Deterministic Inventory Models

Introduction

Inventory decisions balance two opposing costs. Excess inventory ties up capital, occupies space, becomes obsolete, and may deteriorate. Insufficient inventory interrupts production, delays customer deliveries, and can damage service reputation. Deterministic inventory models provide structured methods for deciding how much to order, when to order, and how often to review stock when demand and other relevant conditions are known or treated as constant.

These models are particularly useful in production environments where demand is stable, replenishment lead time is predictable, and purchasing and holding costs can be estimated. They are not complete representations of reality; rather, they are analytical benchmarks that support purchasing, stores, production-planning and materials-control decisions.

Learning Objectives

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

1. Explain the logic and assumptions of deterministic inventory models.
2. Calculate economic order quantity and economic production quantity.
3. Evaluate quantity discounts and identify the least-cost order quantity.
4. Calculate reorder points, minimum and maximum stock levels, and two-bin quantities.
5. Distinguish continuous-review and periodic-review systems.
6. Select an appropriate model while recognising its limitations and operational risks.

1. Inventory Decisions and Cost Structure

The principal inventory decisions are:

• Order quantity: the amount purchased or produced at one time.
• Reorder timing: the point at which replenishment is initiated.
• Review frequency: whether stock is monitored continuously or at fixed intervals.
• Service protection: the stock required to cover expected usage during replenishment.

For a single item, annual relevant cost commonly consists of purchasing cost, ordering or setup cost, and holding cost. If annual demand is D, order quantity is Q, ordering cost per order is S, unit purchase price is C, and annual holding cost per unit is H, then:

The terms represent annual purchase cost, annual ordering cost, and annual cycle-stock holding cost respectively. The equation assumes that each replenishment arrives immediately and that stock falls uniformly from Q to zero. If H is expressed as a percentage i of unit price, then:

All annual quantities must use the same time unit. For example, annual demand and annual holding cost must not be combined with monthly ordering costs unless the data are first converted consistently.

2. Economic Order Quantity

2.1 Model logic

The basic economic order quantity (EOQ) model identifies the order quantity that minimises the sum of ordering and holding costs. Larger orders reduce the number of orders but increase average inventory. Smaller orders reduce average inventory but require more orders.

Ignoring purchase cost when unit price is constant, the relevant cost is:

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

Therefore:

Here, Q* is the economic order quantity in units. The equation is dimensionally consistent: DS/H has units of units², so its square root has units of units.

At the EOQ, annual ordering cost equals annual holding cost:

This equality is a diagnostic property, not a separate assumption.

2.2 EOQ assumptions

The basic EOQ is appropriate when:

• demand is known, constant and evenly distributed;
• replenishment lead time is known and constant;
• the entire order arrives at one time;
• no shortages are permitted;
• unit price is constant, with no quantity discount;
• ordering cost and holding cost are stable;
• the item is independent of other items;
• inventory can be divided into any quantity; and
• capacity, budget and supplier constraints do not bind.

These assumptions rarely hold perfectly. EOQ should therefore be interpreted as a planning benchmark rather than an automatically optimal operational answer.

2.3 Worked example: EOQ

A manufacturer uses 24,000 bearings annually. Each order costs $75 to process, and annual holding cost is $3 per bearing. Find the EOQ, number of orders, cycle time and annual relevant cost.

Orders per year are:

Average time between orders, using a 360-day planning year, is:

Annual ordering cost is approximately:

Annual holding cost is approximately:

The small difference is caused by rounding the EOQ to a whole unit. Using the unrounded value, the two costs are equal at approximately $1,549 each, and total relevant cost is approximately $3,098. Purchase cost is excluded because the unit price is constant and does not affect the choice of Q.

3. Reorder Point and Continuous Review

EOQ determines order size, not order timing. The reorder point (ROP) determines when a replenishment order should be released. If daily demand is constant at d units per day and lead time is L days:

The units of dL are units/day × days = units. If demand or lead time is uncertain, safety stock is added:

Under deterministic conditions, safety stock may be zero. In actual operations, it may be retained to protect against forecast error, supplier delay, transport disruption or quality rejection.

A continuous-review, or Q,r, system continuously monitors inventory position. An order of Q is placed whenever inventory position reaches r, where inventory position is:

A barcode, warehouse-management or enterprise-resource-planning system can support this monitoring. The ROP must be based on inventory position rather than merely physical stock; otherwise, an order already in transit may be ignored and duplicate orders may be generated.

4. Economic Production Quantity

EOQ assumes that the whole order arrives at once. In manufacturing, replenishment may occur gradually while production and consumption occur simultaneously. The economic production quantity (EPQ), also called the production order quantity, accommodates this situation.

Let p be the production rate per unit of time and d the demand or usage rate, with p > d. During production, inventory increases at the net rate p − d. The maximum inventory is less than the production lot because units are being consumed while production continues.

The EPQ is:

The model requires a constant production rate, constant usage rate, no shortages, known setup cost, and p > d. If p becomes equal to or less than d, inventory cannot build and this formula is not applicable.

Maximum inventory and average inventory are:

Worked example: EPQ

A component has annual demand of 18,000 units. Machine setup cost is $120, annual holding cost is $4 per unit, production rate is 60,000 units per year, and usage rate is 18,000 units per year.

Since 1 − 18,000/60,000 = 0.7:

Maximum inventory is:

Average inventory is approximately 435 units. The EPQ exceeds the EOQ that would be calculated for instantaneous delivery because gradual replenishment lowers average inventory for a given lot size.

5. Quantity-Discount Models

Suppliers may reduce unit price when a buyer orders at least a specified quantity. A discount can justify an order larger than the basic EOQ, but only if the reduction in purchase cost exceeds the additional holding and ordering cost.

For each price bracket, calculate:

If holding cost is a percentage of price, use H = iC separately for each bracket. The procedure is:

1. Calculate the EOQ using the holding cost for each price level.
2. Check whether that EOQ lies within the corresponding quantity bracket.
3. If it does not, use the nearest feasible breakpoint in that bracket.
4. Calculate total annual cost for every feasible candidate.
5. Select the candidate with the lowest total cost.

The common error is to select the lowest unit price without calculating total cost. A low price may create excessive holding cost, storage cost and obsolescence exposure.

Worked example: quantity discount

Annual demand is 10,000 units, ordering cost is $50 per order, and holding rate is 20% of unit price. The supplier offers:

The EOQ at each price is:

At $10.00, EOQ ≈ 707 units, which is not feasible in the 1–499 bracket; candidate = 499.

At $9.70, EOQ ≈ 718 units, which is feasible; candidate = 718.

At $9.40, EOQ ≈ 729 units, which is below the bracket minimum; candidate = 1,000.

Total costs are approximately:

Using exact calculation:

• At 499: 100,000 + (10,000/499)50 + (499/2)(2) ≈ $101,503.
• At 718: 97,000 + (10,000/718)50 + (718/2)(1.94) ≈ $98,392.
• At 1,000: 94,000 + 500 + 970 = $95,470.

Therefore, ordering 1,000 units is least costly under the stated assumptions. The result must still be checked against storage capacity, cash availability, shelf life and supplier reliability.

6. Min–Max and Two-Bin Systems

6.1 Min–max system

A min–max system specifies a lower control level and an upper target level. When inventory position reaches or falls below the minimum, the order quantity raises inventory toward the maximum:

For a deterministic item, the minimum is commonly related to lead-time demand and the maximum is minimum plus a replenishment cycle quantity. The exact settings depend on review frequency, lead time and whether safety stock is included.

Min–max systems are practical for stores with many items because they are simple to administer. However, poorly chosen maximum levels can create excess stock, while infrequent record updates can cause inaccurate replenishment.

6.2 Two-bin system

In a two-bin system, stock is divided into two physical containers. The first bin supplies routine demand. When it is empty, the second bin is opened and a replenishment order is placed. The second bin is intended to cover demand during lead time.

If daily usage is d and lead time is L, the reserve-bin quantity is approximately:

The method is highly visible and inexpensive, and it is suitable for low-value, repetitive items such as fasteners. It is less suitable for expensive, bulky, highly variable or rapidly obsolete items. Accurate labelling, standard container sizes and prompt signalling are essential.

7. Periodic-Review Models

A periodic-review, or P,S, system examines stock at fixed intervals of P time units. At each review, the order raises inventory position to a target level S. With deterministic demand d, lead time L, and review period P, the target level is:

If safety stock is required:

The order quantity at a review is:

Periodic review reduces the need for continuous transaction monitoring and allows consolidated orders from a supplier. Its weakness is exposure during both the review interval and lead time. A shortage can occur shortly after a review and remain undetected until the next review. Consequently, periodic-review systems generally require more protection stock than continuous-review systems for the same service objective.

Continuous versus periodic review

8. Model Selection and Operational Governance

Model selection should begin with the item’s demand pattern, value, criticality, lead time and replenishment process. ABC analysis can focus analytical effort: high-value A items usually require accurate records and close review, while low-value C items may use two-bin or simple min–max controls. Criticality may override value; a cheap seal can be more operationally important than an expensive non-critical component.

Before applying a model, managers should validate:

• demand and usage units;
• supplier lead time and minimum order quantity;
• order, setup, transport and receiving costs;
• holding cost, including capital, space, insurance and obsolescence;
• shelf life and quality constraints;
• storage and production capacity;
• supplier price breaks and contract conditions; and
• system data accuracy.

ERP and warehouse-management systems can automate reorder alerts, inventory-position calculations, supplier schedules and exception reports. Artificial intelligence can improve demand classification and detect unusual usage, but it does not eliminate the need for sound master data, human approval and governance. An algorithm trained on distorted historical demand may reproduce poor decisions. Changes to reorder parameters should be traceable, authorised and periodically reviewed.

9. Limitations and Extensions

Deterministic models are limited because demand, lead time, prices and capacity are rarely perfectly stable. The basic EOQ also excludes shortage costs, quantity constraints, multiple-item interactions and supplier risk. It treats holding cost as a known linear cost, although real storage costs may include discrete capacity expansions and nonlinear congestion.

Useful extensions include:

• safety-stock and service-level models for uncertain demand;
• planned backorder models when shortages are permitted;
• multi-echelon models for several stocking levels;
• finite-capacity and lot-sizing models;
• supplier-managed inventory and vendor-replenishment arrangements; and
• simulation when demand, lead time and disruptions interact.

The correct professional response is not to reject simple models, but to understand the conditions under which they are useful and to escalate to more advanced analysis when those conditions are violated.

10. Illustrative Case: Northstar Pumps

Background

Northstar Pumps manufactures industrial water pumps. A standard mechanical seal is used in every finished unit. Annual demand is 12,000 seals. The purchasing department currently orders 2,000 seals each time, although usage is steady at 50 seals per working day. Supplier lead time is eight working days. Ordering cost is $40 per order, and annual holding cost is $2 per seal.

Problem

The large fixed order was introduced informally to reduce purchasing workload. It has created crowded stores and occasional emergency purchases because the reorder trigger is not formally defined.

Analysis

The EOQ is:

The annual ordering and holding costs at the current quantity are:

At EOQ, relevant cost is approximately:

Thus, the proposed quantity reduces relevant annual cost by approximately $280, before considering space benefits. Daily demand is 50 seals, so the deterministic reorder point is:

Northstar should use an inventory-position trigger of 400 seals, not merely a physical on-hand count. If supplier reliability is poor, a separately justified safety stock should be added rather than hidden in an arbitrary large order quantity.

Approach and outcome

The company adopts an order quantity near 700 seals, sets the reorder point at 400 seals, and reviews supplier lead-time performance monthly. Bin labels are standardised, receipts are scanned immediately, and a cycle-count programme is introduced for the seal inventory record.

The expected outcome is lower cycle stock, reduced storage pressure, fewer emergency orders and improved visibility. The result remains conditional on stable demand and reliable replenishment. If demand becomes intermittent or seals have a minimum supplier lot, the parameters must be recalculated.

Learning points

1. Order size and reorder timing are separate decisions.
2. A low administrative workload does not necessarily mean low total cost.
3. Inventory position is more useful than on-hand stock alone.
4. Data discipline is essential for any mathematical model.
5. Parameters should be reviewed when demand, price or lead time changes.

Chapter Summary

Deterministic inventory models provide a foundation for controlling replenishment when demand and lead time are known or sufficiently stable. EOQ balances ordering and holding costs, while EPQ modifies the analysis for gradual internal production. Quantity-discount analysis compares feasible candidates using total cost rather than unit price alone. Reorder points determine when replenishment is released; min–max and two-bin systems provide practical visual and rule-based controls. Continuous-review systems respond quickly but require reliable records, whereas periodic-review systems simplify administration but must protect against demand during both the review period and lead time. All models depend on assumptions, consistent units and accurate data. Professional application requires validation of costs, capacity, service requirements, supplier conditions and operational risk.

Key Terms

Deterministic demand: Demand treated as known and constant over the planning horizon.

Economic order quantity (EOQ): Order size that minimises ordering and cycle-stock holding costs under basic assumptions.

Economic production quantity (EPQ): Production lot size that accounts for gradual replenishment while usage continues.

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

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

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

Cycle stock: Inventory arising from ordering or producing in batches.

Reorder point (ROP): Inventory-position level that triggers replenishment.

Lead time: Time between releasing a replenishment order and receiving usable stock.

Safety stock: Additional inventory held to protect against uncertainty.

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

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

Min–max system: Control system that replenishes stock toward a maximum when it reaches a minimum.

Two-bin system: Visual replenishment method using one bin for use and another as a lead-time reserve.

Continuous review: Inventory monitoring performed continually or after each transaction.

Periodic review: Inventory assessment at predetermined time intervals.

ABC analysis: Classification of items according to their relative annual consumption value or management importance.

Service level: The intended probability or degree of meeting demand without a stockout.

Review Questions

Short Answer Questions

1. What trade-off is balanced by the EOQ model?
2. State four assumptions of the basic EOQ model.
3. How does EPQ differ from EOQ?
4. Define inventory position.
5. What is the main operational difference between continuous and periodic review?

Long Answer Questions

1. Derive the EOQ equation from the annual ordering and holding-cost relationship, and explain the meaning of each variable.
2. Explain the assumptions, uses and limitations of the EPQ model.
3. Describe a systematic procedure for evaluating quantity discounts.
4. Compare min–max, two-bin and continuous-review systems in a production stores environment.
5. Discuss how ERP systems and data governance affect the reliability of deterministic inventory parameters.

Analytical / Application Questions

1. Annual demand is 30,000 units, ordering cost is $60 per order, and annual holding cost is $5 per unit. Calculate EOQ, annual orders and annual ordering cost at EOQ.
2. Daily demand is 80 units and lead time is six working days. Determine the deterministic reorder point. Explain how the answer would change if supplier lead time were uncertain.
3. A factory produces 40,000 units annually and consumes 10,000 units annually. Setup cost is $100 and annual holding cost is $4 per unit. Calculate EPQ and maximum inventory, and explain why maximum inventory is less than the production quantity.

Multiple Choice Questions with Answers

Question 1

Which cost relationship is minimised by the basic EOQ model?

A. Ordering cost plus holding cost

B. Purchase cost plus shortage cost only

C. Labour cost plus transportation cost

D. Selling cost plus ordering cost

Correct Answer: A. Ordering cost plus holding cost

Question 2

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

A. 250 units

B. 500 units

C. 707 units

D. 1,000 units

Correct Answer: C. 707 units

Question 3

Which condition is required for the basic EPQ formula?

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 arrive instantaneously

Correct Answer: B. Production rate must exceed usage rate

Question 4

If daily demand is 40 units and lead time is five days, what is the deterministic reorder point?

A. 8 units

B. 45 units

C. 200 units

D. 400 units

Correct Answer: C. 200 units

Question 5

In a continuous-review system, an order is normally triggered when:

A. Physical stock reaches the supplier’s price break

B. Inventory position reaches the reorder point

C. The review calendar reaches month-end

D. Average inventory becomes equal to annual demand

Correct Answer: B. Inventory position reaches the reorder point

Question 6

When evaluating quantity discounts, the buyer should primarily:

A. Choose the lowest unit price automatically

B. Choose the largest possible order

C. Compare total annual cost for feasible candidates

D. Ignore holding cost because price is discounted

Correct Answer: C. Compare total annual cost for feasible candidates

Question 7

In a two-bin system, the second bin is principally intended to cover:

A. Annual demand

B. Demand during replenishment lead time

C. All possible seasonal demand

D. Supplier production capacity

Correct Answer: B. Demand during replenishment lead time

Question 8

For a deterministic periodic-review system with daily demand d, review period P and lead time L, the target level is generally:

A. d(P − L)

B. dL/P

C. d(P + L)

D. P + L/d

Correct Answer: C. d(P + L)

Question 9

At the unrounded EOQ under the basic model, which relationship normally holds?

A. Ordering cost equals holding cost

B. Purchase cost equals shortage cost

C. Lead-time demand equals annual demand

D. Holding cost equals unit price

Correct Answer: A. Ordering cost equals holding cost

Question 10

Why should inventory position rather than on-hand stock be used for reorder decisions?

A. It excludes all backorders

B. It includes stock already on order and avoids duplicate replenishment

C. It always equals the EOQ

D. It removes the need to know lead time

Correct Answer: B. It includes stock already on order and avoids duplicate replenishment