382. Part-Period Balancing

A lot-sizing heuristic (DeMatteis, 1968) based on the EOQ balance condition: at the EOQ optimum, ordering cost equals holding cost. Replicate this for time-varying demand.

382.1. EOQ balance recap

For stationary demand:

• Setup cost per cycle:
• Holding cost per cycle:

EOQ optimality: these are equal. The Part-Period Balancing analog: equate setup cost to part-periods (cumulative holding cost).

382.2. Algorithm

Economic part-period (EPP):

— the number of unit-periods that would generate holding cost equal to one setup. (One unit held for one period contributes of holding cost; total holding cost on EPP unit-periods is .)

For an order starting in period covering periods:

Extend until part-periods approaches but doesn’t exceed EPP. Stop at the where:

382.3. Worked example

, , . .

Start period 1:

• Part-periods after : 0
• After : ← right at EPP, stop
• (If we went to : , way past.)

Order covers 2 periods.

Start period 3:

• Part-periods after : — below EPP
• After : — past EPP, stop

Take (130 vs 50 — closer to 100? 130 vs 50: . Take .). Note: rules differ; some variants pick “closest” rather than “last not exceeding”.

Start period 6: only one period left. Order covers 1 period.

(Worked answer depends on tie-breaking rule.)

382.4. Variants

• Standard PPB: stop at first where part-periods exceed EPP
• Look-ahead PPB: include the contribution of period before deciding
• Modified PPB: minimize

382.5. Quality

Generally within 5–10% of Wagner-Whitin optimum. Slightly worse than Silver-Meal or LUC in benchmarks, but conceptually simpler — directly mimics EOQ’s balance principle.

382.6. Why it sometimes wins

When demand patterns are close to stationary, PPB hugs the EOQ-like balance more tightly than the period-based or unit-based alternatives.

382.7. See also

• Silver-Meal — period-based
• Least Unit Cost — unit-based
• Wagner-Whitin — optimal
• EOQ — the balance principle PPB emulates