Deterministic Lot-Sizing Models

Introduction

Inventory decisions determine when replenishment occurs and how much material is ordered or produced. In a deterministic environment, demand, replenishment time, costs, and operating conditions are treated as known with sufficient certainty. Deterministic lot-sizing models therefore provide a transparent foundation for understanding the economic trade-offs among ordering, setup, holding, purchasing, production, and shortage costs.

The central managerial problem is not simply to minimize the purchase price of an item. It is to determine a replenishment quantity that minimizes relevant total cost while maintaining feasible operations. Large lots reduce the frequency of orders or production setups, but they increase average inventory and exposure to obsolescence. Small lots reduce inventory, but they increase ordering, setup, transportation, and administrative costs. Quantity discounts and planned shortages add further dimensions to the decision.

These models are stylized. Their value lies in clarifying cost relationships, identifying influential parameters, and creating benchmarks against which more complex policies can be evaluated.

Learning Objectives

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

1. Formulate and interpret the economic order quantity model.
2. Apply the economic production quantity model when replenishment is gradual.
3. Evaluate all-in-unit-cost alternatives under quantity discounts.
4. Formulate a planned-shortage model and interpret its service implications.
5. Assess sensitivity, assumptions, implementation requirements, and managerial limitations.
6. Distinguish a mathematically economical lot size from an operationally and strategically appropriate policy.

1. The Logic of Deterministic Lot Sizing

A lot is the quantity ordered or produced at one replenishment event. Lot-sizing models balance costs that move in opposite directions. Ordering or setup cost per event encourages large lots because a fixed cost is spread over more units. Holding cost encourages small lots because inventory remains in storage and ties up capital. Purchase-price discounts may encourage larger lots, while shortage costs and service expectations generally encourage smaller planned shortages or no shortages at all.

For a repeating cycle with annual demand D, order quantity Q, and constant demand rate, the number of replenishments per year is D/Q. If replenishment is instantaneous and inventory falls linearly from Q to zero, average cycle inventory is Q/2. Thus, the basic relevant annual cost is:

where:

• TC(Q) = relevant annual cost, in currency per year;
• D = annual demand, units per year;
• Q = replenishment quantity, units per order;
• S = ordering or setup cost, currency per order or setup;
• H = annual holding cost, currency per unit per year.

The first term decreases as Q increases, whereas the second increases. This opposing behavior creates an interior cost minimum.

If purchase cost is constant and unavoidable, annual purchasing expenditure DC may be added, where C is unit price. It does not affect the optimal lot size when price is independent of quantity:

The model assumes continuous review, no stockouts, constant demand, immediate replenishment, and a constant unit price. These assumptions must be checked before applying the result.

2. Economic Order Quantity

2.1 Derivation

The economic order quantity (EOQ) minimizes the ordering-plus-holding portion of cost. Differentiating the relevant cost function with respect to Q gives:

Setting the derivative equal to zero produces:

The second derivative is:

for positive D, S, and H. Therefore, the stationary point is a minimum.

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

Consequently, the minimized relevant cost is:

The formula is dimensionally consistent: DS has units of currency per year, and division by H (currency per unit per year) gives units squared; the square root gives units.

2.2 Operating measures

The number of orders per year is:

If the organization operates W working days per year, the time between orders is:

The reorder point under deterministic demand and lead time L is:

where d is demand per unit of time and L is lead time in the same time unit. With no uncertainty and no safety stock, inventory reaches zero exactly when the replenishment arrives. If safety stock SS is used, then:

Safety stock is not part of the classical deterministic EOQ derivation; it is an additional protection against uncertainty.

2.3 Worked example: EOQ

A distributor sells 24,000 units annually. Each order costs $75 to place, and annual holding cost is $3 per unit. Demand is steady, replenishment is instantaneous, and shortages are not permitted.

A practical policy might use 1,095 or 1,100 units, subject to packaging and supplier constraints. At Q = 1,095.45, annual ordering cost is approximately:

Annual holding cost is:

The minimized relevant annual cost is approximately $3,286.34. The number of replenishments is approximately 21.91 per year. The fractional value is an analytical average; actual operations will use an integer number of orders and a feasible lot size.

3. Economic Production Quantity

3.1 Why EOQ may be inappropriate

EOQ assumes that the full lot becomes available at once. In manufacturing, however, production and consumption may occur simultaneously. During a production run, inventory rises only by the difference between production rate and demand rate. The maximum inventory is therefore less than the production lot size.

Let:

• p = production rate, units per time;
• d = demand rate, units per time;
• p > d = necessary feasibility condition;
• Q = production lot size;
• S = setup cost per production run;
• H = annual holding cost per unit.

The fraction of the lot consumed while production is occurring is d/p. Maximum inventory is:

Average inventory is half this amount:

The annual relevant cost becomes:

Minimizing gives the economic production quantity (EPQ):

Because 1 - d/p is less than one, EPQ is larger than EOQ when all other parameters are unchanged. Gradual replenishment reduces average inventory, permitting a larger production lot without increasing holding cost excessively.

3.2 Worked example: EPQ

A plant requires 50,000 units annually. A production run has setup cost of $500. The annual holding cost is $2 per unit. The production rate is 1,000 units per day and demand is 400 units per day.

Since 1 - 0.4 = 0.6,

Maximum inventory is approximately:

Average inventory is approximately 1,936.5 units. Using EOQ without recognizing production replenishment would overstate average inventory and could lead to an unnecessarily small production lot.

The EPQ requires a stable production rate, a setup cost that is reasonably constant, and production capacity sufficient to exceed demand. If production is interrupted, yields are uncertain, or capacity is shared across many products, a finite-capacity scheduling model is more appropriate.

4. Quantity-Discount Models

4.1 Structure of the decision

A supplier may offer a lower unit price when order quantity reaches a threshold. Because purchase cost now depends on Q, the price cannot be ignored. For an all-units discount schedule, every unit in an order receives the price associated with the applicable quantity bracket.

For price level C_j, the annual total cost is:

If holding cost is an annual percentage i of unit price, then:

The unconstrained EOQ for bracket j is:

The correct procedure is:

1. Compute the EOQ for each price level.
2. Check whether that EOQ lies within its price bracket.
3. If it does not, use the nearest feasible breakpoint in that bracket, usually the minimum quantity required to obtain the price.
4. Compute total annual cost for every feasible candidate, including purchase cost.
5. Select the candidate with the lowest total cost.

A common error is to select the lowest unit price without comparing total cost. A discount is economical only if its purchasing savings exceed the additional holding and ordering-related costs.

4.2 Worked example: all-units discount

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

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

This is not feasible in the first bracket, so the best candidate there is 499. For the second bracket, H = 0.20 × 9.80 = $1.96:

This is feasible. For the third bracket, H = 0.20 × 9.50 = $1.90:

This is not feasible because at least 1,000 units are required, so the candidate is 1,000.

Evaluate total cost:

The 1,000-unit candidate is best under the stated assumptions. The result should still be tested against storage capacity, cash-flow constraints, shelf life, and supplier reliability.

Incremental discounts, in which only units above a threshold receive a lower price, require a different purchase-cost calculation. Applying an all-units formula to an incremental schedule can produce an incorrect decision.

5. Planned-Shortage Model

5.1 Rationale and assumptions

Some systems permit temporary shortages. Customers may wait, backorders may be accepted, or production may be delayed. A planned-shortage model trades holding cost against shortage or backorder cost. It is relevant only when shortages are operationally and contractually acceptable.

Let:

• B = shortage cost per unit per year, including backorder, expediting, lost contribution, or service penalties;
• S = ordering cost;
• H = holding cost;
• Q = total cycle quantity;
• S_max = maximum planned shortage;
• I_max = maximum positive inventory.

With a fixed replenishment quantity and a linear shortage-cost model, the optimal maximum shortage and positive inventory are:

Thus, the share of the cycle represented by shortage is larger when shortage cost is relatively low. The optimal total lot size is:

The maximum shortage and inventory can also be written directly as:

These formulas assume linear holding and shortage costs, instantaneous replenishment, constant demand, complete backorder fulfillment, and no fixed service-level requirement. If shortages cause permanent lost sales rather than delayed fulfillment, the model must be reformulated.

As B → ∞, planned shortage approaches zero and the model approaches the no-shortage EOQ logic. As B becomes small relative to H, more shortage is economically tolerated. This does not mean that management should automatically permit shortages: reputational, safety, legal, and strategic consequences may not be captured by B.

6. Sensitivity and Robustness

EOQ has a square-root structure. Therefore, errors in inputs are moderated but not eliminated. If demand changes by a factor k, the optimal quantity changes by √k. A 21 percent increase in demand increases EOQ by only about 10 percent. Similarly, doubling ordering cost increases EOQ by approximately 41 percent, while doubling holding cost reduces it by approximately 29 percent.

At a selected quantity Q, the cost penalty for deviating from EOQ can be expressed relative to minimum relevant cost. Let r = Q/Q*. Then:

This result shows why EOQ is often a useful benchmark even when the exact input estimates are uncertain. If r = 1.2, the ratio is approximately 1.017, so the relevant cost is only about 1.7 percent above the theoretical minimum. However, small cost penalties do not justify ignoring capacity, supplier constraints, or service requirements.

Sensitivity analysis should vary:

• demand volume and seasonality;
• ordering or setup cost;
• holding-cost rate and working-capital cost;
• lead time and transportation constraints;
• price-break thresholds;
• shortage and customer-service costs;
• usable storage and shelf-life limits.

Scenario analysis is particularly important when a parameter is not measured directly. For example, setup cost may include scheduling disruption, quality checks, labor, downtime, and lost capacity—not merely an accounting charge.

7. Integrated Applied Case: Meridian Components

Background

Meridian Components is a fictional manufacturer of a standard coupling used in industrial pumps. Annual demand is 50,000 units. The company currently produces 10,000 units per run. Production capacity is 1,000 units per day, demand is 400 units per day, setup cost is estimated at $500 per run, and annual holding cost is $2 per unit.

Problem

The production manager wants to know whether the current lot size is economically defensible. A sales manager proposes accepting backorders during scheduled production cycles to reduce inventory. The finance manager is concerned about working capital, while operations is concerned about additional setups.

Analysis

The EPQ calculated earlier is approximately 6,455 units. At the current lot size of 10,000 units:

At the EPQ:

The EPQ therefore reduces average cycle inventory by approximately 1,063.5 units. The current policy also creates fewer setups, but the holding-cost component is higher.

Suppose Meridian estimates a credible shortage cost of $12 per unit per year for backordered demand, provided that customers accept a maximum delay of one production cycle. Using the planned-shortage formula with H = 2 and B = 12:

The associated maximum planned shortage is:

This is not an automatic recommendation. The company must verify whether 771 units of backorders can be absorbed without losing customers, violating contracts, or disrupting downstream production.

Approach and solution

Meridian should adopt a staged decision:

1. Use EPQ of approximately 6,455 units as the no-shortage benchmark.
2. Validate setup-cost estimates through time studies and capacity accounting.
3. Pilot a smaller lot size only for customers with explicit backorder agreements.
4. Set a hard service constraint that prevents shortages for safety-critical customers.
5. Recalculate the policy using actual setup time, production yield, demand patterns, and expedited-shipping costs.
6. Review the decision monthly rather than treating the calculated quantity as permanent.

Outcome

Under the stated assumptions, the current 10,000-unit lot is larger than the EPQ benchmark. A planned-shortage policy could reduce positive inventory further, but it exposes Meridian to service and relationship risks not fully represented by a simple shortage cost. The defensible policy is therefore a controlled pilot with customer segmentation and explicit governance, not an unrestricted move to backordering.

Learning points

The case illustrates four principles. First, the replenishment mechanism determines whether EOQ or EPQ is appropriate. Second, shortage models require managerial acceptance of delayed fulfillment. Third, the numerical optimum is conditional on cost estimates and constraints. Fourth, lot sizing is a cross-functional decision involving operations, finance, sales, procurement, and customers.

8. Managerial Implementation and Limitations

Data and governance

Implementation requires consistent definitions. Demand should be measured over a relevant horizon; ordering cost should include administrative and receiving activities; setup cost should include downtime and changeover labor; and holding cost should include capital, space, insurance, damage, obsolescence, and handling. Organizations should document the source, date, owner, and confidence level of each parameter.

Operational constraints

Classical formulas do not directly incorporate minimum order quantities, truckload or pallet multiples, supplier capacity, storage limits, shelf life, order calendars, common components, or production sequencing. The calculated result must therefore be rounded or constrained and then re-evaluated using the actual total-cost function.

Demand and lead-time uncertainty

Deterministic models do not determine safety stock. When demand or lead time is uncertain, reorder-point and safety-stock methods are needed. Using EOQ with a stochastic reorder point is common, but the two decisions should not be confused: EOQ concerns replenishment quantity, whereas safety stock protects against uncertainty.

Strategic and ethical considerations

Minimizing private inventory cost can transfer costs to suppliers, customers, or communities. Excessive order fragmentation may increase transportation emissions and packaging waste. Planned shortages can disadvantage vulnerable customers or critical-service users. Governance should therefore consider service equity, contractual commitments, product safety, environmental impact, and transparency of shortage policies.

Technology and AI

Enterprise resource planning and advanced planning systems can automate lot-size calculations, price-break evaluation, and constraint checks. Machine-learning forecasts may improve demand estimates, but they do not remove the need to specify ordering, holding, setup, and shortage costs. AI-generated recommendations should be auditable: users should be able to inspect parameter sources, assumptions, constraints, and the reason a particular candidate was selected. A sophisticated optimization system with poor cost data can produce a precise but misleading answer.

Best-practice checklist

• State the replenishment process before selecting a formula.
• Use consistent time units and currency units.
• Include purchase cost when price depends on quantity.
• Evaluate feasible candidates, not only unconstrained optima.
• Separate cycle stock from safety stock.
• Test sensitivity to uncertain parameters.
• Validate results against capacity, storage, shelf life, and service requirements.
• Review assumptions after major changes in demand, suppliers, technology, or strategy.

Chapter Summary

Deterministic lot-sizing models explain how organizations balance fixed ordering or setup costs against inventory and shortage costs. The EOQ model applies when replenishment is instantaneous, demand is stable, unit price is constant, and shortages are prohibited. Its optimum is Q* = √(2DS/H), and ordering cost equals holding cost at the optimum.

The EPQ model recognizes gradual production and uses the factor 1 - d/p to account for simultaneous production and consumption. Quantity-discount models require candidate-by-candidate comparison because unit price and sometimes holding cost vary by bracket. Planned-shortage models permit backorders or delayed fulfillment and balance holding cost against shortage cost, but their economic result must be tested against customer, contractual, ethical, and strategic consequences.

All four models are benchmarks rather than automatic policies. Their usefulness depends on valid data, explicit assumptions, feasibility checks, sensitivity analysis, and managerial judgment. Modern planning systems can automate calculations, but responsible decision-making still requires human oversight and transparent governance.

Key Terms

Economic order quantity (EOQ): Replenishment quantity that minimizes ordering and holding costs under the classical instantaneous-replenishment assumptions.

Economic production quantity (EPQ): Production lot size that minimizes setup and holding costs when replenishment occurs gradually.

Lot size: Quantity ordered or produced at one replenishment event.

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

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

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

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

Average inventory: Mean inventory level over a replenishment cycle.

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

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

Safety stock: Additional inventory held to protect against uncertainty.

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

All-units discount: Pricing structure in which all units receive the price associated with the quantity bracket.

Incremental discount: Pricing structure in which only units above a threshold receive the lower price.

Planned shortage: Deliberately permitted temporary negative inventory or backorder.

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

Shortage cost: Economic cost associated with unavailable inventory or delayed fulfillment.

Sensitivity analysis: Examination of how results change when model inputs vary.

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

Deterministic model: Model in which key inputs are treated as known rather than probabilistic.

Review Questions

Short Answer Questions

1. What opposing cost forces are balanced by the EOQ model?
2. Why is average inventory in the classical EOQ model equal to Q/2?
3. Why is EPQ generally larger than EOQ under comparable conditions?
4. Why must purchase cost be included in a quantity-discount analysis?
5. What is the managerial meaning of a planned shortage?

Long Answer Questions

1. Derive the EOQ formula from the annual relevant-cost function and explain why the stationary point is a minimum.
2. Compare EOQ and EPQ assumptions, formulas, and operational applications.
3. Explain the correct procedure for evaluating an all-units quantity-discount schedule.
4. Discuss the assumptions and managerial risks of the planned-shortage model.
5. Evaluate the usefulness and limitations of deterministic lot-sizing models in a modern supply chain.

Analytical / Application Questions

1. Annual demand is 36,000 units, ordering cost is $90, and annual holding cost is $4 per unit. Calculate EOQ, annual ordering cost, annual holding cost, and the number of orders per year.
2. A factory produces at 800 units per day and consumes 300 units per day. Explain how using an instantaneous-replenishment model would distort average inventory, and identify the appropriate model.
3. A supplier offers three price brackets with different minimum quantities. Design a spreadsheet or algorithm that identifies the least-cost feasible candidate, including purchase, ordering, and holding costs. State how you would test the result for storage and service constraints.

Multiple Choice Questions with Answers

Question 1

Which assumption is essential to the classical EOQ model?

A. Demand is probabilistic and highly seasonal

B. Replenishment is instantaneous

C. Shortages are mandatory

D. Unit price decreases continuously with quantity

Correct Answer: B. Replenishment is instantaneous

Question 2

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

A. 500 units

B. 707 units approximately

C. 1,000 units

D. 2,500 units

Correct Answer: B. 707 units approximately

Question 3

At the classical EOQ, which equality holds?

A. Purchase cost equals holding cost

B. Ordering cost equals holding cost

C. Shortage cost equals setup cost

D. Demand equals safety stock

Correct Answer: B. Ordering cost equals holding cost

Question 4

In the EPQ model, why is average inventory lower than Q/2?

A. Demand stops during production

B. Production and demand occur simultaneously

C. Ordering cost is zero

D. Shortages are always planned

Correct Answer: B. Production and demand occur simultaneously

Question 5

For EPQ to be feasible, which condition must hold?

A. Production rate is less than demand rate

B. Production rate equals zero

C. Production rate exceeds demand rate

D. Holding cost must be zero

Correct Answer: C. Production rate exceeds demand rate

Question 6

In an all-units quantity-discount problem, what should be done if the EOQ is outside its price bracket?

A. Use the EOQ without modification

B. Ignore purchase cost

C. Evaluate the nearest feasible breakpoint

D. Automatically choose the lowest price

Correct Answer: C. Evaluate the nearest feasible breakpoint

Question 7

What additional cost must be included when unit price depends on order quantity?

A. Annual purchase cost

B. Only safety-stock cost

C. Only transportation revenue

D. Depreciation of customer equipment

Correct Answer: A. Annual purchase cost

Question 8

In the planned-shortage model, increasing shortage cost generally causes the optimal policy to have:

A. More planned shortage

B. Less planned shortage

C. No ordering cost

D. A lower unit demand rate

Correct Answer: B. Less planned shortage

Question 9

What does the reorder point represent in a deterministic system with demand rate d and lead time L?

A. d ÷ L

B. d + L

C. dL

D. L − d

Correct Answer: C. dL

Question 10

Which statement best describes sensitivity analysis for EOQ?

A. It proves all inputs are exact

B. It examines how the decision changes when inputs vary

C. It eliminates the need for feasibility checks

D. It applies only to shortage models

Correct Answer: B. It examines how the decision changes when inputs vary