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INTERACTION THEORY 
NEW PARADIGM FOR SOLVING THE 
TRAVELING SALESMAN PROBLEM 
(TSP) 
Anang Z.Gani 
Department of Industrial Engineering 
Institut Teknologi Bandung 
Bandung, Indonesia 2012 
ganiaz@gmail.com 
ganiaz@mail.ti.itb.ac.id
2
3
4 
 INTRODUCTION 
 OBJECTIVE 
 BACKGROUND 
 INTERACTION THEORY 
 COMPUTATIONAL 
EXPERIENCES AND 
EXAMPLE 
 CONCLUSION 
AZG 
2012 
(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 
AZG 
2012
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?) 
AZG 
2012 
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 
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. 
AZG 
2012
9 
. "The P versus NP 
Problem" is considered one 
of the seven greatest 
unsolved mathematical 
problems 
AZG 
2012
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! 
AZG 
2012
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) 
AZG 
2012 
11
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) 
AZG 
2012 
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) 
AZG 
2012 
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 
AZG 
2012 
14
15 
TECHNIQUE AND METHOD 
FOR SOLVING TSP 
HEURISTIC EXACT SOLUTION 
• 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 
AZG 
2012
16 
OBJECTIVE FUNCTION 
n 
 
n 
 
z  x(i, j)d(i, j) 
j 1 
i1 
• d(i,j) = (direct) distance between 
city i and city j. 
AZG 
2012
Constraints 
n 
 x(i, j) 
1 , i 1,2, ...,n 
j 1 
n 
 x(i, j) 
1 , j 1,2, ...,n 
i 1 
• Each city must be “exited” exactly once 
• Each city must be “entered” exactly once 
Subtour elimination constraint 
 x(i, j)  S 
 1,  S  {1, 2, ...,n} 
i , jS 
• S = subset of cities 
• |S| = cardinality of S (# of elements in S) 
• There are 2n such sets !!!!!!! 
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 : 
267925490760634893755546189948219873995788690377687 
078048465194329577247030862734015632117088075939986 
913459296483643418942533445648036828825541887362427 
99920969079258554704177287 
AZG 
2012 
Grotschel
19 
AZG 
2012 
 INTERACTION THEORY
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 
AZG 
2012 
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 
AZG 
2012 
21
AZG 
2012 
22 
Two parallel lines
Two parallel lines distorted 
(Hering illusion) 
AZG 
2012 
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 
AZG 
2012 
24
The formula for the interaction 
coefficient ( c) is: 
i,j c= x2/(X.X). 
i,j i,j 
i. .jm 
Xi. =  
j 1 
 
xij (i = 1 ……. m ) 
n 
X.j =  
i 1 
xij (j = 1 ……. n ) 
AZG 
2012 
25
TSP 
INTERACTION 
THEORY 
TSP P=NP 
GENERAL 
AZG 
2012
APPLICATION OF THEORY INTERACTION 
AZG 
2012 
• 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
28 
Transportation 
Problems 
TSP 
(Symmetric & Asymmetric 
Graph 
Network Problems 
Scheduling 
Decision 
Making Clustering 
Layout 
Problems 
Location 
Problems 
Financial 
Analysis 
Assignment 
Problems 
Routing Data Mining 
AZG 
2012
29 
Computer Science 
Transportasi 
Militer 
Ekonomi 
Strategi 
Finansial 
Distribusi / Logistik 
Psikologi 
Kimia 
Fisika 
Biologi 
Operations Research 
Telekomunikasi 
Industri Sosial 
AZG 
2012
90 
80 
70 
60 
50 
40 
30 
20 
10 
0 
49 
45 
17 
86 
38 
47 
16 
8 
48 
84 
61 
64 
19 
82 
85 
83 
5 
91 
60 
93 
98 
100 
11 
7 
18 
99 
37 
96 
59 
92 
62 
6 
89 
88 31 
52 
36 
46 
63 
90 
32 
10 
70 
30 
20 
66 
71 
65 
35 
34 
81 78 
9 
51 
33 
79 
3 
76 77 
50 
1 
69 
27 
101 
53 
28 
26 
12 
68 
80 
29 
24 
54 
55 
4 25 
39 
23 67 
56 
75 
41 
21 
73 72 
74 
22 
40 
58 
94 13 
95 
97 
87 
2 
57 
15 
43 
42 
14 
44 
0 10 20 30 40 50 60 70 80 
Route for 101 cities ( 8 Optimal solutions) 
AZG 
2012 
30
Portrait of Mona Lisa with Solution of a Traveling 
Salesman Problem. Courtesy of Robert Bosch ©2012 
( 7 Optimal solutions) 
AZG 
2012 
31
AZG CONCLUSION 
2012 
• 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 
32
33 
THANK 
YOU 
AZG 
2012 
SIMPLICITY IS 
POWER

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Interaction Theory - New Paradigm in Solving the Traveling Salesman Problem (TSP)

  • 1. INTERACTION THEORY NEW PARADIGM FOR SOLVING THE TRAVELING SALESMAN PROBLEM (TSP) Anang Z.Gani Department of Industrial Engineering Institut Teknologi Bandung Bandung, Indonesia 2012 ganiaz@gmail.com ganiaz@mail.ti.itb.ac.id
  • 2. 2
  • 3. 3
  • 4. 4  INTRODUCTION  OBJECTIVE  BACKGROUND  INTERACTION THEORY  COMPUTATIONAL EXPERIENCES AND EXAMPLE  CONCLUSION AZG 2012 (Keywords: Graph; P vs NP; Combinatorial Optimization; Traveling Salesman Problem; Complexity Theory; Interaction Theory; Linear Programming; Integer Programming ; Network).
  • 5. 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 AZG 2012
  • 6. 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?) AZG 2012 6
  • 7. 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
  • 8. 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. AZG 2012
  • 9. 9 . "The P versus NP Problem" is considered one of the seven greatest unsolved mathematical problems AZG 2012
  • 10. 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! AZG 2012
  • 11. 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) AZG 2012 11
  • 12. 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) AZG 2012 12
  • 13. 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) AZG 2012 13
  • 14. 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 AZG 2012 14
  • 15. 15 TECHNIQUE AND METHOD FOR SOLVING TSP HEURISTIC EXACT SOLUTION • 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 AZG 2012
  • 16. 16 OBJECTIVE FUNCTION n  n  z  x(i, j)d(i, j) j 1 i1 • d(i,j) = (direct) distance between city i and city j. AZG 2012
  • 17. Constraints n  x(i, j) 1 , i 1,2, ...,n j 1 n  x(i, j) 1 , j 1,2, ...,n i 1 • Each city must be “exited” exactly once • Each city must be “entered” exactly once Subtour elimination constraint  x(i, j)  S  1,  S  {1, 2, ...,n} i , jS • S = subset of cities • |S| = cardinality of S (# of elements in S) • There are 2n such sets !!!!!!! AZG 2012
  • 18. 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 : 267925490760634893755546189948219873995788690377687 078048465194329577247030862734015632117088075939986 913459296483643418942533445648036828825541887362427 99920969079258554704177287 AZG 2012 Grotschel
  • 19. 19 AZG 2012  INTERACTION THEORY
  • 20. 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 AZG 2012 20
  • 21. 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 AZG 2012 21
  • 22. AZG 2012 22 Two parallel lines
  • 23. Two parallel lines distorted (Hering illusion) AZG 2012 23
  • 24. 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 AZG 2012 24
  • 25. The formula for the interaction coefficient ( c) is: i,j c= x2/(X.X). i,j i,j i. .jm Xi. =  j 1  xij (i = 1 ……. m ) n X.j =  i 1 xij (j = 1 ……. n ) AZG 2012 25
  • 26. TSP INTERACTION THEORY TSP P=NP GENERAL AZG 2012
  • 27. APPLICATION OF THEORY INTERACTION AZG 2012 • 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
  • 28. 28 Transportation Problems TSP (Symmetric & Asymmetric Graph Network Problems Scheduling Decision Making Clustering Layout Problems Location Problems Financial Analysis Assignment Problems Routing Data Mining AZG 2012
  • 29. 29 Computer Science Transportasi Militer Ekonomi Strategi Finansial Distribusi / Logistik Psikologi Kimia Fisika Biologi Operations Research Telekomunikasi Industri Sosial AZG 2012
  • 30. 90 80 70 60 50 40 30 20 10 0 49 45 17 86 38 47 16 8 48 84 61 64 19 82 85 83 5 91 60 93 98 100 11 7 18 99 37 96 59 92 62 6 89 88 31 52 36 46 63 90 32 10 70 30 20 66 71 65 35 34 81 78 9 51 33 79 3 76 77 50 1 69 27 101 53 28 26 12 68 80 29 24 54 55 4 25 39 23 67 56 75 41 21 73 72 74 22 40 58 94 13 95 97 87 2 57 15 43 42 14 44 0 10 20 30 40 50 60 70 80 Route for 101 cities ( 8 Optimal solutions) AZG 2012 30
  • 31. Portrait of Mona Lisa with Solution of a Traveling Salesman Problem. Courtesy of Robert Bosch ©2012 ( 7 Optimal solutions) AZG 2012 31
  • 32. AZG CONCLUSION 2012 • 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 32
  • 33. 33 THANK YOU AZG 2012 SIMPLICITY IS POWER