Vehicle Routing Problem (VRP): models and algorithms

Agostino Nuzzolo [email protected]

Antonio Polimeni [email protected]

Logistica Territoriale

Vehicle Routing Problem (VRP):

models and algorithms

1

Contents:

Vehicle routing problem VRP definition

VRP formulation

Solution search approach

Example of search tools

2

Vehicle Routing Problem (VRP)

consists of designing optimal delivery or

collection routes from a central depot to a

set of geographically scattered customers

(e.g. retailers or travellers), subject to

various constraints, such as vehicle capacity,

route length, time windows, and so on

3


http://www.jstor.org/stable/2627477

The Truck Dispatching Problem

G. B. Dantzig and J. H. Ramser

Management Science Vol. 6, No. 1 (Oct., 1959),

pp. 80-91

The paper is concerned with the optimum routing

of a fleet of gasoline delivery trucks between a

bulk terminal and a large number of service

stations supplied by the terminal.

4


Veicoli merci

Transit on-demand services - Scuolabus

5

retailer 1

Depot

retailer nretailer i

retailer 3

retailer 2


6

Vehicle Routing Problem (VRP): Example of routes

depot

7


• Homogeneous (or not) vehicle fleet

• Different definition of vehicle capacity (weight, volume, …)

• Pickup or Delivery

• Mixed, both Pickup and Delivery.

• Multi-depot

• …

• Costs (costs for load/unload, empty journey…)

• Customers' priorities

(freight type, delivery time)

8


• Compatibility

(vehicle-freight, vehicle-driver, …)

• Time windows for delivery/pick-up

(rigid or elastic)

• Route constraints

(maximum time, maximum length, …)

9

minimize SvVSiNSjN cij xijv

Vehicle Routing Problem (VRP): general formulation

N: set of clients, contains also the depot

V: set of homogeneous vehicles, |V| = m

cij : cost to move from i to j

xij : binary variable, equal to 1 if the travel

from i to j happens, 0 otherwise

Sv=1,..., m SjN xijv = 1 i N, i ≠ d, j ≠ d

Sv=1,..., m SjN xdjv = m

Sv=1,..., m SjN xjdv = m

SiNSjN rj xijv bv v V

xijv {0, 1} (i,j), v V

d is the depot;

ri freight demand on node i;

bv vehicle capacity, v = 1, 2, …, m;

a client must be reached only once

all vehicles star from depot and return to it

the delivered quantity not exceed the vehicle capacity

the problem variables are binary

10

Route

Path

Depot

Depot

Client

Network node

Vehicle Routing Problem (VRP): Example of routes

11

cij : cost to move from i to j

ji

In a road network, the travel cost is generally a

function of the flow vector: cij (f)

ESEMPIO (formulazione del problema)Definition of Combinatorial Optimization

Combinatorial optimization is the mathematical studyof finding an optimal arrangement, grouping, ordering,or selection of discrete objects usually finite in numbers.

- Lawler, 1976

Constraints can be hard (must be satisfied) or soft (is

desirable to satisfy).

12

Vehicle Routing Problem

Exact approaches

Heuristics approaches

Fisher (1993)Toth & Vigo (2002)Ropke et al. (2007)Baldacci et al. (2008)…

Jones et al. (2002)Montemanni et al. (2005)Bianchi et al. (2005)Hanshar & Ombuki-Berman (2007) Laporte (2007)…

Solution approaches

13

Solution approaches

Saving methods

Branch and Bound: an exact approach to solve the VRP

Genetic Algorithm

14

Solution approaches: B&B steps

0. Initialization: Estimate a value for the Lower Bound

1. Solution tree: Select a node for the branch operation

2. Evaluation: Evaluate the objective function for the current

(partial) solution

3. Test: If the objective function is less than the lower bound, go to

step 1, else quit from the current branch and go to the step1

4. Stop: if all node are explored

15

4

12

3

17

1

13

0

135

2

1

1

1 8

Solution approaches: B&B

16

Number of solutions: n! n = number of nodes

CAVEAT: the number of solution is n! because no constraints are considered

Solution approaches: B&B0 [0]

1 [1] 2 [8] 4 [1]

1 8 1

2 [2]

1

3 [3]

1

4 [4]

1

Search Tree

3 [2]

0 [5]

1

2

lower bound = 5

Initial solution: an initial solution can be used to obtain a

value of the lower bound, in this case, the value is 5.

CAVEAT: the search of the lower bound is as important as finding the

solution

17


1 [1] 2 [8] 4 [1]

1 8 1

2 [2]

1

3 [3]

1

4 [4]

1

Search Tree

3 [2]

0 [5]

1

2

lower bound = 5

18


1 [1] 2 [8] 4 [1]

1 8 1

2 [2]

1

3 [3]

1

4 [4]

1

Search Tree

3 [2]

0 [5]

1

2

lower bound = 5

1 [19] 2 [3] 4 [3]

17 11

19


1 [1] 2 [8] 4 [1]

1 8 1

2 [2]

1

3 [3]

1

4 [4]

1

Search Tree

3 [2]

0 [5]

1

2

lower bound = 5

1 [19] 2 [3] 4 [3]

17 11

4 [16] 1 [4]

13 1

20


1 [1] 2 [8] 4 [1]

1 8 1

2 [2]

1

3 [3]

1

4 [4]

1

3 [2]

0 [5]

1

2

lower bound = 5

1 [19] 2 [3] 4 [3]

17 11

4 [16] 1 [4]

13 1

Search Tree

21

1 [38] 2 [16]

1335


1 [1] 2 [8] 4 [1]

1 8 1

2 [2]

1

3 [3]

1

4 [4]

1

3 [2]

0 [5]

1

2

lower bound = 5

1 [19] 2 [3] 4 [3]

17 11

4 [16] 1 [4]

13 1

1 [36] 2 [14] 3 [2]

35 131

Search Tree

22


1 [1] 2 [8] 4 [1]

1 8 1

2 [2]

1

3 [3]

1

4 [4]

1

3 [2]

0 [5]

1

2

lower bound = 5

1 [19] 2 [3] 4 [3]

17 11

4 [16] 1 [4]

13 1

1 [36] 2 [14] 3 [2]

35 131

4 [39]

35

Search Tree

23


1 [1] 2 [8] 4 [1]

1 8 1

2 [2]

1

3 [3]

1

4 [4]

1

3 [2]

0 [5]

1

2

lower bound = 5

1 [19] 2 [3] 4 [3]

17 11

4 [16] 1 [4]

13 1

1 [36] 2 [14] 3 [2]

35 131

2 [3] 1 [19]

1 17

4 [39]

35

Search Tree

24


1 [1] 2 [8] 4 [1]

1 8 1

2 [2]

1

3 [3]

1

4 [4]

1

3 [2]

0 [5]

1

2

lower bound = 5

1 [19] 2 [3] 4 [3]

17 11

4 [16] 1 [4]

13 1

1 [36] 2 [14] 3 [2]

35 131

2 [3] 1 [19]

1 17

1 [4]

1

0 [5]

4 [39]

35

Search Tree

25

Solution approaches: Meta-heuristics

• Meta-heuristics are iterative generation processes of solutions not

tied to any special problem type and are general methods that can

be altered to fit the specific problem.

• The inspiration from nature is (to name a few):

o Simulated Annealing (SA): molecule/crystal arrangement during cool

down

o Evolutionary Computation (EC): biological evolution

o Tabu Search (TS): long and short term memory

o Ant Colony and Swarms: individual and group behavior using

communication

26

Solution approaches

Saving methods

Branch and Bound

Evolutionary algorithm: genetic algorithm

27

Solution approaches: genetic algorithm

Initial population

Selection

Crossover

Mutation

Population

Test End

Initial population (parameter: population size)

A set of feasible solutions. Examples of

population generation:

1) Random

2) Constructive methods

3) Constructive + improvement methods

28


Initial population

Selection

Crossover

Mutation

Population

Test End

Selection

In the selection procedure the population is

evaluated in order to establish the parents

for the next step (crossover):

1) compute the Fitness Function value

2) compute the selection probability

3) extract the new solutions

29


Initial population

Selection

Crossover

Mutation

Population

Test End

Crossover (parameter: crossover rate)

merging two parents to obtain two children,

example:

1) select the parents

2) for each parent, select a node interchange

set

3) append the interchange node set of a route

at the end of the other and eliminate the

repeated nodes

4) test the solution

5) update the solution cost

30


Initial population

Selection

Crossover

Mutation

Population

Test End

Mutation (parameter: mutation rate)

Mutation allows variations to be introduced

into the solutions, extending the solution

space. Examples of mutation:

1) swapping two nodes in a route

2) reversing a route

31


4 [5]

1 [40]2 [10]

3 [25]

17

28

13

0

20

35

13

15

36

33 23x: node id

[q]: quantity

32

Total number of solutions: Sk(n!/(n-k)!)

n = number of nodes k=1, 2, …n

Vehicle capacity = 40


Initial population: generate a set of initial solutions

Constraint: capacity = 40

0-4-3-2-0 0-1-0

[86] [66]

0-2-4-3-0 0-1-0

[77] [66]

Soluzione 1

Soluzione 2

0-2-3-4-0

[86] [66]

Soluzione 3 0-1-0

33


Selection: selecting the parents

0-4-3-2-0 0-1-0

[86] [66]

0-2-4-3-0 0-1-0

[77] [66]

Soluzione 1

Soluzione 2

0-2-3-4-0

[86] [66]

Soluzione 3 0-1-0

34


Crossover: example of child generation

0-4-3-2-0 0-1-0

[86] [66]

0-2-4-3-0 0-1-0

[77] [66]

Soluzione 1

Soluzione 2

Select the nodes to cross

35



0-4-3-2-2-4-0 0-1-0

[-] [66]

0-2-4-3-3-2-0 0-1-0

[-] [66]

Soluzione 1

Soluzione 2

Crossing the solutions

36



0-3-2-4-0 0-1-0

[-] [66]

0-4-3-2-0 0-1-0

[-] [66]

Soluzione 1

Soluzione 2

Eliminate the duplicated nodes

37



0-3-2-4-0 0-1-0

[61] [66]

0-4-3-2-0 0-1-0

[86] [66]

Soluzione 1

Soluzione 2

Check the feasibility and evaluate objective function

38


Mutation: example node swap

0-3-2-4-0 0-1-0

[61] [66]

0-4-3-2-0 0-1-0

[-] [66]

Soluzione 1

Soluzione 2

In solution 2, swap the node 3 with the node 2

39


Mutation: example node swap

0-3-2-4-0 0-1-0

[61] [66]

0-4-2-3-0 0-1-0

[61] [66]

Soluzione 1

Soluzione 2

After the swap, evaluate the objective function

40

Solution approaches

Saving Methods

Branch and Bound

Genetic Algorithm

41

Solution approaches: saving

Clarke-Wright Algorithm

The CW algorithm is a saving algorithm, an heuristic procedure thatallows to evaluate the ‘saving’ obtainable merging two tours.

0

1

0

2

+

0

21

=

42


Clarke-Wright algorithm

0. Construct a travel cost matrix

1. Assign one round-trip to each location

2. Calculate iteratively the savings for each pair of locations, and enterthem in a savings list

3. Order descending the savings list and merge the locations

43


0

1

0

2

+

0

21

=

Configuration A Configuration B

CA = d(0,1) + d(1,0) + d(0,2) + d(2,0) CB = d(0,1) + d(1,2) + d(2,0)

Saving = CA - CB = d(1,0) + d(2,0) - d(1,2)

44

Clarke-Wright Algorithm

The CW algorithm is a saving algorithm, an heuristic procedure thatallows to evaluate the ‘saving’ obtainable merging two tours.

i j di0 d0j dij sij

1 2 33 23 36 20

0

1

0

2

0

3

0

4

s(i,j) = d(i,0)+d(0,j) – d(i,j)

4

12

3

17

28

13

0

20

35

13

15

36

33 23

33 + 23 – 36


45

i j di0 d0j dij sij

1 2 33 23 36 20

s(i,j) = d(i,0)+d(0,j) – d(i,j)

4 [5]

1 [40]2 [10]

3 [25]

17

28

13

0

20

35

13

15

36

33 23

CAPACITY CONSTRAINT

0

1

0

2

+

0

21

= [q]: quantity

vehicle capacity= 40


33 + 23 – 36

46

Total number of solutions: Sk(n!/(n-k)!) k=1, 2, …n

i j qi qj qi+qj di0 d0j dij sij

1 2 40 10 50 33 23 36 -∞

4 [5]

1 [40]2 [10]

3 [25]

17

28

13

0

20

35

13

15

36

33 23

Check, considering the constraints, if two clients can be inserted in the same route

If the constraint is not satisfied, the saving value is -∞.

0

1

0

2

+

0

21

=


47

CAPACITY CONSTRAINT

[q]: quantity


i j qi qj qi+qj di

0

d0j dij sij

1 3 40 25 65 33 13 17 -∞

3 1 25 40 65 13 33 17 -∞

2 4 10 5 15 23 15 13 25

4 2 5 10 15 15 23 13 25

1 2 40 10 50 33 23 36 -∞

2 1 10 40 50 23 33 36 -∞

3 2 25 10 35 13 23 20 16

1 4 40 5 45 33 15 35 -∞

2 3 10 25 35 23 13 20 16

4 1 5 40 45 15 33 35 -∞

3 4 25 5 30 13 25 28 0

4 3 5 25 30 15 23 28 0

4 [5]

1 [40]2 [10]

3 [25]

17

28

13

0

20

35

13

15

36

33 23


48

[q]: quantity


4 [5]

1 [40]2 [10]

3 [25]

17

28

13

0

20

35

13

15

36

33 23

i j qi qj qi+qj di0 d0j dij sij

2 4 10 5 15 23 15 13 25

4 2 5 10 15 15 23 13 25

3 2 25 10 35 13 23 20 16

2 3 10 25 35 23 13 20 16

3 4 25 5 30 13 25 28 0

4 3 5 25 30 15 23 28 0

1 2 40 10 50 33 23 36 -∞

1 3 40 25 65 33 13 17 -∞

1 4 40 5 45 33 15 35 -∞

2 1 10 40 50 23 33 36 -∞

3 1 25 40 65 13 33 17 -∞

4 1 5 40 45 15 33 35 -∞

Table ordered by descending values


49

[q]: quantity


4 [5]

1 [40]2 [10]

3 [25]

0

4 [5]

1 [40]2 [10]

3 [25]

033

23

13

28

13

0-2-4-3-0

Cost: 77 Cost: 66

0-1-0

Total cost: 77 + 66 = 143


50

4 [5]

1 [40]2 [10]

3 [25]

0

4 [5]

1 [40]2 [10]

3 [25]

033

1313

0-4-2-3-0

Cost: 61 Cost: 66

0-1-0

20

15

Total cost: 61 + 66 = 127


51



Logistica Territoriale



52

http://people.bath.ac.uk/ge277/index.php/vrp-spreadsheet-solver/

Vehicle Routing Problem (VRP): solver

53

Based on local search algorithm

EXCHANGE operator searches all possible pairs of customers in a given

solution and checks if exchanging them would result in a better objective function

value

1-OPT operator examines the possibility of removing every customer within a

given solution and re-inserting it to a different position within the routes to

improve the objective value

2-OPT operator attempts to take a route that crosses over itself and reorder it so

that it does not, and checks if the resulting solution has a better objective value

VEHICLE-EXCHANGE operator attempts to exchange all the customers in the

routes of two vehicles with different types, and is particularly useful for the case

of heterogeneous fleets


54

G. Erdogan (2017). An open source Spreadsheet Solver for Vehicle Routing Problems. Computers & Operations Research,

84.

VRP Solver Console


Initial configuration:

• Number of depots

• Number of customers

• Method for distance/time

evaluation

• Problem parameters

• Solver parameters

55

1. Locations


User location:

• ID and name

• Address (optional, can be used for geocoding)

• Latitude and longitude

• Time windows and service time

• Pickup and delivery amount

• Profit

56

2. Distances


Distance and time evaluation:

• Bird flight

• Euclidean

• On map

Route type

• Shortest

• Fastest

• Fastest with real time traffic

57

3.Vehicles


Vehicle type and characteristics

58

4. Solution


Output

59

5.Visualization


0

12

3

4

5

6

78

9

10

V1 (T1)

V2 (T1)

V3 (T1)

V4 (T2)

V5 (T2)

V6 (T3)

60

5.Visualization


0

12

3

4

5

6

78

9

10

V1 (T1)

V2 (T1)

V3 (T1)

V4 (T2)

V5 (T2)

V6 (T3)

61



Logistica territoriale



62

Vehicle Routing Problem (VRP): models and algorithms

Documents

Transcript of Vehicle Routing Problem (VRP): models and algorithms