| Sq_m | |
|---|---|
| Department | |
| Cross-Dock | 3520 |
| Empty Pallets & Dunnage | 880 |
| Inbound Dock | 2640 |
| Maintenance & Battery Charge | 1320 |
| Outbound Staging — 2-Man Delivery | 5280 |
| Outbound Staging — Parcel | 3520 |
| Oversize/Non-Standard Storage | 2640 |
| Packing / Wrap / Banding | 3520 |
| Pallet Reserve Storage (Bulk) | 46340 |
| QA & Technical Test | 1760 |
| Receiving/Staging | 5280 |
| Returns & WEEE | 2640 |
| Shipping Dock | 3520 |
| Spare Parts & Accessories Cage | 440 |
11 Warehouse Layout Design Optimization
11.1 🎯 Aims
This assignment aims to introduce the concepts of warehouse layout design optimization through mathematical modeling. Frequently, design decisions are made based on experience and rules of thumb. However, mathematical models can provide a more systematic and quantitative approach to layout design, enabling better decision-making and optimization of warehouse operations.
11.2 🧠 Learning Objectives
After completing this assignment, students will be able to:
- Develop a mathematical optimization model for forward-reserve product allocation and functional area sizing problem.
- Develop a mathematical optimization model for warehouse block layout design problem considering different layout shapes (I-shaped, L-shaped, U-shaped) to account for building constraints.
11.3 🔧 Steps
11.3.1 Summary of Tasks
- Product Allocation and Functional Area Sizing (Min. Cost) (5 pts)
- Block Layout Design (6 Departments) (5 pts)
- (Optional) Robust Block Layout Design (All departments) (Bonus 5 pts)
11.3.2 Submission
Upon finishing, submit on Canvas:
- The .ipynb file with your answers.
- A PDF export of your .ipynb file.
11.3.3 Assessment Criteria Summary
- Product Allocation and Functional Area Sizing
- Correct formulation of the optimization model
- Area sizes for cross-docking, reserve, and forward areas
- Optimal product-to-flow assignments
- Total annual cost calculation
- Block Layout Design
- Correct formulation of the block layout optimization model for I-shaped, L-shaped, and U-shaped layouts (for at least 6 departments)
- Visualization of the layouts
Peer Review and Evaluation Rules
For this assignment, peer review will be used only for the Block Layout Design task.
- Each student is assigned two reviews.
- Completing 2 reviews = full points.
- Completing 1 review = half points.
- Completing 0 reviews = no points.
- If a submission has fewer than three reviews, the instructor will assign additional reviewers or provide an instructor review.
- Repair assignments are capped at a maximum grade of 5.5 and are graded directly by the instructor.
- The instructor may randomly audit reviews for fairness.
- Students may appeal reviews within 72 hours of receiving them. Appeals must reference specific rubric rows.
11.4 Product Allocation and Functional Area Sizing
A company is optimizing its warehouse layout to enhance operational efficiency and minimize total annual handling and storage costs. The warehouse manages six product classes, each with unique demand patterns, handling requirements, and storage characteristics.
Table 11.1 shows the costs and space requirements for each product. Notably, only products 3 and 6 have a reserve dwell fraction greater than zero, indicating that they spend on average 20% and 100% of their time in the reserve area, respectively, when they are assigned to flow 3 (Reserve to Forward).
| Metric | Product 1 | Product 2 | Product 3 | Product 4 | Product 5 | Product 6 |
|---|---|---|---|---|---|---|
| Annual demand (units) | 10,000 | 15,000 | 25,000 | 2,000 | 1,500 | 95,000 |
| Order cost ($) | 50 | 50 | 50 | 50 | 50 | 150 |
| Price/unit load ($) | 500 | 650 | 350 | 250 | 225 | 150 |
| Space required (m²) | 10 | 15 | 25 | 10 | 12 | 13 |
| Reserve dwell percentage (%) | 0 | 0 | 20% | 0 | 0 | 100% |
| Yearly carrying cost rate (%) | 10% | 10% | 10% | 10% | 10% | 10% |
When products are assigned to flows 1 (Cross-docking), 2 (Reserve), or 4 (Forward), they are handled as unit loads (e.g., pallets). However, when assigned to flow 3 (Reserve to Forward), products are initially stored in the reserve area as unit loads but are later broken down into cartons. This breakdown process incurs additional handling costs and requires more storage space due to the smaller size of the units. Consider that the break-bulk-factor \(q_{ij}=1\) for all products and flow combinations.
Table 11.2 shows the handling costs per unit load for each product in each material flow.
| Flow/Product | Product 1 | Product 2 | Product 3 | Product 4 | Product 5 | Product 6 |
|---|---|---|---|---|---|---|
| Flow 1 (CD) | 0.0707 | 0.0203 | 0.0267 | 0.3354 | 0.4083 | 0.0726 |
| Flow 2 (R) | 0.0849 | 0.2023 | 0.0420 | 0.5590 | 0.6804 | 0.0871 |
| Flow 3 (RF) | 0.1061 | 0.2023 | 0.0054 | 1.0062 | 1.2248 | 0.1088 |
| Flow 4 (F) | 0.0778 | 0.2023 | 0.0481 | 0.0671 | 0.8165 | 0.0798 |
Table 11.3 shows the yearly cost of storing a product unit load in each material flow.
| Flow/Product | Product 1 | Product 2 | Product 3 | Product 4 | Product 5 | Product 6 |
|---|---|---|---|---|---|---|
| Flow 1 (CD) | 20 | 15 | 4 | 5 | 15 | 20 |
| Flow 2 (R) | 5 | 5 | 20 | 4 | 25 | 5 |
| Flow 3 (RF) | 10 | 10 | 1 | 5 | 45 | 10 |
| Flow 4 (F) | 15 | 10 | 9 | 1 | 30 | 15 |
The warehouse has up to 100,000 m² of storage space available, which must be allocated among the three functional areas: cross-docking, reserve, and forward. The lower and upper bounds for each functional area are shown in Table 11.4.
| Functional Area | Lower bound (m²) | Upper bound (m²) |
|---|---|---|
| Cross-docking | 0 | 15,000 |
| Reserve | 35,000 | 75,000 |
| Forward | 35,000 | 75,000 |
The products are stored in a block layout, with each functional area having a specific number of vertical storage levels, as shown in Table 11.5.
| Functional Area | #Levels |
|---|---|
| Cross-docking | 1 |
| Reserve | 1 |
| Forward | 1 |
NoteEOQ Calculation
The order quantity (EOQ) for the products and the average dwell time are calculated based on the provided data.
11.4.1 Problem Statement
The goal is to determine the optimal allocation of products to material flows and the sizing of functional areas while adhering to constraints on space availability, product-specific requirements, and operational policies. Using the input data provided, address the following tasks:
Determine the optimal allocation of products to the four material flows:
- Flow 1 (CD): Receiving ➤ Cross docking ➤ Shipping
- Flow 2 (R): Receiving ➤ Reserve ➤ Shipping
- Flow 3 (RF): Receiving ➤ Reserve ➤ Forward ➤ Shipping
- Flow 4 (F): Receiving ➤ Forward ➤ Shipping
Calculate the optimal sizes of the cross-docking, reserve, and forward functional areas:
- Cross-docking area
- Reserve area
- Forward area
TipMathematical Model
Refer to the original mathematical model presented by Heragu et al. (2005) or Section 7.5 for guidance on formulating your optimization model.
11.4.2 Deliverables
- Model Implementation: Implement the optimization model using Python and an optimization library such as PuLP or Gurobi. Ensure that your code is well documented and includes comments explaining each step.
- Product Allocation: Optimal assignment of each product to one of the four material flows (CD, R, RF, F). For example:
- Product 1: Flow 2 (Reserve)
- Product 2: Flow 4 (Forward)
- etc.
- Area Sizes: Area sizes for each functional area:
- Cross-docking area: XX,XXX m²
- Reserve area: XX,XXX m²
- Forward area: XX,XXX m²
11.4.3 Assessment Criteria
We will evaluate your submission based on the following criteria: - The EOQ/average dwell time calculations are accurate. - The product-to-flow assignments are valid (each product assigned to exactly one flow). - The area sizes respect the lower and upper bounds for each functional area. - The model correctly minimizes total annual handling and storage costs.
11.4.4 Point Distribution
- Product Allocation: 2.5 pts
- Area Sizes: 2.5 pts
11.5 Warehouse Block Layout Design
Coolblue intends to deploy a new unit-load (white goods) distribution center in the Netherlands to serve the Benelux. Based on historical and projected demand, and on operational requirements (e.g., space, equipment, staffing, technology, product characteristics), the departments, their areas (see Table 11.6), and their adjacency relationships (see Table 11.7 and Table 11.8) are defined.
Coolblue is evaluating three warehouse layout patterns:
- I-Shaped: long rectangular block with inbound and outbound docks at opposite ends.
- L-Shaped: two orthogonal wings with inbound and outbound docks at the ends of each wing.
- U-Shaped: inbound and outbound docks located on the same side, creating a defined flow path.
TipDepartment Descriptions
- Inbound Dock: Trailer apron and doors for receipts; check-in, seal verification, unload to receiving lanes.
- Receiving/Staging: Counting, labeling, exception handling; short buffer before QA/putaway.
- QA & Technical Test: Visual/functional checks for white goods (power-on, cosmetic damage, accessories).
- Cross-Dock: Flow-through for urgent/vendor-prepped units; direct transfer to outbound lanes.
- Pallet Reserve Storage (Bulk): Primary pallet storage for appliances; high cube, stable replenishment.
- Oversize/Non-Standard Storage: Storage for odd-dimension or high-CG units with special handling.
- Packing / Wrap / Banding: Strapping, corner boards, stretch-wrap; labeling and ship docs.
- Outbound Staging — Parcel: Lane accumulation by parcel carrier/route.
- Outbound Staging — 2-Man Delivery: Lane accumulation for bulky deliveries and van loading.
- Shipping Dock: Parcel and linehaul doors; marshal, load sequence, seal.
- Empty Pallets & Dunnage: Pallet stacks, corner boards, wrap, foam; near receiving and pack.
- Maintenance & Battery Charge: MHE charging/maintenance with ventilation and clear aisles.
- Returns & WEEE: Bulky returns triage; compliant WEEE handling for refurb or recycling.
- Spare Parts & Accessories Cage: Small parts and accessories storage to support VAS and repairs.
| Inbound Dock | Receiving/Staging | QA & Technical Test | Cross-Dock | Pallet Reserve Storage (Bulk) | Oversize/Non-Standard Storage | Packing / Wrap / Banding | Outbound Staging — Parcel | Outbound Staging — 2-Man Delivery | Shipping Dock | Empty Pallets & Dunnage | Maintenance & Battery Charge | Returns & WEEE | Spare Parts & Accessories Cage | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| From/To | ||||||||||||||
| Inbound Dock | - | E | U | U | U | U | U | U | U | U | U | U | U | U |
| Receiving/Staging | E | - | A | A | I | U | U | U | U | U | I | U | U | U |
| QA & Technical Test | U | A | - | U | U | U | U | U | U | U | U | U | I | U |
| Cross-Dock | U | A | U | - | U | U | U | A | A | A | U | U | U | U |
| Pallet Reserve Storage (Bulk) | U | I | U | U | - | U | E | U | U | U | U | O | U | U |
| Oversize/Non-Standard Storage | U | U | U | U | U | - | I | U | U | U | U | U | U | U |
| Packing / Wrap / Banding | U | U | U | U | E | I | - | E | E | U | O | U | O | U |
| Outbound Staging — Parcel | U | U | U | A | U | U | E | - | U | E | U | U | U | U |
| Outbound Staging — 2-Man Delivery | U | U | U | A | U | U | E | U | - | E | U | U | U | U |
| Shipping Dock | U | U | U | A | U | U | U | E | E | - | U | U | U | U |
| Empty Pallets & Dunnage | U | I | U | U | U | U | O | U | U | U | - | U | U | U |
| Maintenance & Battery Charge | U | U | U | U | O | U | U | U | U | U | U | - | U | U |
| Returns & WEEE | U | U | I | U | U | U | O | U | U | U | U | U | - | U |
| Spare Parts & Accessories Cage | U | U | U | U | U | U | U | U | U | U | U | U | U | - |
| Meaning | Weight | |
|---|---|---|
| Code | ||
| E | Especially important to be adjacent | 4 |
| A | Absolutely necessary to be near | 3 |
| I | Important to be near | 2 |
| O | Ordinary closeness acceptable | 1 |
| U | Unimportant | 0 |
| X | Undesirable - keep apart | -4 |
11.5.1 Problem Statement
Your task is to design and compare these three layouts using the warehouse block layout model. Each layout should meet the following criteria:
- Accommodate all departments with their target areas.
- Respect fixed locations for inbound and outbound docks.
- Minimize the total unit load travel in the facility (i.e., the product of the flow and distance between departments, assuming unitary costs per unit load per meter).
TipModel Formulation Hints
To model the warehouse block layout problem, consider the following two approaches:
Option 1: Mixed-Integer Quadratic Programming (MIQCP)
- This approach uses a mixed-integer quadratic programming (MIQCP) formulation to handle non-linear relationships (e.g., \(A = L \times W\)) while allowing integer decision variables.
- MIQCP is well-suited for problems involving quadratic terms but may require a powerful solver like Gurobi due to its computational complexity.
Option 2: Mixed-Integer Linear Programming (MILP) with Area Approximation
- This approach uses a mixed-integer linear programming (MILP) formulation with linear approximations for non-linear constraints.
- Techniques such as perimeter-based approximations, aspect ratio constraints, and other techniques to model bilinear terms (e.g., McCormick envelopes) can be employed to simplify the problem.
- While MILP is generally easier to solve than MIQCP, it may involve trade-offs in accuracy.
11.5.2 Deliverables
- SLP Qualitative to Quantitative Conversion: Explain how you converted the qualitative SLP ratings into quantitative flow values.
- Model Implementation: Implement the chosen model (MIQCP or MILP) using Python and Gurobi.
- Layout Designs: Generate the three layout designs (I-shaped, L-shaped, U-shaped) and visualize them using appropriate plotting libraries (e.g., Matplotlib, Plotly).
11.5.3 Assessment Criteria
Robustness: The model should be able to handle at least 6 departments, namely:
- Inbound Dock
- Receiving/Staging
- Pallet Reserve Storage (Bulk)
- Packing / Wrap / Banding
- Outbound Staging (Join areas for 2-Man Delivery and Parcel)
- Shipping Dock
Correctness: For each layout, the model should correctly place all departments, respect fixed locations, and minimize the total unit load travel distance.
Visualization: The layouts should be clearly visualized, with departments labeled and dimensions indicated.
11.5.4 Point Distribution
- I-Shaped Layout: 3 pts
- L-Shaped Layout: 4 pts
- U-Shaped Layout: 3 pts
- Bonus: 5 pts for implementing a solution that can handle all departments: mathematical optimization with MIP gap < 5% or a heuristic.
11.6 References
Heragu, S. S., L. Du, R. J. Mantel, and P. C. Schuur. 2005. “Mathematical Model for Warehouse Design and Product Allocation.” International Journal of Production Research 43 (2): 327–38. https://doi.org/10.1080/00207540412331285841.