Part-Period Balancing

385. 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.

385.1. EOQ balance recap

For stationary demand:

Setup cost per cycle: $𝐾$
Holding cost per cycle: $ℎ \frac{𝑄}{2} \cdot 𝑇_{cycle}$

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

385.2. Algorithm

Economic part-period (EPP):

EPP = \frac{𝐾}{ℎ}

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

For an order starting in period $𝑡$ covering $𝑘$ periods:

Part-periods (𝑘) = \sum_{𝑗 = 1}^{𝑘 - 1} 𝑗 𝑑_{𝑡 + 𝑗}

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

Part-periods (𝑘 + 1) > EPP but Part-periods (𝑘) \leq EPP

385.3. Worked example

$𝑑 = (60, 100, 80, 50, 40, 70)$ , $𝐾 = 100$ , $ℎ = 1$ . $EPP = \frac{100}{1} = 100$ .

Start period 1:

Part-periods after $𝑘 = 1$ : 0
After $𝑘 = 2$ : $1 \cdot 100 = 100$ ← right at EPP, stop
(If we went to $𝑘 = 3$ : $100 + 2 \cdot 80 = 260$ , way past.)

Order covers 2 periods.

Start period 3:

Part-periods after $𝑘 = 2$ : $1 \cdot 50 = 50$ — below EPP
After $𝑘 = 3$ : $50 + 2 \cdot 40 = 130$ — past EPP, stop

Take $𝑘 = 2$ (130 vs 50 — closer to 100? 130 vs 50: $| 130 - 100 | = 30, | 50 - 100 | = 50$ . Take $𝑘 = 3$ .). 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.)

385.4. Variants

Standard PPB: stop at first $𝑘$ where part-periods exceed EPP
Look-ahead PPB: include the contribution of period $𝑡 + 𝑘$ before deciding
Modified PPB: minimize $| Part-periods - EPP |$

385.5. 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.

385.6. 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.

385.7. See also

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