Deterministic Inventory Models

Introduction and Context

Inventory decisions balance two opposing costs. Holding more stock protects customer service and production continuity, but ties up capital and creates storage, insurance, obsolescence, shrinkage and handling costs. Holding too little reduces investment but increases ordering, setup, stockout and expediting costs. Deterministic inventory models provide a disciplined starting point for this trade-off when demand, replenishment time and relevant costs are known with reasonable certainty.

These models are not forecasts of reality. They are analytical representations of reality. Their value lies in identifying economic relationships, establishing a benchmark policy and clarifying which assumptions must be relaxed when operating conditions change. In professional practice, deterministic results are commonly combined with safety stock, service-level analysis, supplier constraints, enterprise-resource-planning data and managerial judgement.

Learning Objectives

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

• explain the logic and assumptions of deterministic inventory models;
• derive and apply the basic economic order quantity (EOQ) model;
• calculate reorder points under known demand and lead time;
• evaluate quantity discounts and incremental holding costs;
• analyse planned shortages and backorders;
• conduct sensitivity analysis and interpret model limitations; and
• translate model outputs into responsible purchasing and supply-chain decisions.

1. The Deterministic Inventory Framework

A deterministic model assumes that relevant quantities are known, or treated as known, rather than random. Typical assumptions include a constant demand rate, known replenishment lead time, fixed ordering or setup cost, and a known unit purchase price. The inventory cycle is often represented as a saw-tooth pattern: inventory falls through demand and rises when an order arrives.

The principal decision variables are:

The total relevant annual cost for a basic replenishment model is usually:

The first term is annual purchase cost, the second is annual ordering cost, and the third is annual cycle-stock holding cost. If unit price is constant and all demand is satisfied, DC does not affect the optimal order quantity, although it remains relevant to the total cost and to supplier negotiations.

2. The Basic EOQ Model

2.1 Assumptions

The classical EOQ model assumes:

1. annual demand D is known, constant and continuous;
2. replenishment is instantaneous, or the model is interpreted using cycle stock only;
3. lead time is known and does not affect the optimal Q;
4. each order incurs a fixed cost S, independent of order size;
5. holding cost H is linear in average inventory;
6. shortages are not permitted;
7. quantity discounts are absent; and
8. the planning horizon is sufficiently long for steady-state averages to apply.

These assumptions make the model transparent. They also explain why EOQ should be treated as a benchmark rather than an automatic purchasing instruction.

2.2 Derivation

Under instantaneous replenishment and no shortages, inventory varies between Q and zero. Average cycle stock is Q/2. The annual number of orders is D/Q. Therefore:

Ignoring the constant purchase-cost term, differentiate with respect to Q:

At the minimum, the derivative is zero:

Solving gives the EOQ:

The second derivative, 2DS/Q³, is positive for positive Q, confirming a minimum. Unit consistency is important: DS has units of currency per year, while H has units of currency per unit per year; their ratio has units of units², so its square root has units of units.

At EOQ, annual ordering cost equals annual holding cost:

This equality is a property of the optimum, not a separate assumption.

2.3 Worked Example: Basic EOQ

A distributor sells 12,000 units annually. Each order costs £75 to process and transport. The annual holding cost is £3 per unit. The purchase price is £20 per unit.

The practical order quantity may be rounded to 775 units, subject to pack-size and supplier constraints. Annual orders are approximately:

The cycle is approximately 23.5 working days if 365 days are used as the annual basis:

Relevant annual cost at the continuous optimum is:

Annual purchase cost is £240,000, so total annual cost including purchasing is approximately £242,323.8. The large purchase-cost component does not change Q because price is constant across order quantities.

2.4 Extensions of EOQ

If replenishment occurs gradually while demand continues, the economic production quantity (EPQ) model is more appropriate. Let p be the production rate and d the demand rate, with p > d. Maximum inventory is less than Q because production and consumption overlap.

This formula applies to an internal or supplier production system with a finite production rate, no shortages and constant rates. As p becomes very large relative to d, the expression approaches the basic EOQ.

If an item is ordered in a fixed multiple, the mathematically optimal Q must be adjusted to an allowable quantity. The best feasible quantity is not necessarily the nearest multiple; the feasible alternatives should be compared using total relevant cost.

3. Reorder Points and Lead Time

EOQ determines how much to order; the reorder point determines when to order. With constant daily demand d and known lead time L in days:

This is the demand expected during replenishment. If the firm uses a review or transaction system in which an order is placed when inventory position reaches R, then inventory position should include on-hand stock plus on-order stock minus backorders.

For a continuous-review system, a practical rule is:

Worked Example: Reorder Point

Daily demand is 40 units and lead time is 6 working days. With no uncertainty or safety stock:

If lead time or demand is uncertain, deterministic R is insufficient. A common service-oriented extension is:

The safety-stock calculation is probabilistic and depends on the chosen service measure, demand and lead-time distributions, and review policy. It should not be inserted into a deterministic model without identifying the new assumptions.

4. Quantity-Discount Models

A supplier may reduce unit price when the buyer orders at least a specified quantity. In that case, purchase price depends on Q and cannot automatically be ignored. The buyer must compare feasible candidate quantities across price brackets.

For each price level j, let Cj be the unit price and Hj the annual holding cost per unit. If holding cost is a percentage i of item value, then:

A common procedure is:

1. calculate the EOQ using each bracket’s H;
2. check whether that EOQ lies within the bracket;
3. if it lies below a bracket minimum, use the minimum qualifying quantity;
4. calculate total annual cost for every feasible candidate; and
5. select the candidate with the lowest total cost.

The purchase-price saving must be large enough to offset additional holding and ordering effects. A lower unit price is therefore not automatically economical.

Worked Example: Quantity Discount

Annual demand is 10,000 units, ordering cost is £60, and holding cost is 20% of unit price. The supplier offers:

For £10.00, H = 0.20 × 10 = £2.00 and EOQ is approximately 775 units, which is not feasible in the first bracket; candidate Q = 499 is used. For £9.80, H = £1.96 and EOQ is approximately 783 units, feasible in the second bracket. For £9.50, H = £1.90 and EOQ is approximately 795 units, below the bracket minimum, so Q = 1,000 is tested.

The relevant costs are approximately:

The 1,000-unit order is economically preferred under the stated assumptions. The decision should nevertheless consider cash availability, storage capacity, product obsolescence and supplier reliability.

5. Planned Shortages and Backorders

Some environments permit customers to wait, or permit a production system to carry a backlog temporarily. A planned-shortage model trades shortage or backorder cost against holding cost. Let π be the annual shortage cost per backordered unit, including contractual penalties, lost contribution, expediting or customer dissatisfaction where these are measurable.

For a replenishment cycle with order quantity Q, maximum on-hand inventory Imax and maximum backlog B, the cycle contains positive inventory and shortage intervals. The standard continuous-review planned-backorder result is:

Here H and π must be expressed in the same annual per-unit basis. Since Imax* + B* = Q*, the order quantity is divided between the maximum positive inventory and maximum backlog. As π becomes very large, B* approaches zero and the model approaches the no-shortage EOQ. As shortage cost declines, planned backlog becomes more attractive.

The model is appropriate only when backorders are operationally and commercially acceptable. It is inappropriate for safety-critical medicines, essential spare parts, regulated products or customer segments with zero-tolerance service expectations unless the shortage-cost representation genuinely captures those consequences.

6. Sensitivity Analysis

Inventory decisions are often made with uncertain estimates of D, S and H. The EOQ is more robust than the individual cost components because it responds to the square root of their ratio:

Thus, a 10% increase in demand increases EOQ by approximately 4.88%, while a 10% increase in holding cost decreases EOQ by approximately 4.65%. Exact changes should be recalculated when precision matters.

The cost curve around EOQ is relatively flat. If Q = kQ*, then the relevant cost divided by the minimum relevant cost is:

For example, ordering 20% more than EOQ means k = 1.2 and the ratio is approximately 1.0167. The relevant cost is only about 1.67% above its minimum, although operational effects may still be significant. This supports managerial flexibility when truckload, pallet, supplier or production constraints make the exact EOQ impractical.

Sensitivity analysis should vary not only numerical parameters but also assumptions. Useful scenarios include:

• demand growth or decline;
• revised carrying-cost estimates;
• supplier lead-time changes;
• minimum order quantities and case packs;
• storage and cash constraints;
• changes in shortage penalties; and
• obsolescence or shelf-life risk.

A spreadsheet or inventory-planning system can calculate candidate policies, but the analyst must document inputs, units, rounding rules and exclusions. AI tools may assist with scenario generation, data validation and anomaly detection, but generated recommendations require human verification, especially where costs, service commitments or regulatory obligations are involved.

7. An Applied Case: Northstar Components

The following is an illustrative case.

Background

Northstar Components supplies a standard sensor to industrial-equipment manufacturers. Annual demand is 18,000 units, demand is reasonably stable, ordering cost is £90 per order, and the supplier’s normal price is £24 per unit. The finance department estimates annual holding cost at £4.80 per unit. Supplier lead time is 8 working days, and the plant operates 250 working days per year.

Problem

Purchasers currently order 1,500 units monthly. The operations manager believes the policy creates excess inventory, while sales argues that frequent orders could increase stockout risk. A supplier proposes £23.40 per unit for orders of at least 2,000 units. Northstar must assess quantity, timing and commercial implications.

Analysis

At the normal price:

The current 1,500-unit policy is not necessarily wrong, but its relevant annual cost is:

At Q = 821.6, relevant cost is approximately £3,942? No: ordering cost is (18,000/821.6)×90 ≈ £1,972, and holding cost is (821.6/2)×4.80 ≈ £1,972, giving approximately £3,944. This reveals a contradiction: the stated current policy cost of £2,880 is lower, so the data should be checked. Recalculation shows the current holding cost is £3,600 and ordering cost is £1,080, total £4,680—not £2,880. The EOQ policy therefore saves approximately £736 of relevant annual cost before practical constraints.

At the discounted price, if holding cost remains £4.80 per unit, the EOQ calculation is unchanged but Q must be at least 2,000. The cost at Q = 2,000 is:

At the normal price and Q ≈ 821.6, total cost is approximately:

The discount appears attractive, but the comparison assumes the quoted price is firm, demand will be consumed before obsolescence, and storage and working-capital costs are fully represented by £4.80.

Daily demand is 18,000/250 = 72 units. With an 8-day deterministic lead time:

This reorder point contains no protection against demand or lead-time uncertainty.

Approach and Outcome

Northstar should test the 2,000-unit discount through a total-cost and feasibility review. It should confirm warehouse capacity, cash-flow impact, shelf life, supplier cancellation terms and the cost of excess inventory. It should also validate the holding-cost estimate: if the £4.80 figure excludes obsolescence and capital cost, the discount may be overstated. Subject to these checks, the 2,000-unit policy has the lower calculated cost, while an inventory-position trigger of 576 units provides the deterministic timing rule.

Learning Points

The case demonstrates four professional practices: compare total cost rather than unit price alone; check arithmetic and data definitions; distinguish order quantity from reorder point; and treat model output as a controlled decision input rather than an unquestioned answer.

8. Managerial Implications, Governance and Limitations

Deterministic models support procurement, production scheduling, spare-parts planning and warehouse design. They make trade-offs visible and facilitate negotiations about minimum order quantities, delivery frequency and supplier pricing. However, several limitations are material:

• demand may be seasonal, intermittent or promotional rather than constant;
• lead times may be variable or correlated with disruptions;
• holding cost is difficult to measure and may include nonlinear obsolescence;
• ordering cost may vary by supplier, transport mode or order complexity;
• capacity, cash and shelf-life constraints can dominate cost minimisation;
• multiple items compete for warehouse and budget resources; and
• service failures may damage reputation in ways not captured by a simple shortage cost.

Governance requires clear ownership of assumptions, version-controlled models, documented data sources and periodic parameter review. Managers should avoid manipulating holding or shortage costs merely to justify a preferred quantity. Ethical inventory decisions also consider product availability, fair allocation during scarcity, supplier payment practices, waste and environmental effects. For regulated or safety-critical goods, compliance and service continuity take precedence over a narrow EOQ saving.

A sound implementation sequence is to classify items, validate demand and cost data, select an appropriate model, calculate a baseline, test sensitivity, impose operational constraints, obtain stakeholder approval, monitor actual performance and revise parameters. Relevant performance measures include inventory turnover, fill rate, stockout frequency, average inventory, order frequency, obsolete inventory and total cost to serve.

Chapter Summary

Deterministic inventory models assume known demand, lead time and cost parameters and provide useful benchmarks for inventory policy. The basic EOQ balances annual ordering cost and holding cost, yielding Q* = √(2DS/H). The reorder point under known demand and lead time is R = dL, while uncertainty requires an explicitly probabilistic safety-stock extension. Quantity-discount analysis compares feasible candidates using purchase, ordering and holding costs; the lowest unit price is not automatically the lowest total cost. Planned-shortage models permit controlled backorders when shortage costs are finite and operationally acceptable. Sensitivity analysis shows that EOQ changes less than proportionately with its inputs and that practical quantities can often be used with little cost penalty. Effective managers combine these calculations with capacity, service, cash, risk, ethical and regulatory considerations.

Key Terms

Deterministic model: An inventory model using known or fixed parameter values.

Economic order quantity (EOQ): The order quantity that minimises relevant ordering and holding costs under basic assumptions.

Ordering cost: A cost incurred each time an order is placed.

Holding cost: The annual cost of carrying one unit in inventory.

Cycle stock: Inventory held to meet demand between replenishments.

Reorder point: Inventory position at which a replenishment order is triggered.

Lead time: Time between placing an order and receiving it.

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

Economic production quantity: Order or production quantity for finite-rate replenishment.

Quantity discount: A lower unit price offered for meeting a quantity threshold.

Planned shortage: An intentionally permitted temporary stockout or backlog.

Backorder: Demand recorded for later fulfilment rather than immediately lost.

Shortage cost: Cost associated with an unfulfilled unit or delayed fulfilment.

Sensitivity analysis: Examination of how results change when inputs or assumptions vary.

Relevant cost: Cost that changes among the alternatives being compared.

Safety stock: Additional inventory held to protect against uncertainty.

Service level: A specified measure of the probability or frequency of meeting demand.

Inventory turnover: A measure relating annual usage or cost of goods sold to inventory investment.

Inventory governance: Controls for assumptions, data, approvals, monitoring and model use.

Review Questions

Short Answer Questions

1. What is the principal trade-off represented by the EOQ model?
2. Why does constant unit purchase price not affect basic EOQ?
3. Distinguish between order quantity and reorder point.
4. State two assumptions of the basic deterministic inventory model.
5. Why must feasible candidates be compared in a quantity-discount problem?

Long Answer Questions

1. Derive the basic EOQ formula and explain the meaning of the cost-balance condition.
2. Explain how finite production rate changes the basic EOQ model.
3. Describe the assumptions, formula and managerial interpretation of a planned-shortage model.
4. Critically evaluate the use of deterministic models in a volatile supply environment.
5. Explain how governance and sensitivity analysis improve inventory-model decisions.

Analytical / Application Questions

1. Annual demand is 20,000 units, ordering cost is £50, and annual holding cost is £2. Calculate EOQ, annual orders and relevant annual cost.
2. Daily demand is 65 units and deterministic lead time is 7 working days. Calculate the reorder point and explain what additional information is required to set safety stock.
3. A supplier offers a discount above a minimum quantity. Design a total-cost comparison procedure and identify at least four non-price constraints that should be checked before implementation.

Multiple Choice Questions with Answers

Question 1

In the basic EOQ model, which two annual costs are balanced at the optimum?

A. Purchase cost and shortage cost

B. Ordering cost and holding cost

C. Transport cost and labour cost

D. Obsolescence cost and selling cost

Correct Answer: B. Ordering cost and holding cost

Question 2

Annual demand is 8,000 units, ordering cost is £50 per order, and annual holding cost is £2 per unit. What is the approximate EOQ?

A. 200 units

B. 447 units

C. 632 units

D. 1,000 units

Correct Answer: C. 632 units

Question 3

If daily demand is 30 units and deterministic lead time is 5 days, with no safety stock, the reorder point is:

A. 6 units

B. 35 units

C. 150 units

D. 600 units

Correct Answer: C. 150 units

Question 4

Why must quantity-discount candidates be evaluated using total cost?

A. Because ordering cost is always zero at a discount

B. Because a lower unit price can be offset by higher holding cost

C. Because EOQ never applies when prices vary

D. Because purchase price is excluded from inventory cost

Correct Answer: B. Because a lower unit price can be offset by higher holding cost

Question 5

Which condition is required for the standard planned-shortage model?

A. Backorders are operationally and commercially acceptable

B. Demand must be completely random

C. Lead time must be infinite

D. Holding cost must be zero

Correct Answer: A. Backorders are operationally and commercially acceptable

Question 6

If demand increases by 21% and all other EOQ inputs remain constant, EOQ changes by approximately:

A. 10% increase

B. 21% increase

C. 42% increase

D. 21% decrease

Correct Answer: A. 10% increase

Question 7

Which item is included in inventory position?

A. Only damaged stock

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

C. Annual demand minus holding cost

D. Only inventory physically in the warehouse

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

Question 8

In the EPQ model, production and demand overlap during replenishment. Compared with instantaneous replenishment, this generally means:

A. Maximum inventory is lower for the same production quantity

B. Demand becomes zero

C. Ordering cost becomes negative

D. Lead time is irrelevant to all decisions

Correct Answer: A. Maximum inventory is lower for the same production quantity

Question 9

A deterministic reorder point without safety stock protects against:

A. Random demand variation

B. Random lead-time variation

C. Expected demand during known lead time

D. Supplier bankruptcy

Correct Answer: C. Expected demand during known lead time

Question 10

The principal purpose of sensitivity analysis is to:

A. Guarantee that the forecast is correct

B. Examine how results change when inputs or assumptions vary

C. Remove all operational constraints

D. Replace managerial judgement completely

Correct Answer: B. Examine how results change when inputs or assumptions vary