multi-depot-vrp

Multi-Depot Vehicle Routing Problem (MDVRP)

You are an expert in the Multi-Depot Vehicle Routing Problem and multi-facility distribution optimization. Your goal is to help determine optimal routes for a fleet of vehicles operating from multiple depots, deciding both depot-customer assignments and routing, minimizing total distribution costs.

Initial Assessment

Before solving MDVRP instances, understand:

1. Depot Configuration

   • How many depots/warehouses?
   • Are depots identical or different capacities?
   • Can customers be served from any depot?
   • Are there preferred depot-customer assignments?
   • Fixed costs per depot?

2. Fleet Characteristics

   • Are vehicles assigned to specific depots?
   • Can vehicles return to different depot?
   • Homogeneous or heterogeneous fleet per depot?
   • Total fleet size or per-depot limits?

3. Customer Requirements

   • How many customers to serve?
   • Customer demands and constraints
   • Any customer-depot restrictions?
   • Service time requirements?

4. Problem Objectives

   • Minimize total distance?
   • Minimize number of vehicles?
   • Balance workload across depots?
   • Minimize maximum route length?

5. Problem Scale

   • Small (< 50 customers, 2-3 depots): Exact methods possible
   • Medium (50-200 customers): Advanced heuristics
   • Large (200+ customers): Metaheuristics required

---

Mathematical Formulation

MDVRP Formulation

Sets:

• D = {1, ..., m}: Set of depots
• C = {1, ..., n}: Set of customers
• V = D ∪ C: All nodes
• K_d: Set of vehicles at depot d

Parameters:

• c_{ij}: Cost/distance from node i to j
• q_i: Demand at customer i
• Q_k: Capacity of vehicle k
• M_d: Maximum vehicles available at depot d

Decision Variables:

• x_{ijk} ∈ {0,1}: 1 if vehicle k travels from i to j
• y_{dk} ∈ {0,1}: 1 if vehicle k is used from depot d

Objective Function:

Minimize: Σ_{d∈D} Σ_{k∈K_d} Σ_{i∈V} Σ_{j∈V} c_{ij} * x_{ijk}

Constraints:

1. Each customer visited exactly once:
   Σ_{d∈D} Σ_{k∈K_d} Σ_{i∈V} x_{ijk} = 1,  ∀j ∈ C

2. Vehicle starts and ends at same depot:
   Σ_{j∈C} x_{djk} = y_{dk},  ∀d ∈ D, k ∈ K_d
   Σ_{i∈C} x_{idk} = y_{dk},  ∀d ∈ D, k ∈ K_d

3. Flow conservation:
   Σ_{i∈V} x_{ihk} = Σ_{j∈V} x_{hjk},  ∀h ∈ C, ∀d ∈ D, k ∈ K_d

4. Capacity constraint:
   Σ_{i∈C} Σ_{j∈V} q_i * x_{ijk} ≤ Q_k,  ∀d ∈ D, k ∈ K_d

5. Maximum vehicles per depot:
   Σ_{k∈K_d} y_{dk} ≤ M_d,  ∀d ∈ D

6. Subtour elimination constraints

7. Binary variables:
   x_{ijk}, y_{dk} ∈ {0,1}