With constant demand and lead time, the deterministic reorder point is calculated as:

A. Ordering cost × holding cost

B. Annual demand ÷ order quantity

C. Demand per time unit × lead time

D. Order quantity ÷ annual demand

Correct Answer: C. Demand per time unit × lead time

Question 6

A periodic-review system must protect against demand during which period?

A. Only the supplier transit time

B. Only the review interval

C. The review interval plus lead time

D. The annual planning horizon

Correct Answer: C. The review interval plus lead time

Question 7

In a min–max policy, the replenishment quantity is commonly calculated as:

A. Minimum − maximum

B. Maximum − inventory position

C. Inventory position − maximum

D. Demand ÷ lead time

Correct Answer: B. Maximum − inventory position

Question 8

A two-bin system is most suitable for:

A. High-value items with highly volatile demand

B. Perishable products requiring a single-period decision

C. Low-value, stable-demand consumables

D. Items with no replenishment lead time

Correct Answer: C. Low-value, stable-demand consumables

Question 9

If underage cost is $12 and overage cost is $8, what is the newsvendor critical fractile?

A. 0.40

B. 0.60

C. 0.67

D. 1.50

Correct Answer: C. 0.67

Question 10

Which statement about safety stock is most accurate?

A. It eliminates all possible stockouts

B. It compensates for uncertainty in demand or lead time

C. It is identical to cycle stock

D. It is required only under EOQ assumptions

Correct Answer: B. It compensates for uncertainty in demand or lead time