Variable Neighborhood Search for the Orienteering Problem ISCIS’06 –The 21st International...

Variable Neighborhood Search for the Orienteering

Problem

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

F. Erdoğan SEVİLGEN Gebze Institute of Technology Department of Computer Engineering

Zülal ŞEVKLİ Fatih University Department of Computer Engineering

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

OUTLINE

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


Definition of the Problem

The Orienteering Problem (OP) is a subset selection version of well-known Traveling Salesman Problem.

The objective of the OP is to construct a path starting at an origin and ending at a destination that maximizes the total profit without violating prescribed limits.


Graphical Representation of the OP

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2020

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Score of the point

Start and end points

Distance Limit=15

MaxScore=120 Distance=14.95


OUTLINE



Literature Review

Heuristic methods

Meta-heuristic methods


Literature Review

Exact methods


OUTLINE



Variable Neighborhood SearchProcedure VNS

Define neighborhood structures Nk (k=1,...,kmax) Generate initial solution s Є S while stopping condition is not met do

k 1

while k ≤ kmax do

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+1end-while

end-whileEnd-Procedure


Variable Neighborhood Search

Nk1

Nk2

Nk3

Shaking

Initial Solution

Local Search

This prosedure continues until a stopping condition is met.

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


Variations of VNSProcedure VNS Define neighborhood structures Nk (k=1,...,kmax) Generate initial solution s S while stopping condition is not met do

k 1while 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+1end-while

end-whileEnd-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

Reduced Variable Neighborhood Search

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


OUTLINE



VNS for the Orienteering Problem

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 3 4 7 1 2 6 5permutation1 2 3 4 5 6 7

3 4 7

Feasible solution

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0 8

The first algorithm solely based on VNS

to solve the OP



Neighborhood Structures – Insert

– Exchange

3 4 7 1 2 6 5

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

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

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.

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

3 4 7 1 562

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



Neighborhood Structures– Path insert

– Path exchange

3 4 7 1 562

3 4 7 1 562



Variations of VNS for the OP


OUTLINE



Experimental Results

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 by Chao [3] • 32 points(dataset 1 includes 18 problems),





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



-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

-VNSwithVND and VNSwithRVNS produce the best known solution for all problems. -The computational time of VNSwithVND is much larger than the others

-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.



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


OUTLINE



Conclusion

VNSwith

RVNS

RVNS

CPU √

RPE √

ARPE √

VNSwith

RVNS

VNSwith

VND

CPU √

RPE √ √

ARPE √ √

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 11 benchmark problems.


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Variable Neighborhood Search for the Orienteering Problem ISCIS’06 –The 21st International...

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