CEBIT EURASIA BILISIM 2004

1
Euclidean 2D Traveling Salesman
Problem TieSp Murat Erentürk, Ender
Özcan merenturk_at_yeditepe.edu.tr ,
eozcan_at_cse.yeditepe.edu.tr
2004

• INTRODUCTION
• The origins of the traveling salesman problem or
  TSP for short are obscure. In the 1920's, the
  mathematician and economist Karl Menger
  publicized it among his colleagues in Vienna. In
  the 1930's, the problem reappeared in the
  mathematical circles of Princeton. TSP is one of
  the most prominent problem in combinatorial
  optimization. Its simple definition along with
  its notorious difficulty has stimulated and still
  stimulates many efforts to find an efficient
  algorithm. Due to the NP-completeness of the TSP,
  only approximate solutions can expected. The
  traditional algorithms fail to make even a good
  approximation, leading researchers to apply
  meta-heuristic approaches.
• DEFINITION OF TSP
• Given a finite number of "cities" along with the
  cost of travel between each pair of them, find
  the cheapest way of visiting all the cities and
  returning to the starting point, assuming that
  the salesperson knows the distance between each
  pair of cities he wants to visit. The travel
  costs are symmetric in the sense that traveling
  from city X to city Y costs just as much as
  traveling from Y to X the "way of visiting all
  the cities" is simply the order in which the
  cities are visited. To put it differently, the
  data consist of integer weights assigned to the
  edges of a finite complete graph the objective
  is to find a Hamiltonian cycle which is a cycle
  passing through all the vertices, of the minimum
  total weight. In this context, Hamiltonian cycles
  commonly called tours. For small number of cities
  the problem is easy to solve, but examples with
  100 or 1000 cities show that a systematic search
  for a solution is very expensive. So far, nobody
  was able to come up with an algorithm for solving
  the traveling salesman problem that does not show
  an exponential growth of run time with a growing
  number of cities. There is a strong belief that
  no algorithm that will not show this behavior,
  but no one was able to prove this (yet) but one
  was able to prove that the traveling salesman
  problem is a kind of prototypical problem for a
  big class of problems (the famous class NP) that
  show this exponential behavior. This is the
  reason why many research groups are interested in
  the traveling salesman problem, since techniques
  developed for this problem can transferred to
  other problems of this class. More formally,
  given N cities, TSP requires a search for a
  permutation
• using a cost matrix Ccij, where cij
  denotes the cost (assumed to be known by the
  salesman) of the travel from city i to j, that
  minimizes the path length
• In where denotes the ith city in the
  tour. The search space of an Euclidean TSP is
  giant containing N! permutation and identified by
  Garey to be NP (Nondeterministic)-hard. Because
  of the difficulties arising due to the nature of
  the problem, there are many exact and
  approximation algorithms used for solving TSP,
  since the traditional methods fail. All the
  reasons mentioned above gained notoriety to TSP
  as a prototype of a hard problem in combinatorial
  optimization.
• APPLICATION AREAS
• Vehicle routing, robot control,
  crystallography, computer wiring, scheduling,
  logistics, etc...
• IMPLEMENTATION
• The cities are given by their positions
  (xi,yi) on the plane and the distance matrix is
  given by the Euclidean distance, which means that
  the distance between the cities can be calculated
  in terms of linear equations. In Figure, position
  of city 1 is denoted by (x1,y1) and position of
  city 2 is denoted by (x2,y2). The distance
  between these cities calculated as shown in
  Equation

TieSp Is a program to solve 2D Euclidean
TSP instances. TieSp developed in Borland C
Builder environment with a including Graphic User
Interface(GUI). Solution steps may observed
depending on the request. META
HEURISTICS BEHIND TieSp 2-OPT method used
as a local heuristics, Genetic Algorithms
Simulated Anneaing used as a meta-heuristics,
Genetic Algorithm combined with Hill Climbing
used as a Memetic Algorithm and Simulated
Annealing with Genetic algorithms used as a
Memeti Annealing. TEST DATA For the
testing purposes, test datas generated
artificialy whose optimal fitness known by the
generators to test the program. Test datas used
during the project given below CONCLUSIONS
The best combination of meta heuristics approach
obtained from the result used for the turkish
cities. And the optimal tour calculated as given
below HAKKARI,
SIRNAK, SIIRT, BITLIS, MUS, BINGÖL, ERZINCAN,
TUNCELI, ELAZIG, DIYARBAKIR, BATMAN, MARDIN,
SANLIURFA, ADIYAMAN, MALATYA, KAHRAMANMARAS,
GAZIANTEP, KILIS, HATAY (Antakya), OSMANIYE,
ADANA, KAYSERI, YOZGAT, NEVSEHIR, NIGDE, IÇEL
(Mersin), KARAMAN, ANTALYA, BURDUR, AFYON,
ISPARTA, KONYA, AKSARAY, KIRSEHIR, KIRIKKALE,
KARABÜK, BARTIN, ZONGULDAK, BOLU, DÜZCE, SAKARYA
(Adapazari), BILECIK, KÜTAHYA, USAK, DENIZLI,
MUGLA, AYDIN, IZMIR, MANISA, BALIKESIR,
ÇANAKKALE, EDIRNE, KIRKLARELI, TEKIRDAG,
ISTANBUL, BURSA, YALOVA, KOCAELI (Izmit),
ESKISEHIR, ANKARA, ÇANKIRI, KASTAMONU, ÇORUM,
SINOP, AMASYA, SAMSUN, TOKAT, SIVAS, ORDU,
GIRESUN, GÜMÜSHANE, TRABZON, BAYBURT, ERZURUM,
RIZE, ARTVIN, ARDAHAN, KARS, AGRI, IGDIR,
VAN Ref E. Ozcan and M. Erenturk, A BRIEF
REVIEW OF MEMETIC ALGORITHMS FOR SOLVING
EUCLIDEAN 2D TRAVELING SALESREP PROBLEM, Proc. of
the 13th TAINN, pp. 281-290, June 2004.