INTERACTION THEORY

NEW PARADIGM FOR SOLVING THE

TRAVELING SALESMAN PROBLEM

(TSP)

Department of Industrial Engineering

Institut Teknologi Bandung

Bandung, Indonesia 2012ganiaz@gmail.com

ganiaz@mail.ti.itb.ac.id

Anang Z.Gani

2

3

4

INTRODUCTION

OBJECTIVE

BACKGROUND

INTERACTION THEORY

COMPUTATIONAL

EXPERIENCES AND

EXAMPLE

CONCLUSION

AZG2012

(Keywords: Graph; P vs NP; Combinatorial Optimization;

Traveling Salesman Problem; Complexity Theory; Interaction

Theory; Linear Programming; Integer Programming ;

Network).

5

The area of Applications :

Robot control

Road Trips

Mapping Genomes

Customized Computer Chip

Constructing Universal DNA Linkers

Aiming Telescopes, X-rays and lasers

Guiding Industrial Machines

Organizing Data

X-ray crytallography

Tests for Microprocessors Scheduling Jobs

Planning hiking path in a nature park

Gathering geophysical seismic data

Vehicle routing

Crystallography

Drilling of printed circuit boards

Chronological sequencing

AZG2012

The problem of TSP is to find the shortest

possible route to visit N cities exactly once and

returns to the origin city.

The TSP very simple and easily stated but it is

very difficult to solve.

The TSP - combinatorial problem

the alternative routes exponentially increases

by the number of cities.

1/2 (N-1)!

4 cities = 3 possible routes

4 times to 16 cities = to 653,837,184,000.

10 times to 40 cities =1,009 x1046

IF 100,000 CITIES...... (possible routes?)

AZG2012

6

SOAL 33 KOTA

ALTERNATIVE RUTE 32!/2 =

131.565.418.466.846.756.083.609.606.080.000.000

KOMPUTER PALING TOP $ 133.000.000 ROADRUNNER CLUSTER DARI UNITED STATES DEPARTMENT OF ENERGY DIMANA 129.6600 CORE MACHINE TOPPED THE 2009 RANKING OF THE 500 WORLD’S FASTES SUPER COMPUTERS, DELIVERING UP TO 1.547 TRILION ARITHMETIC OPERATIONS PER SECOND.

DIPERLUKAN WAKTUN 28 TRILIUN TAHUN

SEDANGKAN UMUR UNIVERS HANYA 14

MILIAR TAHUN

INI MEMANG GILA

AZG

2012

7 (tujuh) problem

matematikapada

millenium ini

1. The Birch and Swinnerton-

Dyer Conjecture

2. The Poincare Conjecture

3. Navier-Stokes Equations

4. P versus NP Problem

5. Riemann Hypothesis

6. The Hodge Conjecture

7. Yang-Mills Theory and The

Mass Gap Hypothesis.

AZG

2012

9

. "The P versus NP

Problem" is considered one

of the seven greatest

unsolved mathematical

problems

AZG2012

10

One important statement about the NP-

complete problem (Papadimitriou & Steiglitz) :

a. No NP-complete problem can be solved by

any known polynomial algorithm (and this

is the resistance despite efforts by many

brilliant researchers for many decades).

b. If there is a polynomial algorithm for any

NP-complete problem, then there are

polynomial algorithms for all NP-complete

problems.

THIS IS CHALLENGE TO PROVE

P= NP MUST BE PURSUED!

AZG2012

TSP dealing with the resources :

1. Time (how many iteration it takes to

solve a problem)

2. space (how much memory it takes to

solve a problem).

THE MAIN PROBLEM :

1. THE NUMBER OF STEPS (TIME) INCREASES

EXPONENTIALLY ALONG WITH THE INCREASE IN

THE SIZE OF THE PROBLEM.

2. HUGE AMOUNT COMPUTER RESOURCES ARE

REQUIRED

NEW PARADIGM (BREAKTHROUGH)

AZG2012

11

PARADIGM

OLD NEW

1. LP & DERIVATIVES2. HEURISTIC (PROBABILISTIC)3. PROCEDURE IS

COMPLICATED4. NEEDS RESOURCES OF TIME

AND MEMORY UNLIMITED5. CHECKING ALL ELEMENTS

6. P = NP VS P ≠ NP ?7. KNOWLEDGE IS HIGH

8. LONG OPERATING TIME

1. INTERACTION THEORY2. DETERMINISTIC3. PROCEDURE IS SO SIMPLE4. RESOURCES NEED IS

LIMITED

5. CHECKING LIMITED ELEMENTS (PRIORITY)

6. P=NP7. SIMPLE ARITHMATIC

8. SHORT OPERATING TIME (EFFICIENT AND EFFECTIVE)

AZG2012

12

SUMMARIZES THE MILESTONES OF SOLVING

TRAVELING SALESMAN PROBLEM.

Year Research Team Size of Instance

1954 G. Dantzig, R. Fulkerson, and S.

Johnson

49 cities

1971 M. Held and R.M. Karp 64 cities

1975 P.M. Camerini, L. Fratta, and F.

Maffioli

67 cities

1977 M. Grötschel 120 cities

1980 H. Crowder and M.W. Padberg 318 cities

1987 M. Padberg and G. Rinaldi 532 cities

(109,5 secon)

1987 M. Grötschel and O. Holland 666 cities

1987 M. Padberg and G. Rinaldi 2,392 cities

1994 D. Applegate, R. Bixby, V.

Chvátal, and W. Cook

7,397 cities

1998 D. Applegate, R. Bixby, V.

Chvátal, and W. Cook

13,509 cities

(4 Years)

AZG2012

13

SUMMARIZES THE MILESTONES OF SOLVING

TRAVELING SALESMAN PROBLEM.

Year Research Team Size of Instance

2001 D. Applegate, R. Bixby, V. Chvátal,

and W. Cook

15,112 cities

(ca. 22 Years)

2004 D. Applegate, R. Bixby, V. Chvátal,

W. Cook and K. Helsgaun

24,978 cities

2006 D. Applegate, R. Bixby, V. Chvátal,

and W. Cook

85,900 cities

2009 D. Applegate, R. Bixby, V. Chvátal,

and W. Cook

1,904,711 cities

2009 Yuichi Nagata 100.000

Mona Lisa

AZG2012

14

15

TECHNIQUE AND METHOD

FOR SOLVING TSP

• NEURAL NETWORK

• GENETIC ALGORITHM

• SIMULATED ANNEALING

• ARTIFICIAL INTELLEGENT

• EXPERT SYSTEM

• FRACTAL

• TABU SEARCH

• NEAREST NEIGBOR

• THRESHOLD ALGORITHM

• ANT COLONY OPTIMIZATION

• LINEAR PROGRAMMING

INTEGER PROGRAMMING

• CUTTING PLANE

• DYNAMIC PROGRAMMING

• THE MINIMUM SPANNING

TREE

• LAGRANGE RELAXATION

• ELLIPSOID ALGORITHM

• PROJECTIVE SCALING

ALGORITHM

• BRANCH AND BOUND

• ASAINMENT

HEURISTIC EXACT SOLUTION

AZG

2012

16

OBJECTIVE FUNCTION

• d(i,j) = (direct) distance between

city i and city j.

z x(i, j)d(i, j)j1

n

i1

n

AZG

2012

Constraints

• Each city must be “exited” exactly once

• Each city must be “entered” exactly once

x(i, j)j1

n

1 , i 1,2,...,n

x(i, j)i1

n

1 , j 1,2,...,n

Subtour elimination constraint

• S = subset of cities

• |S| = cardinality of S (# of elements in S)

• There are 2n

such sets !!!!!!!

x(i, j) Si , jS

1, S {1, 2,...,n}

AZG

2012

18

NUMBER OF LINIER INEQUALITIES

AS CONSTRAINS IN TSP

• If n=15 the number of countraints is 1.993.711.339.620

• If n=50 the number of countraints 1060

• If n=120 the number of countraints 2 x 10179

or to be exact :

26792549076063489375554618994821987399578869037768707804846519432957724703086273401563211708807593998691345929648364341894253344564803682882554188736242799920969079258554704177287

AZG2012

Grotschel

19

INTERACTION THEORY

AZG2012

INTERACTION THEORY

In 1965 Anang Z. Gani [28] did research on the Facilities Planning

problem as a special project (Georgia Tach in 1965)

Supervision James Apple

Later, J. M. Devis and K. M. Klein further continued the original

work of Anang Z. Gani

Then M. P. Deisenroth “ PLANET” direction of James Apple

(Georgia Tech in1971)

Since 1966, Anang Z. Gani has been continuing his research and

further developed a new concept which is called “The Interaction

Theory” (INSTITUT TEKNOLOGI BANDUNG)

The model is the From - To chart the which provides quantitative

information of the movement between departments

AZG2012

20

The model is the From - To chart the which

provides quantitative information of the

movement between departments (common

mileage chart on the road map).

The absolute value or the number of a

element as an individual of a matrix can not

be used in priority setting

the TSP matrix has two values,

1. the initial absolute value (interaction

value)

2. the relative value (interaction coefficient)

DIM = The Delta Interaction Matrix

AZG2012

21

AZG2012

22

Two parallel lines

Two parallel lines distorted

(Hering illusion)

AZG2012

23

1 2 3 4

1 0 700 10 20

2 2 0 800 15

3 4 3 0 10

4 10 2 30 0

RELATIVE VALUE

AZG2012

24

The formula for the interaction

coefficient ( ci,j ) is:

ci,j = xi,j2/(Xi. .X.j).

Xi. =

m

j 1

xij (i = 1 ……. m )

X.j =

n

i 1

xij (j = 1 ……. n )

AZG2012

25

TSP

INTERACTION THEORY

TSP P=NP

GENERAL

AZG

2012

APPLICATION OF THEORY INTERACTION

• Traveling Salesman Problem (Symmetric and Asymmetric, minimum and maximum).

• Transportation Problem.• Logistic.• Assignment problem.• Network problem• Set Covering Problem.• Minimum Spanning Tree (MST)• Decision Making.• Layout Problem.• Location Problem• Financial Analysis.• Clustering.• Data Mining

AZG

2012

28

TSP(Symmetric & Asymmetric

Transportation

ProblemsGraph

Network Problems

Scheduling

Decision

Making Clustering

Layout

Problems

Routing Data Mining

Financial

Analysis

Location

Problems

Assignment

Problems

AZG

2012

29

Computer Science

Transportasi

Militer

Ekonomi

Strategi

Finansial

Distribusi / Logistik

Psikologi

Kimia

Fisika

Biologi

Operations Research

Telekomunikasi

Industri Sosial

AZG2012

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70 80

38

86

16

6185

91

100

98

37

92

5993

9996

6

89

5218

83

60

584

17

45

8

46

47

36

49

64

6390

32

1062

11

19

48

82

788 31

70

30

20

66

71

65

35

34

7881

951

33

793

7776

50

1

69

27

10153

28

26

1280

68

29

24

54

55

254

39

6723

56

75

41

2274

7273

21

4058

1394

95

97

87

2

57

1543

42

14

44

Route for 101 cities ( 8 Optimal solutions)

AZG2012

30

Portrait of Mona Lisa with Solution of a Traveling

Salesman Problem. Courtesy of Robert Bosch ©2012

( 7 Optimal solutions)

AZG2012

31

• The conclusion is that the

Interaction Theory creates a new

paradigm to the new efficient and

effective algorithm for solving the

TSP easily (N=NP).

• Overall, the Interaction Theory

shows a new concept which has

potential for development in

mathematics, computer science

and Operations Research and their

applications

CONCLUSIONAZG2012

32

33

THANK

YOU

AZG2012

SIMPLICITY IS

POWER