Case Study 2: Integer Programming

Distribution Network Design for a Regional Food Bank

ISYE 671 — Linear Optimization and Network Flows

Background

The Northern Illinois Community Food Bank (NICFB) serves 11 counties across northern Illinois. The organization currently operates out of a single aging warehouse in Rockford and distributes food to 85 partner agencies (pantries, shelters, soup kitchens, after-school programs). Over the past five years, demand for food assistance has grown 40%, and the existing facility is at capacity. NICFB has received a $12 million capital grant from a foundation, and the board has decided to redesign its distribution network by opening new warehouse/distribution centers and potentially closing or downsizing the existing one.

You have been hired as a consultant to determine the optimal network configuration: which facilities to open, what capacity to build, which agencies to serve from each facility, and what vehicle fleet to acquire — all while respecting budget constraints, service-level requirements, and the unique operational realities of food banking.

Problem Data

Candidate Facility Locations

NICFB has identified 7 candidate locations for warehouse/distribution centers. Each location can be built at one of three capacity tiers (Small, Medium, Large). The existing Rockford facility can be retained at its current size or closed.

If the Rockford facility (F1) is closed, there is a one-time closure cost of $200,000 (lease termination, moving equipment). The existing facility has fixed annual operating costs of $180,000 regardless of utilization. If F1 is retained, it can additionally be renovated to Medium capacity (15,000 sq ft) for $1.5M or Large (25,000 sq ft) for $3.8M.

Annual operating costs per facility (includes utilities, insurance, basic staffing):

Each square foot of warehouse space can support approximately 25 pounds of food throughput per week.

Partner Agencies

The 85 partner agencies are grouped into 18 demand zones based on geographic proximity. Each zone has the following attributes:

Total weekly demand: approximately 240,000 lbs.

Distance Matrix (miles, facility to zone centroid)

Vehicle Fleet

NICFB must also decide what delivery vehicles to acquire. Vehicles are purchased (one-time capital cost from the $12M grant) and have annual operating costs:

Fleet constraints: - Each facility must have at least 1 refrigerated truck (cold chain requirements). - Total fleet size across all open facilities cannot exceed 25 vehicles (driver availability). - Vehicles are assigned to (home-based at) specific facilities. - Each delivery from a facility to a zone requires one round trip. A vehicle serving a zone within 20 miles can make the round trip and do another delivery the same day (counts as 1 delivery toward its weekly max). Beyond 20 miles, a delivery consumes an entire half-day (counts as 1.5 deliveries toward its weekly max).

Budget and Planning Horizon

The $12 million capital grant must cover all facility construction/renovation, closure costs, and vehicle purchases. NICFB wants to minimize total annual operating cost (facility operating costs + vehicle operating costs + transportation cost) over the planning horizon.

Transportation cost: $2.50 per mile per delivery trip (round-trip distance = 2 × one-way distance from distance matrix).

The planning horizon for evaluating operating costs is 5 years. The objective should account for both one-time capital expenditures and the present value of 5 years of annual operating costs (use a discount rate of 4%).

Service Requirements

1. Coverage: Every demand zone must be served by exactly one facility.
2. Distance: No zone may be assigned to a facility beyond its maximum acceptable distance.
3. Capacity: Total weekly demand assigned to a facility must not exceed its throughput capacity.
4. Delivery feasibility: Each facility must have sufficient vehicle capacity (total lbs per week across its fleet) to deliver all assigned demand, and sufficient delivery slots per week.
5. Equity: No single facility may serve more than 40% of total system demand (to avoid single-point-of-failure risk).
6. Minimum facilities: At least 2 facilities must be open (for redundancy).

Your Assignment

Part 1: Model Formulation

Formulate a mixed-integer programming model. Your formulation should clearly define:

1. Sets and indices for facilities, capacity tiers, zones, and vehicle types.
2. Parameters with clear notation matching the data above.
3. Decision variables: You will need binary variables for facility location and tier selection, integer variables for vehicle fleet composition, and continuous or integer variables for zone assignments and delivery logistics.
4. Objective function: Minimize total cost (capital + discounted annual operating costs). Write out the NPV calculation explicitly.
5. Constraints: All service requirements, budget constraints, capacity constraints, fleet constraints, and logical constraints (e.g., at most one tier per facility, can only assign zones to open facilities).
6. Modeling choices: Document any assumptions. In particular, address how you handle the fixed-charge nature of facility costs, the conditional renovation/closure logic for F1, and the delivery counting rule for distances beyond 20 miles.

Part 2: Implementation and Solution

Implement your model in AMPL with Gurobi in a Google Colab notebook.

1. Solve the model and report: which facilities open at which tier, the zone-to-facility assignment map, the vehicle fleet at each facility, and the total cost breakdown (capital vs. annual operating vs. transportation).

2. Produce a clear summary table and, if possible, a simple visualization (e.g., a map or network diagram using matplotlib/networkx) showing the recommended network.

3. Report the MIP gap achieved and solve time. If the model does not solve to optimality within a reasonable time, discuss what tightening strategies you might employ (e.g., valid inequalities, symmetry breaking).

Part 3: Analysis and Recommendations

1. What-if analysis: Test at least three of the following scenarios:

   • Demand grows 25% across all zones.
   • The capital budget is reduced to $9 million.
   • Maximum acceptable distances are reduced by 5 miles for all zones (stricter service level).
   • One candidate location (your choice) becomes unavailable.
   • The Rockford facility must be retained (political/community pressure).

2. Robustness discussion: How sensitive is the optimal network design to the demand forecasts? If demand shifts geographically (e.g., rural zones grow faster), would the recommended network still perform well? Discuss qualitatively.

3. Board presentation: Write a concise recommendation memo (1–2 pages) suitable for the NICFB board of directors. Include the recommended network design, total investment required, projected annual costs, key trade-offs considered, and risks. Use non-technical language appropriate for a board audience.

Deliverables

• A written formulation document (PDF or markdown).
• A Google Colab notebook (.ipynb) with working AMPL/Gurobi code, results, scenario analysis, and the board memo.