Variable Neighborhood Search for the Orienteering Problem

1
Variable Neighborhood Search for the Orienteering
Problem
Zülal SEVKLI Fatih University Department of
Computer Engineering
F. Erdogan SEVILGEN Gebze Institute of
Technology Department of Computer Engineering
ISCIS06 The 21st International Symposium on
Computer and Information Sciences
2
OUTLINE

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

3
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.

4
Graphical Representation of the OP
Score of the point
Distance Limit15
MaxScore120 Distance14.95
Start and end points
5
OUTLINE

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

6
Literature Review

• Heuristic methods
• Meta-heuristic methods

7
Literature Review

• Exact methods

8
OUTLINE

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

9
Variable Neighborhood Search

• Procedure VNS
• Define neighborhood structures Nk
  (k1,...,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) lt Fitness(s))
• s ? s
• k ? 1
• else
• k ? k1
• end-while
• end-while
• End-Procedure

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Variable Neighborhood 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.
Shaking
Initial Solution
Nk3
Nk2
Nk1
Local Search
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Variations of VNS
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
  (k1,...,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) lt Fitness(s))
• s ? s
• k ? 1
• else
• k ? k1
• 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
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OUTLINE

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

13
VNS for the Orienteering Problem
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.Lmt10
3 4 7 1 2 6 5
permutation
1
2
3
4
5
6
7
0
8
Feasible solution
3
4
7
0
8
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VNS for the Orienteering Problem

• Neighborhood Structures
• Insert
• Exchange

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
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
3
4
7
1
5
6
2
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VNS for the Orienteering Problem

• Neighborhood Structures
• Path insert
• Path exchange

3
4
7
1
5
6
2
3
4
7
1
5
6
2
16
VNS for the Orienteering Problem

• Variations of VNS for the OP

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OUTLINE

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

18
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),
• 66 points(dataset 5 includes 26 problems)
• 64 points(dataset 6 includes 14 problems)

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Experimental Results

• 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
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Experimental Results
-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

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Experimental Results

• 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

22
OUTLINE

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

23
Conclusion

• Proposes VNS for solving the OP problem and
  examines the performance of our approaches based
  on solution quality and execution time.
• VNSwithVND and RVNS and one which is the
  combination of VNS and RVNS are implemented and
  tested on 107 benchmark problems.
• 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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Conclusion

• 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.

VNSwith RVNS RVNS
CPU v
RPE v
ARPE v
VNSwith RVNS VNSwith VND
CPU v
RPE v v
ARPE v v
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