Warehouse Slotting Optimization
Quadratic Assignment Problem (QAP) — NP-Hard
Assigning 500+ SKUs to storage locations is a QAP: items frequently ordered together should be stored together. Optimal slotting reduces picker travel by 20–35%, saving €200,000+ annually in a mid-size distribution center.
Where This Decision Fits
Supply chain operations — the highlighted step is what this page optimizes
The Problem
From warehouse aisles to quadratic assignment
A distribution center has 6 SKUs (products) that must be assigned to 6 storage locations arranged on a 2D grid (2 rows × 3 columns). The objective is to minimize total picker travel distance by placing items that are frequently ordered together in nearby locations.
This maps directly to the Quadratic Assignment Problem (QAP), one of the hardest combinatorial optimization problems. The QAP was introduced by Koopmans & Beckmann (1957) and is NP-Hard—even instances with n=30 remain unsolved to proven optimality. With 6 items, there are 6! = 720 possible assignments.
| Warehouse Domain | QAP Model | |
|---|---|---|
| SKU (product) | Facility i | |
| Storage location | Location j | |
| Co-pick frequency | Flow fik | |
| Aisle distance | Distance djl | |
| SKU → slot assignment | Permutation π | |
| Total picker travel | QAP objective value |
Mathematical Formulation
The Quadratic Assignment Problem
Where fik = co-pick frequency between SKU i and SKU k // symmetric flow matrix
d(j, l) = Euclidean distance between location j and location l // distance matrix
π : {1,...,n} → {1,...,n} // bijection (permutation) mapping SKUs to locations
Linearized ILP (Lawler, 1963) minimize Σi Σj Σk Σl fik · djl · xij · xkl
s.t. Σj xij = 1 ∀ i // each SKU assigned once
Σi xij = 1 ∀ j // each location used once
xij ∈ {0, 1}
Interactive Solver
Choose a scenario, pick an algorithm, and watch the assignment
Warehouse Slotting Solver
Standard ABC classification. High-velocity A-items should be placed closest to the depot (location 0,0).
| Algorithm | Total Cost | % Improvement | Assignment | Time |
|---|---|---|---|---|
| Press Solve to run all algorithms | ||||
Solution Algorithms
Three approaches to the quadratic assignment problem
Random Assignment
A random permutation of SKUs to locations serves as the baseline against which all other methods are measured. With 6 items there are 720 possible assignments; the random solution typically falls near the median of this distribution.
In practice, many warehouses operate with effectively random slotting due to ad-hoc placement decisions over time, making this a realistic baseline.
Greedy (Popular Near Depot)
The greedy heuristic ranks SKUs by total co-pick frequency (popularity) and assigns the most popular items to locations closest to the depot (origin). This implements the classic ABC analysis principle: A-items in golden zone, B-items in middle, C-items in far aisles.
While simple and fast, greedy ignores pairwise interactions—two items that are always ordered together may still end up far apart if they have different total volumes.
Simulated Annealing
Simulated Annealing (SA) explores the solution space by performing random pairwise swaps of SKU assignments. Improving swaps are always accepted; worsening swaps are accepted with probability exp(−Δ/T), where T decreases geometrically over 100 iterations.
This allows SA to escape local optima that trap greedy methods. For the 6-item QAP, SA reliably finds near-optimal solutions; for real-world instances (n > 30), it remains one of the best practical approaches alongside tabu search and genetic algorithms.
Complexity & Hardness
- QAP is NP-Hard: no known polynomial-time algorithm exists; the search space is n! permutations
- Exact solvers: branch-and-bound with Gilmore-Lawler bound can handle n ≤ 30 in hours; instances with n ≥ 40 remain open
- Greedy: O(n² log n) — fast but ignores pairwise structure, typically 15–30% from optimal
- Simulated Annealing: O(n × iterations) — empirically finds solutions within 2–5% of optimal on QAPLIB benchmarks
- Practical note: real warehouse slotting also considers ergonomics, weight, shelf height, and zone constraints beyond the pure QAP formulation
References
Key literature on QAP and warehouse slotting