Apresenta

1
Hybrid Metaheuristics for the Prize Collecting
Traveling Salesman Problem
Antonio Augusto Chaves - Luiz Antonio Nogueira
Lorena National Institute for Space Research -
INPE São José dos Campos, Brazil chaveslorena_at_lac
.inpe.br EvoCOP 2008 - Eighth European
Conference on Evolutionary Computation in
Combinatorial Optimization
2
Introduction

• Objective Solve the Prize Collecting Traveling
  Salesman Problem (PCTSP) using a new hybrid
  metaheuristic, known as Clustering Search (CS).
• The CS consists of detecting promising areas of
  the search space using an algorithm that
  generates solutions to be clustered. These
  promising areas may then be explored through
  local search as soon as they are discovered.
• The commercial solver CPLEX has been used to
  solve a formulation of the PCTSP in order to
  validate the computational results of CS
  algorithm.
• The optimal solution for the PCTSP is hard to
  find due the large number of possible solutions
• It is classified as NP-hard

3
PCTSP

• The PCTSP is a generalization of the Traveling
  Salesman Problem (TSP), where a salesman collects
  a prize pi in each city visited and pays a
  penalty ?i for each city not visited, considering
  travel costs cij between the cities.
• The problem is to minimize the sum of the costs
  of the tour and penalties paid, while including
  in the tour enough cities to collect a minimum
  prize pmin, defined a priori. In this tour, each
  city visited at most once.

Example of a solution with 10 nodes
?i
pi
Balas, E.The prize collecting traveling salesman
problem. Networks 19 (1989) 621636
4
Mathematical Model

• The PCTSP can be formulated as an integer linear
  programming problem as follows

(1)
(2)
(3)
(4)
5
Mathematical Model
(5)
(6)
(7)
(8)
(9)
(10)
(11)
6
Mathematical Model

• CPLEX 10.0 was used to solve the PCTSP.
• CPLEX solved PCTSP small instances in a
  reasonable execution time.
• For the test problems with 80 nodes, the CPLEX
  took several hours execution to find the optimal
  solution,
• For test problems with 100 nodes the CPLEX failed
  in close the gap between lower and upper bounds
  in 100,000 seconds.
• The CPLEX failed in finding a feasible solution
  for test problems with n ? 200 in 100,000
  seconds.
• Due to the limitation of the CPLEX, the study of
  heuristic techniques for solving this problem
  become interesting.

7
Clustering Search Metaheuristic

• Clustering Search (Oliveira and Lorena, 2004)
• Oliveira, A.C.M., Lorena, L.A.N. Detecting
  promising areas by evolutionary clustering
  search. Advances in Artificial Intelligence.
  Springer Lecture Notes in Artificial Intelligence
  Series 3171 (2004) 385394
• The CS employs clustering for detecting promising
  areas of the search space. A clustering process
  is executed simultaneously to a metaheuristic,
  identifying groups of solutions that deserve
  special interest.
• These promising areas should be explored through
  local search methods as soon as they are
  discovered.
• The idea of the CS is to avoid applying a local
  search heuristic to all solutions generated by a
  metaheuristic.
• To detect promising regions becomes an
  interesting alternative preventing the
  indiscriminate application of local search
  heuristics.

8
Example Non-linear optimization
9
Clustering Search

• Iterative clustering behavior

10
Diagram for the CS algorithm
Begin
Clustering Process
LS
Create Clusters
yes
promising cluster?
SM
no
IC
AM
Stop criterion?
Legend
yes
CS flow
Clusters update flow
End
Clustering process flow
11
Search Metaheuristic (SM)

• The search metaheuristic (SM) component works as
  a full-time solution generator. The algorithm is
  executed independently of the remaining
  components and must be able to provide a
  continuous generation of solutions to the
  clustering process. Clusters are simultaneously
  maintained to represent these solutions.

12
CS for the PCTSP SM

• The GRASP/VNS metaheuristic
• The GRASP is basically composed of two phases
• a construction phase, in which a feasible
  solution is generated, and
• a local search phase, in which the constructed
  solution is improved.
• Construction phase The greedy evaluation
  function for adding a node k between the nodes i
  and j is
• g(k) cij ?k cik ckj
• Local Search Phase uses the VNS, which is a
  metaheuristic going on a systematic change of the
  neighborhood within a local search algorithm.

13
CS for the PCTSP SM
procedure GRASP/VNS for (number of iterations is
not satisfied) do s ? while (solution not
built) do compute candidate list (C) RCL
C ? e select at random a value of RCL s
s ? e end while kmax number of
neighborhoods while (stop condition is not
satisfied) do k ? 1 while (k ?
kmax) generate at random s? Nk(s) s
apply VND with s if ( f (s) lt f (s))
then s ? s k ? 1 else k ? k
1 end while end while end for end GRASP/VNS
Construction Phase
Local Search Phase (VNS)
14
CS for the PCTSP SM
procedure GRASP/VNS for (number of iterations is
not satisfied) do s ? while (solution not
built) do compute candidate list (C) RCL
C ? e select at random a value of RCL s
s ? e end while kmax number of
neighborhoods while (stop condition is not
satisfied) do k ? 1 while (k ?
kmax) generate at random s? Nk(s) s
apply VND with s if ( f (s) lt f (s))
then s ? s k ? 1 else k ? k
1 end while end while end for end GRASP/VNS
Construction Phase
Neighborhood structures of VNS

1. Add one node (AD)
2. Drop one node (DR)
3. Swap two nodes (SW)
4. (AD)2
5. (DR)2
6. (SW)2
7. (AD)3
8. (DR)3
9. (SW)3

Local Search Phase (VNS)
15
CS for the PCTSP SM
procedure GRASP/VNS for (number of iterations is
not satisfied) do s ? while (solution not
built) do compute candidate list (C) RCL
C ? e select at random a value of RCL s
s ? e end while kmax number of
neighborhoods while (stop condition is not
satisfied) do k ? 1 while (k ?
kmax) generate at random s? Nk(s) s
apply VND with s if ( f (s) lt f (s))
then s ? s k ? 1 else k ? k
1 end while end while end for end GRASP/VNS
Construction Phase
Neighborhood structures of VND

1. SeqAdd
2. AddDrop
3. SeqDrop

Local Search Phase (VNS)
At each step of VND, the neighborhood Nt(s)
of s is explored completely.
16
Iterative Clustering (IC)

• The iterative clustering (IC) component aims to
  gather similar solutions into groups, identifying
  a representative cluster center for them.The
  clustering is progressively fed by solutions
  generated in each iteration of SM. A distance
  metric must be defined, a priori, allowing a
  similarity measure for the clustering process.

17
CS for the PCTSP IC

• The metric distance is the number of different
  edges between the GRASP/VNS and the center of the
  cluster solutions. A large number of different
  edges between them increases the dissimilarity.
• The assimilation process uses the path-relinking
  method.

18
Analyzer Module (AM)

• The analyzer module (AM) provides an analysis of
  each cluster, indicating a probable promising
  cluster. A cluster density is a measure that
  indicates the activity level inside the cluster.
  Whenever the density reaches a certain threshold,
  that information cluster must be better
  investigated to accelerate the convergence
  process on it.

19
CS for the PCTSP AM

• The AM component is executed whenever a solution
  is assigned to a cluster, verifying if the
  cluster can be considered promising.
• A cluster becomes promising when reaches a
  certain density,
• where, NS is the number of solutions generated
  in the interval of analysis of the
• clusters, Clus is the number of clusters, and
  PD is the desirable cluster density beyond the
  normal density, obtained if NS was equally
  divided to all clusters.
• number of solutions generated at each analysis
  of the clusters NS 200
• maximum number of clusters NC 20
• density pressure PD 2.5
• The center of a promising cluster is improved
  through the LS component.

20
Local Searcher (LS)

• The local search (LS) component is a local search
  module that provides the exploitation of a
  supposed promising search area framed by the
  cluster. This process is executed each time AM
  finds a promising cluster and the local search is
  applied on the center of the cluster.

21
CS for the PCTSP LS

• The LS component was implemented by the 2-Opt
  heuristic.
• The 2-Opt consists in 2-changes over a tour,
  deleting two arcs and replacing them by two other
  arcs to form a new tour. This method continues
  while there is improvement in the tour through
  this movement.

22
CS for the PCTSP
23
Computational Resultshttp//www.lac.inpe.br/lore
na/instancias.html

• The CS was coded in C and it was run on a 3 GHz
  Pentium 4.
• There are no available test problems for the
  PCTSP in the literature. In this paper, test
  problems were randomly generated as in DellAmico
  et al.
• n (20, 40, 60, 80, 100, 200, 300, 400, 500)
  vertices
• travel costs cij ? 1, M with M ? 1000, 10000
• prizes pi ? 1, 100
• penalties ?i ? 1, N with N ? 100, 1000,
  10000.
• The value of minimum prize (pmin) has been
  generated as
• with ? ? 0.2, 0.5, 0.8.

24
PCTSP cij? 1, 1000 ?i ? 1, 100
25
PCTSP cij? 1, 10,000 ?i ? 1, 1000
26
PCTSP cij? 1, 1000 ?i ? 1, 10,000
27
PCTSP cij? 1, 10,000 ?i ? 1, 100
28
Conclusions

• The Clustering Search (CS) uses the concept of
  hybrid algorithms, combining metaheuristics with
  a clustering process.
• The idea of the CS is to avoid applying a local
  search heuristic to all solutions generated by a
  metaheuristic.
• The CS detects the promising regions in the
  search space during the solution generation
  process and applies the local search heuristics
  only in these regions.
• CS algorithm got better results than GRASP/VNS
  without clustering process and it founds good
  values comparing to CPLEX.
• CS has two advantages over CPLEX execution time,
  and the cost of a commercial solver.
• These results validate the CS application to the
  PCTSP.

29
References

• Oliveira, A.C.M., Lorena, L.A.N. Detecting
  promising areas by evolutionary clustering
  search. Advances in Artificial Intelligence.
  Springer Lecture Notes in Artificial Intelligence
  Series 3171 (2004) 385394
• Oliveira, A.C.M., Lorena, L.A.N. Hybrid
  evolutionary algorithms and clustering search. In
  Grosan, C., Abraham, A., Ishibuchi, H., eds.
  Hybrid Evolutionary Systems - Studies in
  Computational Intelligence. Springer SCI Series
  (2007) 81102
• DellAmico, M., Mafioli, F., Sciomanchen, A. A
  lagrangian heuristic for the prize collecting
  traveling salesman problem. Annals of Operations
  Research 81 (1998) 289305
• Feo, T., Resende, M. Greedy randomized adaptive
  search procedures. Journal of Global Optimization
  6 (1995) 109133
• Mladenovic, N., Hansen, P. Variable neighborhood
  search. Computers and Operations Research 24
  (1997) 10971100
• Balas, E. The prize collecting traveling
  salesman problem. Networks 19 (1989) 621636