Deterministic Economic Order Models

Introduction and context

Inventory decisions balance two opposing risks. Excess inventory ties up capital, occupies space, becomes obsolete, and may require insurance and handling. Insufficient inventory causes stockouts, production interruption, expediting, lost sales, and dissatisfied customers. Deterministic economic order models provide a disciplined starting point for this trade-off when demand and replenishment conditions are sufficiently stable and known.

The best-known model is the economic order quantity (EOQ) model. It determines the order or production quantity that minimizes relevant annual inventory cost. The model does not claim that one quantity is universally optimal. Rather, it isolates the economic relationship among demand, ordering or setup cost, and inventory carrying cost. Its value lies both in its answer and in the managerial insight produced by examining its assumptions and sensitivity.

This chapter develops the basic EOQ model, derives the total relevant cost function, explains the reorder point, evaluates sensitivity, and applies the ideas to a practical case.

Learning objectives

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

1. Explain the logic and assumptions of deterministic inventory models.
2. Derive and calculate the basic EOQ.
3. Compute annual ordering, holding, and total relevant costs.
4. Determine reorder points for constant demand and known lead time.
5. Distinguish cycle stock from safety stock and recognize when each is appropriate.
6. Conduct sensitivity analysis and interpret its managerial meaning.
7. Critically evaluate the assumptions, implementation issues, and limitations of EOQ decisions.

1. The inventory decision in a deterministic setting

A deterministic model assumes that relevant quantities are known with certainty over the planning horizon. Demand occurs at a constant rate, replenishment lead time is known, and the order arrives as a complete shipment. The basic model is therefore appropriate for illustrating cycle-stock economics, although real organizations commonly require extensions for uncertainty, quantity discounts, finite production, or multiple items.

An inventory cycle begins when an order arrives and ends when the stock reaches zero immediately before the next order arrives. If annual demand is D units and each order contains Q units, the number of orders per year is:

where N is orders per year. If replenishment is instantaneous and stockouts are not permitted, inventory decreases linearly from Q to zero. Average cycle inventory is therefore:

This is average on-hand cycle stock, not necessarily average total inventory. If safety stock, pipeline inventory, or anticipation inventory exists, those components must be considered separately.

A useful distinction is:

• Cycle stock: inventory created by ordering in batches rather than continuously.
• Safety stock: inventory held to protect against uncertainty in demand or lead time.
• Pipeline inventory: material currently in transit or otherwise ordered but not yet available.
• Anticipation inventory: inventory accumulated for predictable peaks, shutdowns, or seasonal demand.

The basic EOQ model addresses cycle stock only.

2. Cost structure of the basic EOQ model

Let:

Three cost components are relevant in the standard purchasing model.

Ordering cost. Each order costs S, and there are D ÷ Q orders per year. Thus:

As Q increases, fewer orders are placed and annual ordering cost decreases.

Holding cost. Average cycle inventory is Q ÷ 2. Thus:

As Q increases, average inventory and annual holding cost increase.

Purchase cost. If the unit price is constant and every unit demanded is purchased, annual purchase cost is:

For a fixed unit price, this term does not change when Q changes. It is part of total expenditure but is irrelevant to selecting Q. The relevant cost for the EOQ decision is therefore:

where TRC means total relevant annual cost. If purchase cost is included, total annual cost is:

The distinction prevents a common error: omitting purchase cost from a financial budget is incorrect, but including a constant purchase cost in the optimization does not alter the optimal order quantity.

3. Derivation of the economic order quantity

The EOQ minimizes TRC(Q), subject to Q > 0. Differentiating with respect to Q gives:

At the minimum, the first derivative equals zero:

Rearranging:

Therefore:

Q* is the economic order quantity. The second derivative is:

Because D, S, and Q are positive, the second derivative is positive. Hence the stationary point is a minimum.

At Q*, annual ordering cost equals annual holding cost:

This equality is a consequence of the optimum, not a general rule for every order quantity. It is also a useful computational check.

Unit consistency

The numerator 2DS has units:

(units/year) × (currency/order)

Since an order is a counting label, this is effectively currency × units/year. Dividing by H, measured as currency/unit/year, gives units². Taking the square root gives units, as required for Q*.

Economic interpretation

The EOQ is the point at which the marginal annual saving from placing fewer orders equals the marginal annual cost of carrying additional inventory. It does not minimize ordering cost or holding cost separately. It minimizes their sum.

4. Worked example: basic EOQ and cost evaluation

A manufacturer uses 24,000 standard fasteners per year. The administrative and receiving cost is £75 per order. Annual holding cost is £3 per fastener. Demand is constant, replenishment is instantaneous, and shortages are not permitted.

Here, D = 24,000 units/year, S = £75/order, and H = £3/unit/year.

A practical order quantity may be 1,095 or 1,100 units, depending on packaging and supplier constraints. Using Q = 1,095.45 for the theoretical calculation:

Average cycle inventory is:

Annual ordering cost is approximately:

Annual holding cost is:

Thus:

If the unit purchase price were £8, annual purchase cost would be £192,000, and total annual cost would be approximately £195,286.34. The purchase cost is financially important but does not affect Q* while £8 remains constant for all order quantities.

The time between replenishment orders depends on the time unit used for demand. With 365 operating days per year:

This is the order cycle interval, not the lead time. Confusing these two concepts can cause serious timing errors.

5. Reorder point and order timing

EOQ determines how much to order. The reorder point determines when to order. If demand is constant at d units per day and lead time is L days, demand during lead time is dL. The deterministic reorder point is therefore:

If annual demand D is spread uniformly over Y operating days per year:

and therefore:

The units must match: if d is units/day, L must be in days. If the supplier lead time is 8 days and daily demand is 100 units, R = 800 units. An order is released when inventory position reaches 800 units, so that the shipment arrives just as available stock is exhausted.

Inventory position

Reorder decisions should normally use inventory position rather than physical on-hand inventory:

If a firm orders when on-hand stock alone reaches R while substantial material is already in transit, it may over-order. Conversely, ignoring backorders can understate the replenishment requirement.

Safety stock extension

When demand or lead time is uncertain, a deterministic reorder point is not sufficient. A common policy is:

If demand during lead time has standard deviation σ_LT and a target cycle-service factor z is selected, a common approximation is:

Thus:

where μ_LT is expected lead-time demand. This is a stochastic extension, not part of the basic deterministic EOQ. The validity of the formula depends on how demand and lead-time variability are estimated and on the chosen service definition.

6. Sensitivity analysis

EOQ decisions are often more robust than the exact formula suggests. Since:

Q* has elasticity +½ with respect to D and S and −½ with respect to H. In practical terms:

• a 10% increase in demand increases EOQ by approximately 4.88%;
• a 10% increase in ordering cost increases EOQ by approximately 4.88%;
• a 10% increase in holding cost decreases EOQ by approximately 4.65%.

For small percentage changes, the approximation is often stated as half the percentage change. The exact factor for a proportional change k is √k.

The cost curve is relatively flat around its minimum. This means a feasible quantity near EOQ may have almost the same total relevant cost. For example, in the fastener case, choosing Q = 1,200 rather than 1,095.45 gives:

The difference from the theoretical minimum is only about £13.66 per year. A quantity of 1,200 may be preferable if it matches supplier cartons, reduces counting effort, or supports a delivery schedule. Exact mathematical optimality should not be confused with practical superiority.

A useful relative-cost expression is obtained by setting x = Q ÷ Q*. Then:

This expression shows that both over-ordering and under-ordering produce symmetrical cost effects around the optimum in the basic model, although operational consequences may not be symmetrical. A quantity twice EOQ has a relative relevant cost of 1.25; a quantity half EOQ also has a relative relevant cost of 1.25.

7. Model assumptions and critical evaluation

The basic EOQ model rests on assumptions that must be tested rather than accepted automatically.

Known, constant demand. Actual demand may be seasonal, intermittent, promotional, or dependent on a production schedule. Annual average demand can conceal short-term peaks and create stockouts.

Known and constant lead time. Supplier delays, transport disruption, customs, quality holds, and capacity constraints can make a deterministic reorder point unsafe.

Instantaneous replenishment. A complete shipment is assumed to arrive at once. This is plausible for purchasing but not for in-house production, where inventory accumulates during a production run.

No shortages. The standard model assumes service protection has priority over deliberate backorders. If shortage costs are known and backorders are acceptable, a planned-shortage model may be more appropriate.

Constant unit price. Quantity discounts, freight breaks, tariffs, and inflation violate this assumption. In such cases, purchase cost cannot be treated as constant.

Constant ordering and holding costs. Ordering cost may depend on supplier, order complexity, or electronic integration. Holding cost may include nonlinear space costs, expiry, obsolescence, insurance, shrinkage, and damage.

Single-item analysis. Shared warehouse capacity, supplier minimums, transport consolidation, and budget constraints create interactions among items. Independent EOQs may be infeasible collectively.

Unlimited capital and storage. Real firms face working-capital limits, shelf-life restrictions, hazardous-material rules, and finite warehouse capacity.

Order quantity divisibility. The formula returns a continuous number, while real items may require integer units, pallets, batches, or minimum order quantities.

The appropriate managerial response is not to reject EOQ, but to use it as a benchmark and then incorporate the constraints that materially affect the decision.

8. Extensions of deterministic economic order models

Economic production quantity

If replenishment occurs gradually at production rate p while demand occurs at rate d, with p > d, maximum inventory is less than Q because units are consumed during production. The economic production quantity is:

Here S is setup cost and H is annual holding cost. The factor (1 − d ÷ p) adjusts for simultaneous production and consumption. As p approaches d, replenishment becomes slow and the economically appropriate run size increases.

Quantity discounts

With all-units quantity discounts, unit price changes at specified quantity thresholds. The decision procedure is to calculate the EOQ for each price interval, check whether it lies within that interval, evaluate feasible break quantities, and compare total annual cost:

Holding cost may itself depend on price, for example when H is a percentage of unit value. The cheapest unit price is not automatically the cheapest total-cost option.

Planned shortages

Where backorders are possible, the model introduces shortage or backorder cost and determines both order quantity and maximum shortage. This should be used only when the organization can measure the consequences of delay and has a deliberate service policy. Treating unplanned stockouts as if they were costless is poor managerial practice.

Multi-item and constrained decisions

When several items compete for capital or storage, EOQs should be evaluated within a portfolio. A local item optimum may violate a system constraint. Linear or nonlinear optimization, constrained simulation, or policy-based prioritization may be required.

9. Applied case: Meridian Components Ltd.

Background

Meridian Components Ltd. assembles industrial pumps and purchases a bearing used in every finished pump. Annual usage is 18,000 bearings. The purchasing department estimates an administrative cost of $60 per order, while finance estimates annual holding cost at $2.40 per bearing. The supplier quotes a 6-day lead time, and the plant operates 300 days per year. Current practice is to place a standard order of 1,500 bearings whenever the storekeeper believes stock is low.

Problem

Management wants to know whether the current order quantity is economically defensible and when replenishment should be released. It also wants to avoid treating a theoretical result as a rigid instruction because bearings are packed in cartons of 50.

Analysis

The EOQ is:

The nearest carton multiple is 950 bearings. Annual ordering cost at Q = 950 is:

Annual holding cost is:

Thus relevant annual cost is approximately $2,276.84. At the existing quantity of 1,500:

The theoretical improvement is about $243 per year before considering implementation costs. Daily demand is:

With a six-day lead time and no safety stock in the deterministic calculation:

The replenishment order should be released when inventory position reaches 360 bearings, not merely when an arbitrary visual level is observed.

Approach and solution

Meridian adopts a base policy of 950 bearings and a reorder point of 360 bearings for the initial trial. Before final approval, the team checks supplier delivery records, demand variability, carton integrity, storage space, and whether 6 days is a genuine guaranteed lead time. If historical variability is material, safety stock is added using a service-level policy rather than hidden informally in the order quantity.

Outcome

The policy reduces average cycle inventory from 750 to 475 bearings and aligns ordering with carton size. The financial saving is modest because the EOQ cost curve is flat, but the policy improves transparency: purchasing cost, holding cost, service protection, and supplier performance can now be monitored separately.

Learning points

1. EOQ addresses order size, while reorder point addresses order timing.
2. Operationally feasible quantities should be compared with the theoretical optimum.
3. A deterministic reorder point is appropriate only when lead-time demand is reliably predictable.
4. A small annual cost difference may justify selecting a quantity that is easier to handle.
5. Inventory position is the proper control variable when orders are in transit.

10. Implementation and governance

A sound EOQ implementation requires reliable master data. Demand should be defined consistently, ordering cost should include relevant administrative and receiving activities, and holding cost should reflect capital, space, insurance, shrinkage, damage, and obsolescence where material. Data owners should document the period, currency, operating days, and assumptions used.

Inventory policies should be reviewed when demand, price, supplier performance, product life, or warehouse capacity changes. An enterprise resource planning system can automate order proposals, inventory-position calculations, and exception alerts, but automation does not validate the assumptions. AI and advanced analytics may improve demand forecasting, lead-time prediction, and parameter estimation; they do not eliminate the need for explainable policies, human oversight, and controls against biased or stale data.

Governance should also prevent dysfunctional incentives. A buyer rewarded solely for purchase-price reduction may order excessive quantities, while a warehouse manager rewarded solely for low stock may create production interruptions. Performance measures should balance total cost, service, inventory turns, obsolescence, supplier reliability, and working capital.

Chapter Summary

The basic EOQ model determines the order quantity that balances annual ordering cost and annual holding cost. Under constant demand, instantaneous replenishment, no shortages, constant unit price, and known parameters, the economic order quantity is Q* = √(2DS ÷ H). The total relevant cost is the sum of annual ordering and holding costs; purchase cost is added for total expenditure but does not affect the optimum when unit price is constant.

EOQ answers how much to order, whereas the deterministic reorder point answers when to order: R = dL. Inventory position, rather than on-hand stock alone, should guide replenishment. Sensitivity analysis shows that EOQ changes less than proportionally when demand, ordering cost, or holding cost changes, and that feasible quantities near EOQ often have nearly equal cost.

The model is a benchmark, not a universal prescription. Uncertain demand, variable lead time, quantity discounts, finite production rates, shortages, capacity constraints, and multi-item interactions require extensions or alternative methods. Effective managerial practice combines mathematical discipline with operational feasibility, reliable data, service objectives, and periodic review.

Key Terms

EOQ: Economic order quantity that minimizes relevant ordering and holding cost under stated assumptions.

Deterministic model: A model in which relevant demand, lead time, and parameters are treated as known.

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

Setup cost: Cost of preparing equipment or processes for a production run.

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

Cycle stock: Inventory resulting from replenishing in batches.

Safety stock: Inventory held to protect against uncertainty.

Pipeline inventory: Inventory ordered or in transit but not yet available for use.

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

Lead time: Time between order release and receipt of usable inventory.

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

Total relevant cost: Ordering cost plus holding cost when unit purchase price is constant.

Purchase cost: Cost of acquiring the demanded units.

Economic production quantity: Order or run quantity for gradual internal replenishment.

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

Service level: A measure of the degree to which demand is supplied without shortage.

Demand rate: Units required per unit of time.

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

Cycle-service factor: A statistical factor used to relate uncertainty to safety stock.

Master data: Controlled data describing items, costs, demand, suppliers, and lead times.

Review Questions

Short Answer Questions

1. Why does the basic EOQ model exclude constant purchase cost from the optimization?
2. Define cycle stock and distinguish it from safety stock.
3. State the assumptions required for the deterministic reorder-point formula.
4. Why are ordering cost and holding cost equal at the EOQ?
5. What is the difference between inventory position and on-hand inventory?

Long Answer Questions

1. Derive the basic EOQ formula from the total relevant cost function and verify that the stationary point is a minimum.
2. Explain the assumptions of the basic EOQ model and evaluate their practical significance in a manufacturing organization.
3. Discuss how quantity discounts change the EOQ decision process.
4. Compare the basic EOQ model with the economic production quantity model.
5. Explain why sensitivity analysis is important when implementing inventory policies.

Analytical / Application Questions

1. Annual demand is 30,000 units, ordering cost is $90 per order, and annual holding cost is $5 per unit. Calculate EOQ, annual orders, average cycle inventory, and total relevant annual cost.
2. A facility operates 250 days per year, has annual demand of 50,000 units, and lead time of 4 operating days. Calculate the deterministic reorder point and explain how the answer would change if lead time were uncertain.
3. A supplier offers a lower price at a quantity above the EOQ. Design a procedure for deciding whether the discount is economically justified, identifying the cost terms and feasibility checks that must be included.

Multiple Choice Questions with Answers

Question 1

Which expression represents the basic EOQ under constant demand and constant unit price?

A. Q* = √(2DS ÷ H)

B. Q* = √(DH ÷ 2S)

C. Q* = 2DSH

D. Q* = D ÷ (SH)

Correct Answer: A. Q* = √(2DS ÷ H)

Question 2

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

A. Purchase cost equals holding cost

B. Ordering cost equals holding cost

C. Ordering cost equals purchase cost

D. Holding cost is zero

Correct Answer: B. Ordering cost equals holding cost

Question 3

Annual demand is 12,000 units and order quantity is 600 units. How many orders are placed per year?

A. 12

B. 20

C. 600

D. 7,200

Correct Answer: B. 20

Question 4

If daily demand is 80 units and deterministic lead time is 5 days, what is the reorder point without safety stock?

A. 16 units

B. 85 units

C. 400 units

D. 4,000 units

Correct Answer: C. 400 units

Question 5

Which cost is constant with respect to order quantity when unit price does not vary?

A. Annual ordering cost

B. Annual holding cost

C. Annual purchase cost

D. Shortage cost

Correct Answer: C. Annual purchase cost

Question 6

What is the correct inventory-position formula?

A. On-hand inventory − on-order inventory

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

C. On-order inventory − demand rate

D. On-hand inventory + annual demand

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

Question 7

If annual demand increases by 21% and all other parameters remain unchanged, EOQ changes by approximately:

A. 10% increase

B. 21% increase

C. 42% increase

D. 10% decrease

Correct Answer: A. 10% increase

Question 8

Which condition is required for the economic production quantity formula?

A. Production rate is less than demand rate

B. Production rate equals zero

C. Production rate exceeds demand rate

D. Demand must be unpredictable

Correct Answer: C. Production rate exceeds demand rate

Question 9

In the basic EOQ model, average cycle inventory is:

A. Q

B. Q ÷ 4

C. 2Q

D. Q ÷ 2

Correct Answer: D. Q ÷ 2

Question 10

Which situation most directly violates the constant-unit-price assumption?

A. Orders arrive instantaneously

B. The supplier offers all-units quantity discounts

C. Demand is measured in units per year

D. Shortages are prohibited

Correct Answer: B. The supplier offers all-units quantity discounts