The Vehicle Routing Problem Using CP in the real world

The Vehicle Routing Problem

Using CP in the real world

2/58NICTA Copyright 2011 From imagination to impact

Outline

Session 1

• What is the vehicle routing problem (VRP)?

• ‘Traditional’ (Non-CP) solution methods

• A simple CP model

• Playtime

Session 2

• A CP model for the VRP

• Large Neighbourhood Search revisited

• Propagation

• More Playtime


What is the Vehicle Routing Problem

Given a set of customers,

and

a fleet of vehicles to make deliveries,

find

a set of routes that services all customers at minimum cost


Vehicle Routing Problem




Why study the VRP?

• The logistics task is 9% of economic activity in Australia

• Logistics accounts for 10% of the selling price of goods



For each customer, we know

Quantity required

The cost to travel to every other customer

For the vehicle fleet, we know

The number of vehicles

The capacity

We must determine which customers each vehicle serves, and in what order, to minimise cost



Objective function

In academic studies, usually a combination:

First, minimise number of routes

Then minimise total distance or total time

In real world

A combination of time and distance

Must include vehicle- and staff-dependent costs

Usually vehicle numbers are fixed



• Might be more than one ‘capacity’

– e.g. limited weight and volume

• Might be more than one type of vehicle

– Different capacities and costs

– ‘Heterogeneous Vehicles’


MIP formulation

}1,0{

}x{

xq

,0xx

1x

1x

subject to

c:minimise

ijk

jijki

j k j khjkihk

j kijk

i kijk

,ij

ijk

ki

ji kijk

x

S

kQ

hk

i

j

x

Datacij: Cost of travel from i to j

qi: Demand at i

Decision var:

xijk: Travel direct from i to j on vehicle k

Depot



In a classic 2-echelon routing problem:

Stage 1: Production/warehouse to distribution depots

Stage 2: Distribution depots to end customer

VRP is primarily concerned with Stage 2

• Strategic decision

• Also study dynamic version


History: Travelling Salesman Problem

• A travelling salesman has to visit a number of cities. He knows the cost of travel between each pair.What order does he visit the cities to minimise cost?

• Long and well-studied problem (since the 60’s)• A sub-problem in many others• Used in chip fabrication and many other real-world

problems

• TSP = VRP with 1 vehicle of infinite capacity

• In vehicle routing - having decided which vehicle will visit which customers, each vehicle route is a travelling salesman problem


Travelling Salesman Problem

• Exact solution are found regularly for problems with 200-300 cities, and occasionally for problems with 1000 nodes.

• Some larger problems solved(24,798 cities, towns and villages in Sweden)

• But – no constraints on the solution

• Even one constraint, and the whole method is unusable


Routing with constraints

• Each new twist (business rule, practice, limitation) changes which solutions are good, and which are even acceptable

• The types of constraints that must be observed may impact on the way the problem is solved

• Many types of constraint studied – but usually in isolation

• We will look at a few that have been studied in the literature


Time window constraints

Vehicle routing with Constraints • Time Window constraints

– A window during which service can start– E.g. only accept delivery 7:30am to 11:00am

– Additional input data required• Duration of each customer visit• Time between each pair of customers

• (Travel time can be vehicle-dependent or time-dependent)

– Makes the route harder to visualise


Time Window constraints


Pickup and Delivery problems

• Most routing considers delivery to/from a depot (depots)

• Pickup and Delivery problems consider FedEx style

problem:

pickup at location A, deliver to location B

Load profile:


Pickup and Delivery problems

• PDPs have two implied constraints:

– pickup is before delivery

– pickup and delivery are on the same vehicle

• Usually, completely different methods used to solve this sort of problem

• Can be quite difficult

• Standard VRP is in effect a PDP with all stuff picked up at (delivered to) a depot. Not usually solved that way


Pickup and Delivery problem

Interesting variants

• Dial-a-ride problem:

– Passenger transport

– Like a multi-hire taxi

– Pickup passenger A, pickup passenger B, drop off B, pickup up C, drop off A, ….

– Ride-time constraints (e.g. max 1.4 x direct travel time)

• PDP can be used to model cross-docking

– Pick up at Factory, Deliver to DC; Pickup at DC, Deliver to customer

– Constraint: “Deliver to DC” before “Pickup at DC”


Fleet size and mix

• Heterogonous vehicles

– Vehicles of different capacities, costs, speeds etc

• Fleet size and mix problem

– Decide the correct number of each type of vehicle

– Strategic decision

• Can be the most important part of optimization


Other variants

Profitable tour problem

• Not all visits need to be completed

• Known profit for each visit

• Choose a subset that gives maximum return(profit from visits – routing cost)


Other variants

Period Routing

• Routing with periodical deliveries

• Same routes every week / fortnight

• Deliver to different customers with different frequencies: patterns

• 3-part problem

– Choose pattern for each cust

– Choose qty for each delivery

– Design route for each day

M T W T F

? ?

? ?

? ? ?

? ? ? ? ?


VRP meets the real world

Many groups now looking at real-world constraints

Rich Vehicle Routing Problem

• Attempt to model constraints common to many real-life enterprises

– Multiple Time windows

– Multiple Commodities

– Heterogeneous vehicles

– Compatibility constraints • Goods for customer A can’t travel with goods from

customer B• Goods for customer A can’t travel on vehicle C



Other real-world considerations

• Fatigue rules and driver breaks

• Vehicle re-use

• Ability to change vehicle characteristics (composition)

– Add trailer, or move compartment divider

• Use of limited resources

– e.g limited docks for loading, hence need to stagger dispatch times

• Variable loading / unloading times



Yet more constraints

• Only two types of product on each vehicle

• ‘Grand tour’ constraint

– Customers visited in ‘patterns’ (Period Routing)

– Grand tour includes all visits

– Skip visit if not required that day

– Each day must be feasible (incl. time windows)

• Meet ferry

• Blood transport (dynamic time window)

• Promiscuous driver constraint



New data sources

• Routing with time-of-day dependent travel times

– Uses historical data to forecast travel time at different times of day

• Routing with dynamic travel times

– Uses live traffic information feed to update expected travel time dynamically



Stochastic Routing

What if things don’t go according to plan?

• Sources of uncertainty

– Uncertainty in existence (do I even need to visit)

– Uncertainty in quantity (how much is actually required)

– Uncertainty in travel times (traffic)

– Uncertainty in duration (maintenance engineer)

• Optimal solution can be brittle

– If something is not quite right, whole solution falls apart



E.g. garbage collection

• Wet rubbish is heavy

• On rainy days, trucks may have more load than usual(uncertainty in quantity)

• Need stops near dump in case they have to double back

• = Recourse.


Solution Methods


Solving VRPs

VRP is hard

• NP Hard in the strong sense

• Exact solutions only for small problems (20-50 customers)

• Most solution methods are heuristic

• Most operate as

– Construct

– Improve


ILP

Advantages

• Can find optimal solution

Disadvantages

• Only works for small problems

• One extra constraint back to the drawing board

}1,0{

}x{

xq

,0xx

1x

1x

subject to

c:minimise

ijk

jijki

j k j khjkihk

j kijk

i kijk

,ij

ijk

ki

ji kijk

x

S

kQ

hk

i

j

x


ILP – Column Generation

89 76 99 45 32

1 1 1 1 0 0 …

2 0 1 1 0 1 …

3 0 0 0 0 0 …

4 1 0 1 1 0 …

5 1 0 0 0 0 …

Columns represent routes

Rows represent customers

Column/route cost ck

Array entry aik = 1 iff customer i is covered by route k


Column Generation

• Decision var xk: Use column k?

• Column only appears if feasible ordering is possible

• Cost of best ordering is ck

• Best order stored separately

• Master problem at right

}1,0{

1

subject to

min

k

kik

kk

x

ia

xc

i


Column Generation

Subproblem

• i.e. shortest path with side constraints

• If min is –ve add to master problem

• CP!

sconstraint Route

1

1 s.t.

)(min,

iij

jij

jijji

ij

jx

ix

cx


Heuristic methods - Construction

Insert methods

Insert1.dig

Order is important:

Insert2.dig

Compare.dig


Heuristic methods - Construction

Savings method (Clarke & Wright 1964)

• Calculate Sij for all i, j

• Consider cheapest Sij

• If j can be appended to i

– merge them to new i

– update all Sij

• else

– delete Sij

• Repeat

ij


Improvement Methods

2-opt (3-opt, 4-opt…)

• Remove 2 arcs

• Replace with 2 others


Improvement methods

Large Neighbourhood Search

= Destroy & Re-create

• Destroy part of the solution

– Remove visits from the solution

• Re-create solution

– Use insert method to re-insert customers

– Different insert methods lead to new (better?) solutions

• If the solution is better, keep it

• Repeat


Improvement methods

Variable Neighbourhood Search

• Consider multiple neighbourhoods

– 1-move (move 1 visit to another position)

– 1-1 swap (swap visits in 2 routes)

– 2-2 swap (swap 2 visits between 2 routes)

– 2-opt

– 3-opt

– Or-opt size 2 (move chain of length 2 anywhere)

– Or-opt size 3 (chain length 3)

– Tail exchange (swap final portion of routes


Improvement methods

Variable Neighbourhood Search

• Consider multiple neighbourhoods

• Find local minimum in smallest neighbourhood

• Advance to next-largest neighbourhood

• Search current neighbourhood

– If a change is found, return to smallest neighbourhood

– Otherwise, advance to next-largest


Improvement methods

Path Relinking

• Applies where-ever a population of solutions is available

• Take one (good) solution A

• Take another (good) reference solution B

• Gradually transform solution A into solution B

– pass through new solutions “between” A and B

– new solutions contain traits of both A and B

– should be good!


Improvement methods

Genetic Programming

• Simulated Natural Selection


Genetic Programming

• Generate a population of solutions (construct methods)

• Evaluate fitness (objective)• Create next generation:

– Choose two solutions from population– Combine the two (two ways)– (Mutate)– Produce offspring (calculate fitness)– (Improve)– Repeat until population doubles

• Apply selection:– Bottom half “dies”

• Repeat


Genetic Programming

1 4 6 2 3 8 9 5 10 7

1 10 8 2 5 4 9 3 6 7

125

132

6 1 9 7 3 2 10 4 8 5

1 9 7 10 8 6 4 3 2 5

160

198


Genetic Programming

1 4 6 2 3 8 9 5 10 7

1 10 8 2 5 4 9 3 6 7

125

132

6 1 9 7 3 2 10 4 8 5

1 9 7 10 8 6 4 3 2 5

160

198

1 10 8 2 3 9 5 10 7 4 128


Genetic Programming

1 4 6 2 3 8 9 5 10 7

1 10 8 2 5 4 9 3 6 7

125

132

6 1 9 7 3 2 10 4 8 5

1 9 7 10 8 6 4 3 2 5

160

198

1 10 8 2 5 9 5 10 7 4

1 4 6 2 3 9 3 7 8 10

128

206


Genetic Programming

1 4 6 2 3 8 9 5 10 7

1 10 8 2 5 4 9 3 6 7

125

132

6 1 9 7 3 2 10 4 8 5160

1 10 8 2 5 9 5 10 7 4 128


Genetic Programming

Split

• Solution representation does not include route delimiters

• Routes are determined using “Split” routine

– Solves shortest path problem on an auxiliary graph

– O(mn2)

• Allows freer movement in solution space


Scaling

Solving problems with tens of thousands of nodes

• Decompose problem

– Split into smaller problems

• Limit search

– Only consider inserting next to nearby nodes

– Only consdier inserting into nearby routes


Solution Methods

.. and the whole bag of tricks

• Tabu Search

• Simulated Annealing

• Ants

• Bees

• ….


A CP Model for the TSP


Model

Classic TSP

• n customers (fixed in this model)

• fixed locations

– where things happen

– one for each customer + one for the depot

• Know distance/cost C(i,j) between each location pair

– Obeys triangle inequality

• Each customer has an index in N = {1..n}

• Customers are ‘named’ in CP by their index


Variables

Successor variables: si

• si gives direct successor of i, i.e. the index of the next visit on the route that visits i

• si N


Constraints

• AllDifferent ( s )

2

4

1

5

3

6

i si

1 4

2 1

3 6

4 2

5 3

6 5


1

Constraints

• AllDifferent ( s )

• Circuit ( s )

2

45

3

6

i si

1 4

2 3

3 6

4 2

5 1

6 5


Objective

Objective is

• minimise( sumi in N ( C( i, si) ) )


Playtime

Now you have a try…

Documents

The Vehicle Routing Problem Using CP in the real world