00 Interaction Edisi Mgb 11-Dec-2013

INTERACTION THEORY

[NEW PARADIGM] FOR SOLVING THE TRAVELING

SALESMAN PROBLEM (TSP)

11 Desember 2013, Bandung

Balai Pertemuan Ilmiah ITB

Anang Zaini Gani

MAJELIS GURU BESAR

INSTITUT TEKNOLOGI BANDUNG

mailto:[email protected]

SEJARAH SINGKAT

Mengembangkan suatu metode untuk : TSP

dan Facilities Planning. GaTech USA

Interaction Theory, ITB, Bandung, Indonesia

PLANET, GaTech USA

Aplikasi untuk TSP. (the ORSA / TIMS) St.

Louis, MO. USA

Aplikasi untuk Transportation Problem, (the

ORSA / TIMS) Washington DC. USA

Dihadapan Senat Guru Besar ITB

International Workshop on Optimal Network

Topologies (IWONT)

1965 :

1966 :

1969 :

1987 :

1988 :

1992 :

2012 :

3

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INTRODUCTION

OBJECTIVE

BACKGROUND

INTERACTION THEORY

COMPUTATIONAL

EXPERIENCES AND

EXAMPLE

CONCLUSION

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

Traveling Salesman Problem; Complexity Theory; Interaction

Theory; Linear Programming; Integer Programming ;

Network).

8

INTRODUCTION

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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

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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?)

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1 2

4 3

1 2

4 3

1 2

4 3

1 2

4 3

1 – 2 – 3 - 4

1 – 2 – 3 – 3 - 1 1 – 3 – 2 – 4 - 1

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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 WAKTU 28 TRILIUN TAHUN

SEDANGKAN UMUR UNIVERS HANYA 14

MILIAR TAHUN (W. Cook)

INI MEMANG GILA

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7 (tujuh) problem

matematika pada

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.

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CH 150

76

96

89

8

7

84

30

6348

73

34

5916

111105

33

126

52

93

124

35

12

149

62

3113

10

94

88

121

79

150

77133

15

21

78

115

4104

22

72

64

112

136

145

49

144

8014 122

143

100

51079582

103

981

87

137132

65

85142

18

75

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146

97

54

92 46

90

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83141

58

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2981

11025

20

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3823

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131

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138 120

429 28

6372

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99

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8

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57

127

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101

116

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24118

5340

139

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6111

129

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17

60

1314074

130

123

663127

148

10691

119

44

71

128

68

45

147

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150 CITIES

150 cities = 5.7134x10262

Alternative routes

18

CH 150

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96

89

8

7

84

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6348

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5916

111105

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126

52

93

124

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125

149

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3113

10

94

88

121

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150

77133

15

21

78

115

4104

22

72

64

112

136

145

49

144

80

14 122

143

100

51079582

103

98

1

87

137132

65

85142

18

75

26

146

97

54

92 46

90

56

83141

58

55

50

2981

11025

20

51

109

43

67

3823

32

131

134

138 120

429 28

637

2

19

99

114

102

108

70

135

86

41

36

69

57

127

39

101

116

12

24118

5340

139

47

61

11

129

117

17

60

13

14074

130

123

663127

148

10691

119

44

71

128

68

45

147

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150 CITIES

The shortest traveling salesman route going through all 13,509

cities in the United States with a population of at least 500 (as of

1998). Illustration: Courtesy of David Applegate, Robert Bixby,

Vasek Chvatal and William Cook

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http://www.wired.com/images_blogs/wiredscience/2013/01/tsp_map.jpg

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OBJECTIVE

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. "The P versus NP

Problem" is considered one

of the seven greatest

unsolved mathematical

problems

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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!

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BACKGROUND

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The class of problem :

(P problem) solved in polynomial time

(NP Problem). that cannot be solved in

polynomial time

P vs NP

impossible to solve the NP-complete problem in

polynomial time, (P ≠ NP).

OR

NP problem can be solved in P time (P = NP).

until now no-one has been able to prove whether

P ≠ NP or P = NP.

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If the TSP can be solved using an algorithm in

polynomial time, this will prove that NP problem

can be solved in polynomial time (P = NP).

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)

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PARADIGM

OLD NEW

1. LP & DERIVATIVES 2. HEURISTIC (PROBABILISTIC) 3. PROCEDURE IS

COMPLICATED 4. NEEDS RESOURCES OF TIME

AND MEMORY UNLIMITED 5. CHECKING ALL ELEMENTS

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

8. LONG OPERATING TIME

1. INTERACTION THEORY 2. DETERMINISTIC 3. PROCEDURE IS SO SIMPLE 4. RESOURCES NEED IS

LIMITED

5. CHECKING LIMITED ELEMENTS (PRIORITY)

6. P=NP 7. SIMPLE ARITHMATIC

8. SHORT OPERATING TIME

(EFFICIENT AND EFFECTIVE)

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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)

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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)


W. Cook and K. Helsgaun

24,978 cities


and W. Cook

85,900 cities


and W. Cook

1,904,711 cities

2009 Yuichi Nagata 100.000

Mona Lisa

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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

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OBJECTIVE FUNCTION

• d(i,j) = (direct) distance between

city i and city j.

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

n

i1

n

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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}

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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

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Grotschel

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INTERACTION THEORY

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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

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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

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BAHWA SETIAP ELEMEN ATAU UNSUR MEMILIKI SPESIALISASI

UNTUK DAPAT BERKEMBANG & TUMBUH

INTERAKSI TIDAK HANYA TERJADI DUA ELEMEN

INTERAKSI TIDAK HANYA DALAM BENTUK; AKSI REAKSI, JARAK, KEUANGAN, BIAYA,FREKUENSI, KOMUNIKASI & KAITAN KIMIA DAN FISIKA PERSOALAN YANG DIHADAPI NILAI ABSOLUT

ANGKA ABSOLUT INTERAKSI

INTERAKSI KOMBINASI (COMBINED EFFORT)

INTERAKSI KOMBINASI INI DAPAT DIUNGKAPKAN DALAM BENTUK KOEFISIEN INTERKASI

KOEFISIEN INTERAKSI MERUPAKAN NILAI RELATIF DARI COMBINED EFFORT

DENGAN MELUPAKAN NILAI ABSOLUT DAN MENGGUNAKAN KOEFISIEN INTERAKSI

POKOK PIKIRAN TEORI INTERAKSI

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Two parallel lines

Two parallel lines distorted

(Hering illusion)

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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

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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 )

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TSP

INTERACTION THEORY

TSP P=NP

GENERAL

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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

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Formulation of the TSP with Interaction

Theory is very simple.

THE FORMULATION AND THE ALGORITHM

• The main activity of the exsisting

algoritms of TSP is searching to find

the optimal solution from so many

alternatives.

• The selection is related to the priority.

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The algorithm is

divided into two

general phases:

Preparation phase

Processing phase

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1. Preparation phase

consists of main 5

steps:

o Defining distance between cities or the

interaction matrix (IMAT)

oNormalization of IMAT (NIMAT)

oCalculating the interaction coefficient

matrix (ICOM)

oSorting the interaction coefficient as the

sorted ICOM (SICOM)

oPrioritizing the interaction between

cities using the delta interaction matrix

(DIM)

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2. Normalization of IMAT (NIMAT)

Normalization of IMAT is necessary to

normalize the matrix elements with each

element is added in front of the numbers

with the numbers 1 + zero is taken from

the digits of the largest element

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3. The Interaction coefficient

matrix (ICOM)

The interaction coefficient represents the

relative value of interaction between

elements to other elements

The formula of the

interaction coefficient is: ci,j = xi,j2/(Xi. .X.j)

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3. The Interaction Coefficient Matrix (ICOM)

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* c(ij)

< c(ij + 1)

j = 1, ……….,n-1

* c(i1)

= minj c

ij i = 1, ……….,m

Note of the element: The top value is the original ICOM column number

The bottom value is the sorted interaction coefficient

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Note of the element: The top value is the original the DIM column number The bottom value is the sorted incremental value

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2. Processing phase

Processing phase is searching process is

to choose the shortest path.

The searching process is to choose the

shortest path. The searching is related

to the priority

Guide line to use the DIM for

determining the optimal solution (8

Columns)

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COMPUTATIONAL

EXPERIENCES AND

EXAMPLE

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EXAMPLE ASYMETRIC TSP

1. THE INTERACTION MATRIX (IMAT) 7X7

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2. Normalization of IMAT (NIMAT) 7X7

109002

75848 x 75153

= 20.843

3. The Interaction Coefficient

Matrix (ICOM) 7X7

1 2 3 4 5 6 7

1 21.145 20.843 19.016 20.769 20.283 21.075 20.790

2 20.094 21.411 20.481 19.193 20.753 20.449 20.290

3 20.435 19.144 21.337 20.081 20.840 21.095 20.007

4 20.169 20.962 20.499 21.461 19.082 20.506 19.799

5 21.000 19.197 20.586 20.188 21.377 20.395 19.233

6 19.099 20.734 20.838 20.386 20.051 21.232 20.823

7 21.214 20.324 20.079 20.394 21.158 18.584 21.387

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4. The Sorted ICOM (SICOM) 7X7

1 3 5 4 7 2 6 1

19.016 20.283 20.769 20.790 20.843 21.075 21.145

2 4 1 7 6 3 5 2

19.193 20.094 20.290 20.449 20.481 20.753 21.411

3 2 7 4 1 5 6 3

19.144 20.007 20.081 20.435 20.840 21.095 21.337

4 5 7 1 3 6 2 4

19.082 19.799 20.169 20.499 20.506 20.962 21.461

5 2 7 4 6 3 1 5

19.197 19.233 20.188 20.395 20.586 21.000 21.377

6 1 5 4 2 7 3 6

19.099 20.051 20.386 20.734 20.823 20.838 21.232

7 6 3 2 4 5 1 7

18.584 20.079 20.324 20.394 21.158 21.214 21.387

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5. The Delta Interaction

Matrix (DIM) 7X7

1 3 5 4 7 2 6 1

0 1.267 1.753 1.774 1.827 2.059 2.129

2 4 1 7 6 3 5 2

0 901 1.097 1.256 1.288 1.560 2.218

3 2 7 4 1 5 6 3

0 863 937 1.291 1.696 1.951 2.193

4 5 7 1 3 6 2 4

0 717 1.087 1.417 1.424 1.880 2.379

5 2 7 4 6 3 1 5

0 36 991 1.198 1.389 1.803 2.180

6 1 5 4 2 7 3 6

0 952 1.287 1.635 1.724 1.739 2.133

7 6 3 2 4 5 1 7

0 1.495 1.740 1.810 2.574 2.630 2.803

SOLUTION : 1-3-2-4-5-7-6-1 (2758)

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MATRIK BAYS29 -JARAK 509

1 2 3 4 5 6 7 8 9

1 0 107 241 190 124 80 316 76 152

2 107 0 148 137 88 127 336 183 134

3 241 148 0 374 171 259 509 317 217

4 190 137 374 0 202 234 222 192 248

5 124 88 171 202 0 61 392 202 46

6 80 127 259 234 61 0 386 141 72

7 316 336 509 222 392 386 0 233 438

8 76 183 317 192 202 141 233 0 213

9 152 134 217 248 46 72 438 213 0

10 157 95 232 42 160 167 254 188 206

11 283 254 491 117 319 351 202 272 365

12 133 180 312 287 112 55 439 193 89

13 113 101 280 79 163 157 235 131 209

14 297 234 391 107 322 331 254 302 368

15 228 175 412 38 240 272 210 233 286

16 129 176 349 121 232 226 187 98 278

17 348 265 422 152 314 362 313 344 360

18 276 199 356 86 287 296 266 289 333

19 188 182 355 68 238 232 154 177 284

20 150 67 204 70 155 164 282 216 201

MATRIK BAYS29 - KOEFISIEN

1 2 3 4 5 6 7 8 9

1 1201152 1188622 1188630 1188632 1188622 1188616 1188640 1188616 1188623

2 1188622 1201154 1188619 1188626 1188618 1188623 1188644 1188630 1188622

3 1188630 1188619 1201137 1188647 1188620 1188631 1188657 1188639 1188624

4 1188632 1188626 1188647 1201154 1188633 1188636 1188629 1188631 1188636

5 1188622 1188618 1188620 1188633 1201150 1188613 1188649 1188631 1188609

6 1188616 1188623 1188631 1188636 1188613 1201149 1188648 1188623 1188612

7 1188640 1188644 1188657 1188629 1188649 1188648 1201139 1188629 1188652

8 1188616 1188630 1188639 1188631 1188631 1188623 1188629 1201150 1188630

9 1188623 1188622 1188624 1188636 1188609 1188612 1188652 1188630 1201146

10 1188629 1188622 1188630 1188615 1188628 1188629 1188634 1188632 1188632

11 1188640 1188637 1188658 1188620 1188643 1188647 1188623 1188637 1188647

12 1188620 1188626 1188634 1188640 1188616 1188609 1188651 1188626 1188611

13 1188623 1188622 1188636 1188620 1188628 1188627 1188632 1188625 1188632

14 1188642 1188635 1188646 1188619 1188644 1188645 1188630 1188642 1188648

15 1188636 1188630 1188651 1188613 1188636 1188640 1188627 1188636 1188640

16 1188624 1188631 1188644 1188624 1188636 1188635 1188625 1188619 1188640

17 1188646 1188636 1188648 1188622 1188641 1188646 1188635 1188645 1188644

18 1188640 1188631 1188643 1188617 1188641 1188641 1188632 1188641 1188644

19 1188632 1188632 1188645 1188618 1188637 1188636 1188621 1188629 1188641

20 1188627 1188618 1188626 1188618 1188627 1188628 1188637 1188635 1188631

21 1188617 1188615 1188624 1188627 1188616 1188618 1188642 1188626 1188620

22 1188645 1188638 1188649 1188622 1188647 1188648 1188633 1188645 1188651

23 1188627 1188638 1188648 1188634 1188640 1188633 1188618 1188616 1188641

124 > 113 1188622 < 1188623

65

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AZG

2013 Soal 150

101 cities instance generate 31 solutions

657 cities instance gives 4 solutions

Monalisa Instance (100.000 cities), 7 solutions

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Computational experiences

Breakthrough for a TSP algorithm

The process of finding a solution :

•Requires only max 20 columns (DIM)

•a huge saving in time and storage space

1. Length ;Team ;Date

2. 6204999 ;Xavier Clarist ;27.8.2012

3. 6204999 ;Keld Helsgaun ;17.9.2012

4. 6204999 ;Yuichi Nagata ;1.2.2013

5. 6205000 ;Vladimir Shylo ;4.2.2013

6. 6205001 ;Xavier Clarist ;6.8.2012

7. 6205005 ;Keld Helsgaun ;1.8.2012

8. 6205015 ;Ivan Gradinar ;11.4.2013

9. 6205017 ;Vladimir Shylo ;25.1.2013

10. 6205028 ;Xavier Clarist ;30.7.2012

11. 6205064 ;Keld Helsgaun ;16.7.2012

12. 6205118 ;Roman Bazylevych, Bohdan Kuz, Roman Kutelmakh ;3.7.2013

13. 6205251 ;Mohammad Syarwani ;3.7.2013


15. 6205320 ;Ashley Wang ;4.11.2012


17. 6211995 ;Geir Hasle, Torkel Haufmann, Christian Schulz ;4.7.2013

18. 6221125 ;Marco Alves Ganhoto ;3.7.2013

19. 6251141 ;Wenhong Tian ;3.7.2013

20. 6424026 ;David Liu ;2.7.2013

21. 6598272 ;Cyrille Yemeli Tasse ;4.3.2013

Final Leader Board : The winner of the USA TSP Challenge (115.475)

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TSP (Symmetric & Asymmetric

Transportation

Problems Graph

Network Problems

Scheduling

Decision

Making Clustering

Layout

Problems

Routing Data Mining

Financial

Analysis

Location

Problems

Assignment

Problems

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Computer Science

Transportasi

Militer

Ekonomi

Strategi

Finansial

Distribusi / Logistik

Psikologi

Kimia

Fisika

Biologi

Operations Research

Telekomunikasi

Industri Sosial

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0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70 80

38

86

16

61 85

91

100

98

37

92

59 93

99 96

6

89

52 18

83

60

5 84

17

45

8

46

47

36

49

64

63 90

32

10 62

11

19

48

82

7 88 31

70

30

20

66

71

65

35

34

78 81

9 51

33

79 3

77 76

50

1

69

27

101 53

28

26

12 80

68

29

24

54

55

25 4

39

67 23

56

75

41

22 74

72 73

21

40 58

13 94

95

97

87

2

57

15 43

42

14

44

Route for 101 cities ( 31 Optimal solutions)

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Portrait of Mona Lisa with Solution of a Traveling

Salesman Problem. Courtesy of Robert Bosch ©2012

( 7 Optimal solutions)

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• The conclusion is that the

Interaction Theory creates a new

paradigm to the new efficient and

effective algorithm for solving the

TSP easily (P=NP).

• Overall, the Interaction Theory

shows a new concept which has

potential for development in

mathematics, computer science

and Operations Research and their

applications

CONCLUSION AZG

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Four Mathematician Are Hired By The USA Government

To Solve The Most Powerful Problem In Computer

Science History

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THANK

YOU

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