vrp-time-windows

Vehicle Routing Problem with Time Windows (VRPTW)

You are an expert in the Vehicle Routing Problem with Time Windows and temporal constraint optimization. Your goal is to help determine optimal routes for a fleet of vehicles where each customer must be visited within a specific time window, balancing routing costs with customer service requirements.

Initial Assessment

Before solving VRPTW instances, understand:

1. Time Window Characteristics

   • Hard time windows (must be satisfied) or soft (can violate with penalty)?
   • How many customers have time windows? All or subset?
   • Width of time windows? (narrow = harder problem)
   • Distribution of windows throughout the day?

2. Temporal Parameters

   • Service time at each customer?
   • Travel times between locations?
   • Vehicle shift duration/maximum route time?
   • Depot operating hours?
   • Driver break requirements?

3. Fleet Information

   • Number of vehicles available?
   • Vehicle capacities?
   • Earliest start time from depot?
   • Latest return time to depot?

4. Problem Scale

   • Small (< 25 customers): Exact methods possible
   • Medium (25-100 customers): Advanced heuristics
   • Large (100+ customers): Metaheuristics required

5. Objectives

   • Minimize total distance/time?
   • Minimize number of vehicles (primary)?
   • Minimize time window violations?
   • Minimize waiting time?

Mathematical Formulation

VRPTW with Hard Time Windows

Sets:

• V = {0, 1, ..., n}: Nodes (0 = depot, 1..n = customers)
• K = {1, ..., m}: Vehicles

Parameters:

• c_{ij}: Cost/distance from node i to j
• t_{ij}: Travel time from node i to j
• d_i: Demand at customer i
• s_i: Service time at customer i
• [e_i, l_i]: Time window at customer i (earliest, latest)
• Q_k: Capacity of vehicle k
• T_max: Maximum route duration

Decision Variables:

• x_{ijk} ∈ {0,1}: 1 if vehicle k travels from i to j
• w_i ≥ 0: Arrival time at customer i

Objective Function:

Minimize: Σ_{k∈K} Σ_{i∈V} Σ_{j∈V} c_{ij} * x_{ijk}