Modeling Rich Vehicle Routing Problems

TIEJ601 Postgraduate SeminarTuukka Puranen

October 19th 2009

Contents• A tour on combinatorial optimization

problems relevant to logistics design• Vehicle Routing Problem and its variants• The proposed model for VRP– Analysis

• Effects and possibilities of the new model• I will not talk about implementation, system

design, optimization results, algorithms

A Tour on Combinatorial Optimization

Modeling Rich Vehicle Routing Problems

Computational Logistics• Computational– Of or related to computation

• Logistics– Management of the flow of goods, information,

energy, and people between the points of origin and the points of consumption

• Computational Logistics– Information system assisted planning based on

formulating, solving and analyzing computational problems in logistics

Examples of Logistic Problems• Shortest Path Problem• Traveling Salesman Problem• Vehicle Routing Problem• Logistics Network Design Problem– Production and distribution, linear programming

• Network flow problem• K-means problem, coverage problem• Inventory management, storage design• Job scheduling

Traveling Salesman Problem• Given a list of locations (e.g., cities) and

distances between them, find the shortest tour that visits each location

• Mathematically formulated in 1930; one of the most intensively studied problems in optimization

• Note also that usually in our context, TSP contains SPP as a subproblem when solved in a graph, e.g., road network

TSP

Vehicle Routing Problem• Given a list of customers, distances between

them and a set of vehicles, find tours that minimize the total length of the tours, such that one vehicle visits each location

• Formulated in 1959• Typically, one has to serve a scattered set of

customers from a single central depot, such that each vehicle has a limited capacity

VRP

Vehicle Routing Problem Variants• VRP with time windows (VRPTW)• Fleet size and mix VRP (FSMVRP)• Open VRP (OVRP)• Multi-depot VRP (MDVRP)• Periodic VRP (PVRP)• VRP with backhauls (VRPB)• Pickup and delivery problem (PDP)• Dynamic VRP (DVRP)• VRP with stochastic demands (VRPSD)

Pickup and Delivery Problem• Each task consists of two parts– Pickup– Delivery

• VRP (and MDVRP) a special case of PDP• Can be combined with other aspects– Time windows, capacity, fleet size and mix, ...

• Real-life examples include oil transportation, school buses, courier services, …

PDP

0

8 8

6

71

7

6

443

23

12

5

5

A New Way of Describing Vehicle Routing Problems

Modeling Rich Vehicle Routing Problems

Real-life Models• In theory, these simple models work• But if you would ever want to create a system

for solving these problems, you would like to have a bit more expressiveness

• The ‘messy real-life’• Driver breaks, QoS limitations, compartments,

special equipment, service restrictions, …• COMDFSMPDPTW– Hence the name ’Rich VRP’

Real-life Objectives• Minimize distance• Maximize profit• Minimize time• Minimize CO2 emissions

• Minimize effects on congestion• Maximize customer satisfaction• Minimize employee workload

Motivation• A number of different cases have to be

modeled and solved• Time to build only a single solver• A modeling language for describing the

problem in a way that requires no changes on the solution space exploring system– Meaning that feasible region can be defined

without modifying the solver itself– Algorithms and objectives can be tailored, but not

necessarily require it

The Proposed Model• Based partially on an idea of General Pickup

and Delivery Problem (GPDP)– Each vehicle starts from and ends at an arbitrary

point

• Combines concepts from constraint programming and automata theory

• In essence, a labeled network formulation• Objecive is to be able to utilize combinatorial

metaheuristic local and global search

Actors and Activities• Actors and activities are described as nodes in

a network• Each actor corresponds to a vehicle• Each activity corresponds to a task– Usually order pickup and delivery points– Can be used to other tasks, e.g., fleet selection

• A solution is formulated by selecting, ordering and assigning the activities to actors

Actors and Activities Illustrated

1 1

2 2

3

4

3

4

Labels• Each node can have a set of labels that have

an adjoining integral value• There are two rules– Each label must have a nonnegative accumulated

value– Each label must have zero value at the end

A +1 A -1B +1 B -1

1 13 3

Example: Vehicle Capacities

1 13 3

A +1C +10

A -1C -10

X +1C -2

X -1C +2

4 4

Y +1C -5

Y -1C +5

Metrics• If actors and activities are nodes in a network,

we need a way to describe their relation, i.e., arcs– These relations include, for example, distance

• The model can have any number of metrics• Metrics can also be assigned to nodes

1 13 3

time = 5dist = 3

time = 6dist = 4

time = 2dist = 1

time = 2dist = 1

Situation• A situation in given point is defined by– The set of labels and their accumulated values at

that point– The values of each accumulated metric for each

label

1 13 3

A +1 A -1B -1

Situation A = 1 time = 0 dist = 0

A = 1 time = 5 dist = 3B = 1 time = 0 dist = 0

A = 1 time = 7 dist = 4B = 0 time = 2 dist = 1

A = 0 time = 13 dist = 8

time = 5dist = 3

time = 6dist = 4

B +1

Constraints• Constraints impose lower and upper bounds

on metrics• Assigned to given label-metric pair• If that label is present in a situation, its given

accumulated metric value must fall between the defined bounds

• Can be used to model time windows, breaks, QoS requirements

Example: Maximum Travel Time• Assume that we need to – Restrict the length of the shift of the driver– Ensure that the customer sits in the vehicle no

more than given number of minutes

time = 2dist = 1

13

A +1 A -1B -1

A = 1 time = 0 dist = 0

A = 1 time = 5 dist = 3B = 1 time = 0 dist = 0

A = 1 time = 7 dist = 4B = 0 time = 2 dist = 1

A = 0 time = 13 dist = 8

time = 5dist = 3

time = 6dist = 4

B +1(A, time) < 15(B, time) < 5

1 3

Feasibility• A route is feasible when– All labels in every situation are nonnegative– Labels have zero sum at the end– Metric values are within constraints in every

situation

• A solution is feasible when– All routes are feasible

• Note that this does not require visit on every node

Example: Capacity Feasibility

1 13 3

A +1C +10

A -1C -10

X +1C -2

X -1C +2

4 4

Y +1C -5

Y -1C +5

A = 1X = 1Y = 1C = -2

A +1C +10

A -1C -10

X +1C -2

X -1C +2

Y +1C -5

Y -1C +5

Z +1C -5

1 13 34 45

Objective Function• Objective function becomes just a single

metric that has no constraints– Simple multiobjective optimization becomes

natural feature of the system: change an objective to constrained metric and vice versa

• As usual, is used to evaluate the solution at given situation

• A penalty must be assigned for not visiting the nodes since feasibility does not require this

Dynamic Metrics• Sometimes metrics change depending on the

solution structure• Label dependent– Trailers, special equipment, ...– Also in objective function: complex cost structures– Assigning a metric transformation to labels– Keeping track of the active transformation

• Situation dependent– DVRP

Example: Trailer Affects Speed

1 17 3

A +1T +1C +10

A -1T -1C -10

B +1T -1C +5

X -1C +3

3 7

X +1C -3

B -1T +1C -5

(dist, A) = f( p1, p2 )(time, A) = dist * 1,0(time, B) = dist * 1,1

time = Adist = A

time = B, Adist = A

time = B, Adist = A

time = B, Adist = A

time = Adist = A

time = dist =

Analysis on the Proposed ModelModeling Rich Vehicle Routing Problems

Benefits• More expressive model– Expandable

• More implementation friendly formulation– Less work per modeled case– Visual

• Per aspect analysis– Easier to evaluate the cost on complexity– Generate only relevant aspects

• Multidisciplinary research

Variants• VRP with time windows (VRPTW)• Fleet size and mix VRP (FSMVRP)• Open VRP (OVRP)• Multi-depot VRP (MDVRP)• Periodic VRP (PVRP)• VRP with backhauls (VRPB)• Pickup and delivery problem (PDP)• Dynamic VRP (DVRP)• VRP with stochastic demands (VRPSD)

Objectives• Minimize distance• Maximize profit• Minimize time• Minimize CO2 emissions

• Minimize effects on congestion• Maximize customer satisfaction• Minimize employee workload

The ‘Messy Real-life’• Driver breaks• QoS limitations– Maximum waiting time– Maximum ride time

• Fleet selection, special equipment• Service restrictions, preferences• Multiple capacities• Compartment loading decisions• Time dependent continuous demand

Future Research & ConclusionsModeling Rich Vehicle Routing Problems

Future Research• Continuing implementation• Modeling– Compartments– Stochastic metrics, labels– Interroute dependencies, e.g., assisting drivers

• Testing– Modeling complex cases– Benchmarking solution methods

• Multiobjective optimization

Conclusions• A number of combinatorial optimization

problems, starting from TSP, are important in designing logistic operations

• In practice, a more detailed model is often needed

• We proposed a new way for modeling VRPs, which should make it easier to incorporate difficult real-life aspects into optimization problems