Lot-Sizing and Order Policies

Introduction

Material Requirements Planning (MRP) determines what materials are required, in what quantities, and when they are needed. The time-phased net requirements produced by an MRP calculation do not, by themselves, determine the size of each planned order. That decision is the responsibility of the lot-sizing rule.

Lot sizing converts a sequence of net requirements into planned order receipts. A good policy balances opposing objectives: minimizing ordering and setup costs, avoiding excessive inventory, protecting production continuity, and remaining practical to administer. The choice is therefore not merely mathematical. It is a policy decision influenced by demand variability, capacity, supplier relationships, lead time, product structure, and the cost of inventory errors.

In dependent-demand environments, lot sizing is especially important because a large order for one component may create inventory at several levels of the product structure. Conversely, an order that is too small may cause repeated setups, expediting, or disruption of assembly operations.

Learning Objectives

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

1. Explain the role of lot sizing within MRP.
2. Distinguish static and dynamic lot-sizing policies.
3. Apply lot-for-lot, fixed-order quantity, economic order quantity, period order quantity, least unit cost, least total cost, and part-period balancing.
4. Compare policies using ordering, setup, holding, capacity, and service considerations.
5. Evaluate the assumptions and limitations of lot-sizing methods.
6. Recommend an appropriate order policy for a practical MRP environment.

1. Lot Sizing in the MRP Logic

MRP begins with the master production schedule, bill of materials, inventory records, scheduled receipts, and lead times. For an item and a planning period, gross requirements are reduced by available inventory and scheduled receipts to obtain net requirements. A planned order receipt is then created to cover the net requirement, subject to the selected lot-sizing rule. The receipt is offset by lead time to determine the planned order release.

A simplified netting relationship is:

where NR_t is net requirement in period t, GR_t is gross requirement, PAB_t−1 is projected available balance before the period’s receipt, SR_t is a scheduled receipt, and SS_t is the desired safety stock requirement. The relationship assumes that all quantities are expressed in the same unit and that the record uses a consistent inventory convention. Different MRP systems may display safety stock through an equivalent modification of projected balance.

The lot-sizing rule determines the receipt quantity Q_t. Under lot-for-lot, Q_t = NR_t. Under other rules, Q_t may exceed the current net requirement and satisfy future requirements.

Terminology

• Gross requirement: Total demand for an item before considering inventory and scheduled receipts.
• Net requirement: Demand remaining after available supply is considered.
• Planned order receipt: Quantity planned to arrive in a specified period.
• Planned order release: Planned receipt offset by the item’s lead time.
• Lot size: Quantity released or received as one order or production batch.
• Ordering cost: Administrative, procurement, transportation, or transaction cost associated with an order.
• Setup cost: Cost of preparing equipment or a process for a production run.
• Holding cost: Cost of carrying inventory over time.
• Time-phased requirement: Requirement assigned to a particular planning period.
• Part-period: One unit held for one planning period.

2. The Cost Structure of Lot Sizing

Most lot-sizing comparisons use ordering or setup cost and inventory holding cost. If an order is made in advance of need, the excess quantity is carried and incurs holding cost. If a quantity is ordered too late or too small, shortage, expediting, or production-disruption costs may occur.

For a candidate lot-sizing plan, a simplified relevant-cost model is:

where TC is total relevant cost over the planning horizon, S is ordering or setup cost per order, N is the number of orders, h is holding cost per unit per period, I_t is inventory carried during period t, and C_s represents shortage, expediting, transportation, or other policy-specific costs. This expression is applicable when costs are reasonably stable and additive. It is not a complete accounting cost statement; purchase price may be excluded when it is independent of the lot size.

The unit consistency is important. If S is measured in currency per order, h must be currency per unit per period, and I_t must be units carried during that period. If a period-end inventory convention is used, the same convention must be applied to all candidate policies.

In practice, MRP lot sizing often focuses on setup and holding costs but must also consider:

• minimum supplier quantities;
• multiples such as cartons, pallets, or batches;
• storage and shelf-life limits;
• supplier and machine capacity;
• safety stock and service requirements;
• scrap, yield loss, and quality risk;
• transportation economics;
• schedule stability and nervousness.

3. Major Lot-Sizing Policies

3.1 Lot-for-Lot

Lot-for-lot (L4L) creates an order equal to the net requirement in each period:

The policy minimizes planned cycle inventory and is particularly suitable for expensive, perishable, customized, or rapidly changing items. It is also attractive when ordering and setup costs are negligible or when requirements are irregular and closely timed.

Its principal disadvantage is the potential for many orders and setups. A production environment with a substantial setup cost may find L4L uneconomical. It can also create frequent supplier deliveries, administrative effort, and schedule instability.

3.2 Fixed-Order Quantity

A fixed-order quantity policy uses a predetermined quantity whenever a net requirement occurs. If the fixed quantity is Q̄, the receipt may be represented as:

where ceil(x) rounds x upward to the next whole order. This policy is appropriate where production or purchasing is constrained by a standard batch, container, pallet, truckload, or supplier minimum.

The method is simple and operationally convenient. However, it may generate excess inventory because the order cannot be tailored to the exact time-phased requirement. It may also create an order larger than available storage or larger than the item’s remaining useful life.

3.3 Economic Order Quantity

The economic order quantity (EOQ) balances annual ordering cost and annual holding cost under relatively stable independent demand. The classical formula is:

where D is annual demand in units per year, S is ordering or setup cost per order, and H is annual holding cost per unit per year. The units are consistent because DS has currency per year and H has currency per unit per year, producing units squared under the square root.

The classical EOQ assumes constant and known demand, constant lead time, instantaneous replenishment, no quantity discounts, no shortages, and constant ordering and holding costs. It also assumes that the item is managed independently and that the order quantity can be used without violating capacity or minimum/maximum constraints.

The annual relevant cost function is:

where Q is order quantity. The first term is annual ordering cost and the second is annual cycle-stock holding cost, assuming replenishment is instantaneous and average cycle inventory is Q/2. Differentiating with respect to Q and setting the derivative to zero gives the EOQ. The solution is a minimum because the cost curve is convex for positive Q.

In MRP, demand is usually time-phased and dependent rather than smooth and independent. Consequently, EOQ should be treated as a benchmark or fixed-order quantity rather than as an automatic optimum. It can be adapted by using an appropriate annualized component demand, but that approximation may ignore timing and product-structure effects.

3.4 Period Order Quantity

Period order quantity (POQ) converts an order quantity into a time interval. A common approximation is:

where d is average demand per period and POQ is measured in periods. If annual demand is D over T periods, then d = D/T. The calculated period interval is normally rounded according to the organization’s planning calendar.

The order covers net requirements over the selected number of future periods. For example, a two-period POQ beginning in period 3 covers requirements in periods 3 and 4, subject to inventory records and any policy constraints.

POQ responds better than a fixed quantity to variations in time-phased demand because the quantity changes with the requirements in the selected interval. It can nevertheless create substantial inventory when demand is lumpy, and rounding the period interval can materially alter cost.

3.5 Least Unit Cost

The least-unit-cost method tests candidate lots and selects the one with the lowest average relevant cost per unit. For a candidate covering periods from t through k:

Here, the first requirement is assumed to arrive in its requirement period, so it incurs zero holding periods. A requirement in period j is carried for j − t periods. The numerator includes one ordering or setup cost and holding cost for the requirements covered by the candidate lot. The denominator is the total quantity covered.

The method adds successive future requirements while the average cost per unit decreases; it selects the candidate with the lowest value. Exact implementation should specify whether a candidate is stopped when the average rises, or whether all feasible candidates are evaluated. It is a heuristic, not a universal global optimization method.

LUC is useful when demand is uneven and management wants to balance setup cost against the average cost of material carried. It may, however, favor a large lot because the fixed ordering cost is spread over many units, even if the resulting inventory creates operational or obsolescence risk.

3.6 Least Total Cost

Least-total-cost (LTC) evaluates candidate lots using total ordering/setup cost plus holding cost:

The candidate with the lowest total cost is selected. In a sequential heuristic, requirements are added until the total cost begins to increase. The method is conceptually straightforward but must be applied carefully: the comparison concerns the candidate’s relevant costs, not the total cost of all future planning periods counted repeatedly.

LTC differs from LUC because LUC divides by the units covered. A large candidate may have a lower unit cost but a higher total cost. The choice between them reflects managerial priorities: unit-cost efficiency versus total cost for the replenishment decision.

3.7 Part-Period Balancing

Part-period balancing (PPB) seeks a lot size for which cumulative holding cost is approximately equal to the ordering or setup cost. A part-period is one unit carried for one period. The economic part-period is:

where EPP is economic part-periods, S is setup or ordering cost, and h is holding cost per unit per period. The units are part-periods because currency per order divided by currency per unit per period equals unit-periods.

For a candidate lot, cumulative part-periods are:

The lot is selected at or near the point where PP(k) is closest to EPP. A practical rule compares the deviation before and after adding the next requirement and chooses the closer value.

PPB is attractive because it provides an intuitive link between setup and holding costs. It is especially useful for lumpy MRP demand. It remains a heuristic: the closest part-period balance does not guarantee the minimum total cost when constraints, safety stock, capacity, or non-linear costs are present.

4. Worked Example: Comparing Policies

Consider an illustrative component with net requirements over five periods as follows:

Assume an ordering/setup cost of 100 currency units per order and a holding cost of 2 currency units per unit per period. Ignore beginning inventory, scheduled receipts, safety stock, lead time, capacity limits, and shortages. Requirements are assumed to be received in their requirement periods unless covered by an earlier lot.

Lot-for-lot

There is one order in each period, so ordering cost is:

5 × 100 = 500.

No requirement is received early, so holding cost is zero. Total relevant cost is therefore 500 currency units.

Fixed-order quantity of 100 units

Period 1 receives 100 units, satisfying requirements of 40 and 20 and leaving 40 units. Period 3 requires 60, so a second order of 100 is received; it covers period 3 and leaves 40 units for period 4’s requirement of 30. Period 5 requires 50, so a third order is required. Three orders cost 300. The early inventory is 20 units carried from period 1 to period 2, 40 units from period 1 to period 3, and 10 units from period 3 to period 4. Holding cost is:

(20 + 40 + 10) × 2 = 140.

Total relevant cost is 440 currency units, excluding any ending inventory adjustment. The example illustrates that a fixed quantity can reduce ordering cost while creating inventory.

Part-period balancing

Starting in period 1, the economic part-period is:

EPP = 100 / 2 = 50.

Candidate coverage from period 1 gives the following cumulative part-periods:

The value 20 is closer to 50 than 140, so PPB selects a lot covering periods 1 and 2: quantity 40 + 20 = 60. Starting in period 3:

The value 30 is closer to 50 than 130, so the second lot covers periods 3 and 4: quantity 90. The final lot covers period 5: quantity 50. There are three orders, costing 300. Holding cost is (20 + 30) × 2 = 100, so total relevant cost is 400 currency units.

This result is lower than the simplified L4L and fixed-order alternatives, but it is not evidence that PPB is always superior. The result depends on the demand pattern, cost assumptions, and convention for inventory timing. In an actual MRP system, the comparison would also include beginning and ending inventory, safety stock, minimum order quantities, capacity, and the consequences of changing planned orders.

Interpretation

The example demonstrates four important points:

1. L4L minimizes cycle inventory but not necessarily total relevant cost.
2. Fixed quantities are operationally convenient but may leave irregular residual inventory.
3. PPB explicitly balances setup and holding costs and can perform well with lumpy demand.
4. A cost-minimizing result is not automatically the best managerial decision if it increases schedule instability or violates practical constraints.

5. Comparative Evaluation

The choice should be aligned with the item’s operational context. High-value or obsolete-prone items often justify L4L or a short POQ. Stable, repetitive components with high setup costs may suit EOQ or PPB. Items purchased in standard containers may require fixed-order quantities even when another policy has a lower theoretical cost.

6. Advanced Managerial Considerations

Dependent demand and product structure

EOQ’s independent-demand assumptions are often violated in MRP. Component demand is derived from parent schedules and may be intermittent, synchronized, or highly lumpy. A component used in several parents may have aggregate demand that is smoother than the demand of any one parent, but this should be demonstrated from the actual MRP records rather than assumed.

Capacity and setup sequencing

A lot-sizing rule may generate feasible quantities but infeasible capacity requirements. Production planners must check machine hours, labor, tooling, and setup sequence. If setup times are sequence-dependent, the nominal setup cost used in a lot-sizing calculation may be misleading.

Safety stock and service

Lot sizing and safety stock serve different purposes. Lot size addresses replenishment economics; safety stock protects against uncertainty. Combining them without clear logic can produce excess inventory. For critical components, a slightly higher lot may be justified if a stockout would stop a costly assembly line, but that decision should be based on explicit service and downtime costs.

MRP nervousness

Frequent changes to planned orders can destabilize suppliers and production schedules. A mathematically attractive small lot may create repeated rescheduling. Time fences, order modifiers, frozen zones, and planner review can limit nervousness. However, excessive freezing can conceal genuine demand changes and increase shortages.

Quantity discounts and transportation

If purchase price decreases with quantity, the purchase-cost term must be included. The classical EOQ formula is insufficient because purchase cost is no longer independent of quantity. Similarly, a full-truckload policy may be justified by transport economics even when it increases holding cost.

Digital planning and artificial intelligence

Modern MRP and enterprise-resource-planning systems can evaluate multiple lot-sizing rules, constraints, and scenarios rapidly. Machine-learning models may forecast demand or identify anomalous consumption, but they do not remove the need for transparent cost assumptions. AI-generated recommendations should be governed through data-quality controls, explainable policy parameters, approval thresholds, and audit trails. A black-box recommendation that ignores shelf life, supplier minimums, or production capacity is not a reliable lot-sizing policy.

7. Illustrative Case: Meridian Pumps

Background

Meridian Pumps manufactures industrial water pumps. A cast-metal impeller is a dependent-demand component used in three pump families. The MRP system produces irregular weekly net requirements because customer orders are released in batches. The foundry incurs a setup cost of 1,200 currency units per casting run. Holding cost is estimated at 3 currency units per impeller per week. The supplier requires a minimum batch of 50 units, while warehouse capacity is 180 units.

Problem

The production planner currently uses lot-for-lot. This creates frequent foundry setups and sometimes requires expedited subcontracting. Finance argues for EOQ because the annual demand is high. Operations argues that large batches would exceed warehouse capacity and increase the risk of casting defects becoming obsolete after a design change.

Analysis

The planner first reviews the time-phased requirements rather than applying annual EOQ mechanically. The requirements are intermittent and tied to the master schedule. PPB is used as an initial heuristic because it compares setup cost with time-phased holding cost. Candidate quantities are then checked against the 50-unit minimum and 180-unit maximum. The resulting plan is simulated against foundry capacity, warehouse space, lead time, and the planned production schedule.

The review shows that pure L4L creates eight setups in the six-week horizon. A PPB-based plan reduces setups to four while remaining below the warehouse limit. A fixed 200-unit batch would reduce setups further but violates warehouse capacity. An EOQ-derived quantity would also be operationally unsuitable because it assumes smooth annual usage and creates receipts unrelated to the actual pump schedule.

Approach and solution

Meridian adopts a constrained PPB policy with the following governance rules:

• use PPB for the initial candidate lot;
• round upward to the supplier minimum only when necessary;
• never exceed the warehouse limit without approval;
• freeze releases inside the two-week supplier commitment window;
• review exceptional demand and design changes weekly;
• record setup, holding, expediting, and obsolescence costs separately.

Outcome

The policy reduces setup frequency while avoiding the oversized batches proposed by a simple fixed-order rule. More importantly, the organization gains a repeatable decision process: cost heuristics generate candidates, and planners validate them against operational constraints.

Learning points

1. Annual demand alone is insufficient for lumpy dependent demand.
2. A theoretically low cost can be infeasible because of capacity or minimum quantities.
3. Lot sizing should be evaluated as part of an MRP governance system, not as an isolated formula.
4. The best policy may be a constrained or hybrid policy rather than a pure textbook rule.

8. Best-Practice Decision Framework

A professional lot-sizing review can follow these steps:

1. Validate the input record: requirements, inventory, lead time, scheduled receipts, safety stock, and units of measure.
2. Classify the item by value, criticality, variability, shelf life, and supply risk.
3. Identify hard constraints: minimums, multiples, capacity, storage, supplier calendars, and quality limits.
4. Calculate at least two feasible candidate policies, such as L4L, POQ, PPB, or a fixed quantity.
5. Compare relevant ordering, holding, shortage, transportation, and obsolescence costs.
6. Test the plan through a time-phased simulation rather than comparing only average quantities.
7. Assess schedule stability, supplier impact, and production workload.
8. Document the selected policy, parameter values, assumptions, and review frequency.
9. Monitor actual performance and revise parameters when demand, costs, or constraints change.

Challenges and Limitations

No lot-sizing method is universally optimal. Common limitations include inaccurate holding-cost estimates, unstable requirements, incorrect inventory records, hidden setup costs, capacity constraints, and inappropriate annualization of dependent demand. Heuristics such as LUC, LTC, and PPB are efficient and understandable, but they may not find the global optimum across a long horizon with multiple items and shared capacity.

There is also a behavioral limitation. Planners may override a policy because of supplier relationships, fear of shortages, or pressure to reduce visible orders. Such overrides may be sensible, but unrecorded overrides prevent organizational learning. Governance should therefore encourage justified exceptions rather than either blind automation or uncontrolled discretion.

Chapter Summary

Lot sizing determines the quantity and timing of planned MRP receipts after net requirements have been calculated. Lot-for-lot matches each net requirement and minimizes cycle inventory, while fixed-order quantity uses a predetermined batch. EOQ balances annual ordering and holding costs under stable independent-demand assumptions, but its assumptions often fit MRP only approximately. POQ translates an order quantity into a time interval and adapts quantity to time-phased demand.

Least unit cost evaluates average relevant cost per unit, whereas least total cost evaluates the total relevant cost of a candidate lot. Part-period balancing selects a lot near the point where cumulative holding cost equals ordering or setup cost. These methods are useful heuristics, especially for lumpy demand, but they must be tested against capacity, minimum quantities, storage, safety stock, lead time, service, and schedule stability.

Effective lot-sizing is therefore a managerial design problem. Cost calculations provide discipline, but the final policy must reflect the item’s supply risk, product structure, operational constraints, and governance requirements. Hybrid and constrained policies are often more appropriate than unmodified textbook rules.

Key Terms

Lot sizing: Determining the quantity of an MRP planned order.

Lot-for-lot: A policy that orders exactly the current net requirement.

Fixed-order quantity: A policy using a predetermined order or batch size.

Economic order quantity: The order quantity balancing ordering and holding cost under classical assumptions.

Period order quantity: A policy that covers requirements for a specified number of periods.

Least unit cost: A heuristic minimizing relevant cost per unit covered.

Least total cost: A heuristic minimizing total relevant cost for a candidate lot.

Part-period balancing: A heuristic balancing cumulative holding cost against setup cost.

Net requirement: Requirement remaining after available supply is considered.

Planned order receipt: Quantity scheduled to arrive in an MRP period.

Planned order release: Order date obtained by offsetting a receipt by lead time.

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

Setup cost: Cost of preparing production resources for a batch.

Holding cost: Cost of carrying inventory over time.

Economic part-period: Setup cost divided by holding cost per unit-period.

Dependent demand: Demand derived from the planned production of another item.

MRP nervousness: Frequent and disruptive changes to planned orders.

Order modifier: Rule such as minimum, maximum, or multiple that constrains an order quantity.

Review Questions

Short Answer Questions

1. What is the role of lot sizing in an MRP system?
2. State two advantages and two limitations of lot-for-lot.
3. What assumptions underlie the classical EOQ formula?
4. Define a part-period and explain its use in PPB.
5. Why can a fixed-order quantity create excess inventory?

Long Answer Questions

1. Compare L4L, EOQ, POQ, LUC, LTC, and PPB in terms of logic, cost behavior, and operational suitability.
2. Explain why EOQ must be used cautiously for dependent and time-phased MRP demand.
3. Derive the classical EOQ relationship from the annual ordering and holding-cost function.
4. Discuss how capacity, minimum quantities, safety stock, and schedule stability affect lot-sizing decisions.
5. Design a governance framework for selecting and reviewing lot-sizing policies in an ERP-supported manufacturing organization.

Analytical / Application Questions

1. Net requirements are 30, 50, 20, 80, and 40 units over five periods. Ordering cost is 120 currency units per order and holding cost is 2 currency units per unit per period. Apply PPB from period 1 and show the candidate part-period calculations.
2. A supplier offers a fixed batch of 200 units, but warehouse capacity is 150 units and demand is intermittent. Evaluate whether the fixed batch is feasible and recommend an alternative policy.
3. A production manager claims that the policy with the lowest cost per unit must always be best. Critically evaluate this claim using LUC, LTC, capacity, obsolescence, and schedule-stability considerations.

Multiple Choice Questions with Answers

Question 1

Which lot-sizing policy sets the planned order receipt equal to the current net requirement?

A. Lot-for-lot

B. Economic order quantity

C. Fixed-order quantity

D. Period order quantity

Correct Answer: A. Lot-for-lot

Question 2

In the classical EOQ model, which expression represents annual ordering cost?

A. QH/2

B. DS/Q

C. DQ/S

D. Q/DH

Correct Answer: B. DS/Q

Question 3

If ordering cost is 100 currency units per order and holding cost is 2 currency units per unit per period, the economic part-period is:

A. 20 part-periods

B. 50 part-periods

C. 100 part-periods

D. 200 part-periods

Correct Answer: B. 50 part-periods

Question 4

Which policy converts an order quantity into an approximate number of demand periods?

A. Least total cost

B. Lot-for-lot

C. Period order quantity

D. Fixed-order quantity

Correct Answer: C. Period order quantity

Question 5

The least-unit-cost method primarily selects the candidate with the lowest:

A. Number of orders

B. Average relevant cost per unit covered

C. Ending inventory value

D. Supplier lead time

Correct Answer: B. Average relevant cost per unit covered

Question 6

Part-period balancing attempts to make cumulative holding cost approximately equal to:

A. Purchase price

B. Annual demand

C. Ordering or setup cost

D. Safety-stock cost

Correct Answer: C. Ordering or setup cost

Question 7

Which assumption is least consistent with the classical EOQ model?

A. Known and relatively stable demand

B. Constant ordering cost

C. Dependent, highly lumpy demand driven by a master schedule

D. No shortages in the basic model

Correct Answer: C. Dependent, highly lumpy demand driven by a master schedule

Question 8

A fixed-order quantity is most directly justified when:

A. Every period has identical net requirements

B. The supplier or process requires a standard batch or container quantity

C. Holding cost is zero in every situation

D. The item has no lead time

Correct Answer: B. The supplier or process requires a standard batch or container quantity

Question 9

If a candidate lot covers requirements in later periods by receiving them early, those later requirements generally create:

A. Negative ordering cost

B. Holding cost

C. Lower lead time automatically

D. Zero inventory

Correct Answer: B. Holding cost

Question 10

Why may a mathematically low-cost lot-sizing result be rejected by a production planner?

A. It always increases purchase price

B. It may violate capacity, storage, minimum-quantity, or schedule-stability constraints

C. It cannot be entered into an MRP system

D. It eliminates all planned receipts

Correct Answer: B. It may violate capacity, storage, minimum-quantity, or schedule-stability constraints