384. Least Unit Cost

A lot-sizing heuristic similar to Silver-Meal, but minimizing cost per unit produced instead of cost per period covered.

384.1. Algorithm

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

Numerator: total setup + holding cost. Denominator: total units in the order.

Extend $𝑘$ while ${𝐶}_{\text{LUC}}\left(𝑘\right)$ decreases. Stop when it increases. Order covers $𝑘$ periods.

384.2. Compared to Silver-Meal

• Silver-Meal: minimize cost per time period
• LUC: minimize cost per unit

When demand is uniform, the two give similar / identical answers. When demand varies, they diverge:

• Spiky demand peaks: LUC tends to order bigger batches (drives down per-unit cost via fixed-cost amortization)
• Smoother demand: Silver-Meal tends to order more frequently

Neither is strictly better. Both are heuristics, both within  5% of Wagner-Whitin optimum on typical instances.

384.3. Worked example

Same data as Silver-Meal: $𝑑=\left(60,100,80,50,40,70\right)$, $𝐾=100$, $ℎ=1$.

Start period 1:

• ${𝐶}_{\text{LUC}}\left(1\right)=\frac{100}{60}=1.67$
• ${𝐶}_{\text{LUC}}\left(2\right)=\frac{100+100}{60+100}=\frac{200}{160}=1.25$ ← decreasing
• ${𝐶}_{\text{LUC}}\left(3\right)=\frac{100+100+160}{60+100+80}=\frac{360}{240}=1.50$ ← rose, stop. Order 1 covers 2 periods.

(Same answer as Silver-Meal in this case.)

384.4. When LUC favors larger orders

If subsequent demand is high (say ${𝑑}_{𝑡+1}=1000$), Silver-Meal’s per-period cost increases fast (high holding cost on big inventory), but LUC’s per-unit cost may still drop (fixed cost spread over more units).

So LUC tends to bundle into larger orders when downstream demand is high.

384.5. See also

• Silver-Meal — sister heuristic
• Part-Period Balancing
• Wagner-Whitin — optimal
• EOQ — stationary-demand analog