Deterministic Inventory Models

Introduction and Context

Inventory decisions connect purchasing, production, customer service, and working-capital management. An organization must determine how much to order or produce, when to replenish, whether to permit shortages, and whether a supplier’s price schedule changes the preferred quantity. These decisions are difficult because inventory creates both benefits and costs. Stock protects operations against demand and supply uncertainty, but it also ties up capital, occupies space, becomes obsolete, and may require insurance, handling, and control.

Deterministic inventory models assume that the relevant parameters—usually demand, replenishment time, ordering cost, production rate, purchase price, and holding cost—are known with certainty. They are therefore not forecasts of reality in a strict sense. Rather, they are analytical benchmarks. They reveal the economic trade-offs in a controlled environment and provide a foundation for more advanced stochastic, multi-echelon, and technology-enabled systems.

This chapter develops the economic order quantity (EOQ), economic production quantity (EPQ), quantity-discount, planned-shortage, and finite-replenishment models. The focus is not merely on applying formulas, but on understanding their assumptions, deriving their cost structures, testing sensitivity, and judging whether their recommendations are operationally credible.

Learning Objectives

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

1. Explain the cost logic and assumptions underlying deterministic inventory models.
2. Derive and apply the EOQ and EPQ models.
3. Evaluate all-units quantity discounts and identify the least-cost feasible alternative.
4. Formulate planned-shortage and finite-replenishment models.
5. Conduct sensitivity analysis and interpret managerial implications.
6. Recognize model limitations and identify when a more advanced approach is required.

1. Inventory Cost Logic

For a single item, annual relevant inventory cost commonly includes purchasing, ordering or setup, holding, and shortage costs. A general representation is:

where D is annual demand in units/year, C is unit purchase cost, S is ordering cost per order, Q is order quantity in units, and H is annual holding cost per unit. The first term is purchase cost; the second is annual ordering cost; and the third is average cycle-stock holding cost under instantaneous replenishment and no shortages.

The model balances two opposing effects. Increasing Q reduces the number of orders and therefore reduces ordering cost, but it increases average inventory and holding cost. The optimal quantity is the minimum point of the total-cost curve, not necessarily the smallest order or the largest supplier discount.

Assumptions and Units

The standard deterministic models generally assume:

• demand is known, constant, and continuous;
• replenishment lead time is known;
• stockouts are either prohibited or treated by an explicit shortage model;
• costs are constant over the planning horizon;
• one item is analyzed independently;
• ordering and holding costs are separable;
• quantity decisions are divisible, unless a discrete-unit constraint is imposed.

Dimensional consistency is essential. If demand is measured in units/year, ordering cost in currency/order, and holding cost in currency/unit/year, then (D/Q)S and (Q/2)H are both currency/year.

2. Basic Economic Order Quantity

Derivation

With instantaneous replenishment and no shortages, inventory cycles from Q to zero. Average inventory is Q/2, and the number of orders per year is D/Q. Thus:

Since DC is independent of Q, minimize the relevant cost:

Differentiating with respect to Q:

Setting the derivative equal to zero gives:

The second derivative, 2DS/Q³, is positive for positive values, confirming a minimum. At the optimum, ordering cost equals holding cost:

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

The order frequency is N = D/Q* orders/year, the cycle time is T = Q*/D years/order, and, when lead time is L years, the reorder point is:

The reorder point formula applies only to deterministic demand during a known lead time and does not include safety stock. If lead time is expressed in days and demand in units/day, the units must be converted consistently.

Worked Example 1: EOQ

A distributor sells 12,000 units annually. Each order costs $75, the unit price is $20, and annual holding cost is $4 per unit. Assume instantaneous replenishment and no shortages.

The practical order quantity is approximately 671 units. Annual ordering cost is:

Annual holding cost is:

Annual purchase cost is $240,000, so total annual cost is approximately $242,683.28. The number of orders is approximately 17.89 per year, and the cycle time is approximately 0.0559 years, or about 20.4 working days if 365 calendar days are used.

Sensitivity of EOQ

EOQ is relatively insensitive to parameter errors because it is a square-root function. If demand is multiplied by a factor k, EOQ is multiplied by √k. A 21% increase in demand, for example, increases EOQ by only about 10%. Similarly, a 10% increase in ordering cost increases EOQ by about 4.9%, while a 10% increase in holding cost decreases it by about 4.7%.

This robustness does not mean that parameters are unimportant. Purchase-price changes, capacity limits, cash constraints, supplier minimums, and service requirements can make a mathematically small change operationally significant.

3. Economic Production Quantity

EOQ assumes the entire order arrives at once. In manufacturing, inventory may accumulate gradually while production occurs. If production rate p exceeds demand rate d, maximum inventory is lower than the production lot size because demand consumes units during the production run.

During a production run, net inventory accumulates at p − d. A lot of size Q therefore reaches maximum inventory:

Average inventory is half this amount. If K is setup cost per production run and h is holding cost per unit per year, the relevant annual cost is:

Minimizing gives:

This is the economic production quantity. It requires p > d; otherwise inventory cannot recover and the production system cannot satisfy continuing demand under the stated conditions. As p → ∞, the expression approaches the EOQ form with setup cost K and demand d.

Worked Example 2: EPQ

A plant uses 24,000 units per year. It produces at 60,000 units per year, incurs a setup cost of $300 per run, and has a holding cost of $2.50 per unit per year.

Because 1 − d/p = 0.6:

Maximum inventory is approximately:

Average inventory is approximately 929.51 units. Annual setup cost and holding cost are each approximately $2,323.79, and the relevant annual cost is approximately $4,647.58.

EPQ is especially useful where production and consumption overlap, such as internal fabrication, filling operations, and repeated manufacturing campaigns. It should not be used when production is highly unreliable, demand is variable, or setup capacity and sequence dependence dominate the economics without further modeling.

4. Quantity-Discount Models

A supplier may reduce the unit price when the order quantity reaches a threshold. Under an all-units discount, the lower price applies to every unit in the order once the threshold is reached. The purchasing term is therefore quantity-dependent, and the unrestricted EOQ at one price may be infeasible for that price bracket.

For price level C_i, holding cost H_i, and feasible interval [q_i, q_i+1), calculate the candidate:

Then project it into the interval:

In practice, evaluate the EOQ if it lies within the bracket; otherwise evaluate the appropriate breakpoint, usually the minimum quantity qualifying for the discount. The total cost is:

If holding cost is a percentage i of unit price, then H_i = iC_i, and the discount changes both purchasing and holding cost. The common error is to select the lowest unit price without evaluating the cost of carrying the additional inventory.

Worked Example 3: All-Units Discount

Annual demand is 10,000 units and ordering cost is $60. The supplier offers:

The unrestricted candidates are:

• At $50: Q = √(2×10,000×60/12.5) ≈ 309.84, feasible in the first bracket.
• At $49: Q ≈ 313.04, below 500, so the relevant candidate is 500.
• At $47.50: Q ≈ 318.20, below 1,000, so the relevant candidate is 1,000.

Total costs are:

The 1,000-unit candidate is least costly despite its higher average inventory. A real decision would additionally test storage capacity, cash flow, shelf life, and the possibility that the discount is incremental rather than all-units.

5. Planned-Shortage Model

Some systems deliberately permit backorders when the cost of carrying inventory is high and customers will wait. Let B be the maximum backorder level, S_b the shortage or backorder cost per unit per year, and H the holding cost. During a cycle, inventory is positive for part of the cycle and backorders accumulate for another part. The average positive inventory is:

The average backorder is:

The annual relevant cost is:

Minimizing with respect to both Q and B gives:

The maximum positive inventory is Q* − B*. As shortage cost becomes very large, B* approaches zero and the no-shortage EOQ is approached. Planned shortages are appropriate only when customers accept delayed fulfillment, shortage costs are measurable, and service commitments permit backorders. Lost sales, safety risks, reputational damage, or contractual penalties require a different cost structure.

Illustrative Calculation

Suppose D = 8,000 units/year, S = $100, H = $5 per unit-year, and S_b = $20 per unit-year.

Thus maximum positive inventory is 1,600 units. The result is economically attractive only if a 400-unit maximum backorder is operationally and commercially acceptable.

6. Finite-Replenishment and Lead-Time Considerations

EPQ is a finite-replenishment model in which production continues while demand occurs. More generally, if replenishment occurs at a finite rate r and demand occurs at rate d, with r > d, the maximum inventory for a lot Q is:

The corresponding average inventory and optimal lot size follow the EPQ logic. If the replenishment source supplies the entire quantity at a finite but constant rate, the production-rate symbol p is replaced by r.

Lead time affects timing, not the basic EOQ quantity, under deterministic conditions. If demand is d units per day and lead time is L days:

If a fixed safety stock SS is deliberately maintained, then:

Safety stock is not produced by the deterministic EOQ model; it is a response to uncertainty or service policy. Confusing cycle stock with safety stock can lead to excessive inventory and misleading performance claims.

7. Comparative Model Selection

Models should be selected from physical and commercial conditions, not from the availability of a familiar formula. For example, EPQ does not automatically account for machine capacity, sequence-dependent setup, scrap, or quality losses. A quantity-discount calculation does not establish that a supplier will hold price or that the buyer has enough warehouse capacity.

8. Detailed Applied Case: Industrial Filter Assembly

Background

This illustrative case concerns a manufacturer that purchases a standardized filter used in an assembly line. Annual demand is 18,000 filters. The supplier charges $32 per order. The normal price is $14.00 per filter, while orders of at least 2,000 receive an all-units price of $13.60. Annual holding cost is 20% of purchase price. Lead time is five working days, demand is stable at 75 filters per working day, and the plant operates 240 days per year.

Problem

The purchasing manager must determine whether to use the ordinary EOQ or qualify for the discount. Storage space is available for 2,500 filters, and finance has not imposed a formal cash limit.

Analysis

At the normal price, H₁ = 0.20×14 = $2.80. The unrestricted EOQ is:

Its total cost is approximately:

At the discount price, H₂ = 0.20×13.60 = $2.72. The unrestricted discounted EOQ is approximately 650.84, which is below the 2,000-unit threshold. Therefore, 2,000 is the feasible candidate for the discounted bracket:

The discount candidate saves approximately $8,669 annually. Its average cycle stock is 1,000 units, below the 2,500-unit capacity. The deterministic reorder point is 75×5 = 375 filters.

Approach and Governance Checks

The recommendation is to order 2,000 filters, subject to four checks: confirm that the price is an all-units discount; verify that the 2,500-unit space includes inbound and quarantine stock; test shelf life and obsolescence risk; and ensure that the supplier can reliably deliver the larger lot. The manager should also test the result under demand and price uncertainty rather than treating the computed saving as guaranteed.

Outcome and Learning Points

The lower unit price dominates the additional holding cost in this case. The case demonstrates the correct procedure: calculate candidates by price bracket, enforce feasibility, compare total annual cost, and then apply operational constraints. It also shows that an apparently attractive discount can be rejected if storage, cash, deterioration, or supplier reliability changes the effective cost.

9. Sensitivity, Robustness, and Managerial Judgment

A deterministic recommendation should be stress-tested. Useful scenarios include changes in demand, ordering cost, holding-rate percentage, production rate, shortage cost, price thresholds, and capacity. For EOQ:

These elasticities mean that a 1% change in demand or ordering cost changes EOQ by approximately 0.5%, while a 1% change in holding cost changes it by approximately −0.5%.

However, total cost is often flatter near its minimum than the quantity itself. A manager may therefore choose a nearby round quantity that fits pallets, supplier packs, or production schedules with little cost penalty. This is economically defensible if the incremental cost is quantified.

Important sensitivity questions include:

• How much does total cost change if demand is 10% higher or lower?
• Does the discount remain attractive if holding cost includes a realistic capital charge?
• Does the proposed lot exceed available storage or working capital?
• What if production rate falls close to demand rate?
• What is the financial and service consequence of a planned shortage?
• Are ordering costs and setup costs genuinely fixed, or do they vary with quantity and complexity?

10. Limitations, Technology, and Contemporary Practice

Deterministic models omit demand variability, supplier disruption, correlated lead times, capacity competition, multiple items, and strategic service differentiation. They may also understate the cost of obsolescence, contamination, damage, carbon emissions, and working capital. In multi-echelon networks, optimizing each item or location independently can produce a system-wide poor result.

Enterprise resource planning and warehouse-management systems automate demand aggregation, inventory balances, purchase orders, and reorder signals. Analytics and AI can improve parameter estimation, identify anomalous consumption, segment items, and simulate scenarios. They do not remove the need for model governance. An algorithm trained on abnormal demand may recommend excessive stock; a discount engine may ignore shelf life; and an automated order may violate segregation, safety, or approval controls.

Best practice is to document parameter definitions, data sources, units, review frequency, constraints, and exception rules. Separate the mathematical optimum from the approved operating quantity. Maintain audit trails for price assumptions and inventory policies. Where environmental reporting is material, include storage energy, transportation frequency, packaging, waste, and disposal consequences in a broader total-cost or total-impact analysis.

Ethically, managers should not create artificial shortages to manipulate customers, conceal service failures through inappropriate backorder assumptions, or accept supplier terms that compromise product safety and fair competition. Inventory governance should support traceability, responsible use of capital, and reliable service.

Chapter Summary

Deterministic inventory models convert the trade-off between replenishment effort and inventory carrying cost into transparent decision rules. The EOQ model gives Q* = √(2DS/H) when demand is constant, replenishment is instantaneous, and shortages are prohibited. The EPQ model adjusts for gradual production and gives Q* = √[2Kd/{H(1−d/p)}]]. Quantity-discount models require candidate generation by price bracket, feasibility testing, and total-cost comparison; the lowest unit price is not automatically optimal. Planned-shortage models trade holding cost against explicitly valued backorders and should be used only when service conditions permit. Finite-replenishment models recognize that inventory accumulates at a net rate rather than arriving all at once.

The models are benchmarks rather than substitutes for judgment. Correct application requires explicit assumptions, consistent units, operational constraints, sensitivity analysis, and governance. Technology can improve data and execution, but it cannot compensate for poor cost definitions or invalid assumptions.

Key Terms

Deterministic inventory model: A model using known and fixed demand, costs, and replenishment parameters.

Economic order quantity (EOQ): The order size that minimizes ordering and holding cost under basic assumptions.

Economic production quantity (EPQ): The production lot size that minimizes setup and holding cost when replenishment is gradual.

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

Setup cost: Cost incurred to prepare a production process for a run.

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

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

Reorder point: Inventory position at which replenishment is initiated.

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

Safety stock: Additional inventory held against uncertainty.

Quantity discount: A lower unit price available when a quantity condition is met.

All-units discount: A discount applied to every unit once the threshold is reached.

Planned shortage: An intentional period of backordered or unavailable inventory.

Backorder: Customer demand fulfilled later rather than lost immediately.

Finite replenishment: Replenishment occurring over time at a limited rate.

Relevant cost: A cost that changes with the decision under analysis.

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

Inventory position: On-hand inventory plus on-order inventory minus allocated or backordered units.

Review Questions

Short Answer Questions

1. What opposing costs are balanced by the EOQ model?
2. Why must the production rate exceed the demand rate in the EPQ model?
3. What distinguishes an all-units discount from an incremental discount?
4. Why is safety stock not included in the basic deterministic EOQ formula?
5. State two conditions under which planned shortages may be inappropriate.

Long Answer Questions

1. Derive the EOQ formula and explain why the stationary point is a minimum.
2. Compare EOQ and EPQ assumptions, cost structures, and managerial applications.
3. Explain the correct procedure for solving an all-units quantity-discount problem.
4. Derive the planned-shortage cost model and interpret the role of shortage cost.
5. Critically evaluate the usefulness and limitations of deterministic inventory models in an ERP-supported supply chain.

Analytical / Application Questions

1. Annual demand is 16,000 units, ordering cost is $50, and annual holding cost is $2.50 per unit. Calculate EOQ, annual ordering cost, and annual holding cost at EOQ.
2. A firm produces 30,000 units annually for internal use. Its production rate is 90,000 units/year, setup cost is $250, and holding cost is $3 per unit-year. Calculate EPQ and maximum inventory.
3. A supplier offers a substantial discount at a quantity above the unrestricted EOQ. Identify the operational, financial, and governance checks that should be completed before accepting the discount.

Multiple Choice Questions with Answers

Question 1

Under the basic EOQ assumptions, which expression gives the optimal order quantity?

A. √(DS/H)

B. √(2DS/H)

C. 2DS/H

D. DS/(2H)

Correct Answer: B. √(2DS/H)

Question 2

At the EOQ optimum, which relationship holds?

A. Annual ordering cost equals annual holding cost.

B. Purchase cost equals ordering cost.

C. Holding cost is twice ordering cost.

D. Ordering cost is always zero.

Correct Answer: A. Annual ordering cost equals annual holding cost.

Question 3

If demand is 12,000 units/year, ordering cost is $75, and holding cost is $4/unit-year, the EOQ is closest to:

A. 300 units

B. 450 units

C. 671 units

D. 1,200 units

Correct Answer: C. 671 units

Question 4

The EPQ model differs from EOQ primarily because:

A. Demand is assumed to be random.

B. Replenishment occurs gradually while demand continues.

C. Purchase price must always be discounted.

D. Shortages are mandatory.

Correct Answer: B. Replenishment occurs gradually while demand continues.

Question 5

For EPQ to support continuing demand under the standard assumptions, which condition is required?

A. p < d

B. p = 0

C. p > d

D. p = d/2

Correct Answer: C. p > d

Question 6

In an all-units quantity-discount problem, a discounted EOQ below the discount threshold should generally be evaluated at:

A. Zero quantity

B. The minimum quantity qualifying for the discount

C. The unrestricted EOQ at the highest price

D. The maximum possible quantity regardless of cost

Correct Answer: B. The minimum quantity qualifying for the discount

Question 7

Which cost is explicitly added in a planned-shortage model?

A. Backorder or shortage cost

B. Depreciation of production equipment only

C. Supplier advertising cost

D. Employee income tax

Correct Answer: A. Backorder or shortage cost

Question 8

If daily demand is 75 units and deterministic lead time is 5 days, with no safety stock, the reorder point is:

A. 15 units

B. 80 units

C. 375 units

D. 3,750 units

Correct Answer: C. 375 units

Question 9

In the EPQ model, maximum inventory for lot size Q, production rate p, and demand rate d is:

A. Q(1+d/p)

B. Q(1−d/p)

C. Qp/d

D. Q−p−d

Correct Answer: B. Q(1−d/p)

Question 10

Why should a quantity-discount decision compare total cost rather than unit price alone?

A. Holding and ordering costs may offset the purchase-price saving.

B. Unit price is never relevant.

C. Discounts always increase total cost.

D. Ordering cost disappears at larger quantities.

Correct Answer: A. Holding and ordering costs may offset the purchase-price saving.