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Variable Neighborhood Search for the Orienteering Problem

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Variable Neighborhood Search for the Orienteering Problem

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Variable Neighborhood Search for the Orienteering Problem

Zülal ŞEVKLİ

Fatih University

Department of Computer Engineering

F. Erdoğan SEVİLGEN

Gebze Institute of TechnologyDepartment of Computer Engineering

ISCIS’06 –The 21st International Symposium on Computer and Information Sciences

- Defitiniton of the Problem
- Literature Review
- Variable Neighborhood Search (VNS)
- VNS for Orienteering Problem
- Experimental Results
- Conclusion

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

- The Orienteering Problem (OP) is a subset selection version of well-known Traveling Salesman Problem.
- The objective of the OPis to construct a path starting at an origin and ending at adestination that maximizes thetotal profit without violatingprescribed limits.

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

Score of the point

Distance Limit=15

MaxScore=120 Distance=14.95

Start and end points

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

- Defitiniton of the Problem
- Literature Review
- Variable Neighborhood Search (VNS)
- VNS for Orienteering Problem
- Experimental Results
- Conclusion

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

- Heuristic methods
- Meta-heuristic methods

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

- Exact methods

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

- Defitiniton of the Problem
- Literature Review
- Variable Neighborhood Search (VNS)
- VNS for Orienteering Problem
- Experimental Results
- Conclusion

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

Procedure VNS

Define neighborhood structures Nk (k=1,...,kmax)

Generate initial solution s ЄS

while stopping condition is not met do

k 1

while k ≤ kmaxdo

s’ Shake(s), s’ Є Nk (s)

s” LocalSearch(s’), s” Є S

if (Fitness(s”) < Fitness(s))

s s”

k 1

else

k k+1

end-while

end-while

End-Procedure

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

This prosedure continues until a stopping condition is met.

-maximum CPU time allowed, maximum number of iterations ormaximum number of iterations between two improvements.

Shaking

Initial Solution

Nk3

Nk2

Nk1

Local Search

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

Reduced Variable Neighborhood Search (RVNS)

RVNS use only shaking phase while exploring the solutions. RVNS is useful for problems where local search is expensive

Procedure VNS

Define neighborhood structures Nk (k=1,...,kmax)

Generate initial solution s S

while stopping condition is not met do

k 1

while k ≤ kmax do

s’ Shake(s), s’ Nk (s)

s” LocalSearch(s’), s” S

if (Fitness(s”) < Fitness(s))

s s”

k 1

else

k k+1

end-while

end-while

End-Procedure

Variable Neighborhood Descent (VND)

In VND, shaking phase is removed from VNS so that the algorithm explores local optima by using neighborhood structures only. VND can be used as a part of VNS in the local search phase

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

- Defitiniton of the Problem
- Literature Review
- Variable Neighborhood Search (VNS)
- VNS for Orienteering Problem
- Experimental Results
- Conclusion

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

The first algorithm solely based on VNS to solve the OP

- Problem Representation
- Permutation of the control points not including the origin and the destination.
- Feasible OP solution from a permutation, the points are inserted between the origin and destination one by one starting from the first point in the permutation until a prescribed cost limit is exceeded.

Let problem size is 9

Dis.Lmt=10

permutation

3 4 7 1 2 6 5

0

1

2

3

4

5

6

7

8

Feasible solution

0

3

4

7

8

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

- Neighborhood Structures
- Insert
- Exchange

The position of a control point outside of the solution can be changed. This operation does not change the current solution but may affect the results of subsequent operations.

The position of a control point inside the solution can be changed to another location inside the solution.

A control point outside the solution can be inserted into the solution.

A control point inside the solution can be deleted from the solution

3

4

7

1

2

6

5

Two control points in the solution can be swapped

Two control points outside of the solution are swapped.

A control point in the solution can be swapped with a control point outside of the solution. This operation is like performing a deletion followed by an insertion at the same position

2

6

3

4

7

1

5

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

- Neighborhood Structures
- Path insert
- Path exchange

2

6

3

4

7

1

5

2

6

3

4

7

1

5

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

- Variations of VNS for the OP

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

- Defitiniton of the Problem
- Literature Review
- Variable Neighborhood Search (VNS)
- VNS for Orienteering Problem
- Experimental Results
- Conclusion

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

- All three variations of VNS algorithm
- implemented in Borland C
- performed on an Intel P4 2.6 GHz PC with 512 MB memory
- tested 107 benchmark problems in the literature
- Problems are provided by Tsiligirides [23],
- 32 points (dataset 1 includes 18 problems),
- 21 points (dataset 2 includes 11 problems),
- 33 points(dataset 3 includes 20 problems)

- Problems are provided byChao [3]
- 32 points(dataset 1 includes 18 problems),
- 66 points(dataset 5 includes 26 problems)
- 64 points(dataset 6 includes 14 problems)

- Problems are provided by Tsiligirides [23],

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

- Each problem is run 10 times
- Results are compared based on
- computational time (CPU),
- relative percentage error (RPE)
- average relative percentage error (ARPE)

RPE is the error in the best solution in all repetitions with respect to the best known solution. It indicates whether an algorithm finds the best known solution throughout the repetitions

ARPE is the average error of all the solutions in the repetitions. ARPE specifying the robustness of an algorithm, will be smaller if more repetitions find good solutions

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

-VNSwithRVNS achieves the best RPE and ARPE values and comparably good computational time.

-Therefore, it can be claimed that it is the best method among the others.

-VNSwithVND and VNSwithRVNS produce the best known solution for all problems.

-The computational time of VNSwithVND is much larger than the others

- -RVNS is the fastest method.
- RVNS could not find the best known solution for several problems.
- Robustness of RVNS is poorer than robustness of the competitors

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

- The results obtained from VNSwithRVNS are also compared with the results in literature for datasets 5 and 6.
- Better results have been observed for 11 problems in total

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

- Defitiniton of the Problem
- Literature Review
- Variable Neighborhood Search (VNS)
- VNS for Orienteering Problem
- Experimental Results
- Conclusion

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

- Proposes VNS for solving the OP problem and examines the performance of our approaches based on solution quality and execution time.
- VNSwithRVNS outperforms other two methods and it finds the best known solution for all the problems and improves the best solution for 11benchmark problems.

VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3

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VNS for the Orienteering Problem-ISCIS'06 Nov 1-2-3