Deterministic Inventory Models

Introduction

Deterministic inventory models address replenishment decisions when demand, lead time, ordering cost, and relevant operating conditions are known with reasonable certainty. They form the analytical foundation of inventory management because they clarify the trade-off between the cost of ordering or producing frequently and the cost of holding inventory. Although real supply chains are exposed to uncertainty, deterministic models remain valuable for stable items, initial system design, benchmarking, and identifying the economic logic behind more advanced stochastic models.

The central decision is often the order quantity: how much should be purchased or produced each time inventory is replenished? Related decisions include whether quantity discounts should be accepted, whether shortages should be deliberately permitted, and whether several items should be ordered jointly. This chapter develops these models and evaluates their assumptions and managerial limitations.

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 apply the basic economic order quantity (EOQ) model.
3. Apply the economic production quantity (EPQ) model when replenishment is gradual.
4. Evaluate all-units quantity discounts.
5. Analyse planned-shortage models and interpret their managerial implications.
6. Explain joint-ordering decisions for multiple items.
7. Critically assess model assumptions and translate results into practical policy.

1. The Economic Logic of Inventory

Inventory creates two opposing effects. Larger replenishment quantities reduce the number of orders or production runs and therefore reduce ordering or setup cost. However, larger quantities increase average inventory and consequently holding cost. A deterministic inventory model seeks the quantity at which the relevant total cost is minimized.

The principal terminology is as follows:

• Demand rate (D): annual or period demand, expressed in units per year or another consistent period.
• Order quantity (Q): units purchased or replenished at one time.
• Ordering cost (S): fixed cost incurred per order, excluding the purchase price unless explicitly stated.
• Holding cost (H): cost of holding one unit for one year or other period. It may include capital, storage, insurance, deterioration, and obsolescence.
• Lead time: time between placing an order and receiving it.
• Reorder point: inventory position at which replenishment is initiated.
• Cycle stock: inventory arising from ordering in batches rather than continuously.
• Shortage or backorder: demand not immediately satisfied but supplied later.

For models in this chapter, the time unit used for D, H, and lead time must be consistent. For example, if D is annual demand, H must be annual holding cost and lead time must be expressed in years when calculating demand during lead time.

2. The Basic EOQ Model

2.1 Assumptions

The classical EOQ model assumes:

1. Demand is known, constant, and continuous.
2. Replenishment is instantaneous.
3. Lead time is known and constant.
4. No shortages are allowed.
5. The purchase price per unit is constant and independent of quantity.
6. Ordering cost and holding cost are constant.
7. Orders are placed independently for one item.
8. The planning horizon is sufficiently long for steady-state analysis.

Under these assumptions, inventory follows a saw-tooth pattern: it falls steadily from Q to zero and immediately rises to Q when an order arrives. Average cycle stock is Q ÷ 2.

2.2 Cost Function and Derivation

Annual ordering cost is the number of orders multiplied by cost per order:

Annual holding cost is average inventory multiplied by holding cost per unit:

If unit purchase price is C, annual purchasing cost is DC. Therefore, total annual relevant cost is:

The purchase cost is constant when C does not depend on Q, so it does not affect the minimizing order quantity. Differentiating the relevant portion with respect to Q gives:

Setting the derivative equal to zero produces the EOQ:

The second derivative is:

Thus, the stationary point is a minimum. At Q*, annual ordering cost equals annual holding cost:

This equality is a consequence of cost minimization, not an independent assumption.

2.3 Related Operating Measures

Once Q* is calculated:

Cycle time has the same time unit as the inverse of D. If D is annual demand, cycle time is in years; multiply by 365 or 12 to express it in days or months.

With instantaneous replenishment and no safety stock, the reorder point is demand during lead time:

where d is demand per unit of time and L is lead time in the same time unit. If demand and lead time are deterministic, no safety stock is required by the model.

2.4 Worked Example: Basic EOQ

A distributor uses 12,000 units annually. Each order costs £75, and annual holding cost is £3 per unit. The unit purchase price is £20 and does not vary with quantity.

The practical order quantity is approximately 775 units. The number of orders is:

Relevant annual ordering and holding cost at the continuous optimum are each approximately £1,161.90, giving total relevant cost of approximately £2,323.80. Annual purchasing cost is £240,000 and is constant across quantities under the stated assumptions.

If lead time is 8 working days and demand is 48 units per working day:

The firm should initiate replenishment when its inventory position reaches 384 units, assuming no safety stock and reliable lead time.

3. Economic Production Quantity

3.1 Why EOQ Is Not Always Appropriate

EOQ assumes that the entire order arrives immediately. This is inappropriate when an organization manufactures the item internally or when a supplier delivers continuously over a production run. During production, inventory increases because production occurs at rate p while demand occurs at rate d. The production rate must exceed the demand rate: p > d.

3.2 Model and Derivation

Let Q be the production lot size. During production, net inventory increases at rate p − d. The maximum inventory is therefore:

Average inventory is half the maximum inventory:

The annual relevant cost is:

Minimizing this expression gives the economic production quantity:

Equivalently, if annual demand is D and annual production capacity is P, use P in place of p:

The formula applies only when P > D. As production capacity approaches demand, the factor 1 − D ÷ P approaches zero and the calculated lot size becomes very large. This reflects the fact that slow net accumulation requires long production runs to justify setup cost.

3.3 Worked Example: EPQ

A manufacturer has annual demand of 10,000 units, setup cost of $400 per production run, annual holding cost of $8 per unit, and annual production capacity of 50,000 units.

Since 1 − 10,000 ÷ 50,000 = 0.8:

Maximum inventory is:

Average inventory is approximately 447 units. The EPQ is larger than the corresponding EOQ, which would be √(2 × 10,000 × 400 ÷ 8) = 1,000 units, because production replenishment is gradual and therefore produces less average inventory for the same lot size.

4. Quantity-Discount Models

4.1 Rationale

Suppliers may offer an all-units quantity discount: once an order reaches a threshold, every unit in that order is charged at the lower price. The buyer must compare purchase savings with additional holding and ordering costs. The ordinary EOQ cannot be applied mechanically because purchase price now depends on Q.

For price level i, total annual cost is:

where Ci is unit price and Hi is holding cost per unit per year at that price. If holding cost is a percentage I of unit price, then Hi = ICi.

4.2 Solution Procedure

For each price bracket:

1. Calculate the unconstrained EOQ using that bracket’s holding cost.
2. Check whether the EOQ lies within the quantity range for that price.
3. If it is below the bracket minimum, use the minimum quantity that qualifies for the price.
4. Calculate total cost at every feasible candidate quantity, including the EOQ for feasible brackets and price-break quantities.
5. Select the candidate with the lowest total cost.

The price-break quantity is important because the lowest-cost solution may occur at a bracket boundary rather than at an unconstrained EOQ.

4.3 Worked Example: All-Units Discount

Annual demand is 5,000 units, ordering cost is $60, and holding cost is 20% of unit price. Supplier prices are:

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

This is feasible. For the second bracket, H = $1.94:

It is below the bracket minimum, so the candidate is 1,000 units. For the third bracket, H = $1.88:

It is below 2,000, so the candidate is 2,000 units.

Approximate total costs are:

• Q = 548 at $10.00: purchasing $50,000; ordering about $547.45; holding about $547.45; total ≈ $51,094.90.
• Q = 1,000 at $9.70: purchasing $48,500; ordering $300; holding $970; total = $49,770.
• Q = 2,000 at $9.40: purchasing $47,000; ordering $150; holding $1,880; total = $49,030.

The 2,000-unit order is preferred despite its higher holding cost because the purchase-price saving dominates. In practice, the decision should also consider storage capacity, cash-flow constraints, product obsolescence, supplier reliability, and whether the discount is genuinely all-units rather than incremental.

5. Planned Shortages and Backorders

5.1 Concept

The basic EOQ prohibits shortages. In some environments, however, customers accept delayed delivery and the organization deliberately allows a controlled backlog. A planned-shortage model trades shortage cost against holding cost and ordering cost. It is appropriate only when shortages can be measured, communicated, and managed without disproportionate loss of goodwill or operational disruption.

Let B be the maximum planned backorder. During a cycle, inventory rises to Q − B, falls to zero, and then becomes a backlog of B before replenishment arrives. Average positive inventory is (Q − B)² ÷ (2Q), while average backorders are B² ÷ (2Q).

If p is the cost per unit of shortage per year, total relevant annual cost is:

Here p must be a shortage penalty measured in currency per unit per unit time, not merely a one-time lost-margin amount. The model assumes all backorders are eventually filled and that shortage cost is adequately represented by p.

5.2 Optimal Policy

Minimizing the cost function with respect to Q and B gives:

The maximum positive inventory is Q − B. The fraction of the cycle with positive inventory and the fraction with backorders depend on the relative size of H and p. As p becomes very large, B* approaches zero and the model approaches the no-shortage EOQ. As p becomes small, planned backorders become more attractive, although the model’s realism may deteriorate.

5.3 Managerial Interpretation

A planned shortage is not costless merely because it is eventually filled. It may cause lost sales, production interruption, expediting, contractual penalties, safety concerns, and reputational damage. Therefore, p should incorporate measurable financial and service consequences. In customer-facing operations, p is often difficult to estimate and may be nonlinear: the first delayed order may be tolerated, while repeated delays may cause permanent defection.

6. Joint Ordering for Multiple Items

6.1 Why Joint Ordering Matters

Ordering several items from the same supplier can reduce administrative, transportation, and receiving costs. Independent EOQ calculations may therefore be suboptimal because each item is treated as if it incurred a separate fixed order cost. Joint-ordering models are especially relevant for suppliers with delivery charges, minimum shipment values, or consolidated transport schedules.

A simple joint-ordering model uses a common replenishment interval T. For item i, annual demand is Di and holding cost is Hi. If a joint order incurs fixed cost S, annual relevant cost is:

The first term is annual joint ordering cost because 1 ÷ T joint orders occur per year. The second term is total cycle-stock holding cost. Minimization gives:

The order quantity for item i is:

This formulation assumes all items can share the same cycle and that the joint fixed cost is independent of the number of items and shipment size. If item-specific ordering costs, truck-capacity limits, or minimum quantities exist, a more detailed integer or mixed-integer model is required.

6.2 Practical Extensions

A firm may use a major-cycle policy in which some items are ordered every joint cycle and others every second or third cycle. This improves flexibility but creates a combinatorial problem. Modern procurement systems can evaluate supplier calendars, transport consolidation, minimum order values, and service constraints using mixed-integer optimization or heuristic search. The resulting policy should still be validated with operational staff because a mathematically efficient schedule may be difficult to execute.

7. Integrated Illustrative Case: Northstar Components

Background

Northstar Components, an industrial-equipment distributor, imports a standard bearing used by several customers. Annual demand is stable at 18,000 units. The supplier’s order cost is £90, annual holding cost is £4.50 per unit, and lead time is 10 working days. The supplier also offers a price of £11.80 for orders of 2,500 units or more, compared with £12.20 for smaller orders. Northstar’s warehouse can store at most 3,000 units of this bearing.

Problem

The purchasing manager has been ordering 1,000 units monthly. This creates 18 orders per year, but management wants to know whether the order quantity should be changed and whether the discount should be accepted.

Analysis

The ordinary EOQ is:

The discounted price requires at least 2,500 units. At £11.80, annual holding cost is assumed to remain £4.50 because it is estimated directly from storage and capital data rather than as a percentage of price. The relevant candidates are 849 units and 2,500 units.

At Q = 849, annual purchase cost is £219,600, ordering cost is approximately £1,908, and holding cost is approximately £1,910. Total cost is approximately £223,418.

At Q = 2,500, annual purchase cost is £212,400, ordering cost is £648, and holding cost is £5,625. Total cost is approximately £218,673. The discount policy therefore reduces estimated annual cost by about £4,745. The quantity is within the 3,000-unit storage limit.

If demand is 80 units per working day, the deterministic reorder point is 800 units for a 10-day lead time. Northstar should nevertheless review supplier lead-time reliability and customer service requirements before operating without safety stock.

Approach and Outcome

Northstar adopts a 2,500-unit order quantity provisionally, subject to quarterly review. It negotiates scheduled releases or split deliveries so that the price discount is retained without physically receiving all units at once. This arrangement reduces inventory exposure while preserving purchasing savings.

Learning Points

1. A price-break candidate must be compared with the feasible EOQ, not ignored because it exceeds EOQ.
2. Storage capacity can invalidate an otherwise economical quantity.
3. Holding cost should be estimated carefully; using a fixed amount and using a percentage of price can produce different results.
4. Deterministic reorder points are benchmarks, not substitutes for safety-stock analysis when uncertainty exists.
5. Contract design, such as scheduled deliveries, can change the practical economics of a model.

8. Critical Evaluation of Deterministic Models

Deterministic models are transparent and useful, but their assumptions can be restrictive.

Demand and Lead-Time Uncertainty

Demand rarely remains constant, and lead time may vary because of congestion, production delays, customs, or transport disruption. A deterministic reorder point may therefore generate stockouts. In uncertain environments, demand during lead time should be represented by a forecast distribution and supplemented with safety stock based on the target service level.

Nonlinear and Hidden Costs

Holding cost is difficult to measure. Capital cost, warehouse space, damage, insurance, obsolescence, handling, and shrinkage may behave differently as inventory rises. Ordering cost may also decline in stages rather than remain fixed. A single S and H can consequently conceal important cost structures.

Capacity and Operational Constraints

EOQ may recommend a quantity that exceeds shelf space, vehicle capacity, supplier minimums, cash limits, or shelf-life restrictions. These constraints require constrained optimization or managerial adjustment. The final policy should be feasible, not merely mathematically optimal.

Price and Contract Effects

Discount models often assume an all-units price and immediate payment. Real contracts may involve incremental discounts, rebates, freight charges, payment terms, minimum annual volume, or quality costs. The total landed cost should be used rather than catalogue price alone.

Behavioural and Strategic Effects

Customers may react strongly to stockouts, and suppliers may respond to unusually large orders by changing lead times or allocation priorities. A short-term cost minimum can therefore damage long-term relationships. Inventory decisions should be integrated with procurement strategy, customer segmentation, and service-level commitments.

9. Technology, Governance, and Best Practice

Enterprise resource planning and warehouse-management systems can automate demand aggregation, reorder alerts, price-break evaluation, and supplier performance measurement. Analytics platforms can run sensitivity analyses showing how EOQ changes when demand, ordering cost, or holding cost changes. Artificial intelligence can improve demand forecasts and detect abnormal consumption, but it does not remove the need to define costs, constraints, and service objectives correctly.

Good governance requires documented assumptions, version-controlled parameters, approval of price and shortage costs, and periodic review. Managers should test sensitivity rather than treat a single output as exact. Since:

moderate parameter errors often produce smaller percentage errors in Q than in total cost. Nevertheless, inaccurate cost estimates can materially change whether a discount, shortage policy, or joint order is worthwhile.

Best practice includes:

• use consistent time units and landed costs;
• distinguish cycle stock from safety stock;
• compare all feasible candidates under discount schedules;
• check warehouse, cash, shelf-life, and transport constraints;
• validate shortage penalties against customer and operational evidence;
• review parameters after changes in demand, supplier performance, or capital cost;
• use deterministic models as benchmarks when uncertainty is material;
• measure outcomes through total cost, service level, inventory turns, and stockout consequences.

Chapter Summary

Deterministic inventory models establish the economic relationship between replenishment frequency, batch size, holding cost, ordering cost, purchasing price, and shortage cost. The basic EOQ is Q* = √(2DS ÷ H) when demand is constant, replenishment is instantaneous, shortages are prohibited, and price is quantity-independent. The EPQ modifies EOQ for gradual production and uses the factor 1 − d ÷ p to account for simultaneous production and consumption.

Quantity-discount analysis requires evaluating every feasible EOQ and price-break quantity using total cost. Planned-shortage models permit controlled backorders when shortage penalties are sufficiently low and measurable. Joint-ordering models coordinate several items to exploit shared fixed ordering costs. All models are simplifications: they should be tested against uncertainty, capacity, service, contractual, behavioural, and strategic realities. Their greatest managerial value lies in providing a transparent benchmark and a disciplined basis for sensitivity analysis and negotiation.

Key Terms

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

EPQ: Economic production quantity for gradual internal or supplier replenishment.

Demand rate: Quantity consumed per unit of time.

Ordering cost: Fixed cost incurred whenever a replenishment order is placed.

Setup cost: Fixed cost incurred when a production run is initiated.

Holding cost: Cost of carrying one unit for a specified period.

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

Lead time: Time between order placement and receipt.

Reorder point: Inventory position that triggers replenishment.

Price break: Quantity threshold at which a different unit price applies.

All-units discount: Discount applied to every unit when a threshold is reached.

Backorder: Demand fulfilled after a temporary delay.

Planned shortage: Deliberately permitted inventory deficit in an economic model.

Joint ordering: Replenishment of multiple items in a coordinated order.

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

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

Cycle time: Time between successive replenishment receipts.

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

Safety stock: Additional inventory held to protect against uncertainty.

Total landed cost: Purchase, freight, duties, handling, and other acquisition costs.

Review Questions

Short Answer Questions

1. What assumptions distinguish the basic EOQ model from a stochastic inventory model?
2. Why does the purchase cost not affect EOQ when unit price is constant?
3. What is the difference between EOQ and EPQ?
4. Why must price-break quantities be included in quantity-discount analysis?
5. What managerial conditions are necessary before planned shortages can be considered?

Long Answer Questions

1. Derive the EOQ formula from annual ordering and holding costs, and explain why the solution is a minimum.
2. Explain the EPQ model, including the meaning of the factor 1 − d ÷ p.
3. Describe a systematic procedure for solving an all-units quantity-discount problem.
4. Evaluate the assumptions and limitations of deterministic inventory models in a global supply chain.
5. Explain how joint ordering changes the analysis of multiple inventory items.

Analytical / Application Questions

1. Annual demand is 24,000 units, ordering cost is $50, and annual holding cost is $2. Calculate EOQ, annual order frequency, and annual ordering plus holding cost.
2. A manufacturer has annual demand of 8,000 units, production capacity of 32,000 units, setup cost of $300, and holding cost of $6 per unit annually. Calculate EPQ and maximum inventory.
3. A firm is considering a 15% unit-price discount requiring an order quantity twice its EOQ. Identify the quantitative and qualitative factors that should be evaluated before accepting the discount.

Multiple Choice Questions with Answers

Question 1

Under the basic EOQ assumptions, which cost is constant with respect to order quantity?

A. Annual ordering cost

B. Annual holding cost

C. Annual purchase cost

D. Shortage cost

Correct Answer: C. Annual purchase cost

Question 2

If annual demand is 10,000 units, ordering cost is $50, and annual holding cost is $2, the EOQ is approximately:

A. 224 units

B. 707 units

C. 1,000 units

D. 2,236 units

Correct Answer: B. 707 units

Question 3

In the EPQ model, production must satisfy which condition?

A. p < d

B. p = 0

C. p > d

D. p = d always

Correct Answer: C. p > d

Question 4

Why is EPQ generally larger than EOQ when production is gradual?

A. Purchase price is always lower

B. Gradual replenishment creates less average inventory for a given lot size

C. Demand becomes variable

D. Shortages are mandatory

Correct Answer: B. Gradual replenishment creates less average inventory for a given lot size

Question 5

In an all-units quantity-discount problem, what must be evaluated in addition to feasible EOQs?

A. Only the smallest possible order

B. Only the supplier’s largest quantity

C. Price-break quantities

D. Only annual demand

Correct Answer: C. Price-break quantities

Question 6

In the basic deterministic model with constant demand and lead time, the reorder point is calculated as:

A. Q ÷ 2

B. dL

C. DS ÷ Q

D. H ÷ S

Correct Answer: B. dL

Question 7

What happens to the optimal planned backorder as shortage penalty becomes very large?

A. It tends toward zero

B. It becomes equal to annual demand

C. It doubles automatically

D. It becomes independent of shortage cost

Correct Answer: A. It tends toward zero

Question 8

Which cost is represented by S in the joint-ordering model?

A. Unit purchase cost

B. Fixed cost per joint order

C. Shortage cost per unit

D. Annual holding cost per unit

Correct Answer: B. Fixed cost per joint order

Question 9

If annual demand is 12,000 units and EOQ is 600 units, the number of orders per year is:

A. 5

B. 12

C. 20

D. 72

Correct Answer: C. 20

Question 10

Which is a major limitation of deterministic reorder points?

A. They require no demand data

B. They automatically optimize safety stock

C. They may cause stockouts when demand or lead time is uncertain

D. They cannot be used as benchmarks

Correct Answer: C. They may cause stockouts when demand or lead time is uncertain