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Modeling Rich Vehicle Routing Problems

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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

TIEJ601 Postgraduate Seminar

Tuukka 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

Modeling Rich Vehicle Routing Problems

A Tour on Combinatorial OptimizationComputational 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

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

- 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, …

Modeling Rich Vehicle Routing Problems

A New Way of Describing Vehicle Routing ProblemsReal-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

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

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

- 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

- 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

- 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

(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

- 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

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

- 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

(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 =

Modeling Rich Vehicle Routing Problems

Analysis on the Proposed ModelBenefits

- 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

Modeling Rich Vehicle Routing Problems

Future Research & ConclusionsFuture 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

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