Silver-Meal

383. Silver-Meal

A lot-sizing heuristic for time-varying demand (Silver & Meal, 1973). Chooses order intervals to minimize average cost per period covered.

Easier than Wagner-Whitin (which is DP-optimal) and often close in quality.

383.1. Setup

Demand $𝑑_{𝑡}$ over $𝑇$ periods. Setup cost $𝐾$ per order, holding cost $ℎ$ per unit per period.

383.2. Algorithm

Start with a candidate order in period $𝑡$ covering $𝑘$ periods of demand. Average cost per period:

𝐶 (𝑘) = \frac{𝐾 + ℎ \sum_{𝑗 = 1}^{𝑘 - 1} 𝑗 𝑑_{𝑡 + 𝑗}}{𝑘}

(Setup once + holding cost for demand carried $𝑗$ periods.)

Extend $𝑘$ as long as $𝐶 (𝑘)$ decreases. Stop at first $𝑘$ where $𝐶 (𝑘 + 1) > 𝐶 (𝑘)$ . Place an order in $𝑡$ to cover $𝑘$ periods. Start a new order in $𝑡 + 𝑘$ . Repeat until all periods are covered.

383.3. Worked example

Demand: $𝑑 = (60, 100, 80, 50, 40, 70)$ , $𝐾 = 100$ , $ℎ = 1$ .

Start at period 1:

$𝐶 (1) = \frac{100}{1} = 100$
$𝐶 (2) = \frac{100 + 1 \cdot 100}{2} = 100$ ← tied
$𝐶 (3) = \frac{100 + 100 + 2 \cdot 80}{3} = \frac{360}{3} = 120$ ← rose, stop. Order in period 1 covers 2 periods (160 units).

Start at period 3:

$𝐶 (1) = \frac{100}{1} = 100$
$𝐶 (2) = \frac{100 + 50}{2} = 75$ ← decreasing
$𝐶 (3) = \frac{100 + 50 + 80}{3} = 76.67$ ← rose. Order in period 3 covers 2 periods (130 units).

Start at period 5:

$𝐶 (1) = 100$
$𝐶 (2) = \frac{100 + 70}{2} = 85$ ← decreasing
( $𝑡 + 2$ doesn’t exist, $𝑇 = 6$ ). Order in period 5 covers 2 periods (110 units).

Three orders: period 1 (160), period 3 (130), period 5 (110). Total cost computable.

383.4. Compared to alternatives

Heuristic	Quality vs WW optimum	Computational cost
Lot-for-lot	worst; high setup cost	trivial
POQ / fixed-quantity (EOQ)	moderate; ignores demand variation	trivial
Least Unit Cost	usually within 5-10%	$𝑂 (𝑇^{2})$
Silver-Meal	usually within 3-5%	$𝑂 (𝑇^{2})$
Part-Period Balancing	usually within 5%	$𝑂 (𝑇^{2})$
Wagner-Whitin	optimal	$𝑂 (𝑇^{2})$ DP

Silver-Meal is fast and good. Wagner-Whitin is exact and not much slower; usually preferred when implementable.

383.5. Limitations

Greedy — no lookahead beyond current order
Can miss good solutions by stopping too early
Failure mode: when demand has long-running upticks, Silver-Meal may stop short of the truly optimal order span

383.6. See also

Least Unit Cost
Part-Period Balancing
Wagner-Whitin — DP-optimal
MRP — where lot-sizing fits