Extended Lot-Sizing and Deterministic Models

Introduction

Inventory decisions become substantially more difficult when replenishment is produced internally, shortages are permitted, suppliers offer price breaks, several items compete for scarce resources, or demand varies over time. The basic economic order quantity model is therefore only a starting point. This chapter develops deterministic extensions that support production, purchasing, capacity, and time-phased planning decisions.

The central managerial question is not simply “how much should be ordered?” It is: when and in what quantity should inventory be replenished so that the relevant total cost is minimized while service and resource constraints are respected?

Learning objectives

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

1. Derive and apply the economic production quantity model.
2. Analyse planned shortages and distinguish shortage cost from lost-sales consequences.
3. Evaluate all-in-unit quantity-discount alternatives.
4. Formulate constrained multi-item lot-sizing problems.
5. Explain the logic and optimality conditions of the Wagner–Whitin algorithm.
6. Apply practical heuristics when exact dynamic programming is inconvenient.
7. Interpret model assumptions, sensitivity, implementation risks, and governance implications.

1. Model foundations and notation

Deterministic lot-sizing models assume that demand, costs, and lead times are known. They are normative planning models rather than forecasts. Common notation is shown below.

The relevant total cost normally includes purchase or production cost, setup or ordering cost, holding cost, shortage cost, and—where appropriate—transportation, quality, or changeover costs. Costs that do not change between alternatives should be excluded from the comparison.

2. Economic production quantity

2.1 Logic of the model

In the economic order quantity model, replenishment is instantaneous. In many factories, however, items are produced at a finite rate p while customers consume them at rate d. Inventory therefore rises gradually during production. Because production and consumption occur simultaneously, the maximum inventory is less than the production quantity.

Assumptions are:

• demand is constant and known;
• production is continuous at a constant rate p, where p > d;
• no shortages are allowed;
• setup cost is fixed per production run;
• unit production cost is independent of lot size;
• holding cost is proportional to average inventory;
• capacity and quality constraints are ignored in the basic model.

During a production run of Q units, production takes Q/p time units and consumption removes dQ/p units. Thus maximum inventory is:

Average inventory is half of the maximum:

The annual relevant cost is:

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

The expression is dimensionally consistent: SD has currency, while H(1−d/p) has currency per unit squared per year when D and d are expressed consistently.

The number of production runs per year is D/Q_EPQ, and the production-run duration is Q_EPQ/p. The cycle time is Q_EPQ/D.

2.2 Worked example

A plant faces annual demand of 24,000 units. Setup cost is $180 per run, holding cost is $4 per unit per year, and the production rate is 80,000 units per year. Demand rate is 24,000 units per year.

Maximum inventory is:

The finite production rate reduces inventory relative to an instantaneous-replenishment model. If p approaches d, the factor 1−d/p approaches zero and the required lot size grows sharply; production has little opportunity to build inventory.

3. Planned-shortage model

A planned-shortage model permits inventory to become negative, interpreting negative inventory as backorders. It is suitable only when customers accept delayed delivery and when backlog costs can be estimated reliably. It is not automatically appropriate for lost sales, safety-critical items, or contractual service requirements.

Assume instantaneous replenishment, constant demand, fixed ordering cost S, holding cost H per unit per time, and backorder cost π per unit per time. Let Q be the replenishment quantity and B the maximum backorder. The maximum positive inventory is Q − B. Over a cycle, the average positive inventory and average backlog are triangular areas divided by cycle time:

The relevant cost is:

Optimizing with respect to Q and B gives:

The maximum positive inventory is therefore:

As shortage cost π becomes very large, B* approaches zero and the model converges toward the no-shortage EOQ. If π is low, larger planned backlogs are economically indicated. The result must nevertheless be screened against customer tolerance, legal obligations, and operational feasibility.

4. Quantity-discount models

Quantity discounts create a trade-off between lower unit price and higher average inventory. The most common case is an all-units discount: once a qualifying quantity is purchased, every unit in the order receives the lower price.

For price level j, let c_j be the unit price and let Q_j be the EOQ calculated using that price’s holding cost. If holding cost is a percentage i of unit price, then H_j = ic_j. If holding cost is specified independently, use that stated value.

The procedure is:

1. Compute the unconstrained EOQ for each price level.
2. If an EOQ is below the minimum quantity required for that price, replace it with the price-break quantity.
3. Compute total annual cost for every feasible candidate.
4. Select the candidate with the lowest total cost.

For all-units discounts:

For incremental discounts, the purchase-cost term must be calculated in blocks; the all-units formula must not be used.

Example

Annual demand is 10,000 units, ordering cost is $60, and holding cost is 20% of unit price. Prices are $10 for 1–499 units, $9.70 for 500–999, and $9.40 for 1,000 or more.

The EOQ at each price is:

At $10: Q≈245, infeasible for the first break only if the candidate is being evaluated at another price; it is feasible in the 1–499 interval. At $9.70, the EOQ is approximately 249, below the 500-unit threshold, so use 500. At $9.40, the EOQ is approximately 252, below 1,000, so use 1,000.

Candidate costs are approximately:

• Q = 245, c = $10: purchase $100,000 + ordering $2,449 + holding $245 = $102,694.
• Q = 500, c = $9.70: purchase $97,000 + ordering $1,200 + holding $485 = $98,685.
• Q = 1,000, c = $9.40: purchase $94,000 + ordering $600 + holding $940 = $95,540.

The 1,000-unit purchase is preferred in this illustration. The conclusion would change if storage, obsolescence, cash-flow, or shelf-life costs were omitted from the simplified model.

5. Constrained multi-item lot sizing

When several items share a machine, warehouse, budget, or supplier capacity, optimizing each item independently may produce an infeasible plan. A basic multi-item model is:

subject to:

where a_i is the resource consumed per order or setup and K is available resource capacity. Additional constraints may include warehouse space, investment budget, minimum order quantities, service levels, and shared setup families.

A continuous relaxation can be solved using a Lagrangian multiplier λ. For a capacity constraint measured in setup resource, the item-specific decision becomes:

Because Q_i appears on both sides in that particular formulation, a more useful operational approach is to define setup frequency n_i = D_i/Q_i. Then:

and the capacity constraint becomes ∑a_in_i≤K. For a continuous frequency relaxation, the first-order condition is:

λ is adjusted until the capacity constraint is satisfied, after which nearby integer frequencies and minimum quantities must be tested. In practice, mixed-integer programming is preferable when setup, sequence, and capacity constraints are numerous.

Managerial interpretation is important: the best item-specific lot size is not necessarily the best system-wide lot size. A high-frequency item may consume scarce changeover time while a low-frequency item may justify a larger lot because its setup cost is high.

6. Dynamic lot sizing and the Wagner–Whitin model

Static models assume constant demand. In a finite planning horizon, demand may vary by period. Let D_t denote demand in period t, S the setup cost whenever production occurs, and h the holding cost per unit carried from one period to the next. Assume:

• demand is known for each period;
• replenishment arrives at the beginning of a period;
• no shortages are allowed;
• production capacity is unlimited;
• unit variable cost is constant or irrelevant to timing;
• inventory is carried forward at linear holding cost.

The Wagner–Whitin property states that, under these assumptions, an optimal plan places production only when inventory is zero immediately before production. In other words, it is unnecessary to produce in a period while positive inventory remains from an earlier production decision. This property substantially reduces the search space.

Let F_t be the minimum cost of satisfying demand through period t, excluding constant unit costs. If one setup in period k covers demand through period t, the holding cost is:

where demand in period r is carried for t − r periods. The dynamic-programming recursion is:

with F₀=0. Store the minimizing k as the predecessor decision, then backtrack from period T to recover the production schedule.

Small illustration

Suppose three periods have demands 40, 70, and 50 units; setup cost is $100; holding cost is $2 per unit per period. Candidate costs are:

• One setup in period 1 for all demand: 100 + 2(70)(1) + 2(50)(2) = 440.
• Setups in periods 1 and 2: 100 + 100 + 2(50)(1) = 300.
• Setups in periods 1, 2, and 3: 300.

The minimum is $300, with production of 40 units in period 1 and 70 and 50 units in periods 2 and 3. In this particular data set, producing period-3 demand early costs exactly the additional setup, so two or three setups tie. A complete dynamic-programming table is required to verify all alternatives; this example illustrates why backtracking and explicit cost comparison matter.

7. Heuristic approaches

Exact dynamic programming is valuable but may become computationally expensive when capacity, multiple items, sequence-dependent setups, or rolling horizons are introduced. Common heuristics include:

• Lot-for-lot: produce exactly the requirement of each period. It minimizes inventory but may create many setups.
• Fixed-period requirements: cover demand for a selected number of future periods.
• Part-period balancing: extend a lot while the additional holding cost is close to the setup cost.
• Silver–Meal heuristic: choose the number of periods that minimizes average cost per period for a replenishment cycle.
• Least-unit-cost heuristic: extend coverage while average cost per unit decreases.

For Silver–Meal, if a lot beginning in period k covers through t, average cost is:

Extend t while average cost falls; stop before the first increase. These methods are fast and transparent but are not guaranteed to produce the global optimum. Their quality should be benchmarked against dynamic programming on representative instances.

8. Illustrative case: Apex Components

Background

Apex Components produces a precision bracket used in three assemblies. Demand is seasonal, and the heat-treatment furnace is shared by six products. Each run requires a costly setup and consumes limited furnace time.

Problem

The production planner initially applies EPQ independently to every item. The resulting plan exceeds furnace capacity by 14%, creates excess inventory before the seasonal peak, and occasionally backlogs a low-volume customer order because the schedule is overloaded.

Analysis

The planner first estimates demand by period, setup times, holding costs, and customer-approved backlog costs. The items are then separated into two decision groups:

1. stable-demand items, for which EPQ provides a useful baseline;
2. seasonal items, for which time-phased Wagner–Whitin analysis is more appropriate.

A capacity constraint is added to the multi-item model. Setup families are sequenced to reduce furnace cleaning and temperature-change losses. Candidate schedules are evaluated using total relevant cost plus a documented penalty for service violations.

Approach and solution

A mixed-integer planning model is used for the monthly master schedule, while a rolling Wagner–Whitin calculation generates initial lot-sizing proposals. The proposals are adjusted for furnace capacity, then tested through discrete-event simulation. A modest increase in setup frequency for the highest-margin item is accepted because it prevents a much larger service failure. The final schedule uses smaller lots before the peak and larger lots for stable items after capacity is available.

Outcome and learning points

The illustrative solution reduces average inventory by 11%, eliminates the capacity violation, and maintains the required service level. The key lesson is that an unconstrained optimum is not a feasible production plan. Lot-sizing must be integrated with capacity, sequencing, customer service, and financial consequences.

9. Limitations, governance, and contemporary practice

Deterministic models can create false precision. Forecast error, disruptions, supplier unreliability, variable processing times, and correlated demand make the inputs uncertain. Sensitivity analysis should vary demand, setup cost, holding cost, production rate, and shortage penalty. Scenario analysis can compare normal, peak, disruption, and recovery conditions.

Governance controls should include versioned master data, approval of price and cost assumptions, audit trails for manual overrides, and explicit documentation of service constraints. A shortage penalty must not be used to justify unsafe or discriminatory allocation of essential products. For regulated goods, expiry, traceability, recall exposure, and minimum service obligations may dominate purely financial optimization.

Enterprise-resource-planning and advanced-planning systems can automate calculations, but technology does not correct poor data. AI-based demand forecasts may improve inputs, yet lot-sizing decisions should remain explainable, constraint-aware, and subject to human review. A useful practice is to compare an optimization recommendation with a robust plan that remains acceptable under plausible forecast errors.

Best-practice checklist

• Define the demand horizon, time bucket, and units consistently.
• Distinguish holding, shortage, setup, purchase, and quality costs.
• Test feasibility before comparing cost.
• Use EPQ only when production is finite and stable.
• Evaluate every feasible price-break candidate.
• Use dynamic programming for small, uncapacitated time-phased problems.
• Use mixed-integer optimization or decomposition for complex constraints.
• Benchmark heuristics against exact solutions on sample cases.
• Perform sensitivity and scenario analysis.
• Document assumptions, overrides, and governance responsibilities.

Chapter Summary

Extended lot-sizing models adapt basic inventory logic to production, shortages, discounts, shared constraints, and changing demand. EPQ accounts for gradual replenishment and yields a lot size larger than the instantaneous-replenishment result when production is finite. Planned-shortage models balance holding and backlog costs but require genuine customer acceptance of delayed delivery. Quantity-discount analysis must compare feasible price-break candidates using the correct discount structure. Multi-item models reveal the importance of shared capacity and setup resources. For deterministic, uncapacitated, time-phased demand, the Wagner–Whitin dynamic program provides an exact solution under its stated assumptions. Heuristics offer speed and practicality but should be validated against exact or benchmark solutions. In professional practice, feasibility, service, data governance, and sensitivity analysis are as important as nominal cost minimization.

Key Terms

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

Production rate: Rate at which a manufacturing system produces units.

Planned shortage: Intentionally permitted temporary negative inventory or backlog.

Backorder cost: Cost incurred per backordered unit per time period.

All-units discount: Pricing rule in which all units receive the lower price after a threshold is reached.

Incremental discount: Pricing rule in which only units in specified quantity blocks receive discounted prices.

Price-break quantity: Minimum quantity required to qualify for a discount.

Multi-item lot sizing: Coordinated lot-size decisions for several items.

Setup cost: Fixed cost incurred when starting an order or production run.

Capacity constraint: Limit on a resource available for production or replenishment.

Dynamic lot sizing: Lot sizing with demand that varies across discrete time periods.

Wagner–Whitin property: Optimality condition eliminating production while positive inventory remains.

Dynamic programming: Method that solves a multistage optimization problem through recursive subproblems.

Part-period balancing: Heuristic that compares holding cost with setup cost over future periods.

Silver–Meal heuristic: Heuristic minimizing average cost per period for a replenishment cycle.

Backtracking: Recovering decisions from stored dynamic-programming predecessor choices.

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

Robust plan: Plan designed to remain acceptable across plausible uncertain conditions.

Review Questions

Short Answer Questions

1. Why is maximum inventory in EPQ less than the production lot size?
2. What managerial conditions justify a planned-shortage model?
3. Distinguish all-units and incremental quantity discounts.
4. State the main assumptions behind the Wagner–Whitin property.
5. Why can independently optimized item lot sizes be infeasible?

Long Answer Questions

1. Derive the EPQ formula and explain the significance of the factor 1−d/p.
2. Explain the planned-shortage model, including the roles of holding cost and backlog cost.
3. Describe a rigorous procedure for solving all-units quantity-discount problems.
4. Explain how dynamic programming solves deterministic time-phased lot sizing.
5. Critically compare exact dynamic programming with lot-sizing heuristics.

Analytical / Application Questions

1. A plant has annual demand of 30,000 units, setup cost of $250, holding cost of $5 per unit-year, demand rate of 30,000 units/year, and production rate of 90,000 units/year. Calculate the EPQ and maximum inventory.
2. A company has four items sharing a machine-hour constraint. Explain how setup frequency, holding cost, and resource consumption should be incorporated into a coordinated model.
3. Monthly demand is known but highly seasonal, while setup capacity is limited. Recommend a planning approach combining an exact model, a heuristic, and sensitivity analysis; justify each component.

Multiple Choice Questions with Answers

Question 1

In the EPQ model, why is maximum inventory less than Q?

A. Some units are defective

B. Demand occurs while production is continuing

C. The supplier applies a discount

D. Shortages are mandatory

Correct Answer: B. Demand occurs while production is continuing

Question 2

Which condition is required for the standard EPQ formula?

A. Production rate is less than demand rate

B. Production rate equals demand rate

C. Production rate exceeds demand rate

D. Demand is completely random

Correct Answer: C. Production rate exceeds demand rate

Question 3

If shortage cost π increases substantially in the planned-shortage model, the optimal maximum backlog generally:

A. Increases without limit

B. Decreases toward zero

C. Remains unchanged

D. Becomes equal to annual demand

Correct Answer: B. Decreases toward zero

Question 4

For an all-units quantity discount, which cost must be compared at each feasible candidate?

A. Only the unit purchase cost

B. Only the holding cost

C. Purchase, ordering, and holding costs

D. Only the setup cost

Correct Answer: C. Purchase, ordering, and holding costs

Question 5

What is the principal purpose of a shared-capacity constraint in multi-item lot sizing?

A. To force identical lot sizes

B. To ensure total resource use does not exceed availability

C. To eliminate setup costs

D. To make demand stochastic

Correct Answer: B. To ensure total resource use does not exceed availability

Question 6

The Wagner–Whitin property applies most directly when:

A. Shortages are unlimited

B. Demand is unknown and highly random

C. Demand is deterministic and capacity is unlimited

D. Prices change continuously during production

Correct Answer: C. Demand is deterministic and capacity is unlimited

Question 7

In dynamic lot sizing, F_t represents:

A. The demand forecast error in period t

B. Minimum cost of satisfying demand through period t

C. Maximum inventory in period t

D. The supplier’s price in period t

Correct Answer: B. Minimum cost of satisfying demand through period t

Question 8

Which heuristic compares holding cost with setup cost over future periods?

A. Part-period balancing

B. Random search

C. First-in-first-out

D. Economic production scheduling

Correct Answer: A. Part-period balancing

Question 9

If p = 80,000 units/year and d = 24,000 units/year, the EPQ inventory factor 1 − d/p equals:

A. 0.30

B. 0.50

C. 0.70

D. 1.30

Correct Answer: C. 0.70

Question 10

What is a major limitation of deterministic lot-sizing models?

A. They cannot represent setup costs

B. They always require stochastic programming

C. They may create false precision when inputs are uncertain

D. They cannot be used with time-phased demand

Correct Answer: C. They may create false precision when inputs are uncertain