Inventory Costs and Deterministic Control Models

Introduction

Inventory is the stock of materials, components, work in process, finished goods, maintenance supplies and other items held to support operations. Inventory enables an organisation to separate purchasing, production and customer demand in time and location. However, inventory is not free. Capital is committed, storage capacity is consumed, items may deteriorate, and excessive stock can conceal process problems. Conversely, insufficient stock may interrupt production or cause lost sales.

Inventory management is therefore a cost-and-service decision. The central managerial question is not “How can inventory be eliminated?” but “What quantity and timing of replenishment provide the required service at an acceptable total cost?” Deterministic control models provide a disciplined answer when demand, lead time and other relevant conditions are known or reasonably stable.

This chapter examines ordering, purchasing, holding, shortage and obsolescence costs. It then develops the economic order quantity (EOQ), economic production quantity (EPQ), quantity-discount, reorder-point and periodic-review models. Each model is useful only when its assumptions are understood. In practice, the models should support managerial judgement rather than replace it.

Learning objectives

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

1. Explain the major categories of inventory cost.
2. Distinguish independent and dependent demand and deterministic from probabilistic control.
3. Derive and apply the EOQ model.
4. Apply the EPQ model when replenishment is gradual.
5. Evaluate quantity discounts using total relevant cost.
6. Calculate reorder points and explain the effect of lead time and safety stock.
7. Compare continuous-review and periodic-review systems.
8. Identify the assumptions, limitations, governance issues and practical uses of inventory models.

1. Inventory as an economic system

Inventory performs several operational functions:

• Cycle stock results from ordering or producing in batches.
• Safety stock protects against uncertainty in demand or lead time.
• Anticipation stock is accumulated for predictable seasonal demand, shutdowns or price changes.
• Pipeline inventory is stock moving between locations or suppliers.
• Decoupling stock permits adjacent processes to operate at different rates.

Independent demand originates outside the immediate production system, such as customer demand for finished goods. Dependent demand is calculated from the demand for another item, as in components required by a bill of materials. EOQ-type models are most directly applicable to independent-demand items, while material requirements planning is normally used for dependent-demand items.

A deterministic model assumes that relevant quantities, such as demand and lead time, are known. A probabilistic model represents uncertainty using probability distributions or service-level requirements. Deterministic models remain valuable for stable, high-volume items and as baseline policies, but uncertainty must be incorporated through safety stock, scenario analysis or more advanced models.

2. Inventory cost structure

A sound inventory decision includes all costs that change as the policy changes. The main categories are as follows.

2.1 Purchase or acquisition cost

Purchase cost is the supplier price paid for each unit. It may include transportation, duties, inspection, receiving and preparation costs. If unit price is constant and every policy purchases the same annual quantity, purchase cost does not affect the EOQ quantity. It does, however, affect the total cost and becomes decisive when quantity discounts are offered.

2.2 Ordering or setup cost

Ordering cost, denoted by S, is the cost incurred each time an order is placed. It may include requisition processing, supplier communication, approval, receiving, inspection and invoice matching. In manufacturing, the analogous cost is setup cost: cleaning, tooling, programming and first-off inspection.

If annual demand is D units and order quantity is Q units, the annual number of orders is:

where N is orders per year. The annual ordering cost is:

Units are orders/year × currency/order = currency/year.

2.3 Holding or carrying cost

Holding cost is the cost of retaining inventory. It commonly includes:

• cost of capital or opportunity cost;
• warehouse space, handling and utilities;
• insurance, taxes and security;
• deterioration, damage, shrinkage and expiry;
• obsolescence and write-down risk.

If inventory is replenished instantaneously and demand is constant, cycle inventory falls linearly from Q to zero. Average cycle inventory is Q/2. If H is annual holding cost per unit, annual holding cost is:

If H is expressed as a fraction i of unit price C, then:

The fraction i must be annual and dimensionless. Average inventory is measured in units, so C_h is currency/year.

2.4 Shortage costs

A shortage occurs when available inventory cannot satisfy an immediate requirement. Shortage costs can include lost margin, emergency freight, production downtime, expediting, substitution, contractual penalties and damage to customer relationships. Some shortages are backordered and fulfilled later; others result in lost sales.

Shortage cost is difficult to estimate because the visible financial cost may be smaller than the long-term reputational or operational cost. A policy that minimises accounting cost but repeatedly disrupts critical operations is not necessarily optimal.

2.5 Obsolescence and deterioration

Obsolescence arises when technology, regulations, fashion, specifications or customer requirements change. Perishable inventory may lose value through age, expiry or contamination. These costs are often nonlinear: the first months may create little risk, while ageing beyond a threshold can cause a large write-off.

Obsolescence should not automatically be treated as a fixed percentage of inventory value. A more realistic analysis may segment items by life cycle, expiry date, technological risk and salvage value. For an item with unit value V, probability p of becoming obsolete during the relevant period, and recoverable salvage value R, an expected obsolescence cost per unit can be approximated by:

This is an expected-value approximation and is useful only when p and R are credible estimates.

2.6 Total relevant annual cost

For the basic EOQ model with constant unit price and no shortages:

D C is annual purchase cost, (D ÷ Q)S is annual ordering cost and (Q ÷ 2)H is annual holding cost. Costs must use the same time basis, normally currency per year. Fixed costs that do not change between alternatives may be excluded from comparison, but they should not be forgotten when reporting total expenditure.

3. Economic order quantity

3.1 Model assumptions

The classical EOQ model assumes:

1. Demand is known, constant and continuous.
2. Lead time is known and constant.
3. Replenishment is instantaneous.
4. The entire order arrives at once.
5. Unit purchase price is constant.
6. No shortages are permitted.
7. Ordering cost and holding cost are constant.
8. The item is independent of other items and capacity is sufficient.

These assumptions describe a baseline, not every real inventory system.

3.2 Derivation

Ignoring the constant purchase term DC, the variable annual cost is:

The first term decreases as Q increases, while the second increases. Differentiating with respect to Q gives:

At the minimum, the derivative equals zero:

Therefore:

Q* is the economic order quantity in units per order. The second derivative, 2DS ÷ Q³, is positive for Q > 0, confirming a minimum.

At EOQ, annual ordering cost equals annual holding cost:

This equality is a useful check, not a separate decision rule.

3.3 Worked example: basic EOQ

A maintenance department uses 12,000 bearings per year. Ordering cost is $45 per order and annual holding cost is $3.60 per bearing.

The recommended order quantity is approximately 548 bearings. The order frequency is:

Annual ordering cost is approximately 21.91 × $45 = $986.07. Annual holding cost is approximately (547.7 ÷ 2) × $3.60 = $986.07. The relevant annual cost is about $1,972.14, excluding the constant purchase cost.

The calculation is a baseline. If bearings are sold in cartons of 100, the organisation must compare feasible quantities such as 500 and 600, including any operational or supplier constraints.

4. Economic production quantity

EOQ assumes that replenishment is instantaneous. In an internal production system, inventory may accumulate while production continues. Let p be the production rate in units per year and d the usage rate in units per year, with p > d. During production, inventory increases at the net rate p − d. The maximum inventory is less than Q because demand occurs while the batch is being produced.

The fraction of the batch remaining after production is:

Thus maximum inventory is Q(1 − d/p), and average inventory is half this amount. The EPQ is:

Here S is setup cost per production run, H is annual holding cost per unit, D is annual demand, p is annual production rate and d is annual usage rate. The formula requires p > d. As p approaches d, replenishment becomes very slow and the calculated batch size increases sharply. If p ≤ d, the system cannot meet ongoing demand and this model is inapplicable.

Worked example: EPQ

A factory consumes 20,000 components per year. It produces them at 100,000 units per year. Setup cost is $80 per run and annual holding cost is $4 per unit.

Maximum inventory is 1,000(1 − 0.2) = 800 units, and average inventory is 400 units. The EPQ exceeds the instantaneous-receipt EOQ because production replenishment is gradual.

5. Quantity-discount decisions

Suppliers may reduce unit price when the order quantity reaches a breakpoint. A discount can reduce purchase cost but increase holding cost and possibly obsolescence risk. The correct method is to calculate total annual cost for each feasible price bracket.

For price level C_j, with minimum quantity L_j and holding cost H_j, the candidate EOQ is:

If Q_j* lies within the price bracket, it is a candidate. If it lies below the bracket minimum, the minimum quantity L_j is the candidate for that bracket. The total cost is:

The lowest feasible total cost is selected. When holding cost is a percentage of price, a lower price also lowers H, although this must be applied consistently with the organisation’s accounting policy.

Worked example: all-units discount

Annual demand is 10,000 units, ordering cost is $50, and holding cost is 25% of unit price. The supplier offers:

For $10.00, H = $2.50 and EOQ = √(1,000,000 ÷ 2.50) ≈ 632.46, which is not feasible in the first bracket; use Q = 499 or, if brackets are interpreted as upper limits, the greatest feasible quantity 499.

For $9.80, H = $2.45 and EOQ ≈ 639.94, which is feasible. For $9.50, H = $2.375 and EOQ ≈ 648.89, below the 1,000-unit minimum; use Q = 1,000.

Relevant total costs are:

• Q = 499 at $10.00: $100,000 + (10,000 ÷ 499)×$50 + (499 ÷ 2)×$2.50 ≈ $101,871.48.
• Q ≈ 639.94 at $9.80: $98,000 + (10,000 ÷ 639.94)×$50 + (639.94 ÷ 2)×$2.45 ≈ $99,565.27.
• Q = 1,000 at $9.50: $95,000 + (10,000 ÷ 1,000)×$50 + (1,000 ÷ 2)×$2.375 = $96,687.50.

The $9.50 price with 1,000 units is preferable because its total cost is lowest, despite higher cycle inventory. The decision should still be reviewed for shelf-life, storage capacity and cash-flow effects.

6. Reorder point and continuous review

EOQ determines how much to order; it does not determine when to order. Under continuous review, inventory position is monitored and an order is placed when it reaches the reorder point.

Inventory position includes on-hand inventory plus on-order inventory minus backorders. If demand is constant at d units per time period and lead time is L periods, the deterministic reorder point is:

Units must be consistent: if d is units per day, L must be days. If safety stock SS is required:

Safety stock is not a substitute for correcting unreliable suppliers, inaccurate records or unstable processes. It is a deliberate protection against uncertainty or service consequences.

Under uncertainty, a common approximation is:

where μ_LT is expected demand during lead time, σ_LT is the standard deviation of lead-time demand, and z is the standard-normal service factor. This formula assumes that lead-time demand is approximately normal and that the selected service level is appropriate.

If daily demand has mean μ_d and standard deviation σ_d, lead time is constant L days, and daily demands are independent:

If both demand and lead time vary independently, with mean lead time μ_L and standard deviation σ_L, an often-used approximation is:

This formula excludes the interaction term associated with simultaneous variability and is therefore an approximation. It should not be used blindly for highly skewed demand, correlated demand and lead time, or intermittent demand.

7. Periodic-review systems

In a periodic-review, or fixed-interval, system inventory is checked every T time units. The order quantity varies and raises the inventory position to a target level. The policy is often called an order-up-to or base-stock policy.

With deterministic demand and known lead time L, the protection interval is T + L. If demand rate is d, the order-up-to level is:

The replenishment quantity at a review is:

where IP is inventory position immediately before ordering; if q ≤ 0, no order is placed. The formula assumes that demand is constant, replenishment is reliable and the review schedule is fixed.

With uncertain demand and constant review and lead times, a service-level version is:

For independent daily demand with mean μ_d and standard deviation σ_d:

Periodic review is convenient when many items are ordered from the same supplier or when continuous monitoring is expensive. Its disadvantage is exposure to demand during the interval between reviews, which generally requires more protection than an equivalent continuous-review system.

8. Choosing and implementing a control model

Model selection should begin with the operational decision, not with a formula. A practical process is:

1. Classify the item by value, criticality, demand pattern, shelf life and supply risk.
2. Clean demand, lead-time, price and inventory-record data.
3. Define service requirements and shortage consequences.
4. Select a continuous- or periodic-review structure.
5. Estimate relevant costs using activity-based evidence where possible.
6. Calculate the baseline quantity and reorder parameters.
7. Check supplier minimums, packaging, capacity, cash flow and shelf life.
8. Simulate or test the policy against historical and stressed demand.
9. Establish ownership, review frequency and exception thresholds.
10. Monitor outcomes and revise parameters when conditions change.

Technology supports this process through enterprise resource planning, barcode or RFID capture, supplier portals and inventory dashboards. Artificial intelligence can assist with anomaly detection, demand classification and parameter recommendations, but it does not remove the need for data governance. An opaque recommendation that creates excessive expiry or systematically disadvantages a supplier should not be accepted without review.

Useful performance measures include inventory turns, days of supply, fill rate, cycle service level, stockout frequency, backorder age, write-offs, forecast bias, supplier lead-time adherence and total inventory cost. No single metric is sufficient. High inventory turns may indicate efficiency or dangerously low stock; high fill rate may be purchased through excessive inventory.

9. Illustrative case: Northstar Medical Devices

Background

Northstar Medical Devices uses sterile valve assemblies in a regulated production line. Annual demand is 24,000 assemblies. Demand is stable, but the assemblies have a two-year shelf life and a high shortage consequence because line stoppage delays hospital equipment.

Problem

Purchasing placed large quarterly orders to obtain administrative convenience. Average inventory was high, expiry write-offs increased and occasional supplier delays still caused shortages. Management needed a policy that balanced cost, availability and expiry risk.

Analysis

The purchasing team estimated ordering cost at $120 per order and annual holding cost at $6 per assembly, including capital, storage and expected expiry exposure. The baseline EOQ was:

Supplier lead time was 10 working days. Average daily demand was 96 assemblies, giving deterministic lead-time demand of 960 assemblies. Because demand and lead time were not perfectly certain, the team added a policy safety stock of 240 assemblies after analysing historical deviations. The reorder point became:

The team also imposed a maximum age rule: receipts had to be issued using first-expiry-first-out, and orders could not exceed the quantity that could be consumed before the expiry horizon without approval.

Approach and solution

Northstar introduced continuous review for the valve assemblies, electronic lot tracking, supplier lead-time reporting and monthly parameter review. The buyer compared the EOQ with the supplier’s case-pack size and selected 1,000 units per order, which was operationally feasible and close to the calculated optimum. A supplier agreement included advance notification of delays and a documented escalation process.

Outcome

In the illustrative scenario, average cycle stock fell substantially from the previous quarterly policy. Expiry write-offs declined, while the safety stock and supplier escalation process protected the production line. The organisation did not judge success by inventory reduction alone; it evaluated total cost, service and compliance together.

Learning points

1. EOQ is a starting point, not an automatic order instruction.
2. Shelf life can make a mathematically attractive quantity unacceptable.
3. Safety stock should be linked to service consequences and evidence.
4. Lot traceability and issue rules are part of inventory control.
5. Regulated materials require documented approvals and audit trails.

10. Limitations, governance and ethical considerations

Classical models can produce misleading results when demand is intermittent, prices change rapidly, capacity is constrained, lead times are correlated with demand, or shortage costs are nonlinear. They also assume that cost estimates are objective, although organisations may omit environmental impacts, worker safety, disposal costs or supplier risks.

Inventory decisions should comply with product safety, hazardous-material, customs, accounting and record-retention requirements. In regulated industries, unauthorised substitutions, expired materials and incomplete traceability can create legal and safety exposure. Buyers should avoid using inventory algorithms to impose unrealistic delivery requirements on vulnerable suppliers. Ethical procurement includes transparency, fair dealing and consideration of environmental waste.

Contemporary practice increasingly combines deterministic baselines with scenario analysis, simulation, postponement, supplier collaboration, digital twins and real-time visibility. The goal is not maximum automation but better decisions. Parameters should be version-controlled, assumptions documented and exceptions reviewed by accountable managers.

Best-practice checklist

• Use total relevant cost rather than unit price alone.
• Separate purchase, ordering, holding, shortage and obsolescence costs.
• Match time units in every formula.
• Recalculate parameters when demand, price or lead time changes materially.
• Use EOQ for stable replenishment, EPQ for gradual internal production and discount analysis for price breaks.
• Add safety stock only for a defined uncertainty or service requirement.
• Validate model quantities against pack sizes, capacity, shelf life and cash constraints.
• Integrate physical controls such as cycle counting, lot traceability and first-expiry-first-out.
• Monitor service and total cost together.
• Document assumptions and maintain human accountability for automated recommendations.

Chapter Summary

Inventory exists to support service and operational continuity, but it creates ordering, purchasing, holding, shortage and obsolescence costs. The basic EOQ model balances annual ordering cost against annual holding cost and gives Q* = √(2DS ÷ H) under constant-demand, instantaneous-replenishment assumptions. The EPQ model adjusts for gradual production using the factor 1 − d ÷ p. Quantity-discount decisions require comparison of total annual cost at every feasible price bracket, not selection of the lowest unit price alone.

EOQ determines how much to order; reorder-point analysis determines when to order. Under deterministic conditions, reorder point equals demand during lead time, while uncertain systems may add safety stock. Continuous review reacts whenever inventory position reaches the reorder point. Periodic review checks at fixed intervals and protects demand during both the review interval and lead time.

These models are useful baselines, but their limitations matter. Accurate implementation requires reliable data, consistent units, service-level decisions, capacity and shelf-life checks, governance, monitoring and professional judgement. Effective inventory management balances cost, availability, risk, compliance and sustainability.

Key Terms

Inventory: Stock held to support production, maintenance or customer service.

Cycle stock: Inventory created by ordering or producing in batches.

Safety stock: Additional inventory held to protect against uncertainty.

Pipeline inventory: Stock currently moving through the supply chain.

Ordering cost: Cost incurred each time a replenishment order is placed.

Setup cost: Cost of preparing a production process for a batch.

Holding cost: Annual cost of owning, storing and managing one unit.

Shortage cost: Cost arising when required inventory is unavailable.

Obsolescence: Loss of value because an item is no longer usable or saleable.

Deterministic model: Model based on known and stable parameter values.

Probabilistic model: Model that explicitly represents uncertainty.

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

EPQ: Production batch quantity that minimises setup and holding cost when replenishment is gradual.

Quantity discount: Lower unit price offered at specified order quantities.

Reorder point: Inventory-position level that triggers replenishment.

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

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

Periodic review: Control system in which inventory is reviewed at fixed time intervals.

Service level: Defined probability or proportion of demand met without shortage, depending on the selected measure.

First-expiry-first-out: Issue principle that uses the stock with the earliest expiry date first.

Review Questions

Short Answer Questions

1. Distinguish ordering cost from holding cost.
2. Why does the basic EOQ model exclude constant purchase cost from the quantity calculation?
3. What is the difference between cycle stock and safety stock?
4. Define inventory position.
5. State two assumptions of the EPQ model.

Long Answer Questions

1. Explain the major categories of inventory cost and discuss why shortage and obsolescence costs are difficult to estimate.
2. Derive the EOQ formula and explain the economic meaning of the cost balance at EOQ.
3. Compare continuous-review and periodic-review inventory systems.
4. Explain how quantity discounts should be evaluated using total annual cost.
5. Discuss the limitations of deterministic inventory models in a volatile supply chain.

Analytical / Application Questions

1. Annual demand is 18,000 units, ordering cost is $75 per order and annual holding cost is $5 per unit. Calculate EOQ, annual ordering cost and annual holding cost at EOQ.
2. Daily demand is 80 units and lead time is 7 working days. Calculate the deterministic reorder point. If management requires 160 units of safety stock, calculate the revised reorder point and explain the purpose of the additional stock.
3. A supplier offers a lower price at a quantity that is above the calculated EOQ. Design a total-cost comparison that includes purchase, ordering, holding, storage capacity and obsolescence risk. Explain what additional information is needed before approving the discount.

Multiple Choice Questions with Answers

Question 1

Which cost generally increases as order quantity increases in the basic EOQ model?

A. Annual ordering cost

B. Annual holding cost

C. Annual demand

D. Supplier lead time

Correct Answer: B. Annual holding cost

Question 2

The basic EOQ formula is Q* = √(2DS ÷ H). What does S represent?

A. Annual demand

B. Unit purchase price

C. Ordering cost per order

D. Safety stock

Correct Answer: C. Ordering cost per order

Question 3

A company has annual demand of 8,000 units, ordering cost of $20 and annual holding cost of $5 per unit. What is the EOQ?

A. 80 units

B. 179 units

C. 400 units

D. 1,600 units

Correct Answer: B. 179 units

Question 4

Which condition is required for the standard EPQ model?

A. Production rate must be less than usage rate

B. Production rate must exceed usage rate

C. Demand must be zero during production

D. Replenishment must be instantaneous

Correct Answer: B. Production rate must exceed usage rate

Question 5

A deterministic reorder point with constant demand and lead time is calculated as:

A. Order quantity plus holding cost

B. Annual demand divided by unit price

C. Demand rate multiplied by lead time

D. Ordering cost divided by demand

Correct Answer: C. Demand rate multiplied by lead time

Question 6

Why can the lowest unit-price discount fail to produce the lowest total cost?

A. A lower price always increases demand

B. Larger quantities may increase holding and obsolescence costs

C. Ordering cost becomes zero

D. Lead time is always shorter

Correct Answer: B. Larger quantities may increase holding and obsolescence costs

Question 7

In a periodic-review system, the order-up-to level must normally protect demand during:

A. Only the supplier’s production setup

B. Only the time between reviews

C. The review interval plus lead time

D. Only the time after receipt

Correct Answer: C. The review interval plus lead time

Question 8

If annual demand is 10,000 units and order quantity is 500 units, how many orders are placed per year?

A. 5

B. 20

C. 500

D. 5,000

Correct Answer: B. 20

Question 9

Which practice is most directly appropriate for stock with expiry dates?

A. First-expiry-first-out issuing

B. Random issuing

C. Always issuing the newest stock first

D. Ignoring lot records

Correct Answer: A. First-expiry-first-out issuing

Question 10

At the EOQ in the basic model, which relationship holds?

A. Purchase cost equals shortage cost

B. Ordering cost equals holding cost

C. Demand equals lead time

D. Safety stock equals zero in every system

Correct Answer: B. Ordering cost equals holding cost