Deterministic Inventory Models

Introduction

Deterministic inventory models are quantitative decision models in which demand, replenishment lead time, costs and other relevant parameters are treated as known or fixed. They provide a disciplined basis for deciding how much to order or produce and, in some models, when to replenish. Although actual supply chains contain uncertainty, deterministic models remain important because they clarify cost trade-offs, establish benchmark policies and support more advanced stochastic and simulation models.

The central economic problem is the balance between ordering or setup costs and inventory carrying costs. Large replenishment quantities reduce the frequency of orders but increase average inventory. Small quantities reduce average inventory but increase ordering or setup frequency. The classical economic order quantity (EOQ) model identifies the quantity at which these opposing costs are balanced.

Deterministic models are especially useful for stable, repetitive-demand items such as standard components, packaging materials, maintenance supplies and regularly consumed commodities. They are less suitable when demand is highly volatile, replenishment is unreliable, prices change rapidly or stockout consequences cannot be represented adequately by a simple cost.

Learning objectives

After studying this chapter, the reader should be able to:

1. Explain the logic and assumptions of deterministic inventory models.
2. Derive and apply the basic EOQ model.
3. Analyse finite-production-rate and economic production quantity models.
4. Evaluate all-unit quantity discounts.
5. Determine optimal policies when planned shortages are permitted.
6. Formulate constrained multi-item inventory decisions.
7. Conduct sensitivity analysis and interpret managerial limitations.
8. Translate model results into implementable inventory policies.

1. Foundations of deterministic inventory analysis

A periodic inventory cycle normally contains a replenishment event followed by inventory depletion. If demand is constant and replenishment is instantaneous, inventory falls linearly from a maximum level to zero and is then immediately restored. If replenishment takes place gradually, as in internal production, inventory rises and falls within the same cycle. If shortages are allowed, inventory may become negative in the analytical representation, indicating backorders rather than physical stock.

The principal variables are shown below.

A basic deterministic model usually assumes:

• demand is known and constant;
• replenishment lead time is known and constant;
• all units in an order are received together, unless a production model is used;
• shortages are either prohibited or explicitly assigned a cost;
• relevant costs are identifiable and stable;
• the item is managed independently, unless a multi-item constraint is introduced;
• the planning horizon is sufficiently long for average annual cost to be meaningful.

These assumptions are not descriptions of reality; they are modelling choices. Their usefulness depends on whether the resulting policy is robust when the assumptions are relaxed.

2. The basic economic order quantity model

2.1 Cost structure

For an item purchased at a constant unit price, annual relevant cost is the sum of ordering cost and holding cost. With instantaneous replenishment and no shortages, average cycle inventory is Q ÷ 2. The number of orders per year is D ÷ Q. Therefore:

Here TC(Q) is annual total relevant cost in currency/year. The purchase term DC is constant when unit price does not depend on Q, but it is retained for completeness.

The ordering component decreases as Q increases, whereas the holding component increases. Differentiating with respect to Q gives:

Setting the derivative equal to zero produces the EOQ:

The second derivative is:

for positive D, S and Q. Thus the stationary point is a minimum. At the optimum, annual ordering cost equals annual holding cost:

This equality is a useful checking relationship, not a general rule that can be applied when discounts, shortages or finite production are present.

2.2 Reorder point

When demand is constant and lead time is L years, the reorder point is:

where T is the number of operating time units per year and d is demand per time unit. If D is annual demand and L is expressed in days, then a 365-day convention gives R = D(L ÷ 365). A different operating-calendar convention must be used consistently.

Under deterministic conditions, no safety stock is required because demand during lead time is known exactly. In practice, safety stock may be added to protect against forecast error or lead-time variation, but that converts the problem partly into a probabilistic one.

2.3 Worked example: EOQ

A manufacturer uses 24,000 bearings per year. The ordering cost is $75 per order and annual holding cost is $6 per bearing. Determine EOQ, orders per year and relevant annual cost.

The practical order quantity is approximately 775 units. Orders per year are:

Annual ordering cost is approximately 31.0 × $75 = $2,324. Annual holding cost is approximately (774.6 ÷ 2) × $6 = $2,324. Thus the relevant annual cost is approximately $4,648, excluding the constant purchase cost of $24,000 × unit price.

If lead time is 5 working days and annual demand is based on 250 working days, daily demand is 24,000 ÷ 250 = 96 units. The reorder point is 96 × 5 = 480 units, assuming replenishment is triggered continuously and demand is uniform.

3. Economic production quantity

The EOQ assumes that the entire quantity becomes available at once. This is inappropriate when an organisation produces the item internally and production occurs while demand continues. The economic production quantity (EPQ), also called the finite-production-rate model, recognises that inventory accumulates only at the net rate p − d.

The model requires p > d. During a production run, inventory increases at p − d; after production stops, it decreases at d. The maximum inventory is therefore:

Average inventory is half the maximum inventory:

Annual setup cost is (D ÷ Q)S. Consequently:

Minimisation gives:

Since 0 < 1 − d/p < 1, EPQ is larger than EOQ for identical D, S and H. The larger quantity compensates for the fact that inventory is not present at its full level throughout the production cycle.

Worked example: EPQ

Annual demand is 10,000 units, setup cost is $200, annual holding cost is $4 per unit, and the production rate is 40,000 units per year. Demand is therefore d = 10,000 units/year and p = 40,000 units/year.

Maximum inventory is:

Average inventory is approximately 433 units. The production run lasts Q ÷ p = 1,154.7 ÷ 40,000 years, or about 7.2 working days if 250 working days constitute a year.

4. Quantity-discount models

Suppliers may reduce the unit price when the purchaser orders at least a specified quantity. A discount can make a larger order economical even though it increases average inventory. The decision must compare total annual cost at feasible quantities under each price bracket.

For an all-unit discount, every unit in the order is charged at the price corresponding to the order quantity. For price level i, with unit price Ci and holding cost Hi, the annual cost is:

If holding cost is a fixed amount per unit per year, use Hi = H. If carrying cost is a percentage ki of unit price, use Hi = kiCi. The EOQ for price level i is:

The procedure is:

1. Compute the EOQ for each price level.
2. Check whether that EOQ lies within the quantity interval for its price.
3. If it is below the interval, use the smallest feasible quantity in that interval.
4. Compute total annual cost for each feasible candidate, including purchase cost.
5. Select the candidate with the lowest total cost.

A common error is to select the largest discount merely because its unit price is lowest. The correct comparison includes purchase, ordering and holding costs.

Worked example: all-unit discount

Annual demand is 12,000 units, ordering cost is $50, and holding cost is 20% of unit price. The supplier offers:

For the first bracket, H = 0.20 × 10 = $2.00 and EOQ = √[(2 × 12,000 × 50)/2] ≈ 775 units. This is feasible.

For the second bracket, H = $1.96 and the calculated EOQ is approximately 782 units, below the bracket. The feasible candidate is therefore 1,000 units.

For the third bracket, H = $1.90 and the calculated EOQ is approximately 795 units, below the bracket. The feasible candidate is therefore 2,000 units.

Candidate costs are:

• Q = 775: purchase = $120,000; ordering ≈ $774; holding ≈ $775; total ≈ $121,549.
• Q = 1,000: purchase = $117,600; ordering = $600; holding = $980; total = $119,180.
• Q = 2,000: purchase = $114,000; ordering = $300; holding = $1,900; total = $116,200.

The best candidate is 2,000 units. The discount saving exceeds the additional inventory cost under the stated assumptions.

5. Planned shortages and backorders

Some organisations permit customers or internal users to wait for replenishment rather than maintaining sufficient stock to avoid every shortage. A planned shortage model is appropriate only when backorders are operationally and commercially acceptable. Lost sales, production stoppages and safety-critical stockouts should not be represented casually as ordinary backorders.

Let B be the maximum backorder during a cycle. The maximum positive inventory is Q − B. Average on-hand inventory is (Q − B)² ÷ (2Q), and average backorder is B² ÷ (2Q). If H is annual holding cost per unit and P is annual shortage cost per backordered unit, annual cost is:

For a fixed Q, differentiating with respect to B gives:

The optimal maximum backorder is:

Substitution into the cost function gives the optimal order quantity:

The maximum on-hand inventory is Qs − B. As P becomes very large, B* approaches zero and the model approaches the no-shortage EOQ. As P becomes small relative to H, larger shortages become economically attractive. The model assumes that shortage cost is measurable and that all backorders are eventually filled.

Worked example: planned shortage

Suppose D = 18,000 units/year, S = $40/order, H = $5/unit/year and P = $20/unit/year.

The maximum backorder is:

Maximum positive inventory is approximately 678.8 units. This policy is economically attractive only if customers tolerate delayed fulfilment and the estimated shortage cost of $20 per unit-year is credible.

6. Constrained multi-item inventory models

The independent EOQ model may prescribe quantities that exceed available warehouse space, working capital, supplier capacity or a total inventory budget. Items then compete for a scarce resource. A constrained problem cannot be solved by optimising every item separately.

Let item i have demand Di, ordering cost Si, holding cost Hi and order quantity Qi. A common continuous relaxation is:

subject to a resource constraint such as:

where ai is the resource consumed per unit of average inventory and K is the available resource. For a shelf-space constraint, ai may be cubic metres per unit; for a value constraint, ai may be unit value. Purchase cost can be added when it varies by quantity or is itself relevant to the budget.

Using a Lagrange multiplier λ for the constraint, the first-order condition for item i is:

Therefore:

The multiplier λ is selected so that the resource constraint is satisfied, usually by numerical search. If the unconstrained EOQs already satisfy the constraint, λ = 0 and no adjustment is required. If the constraint binds, λ > 0 and each item behaves as if its holding cost were increased by λai.

This formulation assumes divisible quantities and a common linear constraint. Real decisions may require integer quantities, supplier minimums, case-pack sizes, order-cycle coordination or multiple constraints. Such variants may require nonlinear programming, mixed-integer optimisation or heuristic policies.

Managerial interpretation

The multiplier λ represents the marginal annual cost of one additional unit of scarce resource. Items with high demand and ordering cost tend to receive larger quantities; items that consume substantial space or capital are penalised through λai. The model does not mean that all items should have the same days of supply. It allocates the scarce resource according to the stated cost structure.

7. Sensitivity, robustness and model risk

EOQ is relatively insensitive to moderate parameter error because it is a square-root function. If Q* = √(2DS/H), then proportional changes satisfy approximately:

For example, a 20% increase in demand increases EOQ by only about 10%, holding other variables constant. Exact recalculation is preferable when changes are material, but the relationship helps managers understand robustness.

The total cost curve is also relatively flat near its minimum. Consequently, a practical quantity may differ from theoretical EOQ because of carton sizes, truckload economics, supplier minimums or administrative convenience without causing a large cost penalty. The model should be used to identify an economically sensible range, not to create false precision.

Important sources of model risk include:

• Demand instability: seasonal, intermittent or promotional demand violates constant-demand assumptions.
• Lead-time uncertainty: deterministic reorder points may cause stockouts when deliveries are late.
• Cost estimation error: ordering cost often includes hidden labour, approval and receiving costs; holding cost should include capital, storage, insurance, deterioration and obsolescence.
• Price dynamics: temporary promotions, inflation and exchange-rate changes can dominate nominal EOQ savings.
• Service requirements: a single shortage cost may not capture customer loyalty, regulatory consequences or production downtime.
• Item interactions: common suppliers, shared transport and component dependencies create economies or constraints absent from independent models.
• Quality and shelf life: large lots may increase deterioration, expiry and quality risk.
• Integer and operational restrictions: orders may need to conform to pallets, batches or production campaigns.

Sensitivity analysis should vary demand, costs, lead time, production rate, discount thresholds and shortage penalties. Scenario analysis is usually more informative than reporting one supposedly optimal number.

8. Implementation in contemporary inventory systems

Enterprise resource planning and warehouse-management systems can calculate reorder points, monitor on-hand and on-order balances, and enforce order multiples. The model is valuable only when master data are reliable. Unit-of-measure conversions, obsolete suppliers, incorrect lead times and duplicated item records can make a mathematically correct policy operationally wrong.

Analytics and AI can support deterministic models by detecting demand regimes, estimating transaction costs, identifying abnormal consumption and simulating alternative discount or capacity scenarios. AI should not be treated as a substitute for model governance. A responsible process records data sources, assumptions, version history, exception rules and human approval. Automated recommendations should be explainable enough for procurement, finance, operations and audit personnel to challenge them.

Contemporary practice often combines deterministic benchmarks with probabilistic safety-stock calculations, supplier-risk assessments, multi-echelon optimisation and scenario simulation. The EOQ may remain the baseline lot-size calculation even when a more advanced system eventually modifies it.

9. Illustrative case: packaging material at Meridian Foods

Background

Meridian Foods uses 60,000 corrugated cartons annually for a stable product line. The procurement team currently orders 10,000 cartons per release. Ordering cost is estimated at $120 per order, and annual holding cost is $0.80 per carton. The supplier’s normal lead time is 8 working days. A supplier offers a price reduction for orders of at least 20,000 cartons, but the larger quantity requires additional warehouse space.

Problem

Management must determine whether the current order quantity is economically defensible and whether the discount should be accepted. Warehouse capacity limits average inventory of this carton to 12,000 units. Demand is sufficiently stable for a deterministic benchmark, but a late delivery could interrupt packaging.

Analysis

The unconstrained EOQ is:

The current quantity of 10,000 is therefore much larger than the basic EOQ. Its average cycle inventory is 5,000 cartons, compared with approximately 2,121 cartons at EOQ. If the 20,000-carton discount creates average inventory of 10,000 cartons, it remains below the stated 12,000-carton capacity, but its holding cost must be compared with the purchase saving.

The team should calculate total annual cost at the normal feasible EOQ, the current 10,000-carton quantity and the 20,000-carton discount quantity. It should also verify pallet dimensions, shelf-life, fire regulations, receiving labour and the probability and consequence of delivery delay. The reorder point under deterministic demand is daily demand multiplied by eight working days. With 250 working days, daily demand is 240 cartons, so the benchmark reorder point is 1,920 cartons before any risk buffer.

Approach and solution

Meridian should use the EOQ as a benchmark rather than automatically replacing the current policy. It should evaluate the discount using the supplier’s exact price schedule and holding-cost definition, then run scenarios for demand growth, capacity usage and late deliveries. If the discount quantity is selected, the organisation should reserve capacity, set a maximum stock rule and verify that the discount does not encourage obsolete packaging.

Outcome

The case illustrates that a quantity decision has three layers: mathematical cost optimisation, physical feasibility and service-risk governance. The theoretical EOQ identifies an opportunity to reduce order size, while the discount and capacity analysis determines whether a larger quantity is justified. The final policy should be documented with its assumptions and reviewed when demand, price or supplier performance changes.

Learning points

1. An existing quantity may reflect historical convenience rather than economic logic.
2. Discount decisions require total-cost comparison, not unit-price comparison.
3. Capacity is a binding constraint only when measured consistently with the inventory policy.
4. A deterministic reorder point should not be mistaken for protection against uncertainty.
5. Data governance and periodic review are part of inventory control, not administrative extras.

10. Best-practice decision process

A professional application of deterministic models should follow these steps:

1. Define the item, planning horizon, service requirement and decision variable.
2. Establish demand, lead time, price, ordering, setup, holding and shortage-cost data.
3. Select the model whose assumptions best fit the operating process.
4. Check units and time bases before calculating.
5. Calculate the unconstrained benchmark.
6. Apply minimums, multiples, discounts, capacity and budget constraints.
7. Compare total cost and operational feasibility.
8. Conduct sensitivity and scenario analysis.
9. Implement through clear reorder, approval and exception rules.
10. Review the policy when demand, prices, capacity, technology or supplier performance changes.

Chapter Summary

Deterministic inventory models provide structured benchmarks for lot-size and replenishment decisions when key parameters are known or treated as fixed. The basic EOQ balances ordering and holding costs and is given by Q* = √(2DS/H). The EPQ modifies the model for finite production by recognising that inventory accumulates at the net rate p − d. Quantity-discount analysis requires comparison of feasible candidate quantities using total annual cost, including purchase cost. Planned-shortage models trade holding cost against a measured backorder cost and are appropriate only when delayed fulfilment is acceptable. Constrained multi-item models allocate scarce space, capital or another resource across items, often using a Lagrange multiplier or numerical optimisation.

The models are not forecasts of reality. Their validity depends on demand stability, cost quality, lead-time behaviour, service expectations and operational restrictions. Sensitivity analysis, scenario testing, data governance and managerial judgement are therefore essential. In practice, deterministic models are best used as transparent baselines within ERP, analytics and broader inventory-control systems.

Key Terms

EOQ: Economic order quantity that minimises ordering and holding cost under the basic model.

EPQ: Economic production quantity for replenishment produced at a finite rate.

Ordering cost: Cost incurred each time a purchase order is placed.

Setup cost: Cost incurred when production equipment or processes are prepared for a run.

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

Cycle stock: Inventory held because replenishment occurs in batches.

Reorder point: Inventory position at which replenishment is initiated.

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

All-unit discount: Pricing arrangement in which every unit receives the price associated with the order quantity.

Planned shortage: Deliberately accepted temporary unavailability represented by backorders.

Backorder: Demand delayed until inventory becomes available.

Finite production rate: Replenishment rate that is greater than demand but not instantaneous.

Average inventory: Mean inventory level over a replenishment cycle.

Relevant cost: Cost that changes as a result of the decision being evaluated.

Capacity constraint: Limit on a resource such as space, capital or supplier capability.

Lagrange multiplier: Shadow-price measure of the marginal cost of a binding constraint.

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

Deterministic assumption: Treatment of a variable as known and fixed for modelling purposes.

Model risk: Risk that assumptions, data or structure make a model unsuitable for the decision.

Review Questions

Short Answer Questions

1. What economic trade-off is balanced by the basic EOQ model?
2. Why is average inventory in the basic EOQ model equal to Q ÷ 2?
3. State the condition that must hold between production rate and demand rate in the EPQ model.
4. Why must total cost, rather than unit price alone, be used in quantity-discount analysis?
5. What does a positive Lagrange multiplier indicate in a constrained multi-item model?

Long Answer Questions

1. Derive the basic EOQ formula from the annual ordering and holding-cost functions, and explain why the stationary point is a minimum.
2. Compare the assumptions, cost structures and managerial uses of EOQ and EPQ.
3. Explain the complete procedure for evaluating an all-unit quantity discount, including feasibility checks.
4. Derive the planned-shortage model’s expressions for maximum backorder and optimal order quantity, and discuss when planned shortages are inappropriate.
5. Critically evaluate the limitations of deterministic inventory models in a volatile supply chain.

Analytical / Application Questions

1. Annual demand is 36,000 units, ordering cost is $90 and annual holding cost is $4.50 per unit. Calculate EOQ, orders per year and annual ordering plus holding cost. State the assumptions used.
2. A firm produces 20,000 units per year of an item demanded at 5,000 units per year. Setup cost is $300 and holding cost is $6 per unit-year. Calculate EPQ and maximum inventory, then explain why EPQ differs from EOQ.
3. Three items compete for warehouse capacity. Explain how you would formulate a constrained multi-item model, identify the variables and constraint, and interpret the shadow price of warehouse capacity.

Multiple Choice Questions with Answers

Question 1

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

A. Purchase cost and shortage cost

B. Ordering cost and holding cost

C. Labour cost and transport cost

D. Setup cost and purchase cost

Correct Answer: B. Ordering cost and 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. 224 units

B. 500 units

C. 707 units

D. 1,000 units

Correct Answer: C. 707 units

Question 3

The EPQ model differs from the basic EOQ model because it assumes that replenishment is:

A. Instantaneous and unlimited

B. Gradual at a finite production rate

C. Random and normally distributed

D. Prohibited during demand

Correct Answer: B. Gradual at a finite production rate

Question 4

For the EPQ model to be valid, which condition must hold?

A. p < d

B. p = 0

C. p > d

D. p is unrelated to d

Correct Answer: C. p > d

Question 5

When evaluating an all-unit quantity discount, a calculated EOQ falls below the quantity range for a price bracket. Which quantity should normally be evaluated for that bracket?

A. Zero units

B. The largest quantity in the bracket

C. The smallest feasible quantity in the bracket

D. The EOQ from the highest-price bracket only

Correct Answer: C. The smallest feasible quantity in the bracket

Question 6

In the planned-shortage model, increasing the shortage cost P generally causes the optimal maximum backorder to:

A. Increase without limit

B. Decrease

C. Remain exactly unchanged

D. Become equal to annual demand

Correct Answer: B. Decrease

Question 7

If deterministic annual demand is D and lead time is L years, with constant demand, the reorder point is:

A. D + L

B. D ÷ L

C. D × L

D. L ÷ D

Correct Answer: C. D × L

Question 8

In a constrained multi-item model, a positive shadow price for warehouse space means that:

A. Warehouse space has no economic value

B. The constraint is binding and additional space has marginal value

C. All items must be ordered at EOQ

D. Holding cost is zero

Correct Answer: B. The constraint is binding and additional space has marginal value

Question 9

If demand increases by 20% while ordering and holding costs remain unchanged, EOQ increases by approximately:

A. 5%

B. 10%

C. 20%

D. 40%

Correct Answer: B. 10%

Question 10

Which situation most directly violates the constant-demand assumption of a basic deterministic EOQ model?

A. A stable annual usage rate

B. A fixed supplier lead time

C. Highly seasonal and unpredictable consumption

D. A constant ordering cost

Correct Answer: C. Highly seasonal and unpredictable consumption