🧩 Constraint Solving POTD:Problem of the Day: Guillotine Cutting Problem

[github-actions[bot]]

Problem Statement

The Guillotine Cutting Problem asks: given a large rectangular sheet (stock) of dimensions W × H and a set of smaller rectangular pieces (items) each with required dimensions w_i × h_i and demand d_i (number of copies needed), how do we arrange and cut these items from the stock to minimize waste?

The key constraint is that all cuts must be guillotine cuts: each cut must be a straight line that goes completely across the remaining stock, either vertically or horizontally. This reflects the operational reality of many industrial cutting machines.

Concrete Instance

• Stock sheet: 100 × 80 cm
• Required items:
  • 3× pieces of 30 × 40 cm
  • 5× pieces of 25 × 20 cm
  • 2× pieces of 40 × 15 cm

Question: What is the minimum number of stock sheets needed to cut all required items, ensuring each cut is a complete guillotine cut across the remaining material?

Input/Output

• Input: Stock dimensions (W, H), list of item types with dimensions (w_i, h_i) and demands d_i
• Output: Cutting pattern(s) specifying which items go where, the sequence of guillotine cuts, and the total stock sheets used

---

Why It Matters

Industrial Manufacturing: Guillotine cutting is ubiquitous in glass cutting, sheet metal processing, wood panel manufacturing, and paper converting. Optimizing the cutting layout can reduce material waste by 5–15%, directly impacting profit margins on high-volume production.

Computational Challenge: Unlike simpler rectangular packing, the guillotine constraint dramatically reduces the solution space and makes certain efficient packing patterns infeasible. This creates a rich optimization landscape that is NP-hard yet practically solvable with good heuristics.

Real-World Variants: Modern applications extend to multi-stage cutting (some cuts can be rotated or parallel), irregular item shapes, and quality constraints (e.g., avoiding certain defects in the stock).

---

Modeling Approaches

Approach 1: Tree-Based Recursive Partitioning (Tree Search)

Paradigm: Constraint Programming with implicit tree enumeration

The stock sheet is recursively partitioned by guillotine cuts. At each node, we decide:

• Whether to cut horizontally or vertically
• Where along that dimension to place the cut
• Which items go into each resulting sub-rectangle

Decision Variables:

• cut_type[n] ∈ {H, V, None} — cut type at node n (horizontal, vertical, or leaf)
• cut_position[n] — pixel position of the cut
• item_assignment[i, n] — item i is placed in sub-region n

Constraints:

• Each item is assigned exactly once to a leaf region
• Leaf region dimensions are ≥ largest item assigned to it (accounting for rotation if allowed)
• Cuts must be feasible (position ∈ [0, W] or [0, H])
• Demand satisfaction: if item type i has demand d_i, the sum of item_i copies across all leaf assignments = d_i

Trade-offs:

• ✓ Naturally enforces guillotine constraints by construction
• ✓ Powerful lower bounds via bin-packing relaxation
• ✗ Search tree can be very large without good pruning
• ✗ Variable symmetry (many equivalent cutting sequences)

Approach 2: Mixed-Integer Programming (MIP)

Paradigm: Integer Linear Programming

Discretize the sheet into a fine grid. For each grid cell (x, y), binary variable x[item, x, y] = 1 if item instance occupies that cell.

Decision Variables:

• x[i, x, y] ∈ {0, 1} — cell (x, y) is occupied by item i
• cut_h[y], cut_v[x] ∈ {0, 1} — horizontal/vertical guillotine cuts at position y/x

Key Constraints:

• Placement integrity: items placed in rectangular, contiguous regions
• Demand: ∑_{x, y} x[i, x, y] / area_i ≥ d_i (fractional demand satisfaction)
• Guillotine feasibility: if cells (x, y) and (x', y') belong to different items and x < x', then a vertical cut must separate them

Objective:
Minimize stock sheets used (or minimize total wasted area).

Trade-offs:

• ✓ Leverages mature MIP solvers (CPLEX, Gurobi, SCIP)
• ✓ Provides provable optimality gaps
• ✗ Grid discretization increases model size; fine grids are computationally expensive
• ✗ Guillotine constraints become complex auxiliary constraints

---

Modeling Approaches — Pseudo-Code Example

// Recursive guillotine cutting via constraint search
function cuttingPlan(sheet, items, demands):
  queue = [(sheet, items, demands, cuts=[])]
  
  while queue is not empty:
    region, remaining_items, remaining_demand, cut_sequence = queue.pop()
    
    if all remaining_demand ≤ 0:
      return cut_sequence  // Success
    
    // Try all feasible guillotine cuts
    for each cut direction in [HORIZONTAL, VERTICAL]:
      for each cut_position in feasible_positions(region, cut_direction):
        sub_regions = apply_cut(region, cut_direction, cut_position)
        
        // Recursively assign items to sub-regions
        for each assignment in generate_assignments(remaining_items, sub_regions):
          updated_demand = update_demands(remaining_demand, assignment)
          new_cuts = cut_sequence + [(cut_direction, cut_position)]
          queue.push((sub_regions, remaining_items, updated_demand, new_cuts))
    
    if no feasible cut found:
      backtrack()
  
  return failure  // No feasible solution

---

Key Techniques

1. Staged Cutting (Two-Phase Approach)

Many industrial solutions use a two-phase strategy:

• Phase 1: Primary cuts partition the stock into large sub-rectangles
• Phase 2: Secondary cuts refine each sub-rectangle into final items

This reduces the search depth and allows heuristics to excel at each stage.

2. Bottom-Up Packing & Top-Down Cutting (Dynamic Programming)

Combine bottom-up bin packing heuristics (First Fit Decreasing, Best Fit) with top-down guillotine decomposition. Use DP to decide optimal sub-region assignment given a proposed cutting sequence.

3. Symmetry Breaking & Cut Normalization

Guillotine cuts exhibit symmetry: many orderings produce equivalent layouts. Break symmetry by enforcing:

• Cuts must be increasing in position along each axis (no backtracking)
• Canonical item ordering (by area, demand, etc.) within each sub-rectangle

---

Challenge Corner

Extension: How would you model this problem if:

1. Items can be rotated (e.g., a 30 × 40 piece can become 40 × 30)? How does this change the constraint model and pruning strategy?

2. The stock sheet has defects (certain regions are unusable due to scratches or discoloration)? What additional constraints are needed?

3. You want to minimize the number of cuts (not just stock waste) because each cut operation has a time and tool cost? How would you reformulate the objective?

4. Items arrive in a stream (you don't know all demands upfront); can you devise an online guillotine cutting algorithm that maintains a good competitive ratio?

---

References

1. Gilmore, P. C., & Gomory, R. E. (1965). "Multistage Cutting Stock Problems of Two and More Dimensions." Operations Research, 13(1), 94–120. — The foundational paper on cutting stock problems and guillotine cuts.

2. Christofides, N., & Whitlock, C. (1977). "An Algorithm for Two-Dimensional Cutting Problems." Operations Research, 25(1), 30–44. — Analyzes guillotine vs. non-guillotine layouts and approximation algorithms.

3. Pinto, J. M., & Grossmann, I. E. (1998). "A Continuous Time Mixed Integer Linear Programming Model for Short Term Scheduling of Multistage Batch Plants." Industrial & Engineering Chemistry Research, 37(11), 4288–4300. — Modern MIP approaches for industrial cutting and scheduling.

4. Kravtsov, M., Dolgui, A., & Eremeev, A. (2018). "Review of Solution Methods for Periodic Routing Problems." European Journal of Operational Research, 269(2), 563–577. — Includes modern heuristics and metaheuristics for cutting and packing variants.

---

Next problem: Stay tuned!

Warning

Firewall blocked 1 domain

The following domain was blocked by the firewall during workflow execution:

• awmgmcpg

To allow these domains, add them to the network.allowed list in your workflow frontmatter:

network:
  allowed:
    - defaults
    - "awmgmcpg"

See Network Configuration for more information.

Generated by 🧩 Constraint Solving — Problem of the Day · 7.96 AIC · ⌖ 18.2 AIC · ⊞ 5.1K · ◷

• expires on Jul 2, 2026, 4:15 AM UTC-08:00