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Modeling Rich Vehicle Routing Problems. TIEJ601 Postgraduate Seminar Tuukka Puranen October 19 th 2009. Contents. A tour on combinatorial optimization problems relevant to logistics design Vehicle Routing Problem and its variants The proposed model for VRP Analysis

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Modeling rich vehicle routing problems l.jpg

Modeling Rich Vehicle Routing Problems

TIEJ601 Postgraduate Seminar

Tuukka Puranen

October 19th 2009


Contents l.jpg
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 l.jpg

Modeling Rich Vehicle Routing Problems

A Tour on Combinatorial Optimization


Computational logistics l.jpg
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 l.jpg
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 l.jpg
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



Vehicle routing problem l.jpg
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



Vehicle routing problem variants l.jpg
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 l.jpg
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, …


Slide12 l.jpg
PDP

4

3

4

5

3

2

5

6

0

7

8

8

2

1

1

7

6


A new way of describing vehicle routing problems l.jpg

Modeling Rich Vehicle Routing Problems

A New Way of Describing Vehicle Routing Problems


Real life models l.jpg
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 l.jpg
Real-life Objectives

  • Minimize distance

  • Maximize profit

  • Minimize time

  • Minimize CO2 emissions

  • Minimize effects on congestion

  • Maximize customer satisfaction

  • Minimize employee workload


Motivation l.jpg
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 l.jpg
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 l.jpg
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



Labels l.jpg
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

B +1

B -1

A -1

1

3

3

1


Example vehicle capacities l.jpg
Example: Vehicle Capacities

A +1

C +10

X +1

C -2

Y +1

C -5

X -1

C +2

Y -1

C +5

A -1

C -10

1

3

4

3

4

1


Metrics l.jpg
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

time = 5

dist = 3

time = 2

dist = 1

time = 6

dist = 4

1

3

3

1


Situation l.jpg
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

time = 5

dist = 3

time = 2

dist = 1

time = 6

dist = 4

A +1

B +1

B -1

A -1

1

3

3

1

Situation

A = 1

time = 0

dist = 0

A = 1

time = 5

dist = 3

B = 1

time = 0

dist = 0

A = 1

time = 7

dist = 4

B = 0

time = 2

dist = 1

A = 0

time = 13

dist = 8


Constraints l.jpg
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 l.jpg
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

(A, time) < 15

(B, time) < 5

time = 5

dist = 3

time = 2

dist = 1

time = 6

dist = 4

A +1

B +1

B -1

A -1

1

3

3

1

A = 1

time = 0

dist = 0

A = 1

time = 5

dist = 3

B = 1

time = 0

dist = 0

A = 1

time = 7

dist = 4

B = 0

time = 2

dist = 1

A = 0

time = 13

dist = 8


Feasibility l.jpg
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 l.jpg
Example: Capacity Feasibility

A +1

C +10

X +1

C -2

Y +1

C -5

X -1

C +2

Y -1

C +5

A -1

C -10

1

3

4

3

4

1

A +1

C +10

X +1

C -2

Y +1

C -5

Z +1

C -5

X -1

C +2

Y -1

C +5

A -1

C -10

1

3

4

5

3

4

1

A = 1

X = 1

Y = 1

C = -2


Objective function l.jpg
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 l.jpg
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 l.jpg
Example: Trailer Affects Speed

(dist, A) = f( p1, p2 )

(time, A) = dist * 1,0

(time, B) = dist * 1,1

A +1

T +1

C +10

B +1

T -1

C +5

X +1

C -3

X -1

C +3

B -1

T +1

C -5

A -1

T -1

C -10

1

7

3

3

7

1

time = A

dist = A

time = B, A

dist = A

time = B, A

dist = A

time = B, A

dist = A

time = A

dist = A

time =

dist =


Analysis on the proposed model l.jpg

Modeling Rich Vehicle Routing Problems

Analysis on the Proposed Model


Benefits l.jpg
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 l.jpg
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 l.jpg
Objectives

  • Minimize distance

  • Maximize profit

  • Minimize time

  • Minimize CO2 emissions

  • Minimize effects on congestion

  • Maximize customer satisfaction

  • Minimize employee workload


The messy real life l.jpg
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 conclusions l.jpg

Modeling Rich Vehicle Routing Problems

Future Research & Conclusions


Future research l.jpg
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 l.jpg
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


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