Week 6 Lecture Notes

Vehicle Routing in Transportation NetworksENGG1400

David Rey

Research Center for Integrated Transport Innovation (rCITI)School of Civil and Environmental Engineering

UNSW Australia

The Vehicle Routing Problem Mathematical Formulation Extensions of the VRP Applications

Outline

1 The Vehicle Routing Problem

2 Mathematical Formulation

3 Extensions of the VRP

4 Applications

Outline

4 Applications

History

• The Vehicle Routing Problem was first introduced by Dantzigand Ramser in 1959 as a combinatorial optimization problem.Generally the context of VRP is that of delivering goodslocated in a central depot to a list of customers which haveplaced orders for these goods.

• This problem can be encountered in the fields ofTransportation, Distribution and Logistics. This problem isgenerally hard to solve for a large number of customers.

Introduction of Vehicle Routing in Networks

• Finding the shortest tour to visit all nodes in a network

• The tour starts and ends at the depot

• In its most simple version (one vehicle, no capacity), the VRPis equivalent to the Travelling Salesman Problem (TSP)

• TSP is a NP-hard optimization problem, hence VRP is alsoNP-hard

A NP-hard problem is a problem for which there is no known polynomial algorithm

Vehicle Routing is a Hard Problem

→ Why is it hard?

3 nodes VRP:

Depot (0)

→ Why is it hard?

3 nodes VRP:

Depot (0)

• 0-123-0

• 0-132-0

• 0-213-0

• 0-231-0

• 0-312-0

• 0-321-0

→ Why is it hard?

3 nodes VRP:

Depot (0)

• 0-123-0

• 0-132-0

• 0-213-0

• 0-231-0

• 0-312-0

• 0-321-0

For N nodes there are N!combinations

N N! = 1× 2× . . .× N

3 64 245 12010 3628800100 9.332622× 10157

Outline

4 Applications

Mathematical Formulation of a basic VRP

Sets and Parameters

• G = (N,A) complete network, i.e. every pair of nodes islinked by two arcs (1 to 2 and 2 to 1)

• [tij ] is the cost matrix where i , j ∈ N and (i , j) ∈ A

• K is the set of available vehicles

Mathematical Formulation of a basic VRP

Sets and Parameters

• G = (N,A) complete network, i.e. every pair of nodes islinked by two arcs (1 to 2 and 2 to 1)

• [tij ] is the cost matrix where i , j ∈ N and (i , j) ∈ A

• K is the set of available vehicles

Decision Variables

xkij ≡

1 if (i , j) ∈ A belongs to the tour of vehicle k ∈ K

0 otherwise

Objective

The most common objective is to minimize the total transportationcost (travel time or other) defined by the cost matrix [tij ].

Objective function

z = min∑

(i ,j)∈A

tijxkij

z is equal to the sum of the transportation cost of every tour.

Notation:∑

(i ,j)∈A ⇔ “sum over all arcs in the network”

Vehicle Routing Constraints (1)

Flow Conservation Constraints

The nb. of vehicles entering and the nb. of vehicles exiting eachnode must the same (similar to Kirchhoff’s circuit laws inelectricity).

(i ,j)∈A

xkij −∑

(j ,i)∈A

xkji = 0 ∀i ∈ N, ∀k ∈ K

WRONG!!

Depot Constraints

Let 0 represent the depot node, each vehicle must leave and comeback to the depot.

(0,j)∈A

xk0j = 1 and∑

(i ,0)∈A

xki0 = 1 ∀k ∈ K

Depot (0)

vehicle 1 tour

vehicle 2 tour

Depot (0)

WRONG!!

Node Degree Constraints

Let Nc = N \ {0} be the set of customer nodes. Each node mustbe visited exactly once and by one vehicle only.

(i ,j)∈A

xkij = 1 and∑

(j ,i)∈A

xkji = 1 ∀i ∈ Nc

Depot (0)

vehicle 1 tour

vehicle 2 tour

Depot (0)

4WRONG!!

Model for the basic VRP?

z = min∑

(i ,j)∈A

tijxkij

subject to∑

(i ,j)∈A

xkij =1 ∀i ∈ Nc

(j ,i)∈A

xkji =1 ∀i ∈ Nc

(0,j)∈A

xk0j =∑

(i ,0)∈A

xki0 =1 ∀k ∈ K

(i ,j)∈A

xkij −∑

(j ,i)∈A

xkji =0 ∀i ∈ N, ∀k ∈ K

xkij ∈{0, 1} ∀(i , j) ∈ A, ∀k ∈ K

Model for the basic VRP?

z = min∑

(i ,j)∈A

tijxkij

subject to∑

(i ,j)∈A

xkij =1 ∀i ∈ Nc

(j ,i)∈A

xkji =1 ∀i ∈ Nc

(0,j)∈A

xk0j =∑

(i ,0)∈A

xki0 =1 ∀k ∈ K

(i ,j)∈A

xkij −∑

(j ,i)∈A

xkji =0 ∀i ∈ N, ∀k ∈ K

xkij ∈{0, 1} ∀(i , j) ∈ A, ∀k ∈ K

→ The solutions of this program can contain subtours

Example

Assume that there is a single vehicle to route:

Depot (0)

Example

The previous model for the VRP may produce to subtours

Depot (0)

vehicle 1 tour

subtour

Observe that all constraints are satisfied: Flow conservationconstraints, depot constraints, node degree constraints.

Subtour Elimination

It is necessary to eliminatesubtours among customersnodes

Let S ⊆ Nc , S has to betraversed by less than |S | − 1 arcs

S={1,2,3}:1

Subtour Elimination

S={1,2,3}:1

3x12 =1

x23 =1

Subtour Elimination

S={1,2,3}:1

3x12 =1

x23 =1

Subtour Constraints∑

(i ,j)∈A:i ,j∈S

xkij ≤ |S | − 1 ∀S ⊆ Nc , ∀k ∈ K

Model for the basic VRP

z = min∑

(i ,j)∈A

tijxkij

subject to ∑

(i ,j)∈A

xkij =1 ∀i ∈ Nc

(j ,i)∈A

xkji =1 ∀i ∈ Nc

(0,j)∈A

xk0j =∑

(i ,0)∈A

xki0 =1 ∀k ∈ K

(i ,j)∈A

xkij −∑

(j ,i)∈A

xkji =0 ∀i ∈ N, ∀k ∈ K

(i ,j)∈A:i ,j∈S

xkij ≤|S | − 1 ∀S ⊆ Nc , ∀k ∈ K

xkij ∈{0, 1} ∀(i , j) ∈ A, ∀k ∈ K

Example

Network and costs:

Depot (0)

Example

1 vehicle:

Depot (0)

vehicle 1 tour

x 0 1 2 3 4

0 - 1 0 0 01 0 - 1 0 02 0 0 - 1 03 0 0 0 - 14 1 0 0 0 -

Optimal tour: z⋆ = 12,

Path → 0-1234-0

Example

2 vehicles:

Depot (0)

vehicle 1 tour

vehicle 2 tour Optimal tours: z⋆ = 5 + 5 = 10

k = 0 → 0-12-0,k = 1 → 0-43-0,

Outline

4 Applications

Capacitated VRP (CVRP)

Assume every node is a customer and every visit corresponds to aproduct delivery.

If every costumer has a specific demand, qi ≥ 0, then vehicleshave a limited capacity, ck ≥ 0, and the total demand for everytour is bounded by this capacity.

Capacity constraints∑

(i ,j)∈A

xkij qi ≤ ck ∀k ∈ K

Example: CVRP

With capacity and demand:

Depot (0)

(80) (50)

vehicle capacity = 130customer demand = (i)

Example: CVRP

Previous solution:

Depot (0)

vehicle 1 tour

vehicle 2 tour

(80) (50)

Previous cost: z⋆ = 5 + 5 = 10

k = 0 → 0-12-0,k = 1 → 0-43-0,

...but infeasible (capacityviolation)!

Example: CVRP

New solution:

Depot (0)

vehicle 1 tour

vehicle 2 tour

(80) (50)

Optimal tours:z⋆ = 10 + 10 = 20

k = 0 → 0-14-0,k = 1 → 0-23-0,

...more expensive but feasible.

Major VRP Variants• Vehicle capacity and customer demand (CVRP)

• Heterogeneous fleet of vehicles (HVRP)

• Tour duration constraint (DVRP)

• Time windows for deliveries (VRPTW)

• Vehicle routing over a period (PVRP)

• Multi-Depot scenario (MDVRP)

• Arc Routing Problem (ARP)

Solution Methods

1 Heuristics• Problem-Specific (Tree Network, Schedule Assignment,...)• Nodes Clustering• Minimum Spanning Tree algorithms can be used to determine

bounds on TSP relaxation

2 Metaheuristics• Genetic Algorithms• Ant Colony Optimization• Tabu Search• Swarm Particles

3 Exact Algorithms• Branch & Bound & Cut (CPLEX)• Branch & Price• Decomposition Methods

In the VRP Workshop

Solution approach:

• Start by solving a VRP model without any subtour eliminationconstraints

• Study how we can incrementally add subtour eliminationconstraints to a VRP model:

1 detect existing subtours2 add the corresponding constraints

• Exact solution method but could be relatively slow in complexVRP instances

Outline

4 Applications

Fuel Delivery

• Capacitated

• Time Windows: operate at night time

Waste Management

• Capacitated

• Arc Routing: visit streets instead of points

Post Logistics

• Mail pick-up and delivery

• Node and Arc Routing: visit streets and points

Ready-Mixed Concrete Delivery

• Multiple visits

• Tour duration: perishable good

Medical Supply

• Periodic visits

• Emergency dispatching

Bibliography

• The Vehicle Routing Problem: An overview of exact andapproximate algorithms, G.Laporte (EJOR, 2001)

• The Vehicle Routing Problem, P. Toth and D. Vigo (SIAM,2002)

• The Vehicle Routing Problem: Latest Advances and NewChallenges, B. Golden, S. Raghavan, E. Wasil (SpringerOR/CS, 2008)

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