4. Deterministic Inventory Models

Introduction and Context

Inventory decisions balance two opposing economic forces. Holding additional stock improves availability and protects operations from interruption, but it ties up capital, occupies space, increases insurance and handling costs, and exposes the firm to obsolescence and deterioration. Ordering or producing too frequently reduces average inventory but increases setup, ordering, receiving, and administrative costs. Deterministic inventory models provide a disciplined way to examine this trade-off when demand, lead time, costs, and replenishment conditions are known or treated as constant.

These models are particularly useful as benchmark tools. They do not claim that real supply chains are perfectly predictable. Rather, they identify the economically attractive policy under specified assumptions. Managers can then modify the result in response to uncertainty, service requirements, supplier reliability, capacity restrictions, sustainability objectives, and strategic considerations.

The models developed in this chapter include the basic economic order quantity, economic production quantity, quantity-discount models, planned-shortage models, and multi-item models subject to constraints. Together, they form a foundation for inventory policy design in manufacturing, distribution, retail, healthcare, spare-parts management, and logistics services.

Learning Objectives

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

1. Explain the assumptions and logic of deterministic inventory models.

2. Derive and apply the basic EOQ model.

3. Distinguish purchasing replenishment from production replenishment.

4. Evaluate all-unit quantity discounts and identify feasible order quantities.

5. Analyse planned shortages and the trade-off between shortage and holding costs.

6. Formulate multi-item inventory decisions under budget, space, or service constraints.

7. Conduct sensitivity analysis and interpret model outputs for managerial decisions.

8. Identify limitations, implementation risks, and appropriate uses of these models.

The Economic Logic of Inventory Models

A deterministic model begins with a demand rate, usually represented by annual demand \(D\). If an item is ordered in lots of size \(Q\), the number of orders per year is \(D/Q\). When each order incurs a fixed ordering cost \(S\), annual ordering cost is:

\[

OC=\frac{D}{Q}S

\]

With instantaneous replenishment and no safety stock, average cycle inventory is \(Q/2\). If the annual holding cost per unit is \(H\), annual holding cost is:

\[

HC=\frac{Q}{2}H

\]

The relevant annual variable cost is therefore:

\[

TC(Q)=\frac{D}{Q}S+\frac{Q}{2}H

\]

Purchase expenditure \(DC\), where \(C\) is unit price, is constant when price does not depend on order quantity. It does not affect the optimal lot size, although it is included when total annual cost is reported.

The model assumes that demand is continuous and uniform, replenishment is instantaneous, shortages are not permitted, unit price is constant, ordering cost is fixed, holding cost is linear, and one item is considered independently. These assumptions make the model tractable but must be examined before implementation.

Important terminology includes cycle stock, the inventory created by ordering in batches; reorder point, the inventory position at which replenishment is initiated; lead time, the elapsed time between placing and receiving an order; inventory position, on-hand inventory plus on-order inventory minus backorders; and stockout, a condition in which available inventory cannot satisfy demand immediately.

The Basic Economic Order Quantity

The EOQ is the order quantity that minimizes the sum of ordering and holding costs. Differentiating total relevant cost with respect to \(Q\), or balancing the two cost components, gives:

\[

EOQ=Q^*=\sqrt{\frac{2DS}{H}}

\]

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

\[

\frac{D}{Q^}S=\frac{Q^}{2}H

\]

The minimum relevant annual cost is:

\[

TRC^*=\sqrt{2DSH}

\]

If lead time is constant and demand per time unit is \(d\), the reorder point without safety stock is:

\[

ROP=dL

\]

where \(L\) is lead time in the same time units as \(d\). The reorder point determines when to order; EOQ determines how much to order. Confusing these two decisions is a common operational error.

Worked Example: Basic EOQ

A distributor uses 24,000 units annually. The ordering cost is $75 per order and annual holding cost is $3 per unit. The EOQ is:

\[

Q^*=\sqrt{\frac{2(24,000)(75)}{3}}=1,095\text{ units approximately}

\]

The firm places approximately \(24,000/1,095=21.9\), or 22 orders per year. Average cycle inventory is about 548 units. Annual ordering and holding costs are each approximately $1,643, and their sum is about $3,286. If daily demand is 80 units and lead time is five working days, the reorder point is 400 units, provided no safety stock is required.

The result should not be interpreted as a command to order exactly 1,095 units regardless of operational conditions. Packaging multiples, truckload economics, supplier minimums, warehouse capacity, and review practices may require a nearby feasible quantity.

Economic Production Quantity

EOQ assumes that the entire order arrives at one time. Manufacturing replenishment often occurs gradually while demand continues. The economic production quantity (EPQ), also called the economic lot size, accounts for this condition.

Let \(p\) be the production rate and \(d\) the demand rate, with \(p>d\). During a production run, inventory increases at net rate \(p-d\). The maximum inventory is less than the production quantity because some units are consumed while production continues. The EPQ is:

\[

Q^*=\sqrt{\frac{2DS}{H}\left(\frac{p}{p-d}\right)}

\]

Maximum inventory is:

\[

I_{max}=Q^*\left(1-\frac{d}{p}\right)

\]

Average inventory is \(I_{max}/2\). As production rate becomes very large relative to demand, the factor \(p/(p-d)\) approaches one and EPQ approaches EOQ. When production rate is only slightly greater than demand, replenishment is slow and the optimal lot size becomes larger because setup costs must be spread over a longer production cycle.

EPQ is relevant to batch manufacturing, packaging lines, printing, food processing, and internal component production. Its assumptions require stable production and demand rates, reliable quality, negligible machine breakdowns, and a setup cost that does not vary with lot size. In practice, capacity loss, changeover sequence, campaign production, and finite storage may require a production-scheduling model rather than a stand-alone EPQ calculation.

Quantity-Discount Decisions

Suppliers may offer lower unit prices for larger orders. A quantity discount creates a trade-off: the purchase saving may exceed the additional holding and ordering cost, or it may not. Under an all-unit discount, the applicable price applies to every unit in the order once the price-break quantity is reached.

For each price level \(C_j\), calculate:

\[

Q_j^*=\sqrt{\frac{2DS}{H_j}}

\]

If holding cost is expressed as a percentage \(i\) of unit price, then \(H_j=iC_j\). The calculated quantity is feasible only if it lies within the price interval. If it does not, use the nearest feasible boundary, normally the minimum quantity required to obtain that price. Then compare total annual cost:

\[

TC_j=DC_j+\frac{D}{Q_j}S+\frac{Q_j}{2}H_j

\]

The lowest total cost identifies the economically preferred alternative, subject to supplier, cash-flow, quality, and storage constraints.

Discount Illustration

Suppose annual demand is 10,000 units, ordering cost is $50, and holding cost is 20 percent of price. A supplier charges $10 for orders below 1,000 units and $9.70 for orders of 1,000 or more. The EOQ at $10 is:

\[

\sqrt{\frac{2(10,000)(50)}{2}}=707\text{ units}

\]

This is feasible at the $10 price. At $9.70, holding cost is $1.94 and the calculated EOQ is approximately 718 units, which is not sufficient to qualify for the discount. Therefore, the candidate at the discounted price is 1,000 units. Total costs for both candidates must be compared, not merely the purchase prices. The discount is attractive only if its purchase saving exceeds the added inventory-related cost.

A frequent mistake is to calculate one EOQ using the cheapest price and accept it automatically. Correct analysis requires evaluating every relevant price bracket and every feasible candidate.

Planned Shortages and Backordering

Some organisations deliberately permit shortages when customers are willing to wait, when backorders can be recorded, or when the cost of carrying inventory is exceptionally high. Planned shortages do not mean that stockouts are costless. A shortage cost \(\pi\) per unit per year must represent backorder administration, expediting, lost goodwill, contractual penalties, or lost contribution margin.

Under a continuous-review model with instantaneous replenishment and backorders, the optimal order quantity is:

\[

Q^*=\sqrt{\frac{2DS}{H}\left(\frac{H+\pi}{\pi}\right)}

\]

The maximum inventory and maximum shortage are:

\[

I_{max}=Q^*\frac{\pi}{H+\pi}

\]

\[

B_{max}=Q^*\frac{H}{H+\pi}

\]

The proportion of the cycle spent with positive inventory and shortage depends on these values. As shortage cost increases, planned shortage falls and the model approaches the no-shortage EOQ. As shortage cost decreases, the organisation accepts more backorders and carries less stock.

This model is appropriate only where shortages can genuinely be controlled and customers will accept delayed fulfilment. It is unsuitable for emergency medical supplies, safety-critical parts, or markets where a stockout immediately causes permanent customer defection. In such contexts, shortage cost is not easily measured and the service consequences may be nonlinear.

Multi-Item Models under Constraints

Independent EOQ calculations can produce an impractical portfolio. The total order value may exceed a purchasing budget, or the combined average inventory may exceed warehouse space. Multi-item models treat the items jointly.

Assume item \(i\) has demand \(D_i\), ordering cost \(S_i\), holding cost \(H_i\), and decision quantity \(Q_i\). Without constraints, each item has:

\[

Q_i^*=\sqrt{\frac{2D_iS_i}{H_i}}

\]

Suppose an annual investment constraint is imposed:

\[

\sum_i C_i\frac{Q_i}{2}\leq B

\]

where \(B\) is the permitted average inventory investment. A common analytical approach uses a Lagrangian multiplier. The constrained quantity becomes:

\[

Q_i(\lambda)=\sqrt{\frac{2D_iS_i}{H_i+\lambda C_i}}

\]

The multiplier \(\lambda\) is selected so that the budget constraint is satisfied, usually through iteration. The interpretation is managerial: \(\lambda\) represents the marginal cost of scarce investment capacity. Items consuming substantial capital or having high holding costs receive stronger downward adjustment.

For a warehouse-space constraint, if each unit occupies \(v_i\) cubic metres and available average space is \(V\), the constraint is:

\[

\sum_i v_i\frac{Q_i}{2}\leq V

\]

The same logic applies, replacing capital consumption with space consumption. Other constraints may involve order frequency, supplier minimums, vehicle capacity, shelf life, service levels, or a fixed number of replenishment events.

A practical procedure is to calculate unconstrained EOQs, test all constraints, identify the binding constraint, adjust quantities using an optimization method or spreadsheet solver, and validate the result operationally. Rounding must be performed carefully because feasible integer quantities can alter the solution.

Sensitivity Analysis and Managerial Interpretation

EOQ is relatively insensitive to moderate estimation error because it is a square-root function. If demand or ordering cost is multiplied by a factor, the EOQ changes by the square root of that factor. For example, a 21 percent increase in demand raises EOQ by only about 10 percent. A 25 percent increase in holding cost reduces EOQ by about 11 percent.

This robustness does not mean that input accuracy is irrelevant. The total cost curve is flat near the optimum, but errors in lead time, demand timing, supplier reliability, or shortage consequences can substantially affect service performance. Managers should therefore distinguish between cost sensitivity and service sensitivity.

Useful sensitivity tests include:

• varying demand, ordering cost, and holding cost;
• testing alternative lead times and replenishment frequencies;
• examining price-break thresholds;
• changing shortage penalties and service requirements;
• imposing realistic minimums, multiples, and capacity limits; and
• comparing deterministic results with scenario or simulation results.

A sound policy reports a range or recommended operating band rather than presenting the EOQ as a false precision point. The final choice should also consider working capital, supplier collaboration, carbon emissions from ordering and transport, and the strategic cost of supply interruption.

Digital Tools and Contemporary Practice

Enterprise resource planning and warehouse-management systems provide demand histories, purchase prices, lead times, open orders, receipts, and inventory positions. These data support automated EOQ calculations and reorder alerts. Spreadsheet models remain valuable for transparent sensitivity analysis, while optimization solvers can handle many items and constraints.

Advanced systems may combine deterministic benchmarks with probabilistic safety-stock calculations, supplier-performance data, demand sensing, and simulation. Machine learning can improve forecasts or identify changing order patterns, but it does not remove the need to define ordering cost, holding cost, service objectives, and constraints. Automated recommendations can reproduce master-data errors, such as incorrect pack sizes or obsolete prices, at scale. Governance therefore requires data ownership, approval rules, exception monitoring, audit trails, and periodic parameter review.

Sustainability considerations are increasingly integrated into lot-sizing decisions. Larger lots may reduce ordering and transport events but increase obsolescence, energy use for storage, and disposal risk. The economically optimal quantity should therefore be tested against environmental and social objectives where these are material to the organisation.

An Illustrative Applied Case: MedEquip Distribution Centre

Background

MedEquip is a fictional regional distributor of diagnostic cartridges. One cartridge is purchased from an approved supplier and supplied to hospitals. Annual demand is 18,000 units, ordering cost is $90 per order, and annual holding cost is $4.50 per unit. Supplier lead time is four working days, and average daily demand is 75 units.

Situation and Analysis

The basic EOQ is:

\[

Q^*=\sqrt{\frac{2(18,000)(90)}{4.5}}=849\text{ units approximately}

\]

The corresponding number of orders is about 21.2 annually, and average cycle stock is approximately 425 units. With deterministic lead time and demand, the reorder point is 300 units. However, hospitals regard delayed delivery as serious, so the operations manager proposes safety stock. Safety stock is not determined by the deterministic EOQ; it requires a service-level and demand/lead-time variability analysis.

The warehouse then reports that cartridges must be stored in cartons of 100 units and that average available space permits no more than 500 cycle-stock units. The rounded order quantity of 900 produces 450 average units and is feasible. A proposed order quantity of 1,200 would produce 600 average units and violate the space constraint, even though it might reduce order frequency.

A supplier discount of $0.18 per unit is offered at 1,000 units. Management compares the purchase saving with the additional holding cost and the space impact. Because the discounted quantity is feasible but increases average inventory to 500 units, it is acceptable only if the total annual cost and service implications are better. The final decision is 1,000 units if total cost is lower and the warehouse can reliably accommodate the resulting inventory; otherwise, 900 units is retained.

Outcome and Learning Points

The case demonstrates that EOQ is the starting point rather than the complete decision. The organisation must separate lot size from reorder point, test packaging and space constraints, evaluate discounts through total cost, and treat safety stock as a separate response to uncertainty. It also illustrates the need for cross-functional agreement among purchasing, finance, warehouse operations, and clinical-service stakeholders.

Limitations, Risks, and Professional Considerations

Deterministic models can mislead when demand is intermittent, lead time is variable, prices change frequently, or costs are poorly measured. Holding cost should include capital, space, insurance, handling, damage, shrinkage, obsolescence, and relevant taxes—not merely warehouse rent. Ordering cost should include administrative and receiving effort where material.

Other common mistakes include using annual demand with monthly holding costs, ignoring units of time, treating purchase cost as irrelevant when discounts exist, using EOQ when supplier minimums dominate, and rounding without recalculating cost. A mathematically optimal quantity may also be ethically inappropriate if it encourages shortages of essential medicines or safety-critical components.

Professional practice requires transparent assumptions, documented data sources, periodic review, segregation of duties in purchasing, and compliance with contracts and regulated-product requirements. Inventory policies should not create incentives to manipulate demand records, conceal shortages, or over-order merely to obtain performance bonuses. In regulated environments, traceability, expiry control, recall capability, and product integrity may outweigh a small calculated cost saving.

Best-Practice Decision Sequence

A robust implementation follows a sequence: define the item and decision horizon; validate demand, cost, lead-time, and unit data; select the model matching the replenishment process; calculate the unconstrained solution; test discounts and constraints; perform sensitivity analysis; validate service, financial, and operational consequences; pilot the policy; and monitor actual performance. Parameters should be reviewed when suppliers, prices, demand patterns, packaging, or warehouse capacity change.

Chapter Summary

Deterministic inventory models quantify the trade-off between ordering or setup cost and holding cost. The basic EOQ is \(\sqrt{2DS/H}\), while EPQ adjusts for gradual production. Quantity-discount analysis compares feasible quantities across price brackets using total annual cost. Planned-shortage models trade holding cost against explicit shortage cost and are suitable only when delayed fulfilment is acceptable. Multi-item models extend lot-sizing decisions to budget, space, and other constraints. Sensitivity analysis reveals that EOQ is usually cost-robust but that service outcomes may remain sensitive to uncertainty. Effective managerial use requires validated data, operational feasibility, ethical judgement, digital governance, and periodic review.

Key Terms

EOQ: Order quantity that minimizes annual ordering and holding costs under the basic deterministic assumptions.

Economic production quantity: Optimal production lot size when replenishment occurs gradually at a finite production rate.

Cycle stock: Inventory held because replenishment occurs in batches rather than continuously.

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

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

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

Lead time: Time between initiating replenishment and receiving usable inventory.

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

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

Quantity discount: Lower unit price offered when an order reaches a specified quantity.

All-unit discount: Pricing arrangement in which the discount applies to every unit once the threshold is reached.

Planned shortage: Deliberately permitted temporary unavailability supported by a measurable shortage cost.

Backorder: Customer demand recorded for later fulfilment rather than immediately lost.

Shortage cost: Economic cost associated with delayed or lost fulfilment.

Average inventory: Mean inventory level over a replenishment cycle.

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

Sensitivity analysis: Examination of how changes in assumptions affect a model’s result.

Constraint: Limitation on resources such as capital, space, capacity, or order frequency.

Lagrangian multiplier: Analytical device used to value and incorporate a binding resource constraint.

Master data: Controlled item, supplier, cost, packaging, and lead-time information used by information systems.

Review Questions

Short Answer Questions

1. State the assumptions of the basic EOQ model.

2. Distinguish between EOQ and reorder point.

3. Why is EPQ generally larger than EOQ when the production rate is finite?

4. What is meant by an all-unit quantity discount?

5. Define planned shortage and backorder.

Long Answer Questions

1. Derive the basic EOQ formula and explain the cost-minimization logic.

2. Explain the EPQ model and discuss the effect of the production-to-demand rate ratio.

3. Describe the correct procedure for analysing quantity discounts.

4. Explain the planned-shortage model and the role of shortage cost.

5. Discuss how budget and warehouse-space constraints change multi-item inventory decisions.

Analytical / Application Questions

1. Annual demand is 36,000 units, ordering cost is $80, and annual holding cost is $5 per unit. Calculate EOQ, annual order frequency, and average cycle inventory.

2. A supplier offers a discount at a quantity that is above the calculated EOQ. Explain which candidate quantities should be compared and what additional operational factors must be assessed.

3. A warehouse has adequate funds but insufficient space for the independent EOQs of 40 stock-keeping units. Propose an analytical and managerial procedure for developing a feasible portfolio policy.

22. Multiple Choice Questions with Answers

Question 1

Which condition is central to the basic EOQ model?

A. Demand and replenishment are treated as known and constant

B. Demand is highly seasonal and uncertain

C. Shortages are mandatory in every cycle

D. Unit price changes continuously with quantity

Correct Answer: A. Demand and replenishment are treated as known and constant

Question 2

A firm’s annual demand is 20,000 units, ordering cost is $50, and annual holding cost is $2 per unit. What is the approximate EOQ?

A. 500 units

B. 707 units

C. 1,000 units

D. 2,000 units

Correct Answer: C. 1,000 units

Question 3

At the basic EOQ, which relationship holds?

A. Purchase cost equals holding cost

B. Ordering cost equals holding cost

C. Shortage cost equals purchase cost

D. Reorder point equals average inventory

Correct Answer: B. Ordering cost equals holding cost

Question 4

Why does EPQ differ from EOQ?

A. EPQ assumes replenishment occurs gradually while production continues

B. EPQ excludes setup cost

C. EPQ assumes unit price is always discounted

D. EPQ applies only when shortages are planned

Correct Answer: A. EPQ assumes replenishment occurs gradually while production continues

Question 5

Under an all-unit quantity discount, the discounted unit price applies when:

A. Only the units above the threshold are purchased

B. The order reaches the specified price-break quantity

C. Annual demand falls below the threshold

D. The supplier has excess warehouse capacity

Correct Answer: B. The order reaches the specified price-break quantity

Question 6

When analysing a quantity discount, why must total annual cost be calculated for each feasible candidate?

A. The lowest unit price always has the lowest total cost

B. Purchase savings must be compared with added holding and ordering costs

C. Ordering cost becomes zero at every discount level

D. EOQ cannot be calculated at any price level

Correct Answer: B. Purchase savings must be compared with added holding and ordering costs

Question 7

If shortage cost increases substantially in a planned-shortage model, the optimal policy will generally:

A. Permit a larger maximum shortage

B. Carry less inventory and order less frequently

C. Permit a smaller shortage and move toward a no-shortage policy

D. Ignore holding cost

Correct Answer: C. Permit a smaller shortage and move toward a no-shortage policy

Question 8

Which expression represents the deterministic reorder point when demand and lead time are constant?

A. Ordering cost divided by demand

B. Demand per time unit multiplied by lead time

C. EOQ divided by holding cost

D. Annual demand multiplied by unit price

Correct Answer: B. Demand per time unit multiplied by lead time

Question 9

A multi-item inventory model is most likely required when:

A. Only one item is stocked with unlimited resources

B. Items jointly compete for budget or warehouse space

C. Demand is zero for all items

D. There are no ordering costs

Correct Answer: B. Items jointly compete for budget or warehouse space

Question 10

Which statement best describes EOQ sensitivity?

A. A 25 percent increase in holding cost causes a 25 percent decrease in EOQ

B. EOQ changes in proportion to every input

C. Because EOQ is a square-root function, moderate input errors often cause smaller proportional changes in quantity

D. EOQ is unaffected by holding cost

Correct Answer: C. Because EOQ is a square-root function, moderate input errors often cause smaller proportional changes in quantity