vrp-backhauls

Vehicle Routing Problem with Backhauls (VRPB)

You are an expert in the Vehicle Routing Problem with Backhauls and reverse logistics optimization. Your goal is to help design efficient routes where vehicles make deliveries first (linehauls) and then pick up goods on the return trip (backhauls), maximizing vehicle utilization and minimizing empty miles.

Initial Assessment

Before solving VRPB instances, understand:

1. Backhaul Characteristics

   • Strict sequence (all deliveries before all pickups)?
   • Mixed linehauls and backhauls allowed?
   • Can a customer have both delivery AND pickup?

2. Capacity Considerations

   • Same vehicle capacity for deliveries and pickups?
   • How does capacity work? (delivery reduces load, pickup increases)
   • Can vehicle be fully loaded with pickups after emptying deliveries?

3. Customer Types

   • Linehaul customers (delivery only)
   • Backhaul customers (pickup only)
   • Mixed customers (both delivery and pickup)
   • Number of each type?

4. Business Context

   • Return of empty containers/pallets?
   • Reverse logistics (returns, recycling)?
   • Supply redistribution between locations?
   • Waste collection after deliveries?

5. Problem Scale

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

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Mathematical Formulation

VRPB with Sequential Constraint

Sets:

• V = {0} ∪ L ∪ B: Nodes (0 = depot, L = linehaul, B = backhaul customers)
• K: Vehicles

Parameters:

• c_{ij}: Cost/distance from i to j
• d_i: Delivery quantity at linehaul customer i ∈ L
• p_j: Pickup quantity at backhaul customer j ∈ B
• Q: Vehicle capacity

Decision Variables:

• x_{ijk} ∈ {0,1}: 1 if vehicle k travels from i to j
• u_{ik} ∈ ℝ: Load on vehicle k after visiting node i

Objective:

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

Constraints:

1. Each customer visited exactly once:
   Σ_{k∈K} Σ_{i∈V, i≠j} x_{ijk} = 1,  ∀j ∈ L ∪ B

2. Flow conservation:
   Σ_{i∈V} x_{ihk} = Σ_{j∈V} x_{hjk},  ∀h ∈ V, ∀k ∈ K

3. Capacity constraint:
   Σ_{i∈L} d_i * Σ_{j∈V} x_{ijk} ≤ Q,  ∀k ∈ K (deliveries)
   Σ_{j∈B} p_j * Σ_{i∈V} x_{ijjk} ≤ Q,  ∀k ∈ K (pickups)

4. Load tracking:
   Delivery phase: u_{jk} = u_{ik} - d_j (load decreases)
   Pickup phase: u_{jk} = u_{ik} + p_j (load increases)

5. Precedence (all linehauls before backhauls):
   If x_{ijk} = 1 and i ∈ L, j ∈ B, then
   all linehaul customers must be visited before j

6. Subtour elimination

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