438. Joint Replenishment

438.1. Joint replenishment (multi-item EOQ with shared setup)

Relax one dimension from basic EOQ: number of items is no longer one. Multiple items share a single supplier (or a single delivery truck, or a single setup operation), so the order cost is paid once per coordinated order — not separately per item.

Single supplier, multiple SKUs: order them all together on the same cadence, save on setup.

438.1.1. Setup

items indexed :

• = annual demand for item
• = holding cost per unit per year for item
• Common setup cost: (paid once when any order goes out, regardless of how many items)
• Decision: a common cycle time — every years, order each item in the quantity needed for one cycle, .

This is the simplest joint-replenishment formulation. (Variants with item-specific minor setups in addition to a major exist — see Roundy / Federgruen-Zheng — but we’ll keep it minimal here.)

438.1.2. Cost model

Per cycle:

• One setup cost:
• Holding for item over the cycle: . Wait — average inventory , time per cycle , so per-cycle holding cost is .

Per unit time, divide by :

The single decision trades off setup cost (decreases with ) against aggregate holding (increases with ).

438.1.3. Derive

Solve for :

438.1.4. Final formulas

Sanity check: for a single item (), , — exactly basic EOQ ✓.

The aggregation comes through the single — items with small contribute little to the optimal cycle, so they’re ordered alongside the big- items at no extra setup cost.

438.1.5. Why is this cheaper than independent EOQs?

Order each item independently with its own EOQ: each pays per order, total cost is .

Joint replenishment: .

By Cauchy-Schwarz / Jensen, , so:

Joint always wins. The savings grow with the number of items and the unevenness of across them.