Chapter 3 of 4

Lot Sizing and Replenishment Policies

3 sections · ~5 min read · 3 practice questions · part of Demand & Inventory Planning

Learning objectives

  • Compute the Economic Order Quantity and explain its assumptions
  • Compare continuous review (s, Q) and periodic review (R, S) policies
  • Choose lot sizes that respect capacity, minimum order quantities, and freight constraints

Economic Order Quantity

The Economic Order Quantity (EOQ) minimizes the sum of ordering cost and holding cost for an item with constant demand, no stockouts, and instantaneous replenishment. The formula is EOQ = sqrt(2 x D x K / h), where D is annual demand, K is the cost to place one order (administrative plus inbound transport setup), and h is the annual holding cost per unit (typically a percentage of unit cost times unit cost). EOQ is a planning target, not a literal order size. It assumes demand is known and steady, holding cost scales linearly with average inventory, and ordering cost is the same regardless of quantity, none of which is strictly true. Its value is in setting a defensible starting point and making the trade-off between ordering and holding explicit.

Two replenishment policies

In a continuous review (s, Q) policy, place a fixed order Q when inventory position reaches the reorder point s. Inventory position is on-hand stock plus outstanding orders minus backorders. In a periodic review (R, S) policy, review that position every R periods and order enough to bring it up to S, with no order when the position is already at or above S. The target covers demand during the review interval and the replenishment lead time. Periodic review makes it easier to combine orders across items; continuous review reacts sooner when stock runs low. Compare the monitoring effort, ordering costs, lead times, and service needs before choosing a policy. An (s, S) policy is different: an order is triggered only at or below s and raises the position to S.

Practical adjustments

Real replenishment rarely follows the textbook. Minimum order quantities from suppliers, full-truck or LTL freight tiers, case-pack constraints, shelf life, and production batch sizes all push the order size away from the EOQ. A common adjustment is to compute the unconstrained EOQ and then round up to the next feasible lot (MOQ, case pack, truckload), and to recompute the true holding cost at that rounded quantity. If the rounded quantity doubles the EOQ, the extra inventory cost is usually modest; if it forces an order ten times the EOQ, the firm should negotiate the MOQ down, change the case pack, or split the truck across SKUs. Capacity constraints inside the plant impose a similar ceiling, and the right answer there is often to desynchronize orders across SKUs so that the line sees a smoother schedule.

Worked example

Problem

Annual demand D = 24,000 units. Ordering cost K = $150 per order (administrative + transport setup). Unit cost = $20 and annual holding cost rate = 25 percent, so h = $5 per unit per year. Compute (a) EOQ, (b) number of orders per year, (c) time between orders in weeks, and (d) annual ordering plus holding cost at the EOQ.

Step by step

  1. EOQ = sqrt(2 x D x K / h). Numerator: 2 x 24,000 x 150 = 7,200,000. Divide by h = 5: 1,440,000. Square root: EOQ = 1,200 units.
  2. Number of orders per year = D / EOQ = 24,000 / 1,200 = 20 orders.
  3. Weeks between orders = 52 / 20 = 2.60 weeks, about every 18 days.
  4. Annual ordering cost = 20 x $150 = $3,000.
  5. Annual holding cost at EOQ = (EOQ / 2) x h = 600 x $5 = $3,000.
  6. Total relevant cost = $3,000 + $3,000 = $6,000 per year (excluding purchase cost).

Answer. EOQ = 1,200 units. Orders per year = 20. Cycle = 2.60 weeks. Annual ordering + holding cost = $6,000.

Practice

Work each question before opening the solution.

  1. A supplier imposes a minimum order of 2,000 units, double the EOQ. Estimate the additional annual holding cost and propose one operational response.

    Show solution for question 1

    Average inventory at MOQ = 1,000 units vs 600 at EOQ. Extra holding cost = 400 x $5 = $2,000 per year. Response: ask supplier to relax MOQ, share a truckload with another SKU, or accept the cost if service benefits (fewer orders, lower transport unit cost) exceed it.

  2. An item moves from continuous (s, Q) to periodic review with R = 4 weeks. Approximately how much additional safety stock is needed if sigma_LT is unchanged?

    Show solution for question 2

    The order-up-to level must now cover demand over R + L instead of just L, so safety stock scales with sqrt((R+L) / L). If R = 4 and L = 3, the new factor is sqrt(7/3) = sqrt(2.333) = 1.528, so safety stock rises by about 53 percent.

  3. Why does an EOQ computed at the list price usually under-state the optimal order size when a quantity discount is available?

    Show solution for question 3

    Two effects both push the optimum upward. First, the discounted price lowers unit cost and therefore lowers h, and EOQ varies as 1/sqrt(h), so the EOQ at the break price is itself larger. Second, the purchase-cost saving from reaching the break is often far larger than the extra holding cost of the bigger lot, so the total-cost minimum sits at the break quantity rather than at any interior EOQ. The correct procedure is to compute EOQ at each price level, discard any EOQ that falls outside its own price band, evaluate total cost (purchase + ordering + holding) at each feasible EOQ and at each break quantity, and take the lowest.