Inventory & Lot Sizing

Mathematical Formulation TC(Q) = (D/Q) · K + (Q/2) · h
Q* = √(2DK / h)
TC* = √(2DKh)

With Planned Backorders Q* = √(2DK/h · (h+b)/b)

DP Formulation (ZIO Property) f(t) = min_{1 ≤ i ≤ t} { f(i−1) + C(i, t) },   f(0) = 0
C(i, j) = K_i + ∑_k=i+1..j h_k−1 · ∑_ℓ=k..j d_ℓ

Flow Balance I_t = I_t−1 + x_t − d_t ≥ 0   ∀ t

Silver-Meal Intuition Starting from the first uncovered period, extend the order to cover additional periods as long as the average cost per period decreases. When it increases, place the order and start a new one. Part-Period Balancing uses the EOQ insight that total cost is minimized when cumulative holding cost approximately equals the ordering cost.

DP Recurrence f(j) = min_{1 ≤ i ≤ j} { f(i−1) + K_i + ∑_k=i..j−1 h_k · ∑_ℓ=k+1..j d_ℓ }
f(0) = 0

ZIO Property I_t−1 · x_t = 0   ∀ t   (order only when inventory is zero)

MILP Formulation min ∑_t [ K_t y_t + v_t x_t + h_t I_t ]
s.t. I_t−1 + x_t = d_t + I_t   ∀ t   (flow balance)
     x_t ≤ C_t · y_t   ∀ t   (capacity linking)
     x_t, I_t ≥ 0,   y_t ∈ {0, 1}

Lot-Shifting Heuristic Phase 1: Start with lot-for-lot baseline (x_t = d_t).
Phase 2: Iterate backward, shifting production to earlier periods with spare capacity when the setup cost saved exceeds the incremental holding cost. This reduces setup frequency while respecting capacity constraints.

Echelon Base-Stock Policy L_ℓ = ∑_k=1..ℓ τ_k   (cumulative lead time)
σ_Lℓ = σ_D · √L_ℓ
S_ℓ = μ_D · L_ℓ + z_SL · σ_D · √L_ℓ
where z_SL = Φ⁻¹(SL)

Why Multi-Echelon Matters Setting safety stock independently at each echelon leads to double-counting of uncertainty buffers. Echelon-based policies coordinate inventory across the chain: the upstream echelon only buffers against the incremental lead-time uncertainty it adds, not the total. Roundy's powers-of-two policy achieves near-optimal cost with the practical constraint of nested reorder intervals.

Demand During Lead Time σ_DDLT = √(μ_L · σ_D² + μ_D² · σ_L²)

Safety Stock and Reorder Point SS = z · σ_DDLT = Φ⁻¹(SL) · σ_DDLT
ROP = μ_D · μ_L + SS

When Lead-Time Variability is Negligible When σ_L ≈ 0, the DDLT formula simplifies to σ_DDLT = σ_D · √μ_L. For fill-rate (Type II) service, the safety stock calculation requires the unit normal loss function G(z) rather than Φ⁻¹(SL).

MILP Formulation min ∑_k y_k
s.t. ∑_k x_ik = 1   ∀ i   (each item assigned once)
     ∑_i w_i x_ik ≤ W · y_k   ∀ k   (weight capacity)
     ∑_i v_i x_ik ≤ V · y_k   ∀ k   (volume capacity)
     x_ik ∈ {0,1},   y_k ∈ {0,1}

FFD Dual-Capacity Strategy Sort items by weight descending. For each item, assign it to the first vehicle with sufficient remaining weight and volume capacity. If no vehicle fits, open a new one. This extends the classic FFD heuristic for 1D bin packing to handle the two-dimensional feasibility check at each assignment step.