" BOARD OF PROFESSOR'S OF INSTITUT TEKNOLOGI BANDUNG"
(MAJELIS GURU BESAR ITB)
Ladies and Gentlemen . . .
Assalamualaikum warahmatullahi ta'ala wabarakatuh.
Gratitude we pray to the Almighty for health and opportunity
that we have this morning so that we can arrive at a scientific meeting hall ITB.
First of all we would like to thank you with regard to the presence of scientific discussion that we held this morning.
In 2000, if I am not mistaken, The Clay Foundation in the U.S. announced a prize of $ 1 million
will be awarded for a solution to each of the seven most difficult problems in mathematics.
They represent the hardest and most important unsolved problems of mathematics in the world.
So $1,000,000 is the prize offered for the so-called millennium problems.
One of which will be discussed at this point by Professor Anang Zaini Gani.
Prof . Anang is our senior Professor of Industrial Engineering. I know him very well
he is expert in Operations Research especially in Combinatorial Problem.
He had talked about many things related to the Traveling Salesman Problem.
Henceforth, I ask Prof. Eddy Baskoro to lead this meeting.
We hope today we got an invaluable lesson from Prof. Gani
I hope this scientific presentation giving us all a lesson.
Assalamualaikum warahmatullahi wabarakatuh
Good morning ...
Members of the Board of Professor of Institut Teknologi Bandung,
especially to Professor Harijono and Professor Hendra Chairman and Secretary of Board of Professor of Institut Teknologi Bandung.
and Professor Anang Gani which will give research presentations on the Interaction Theory.
Ladies and Gentlemen ...
Prof. Harijono has mentioned about TSP or Traveling Salesman Problem in the introduction.
The TSP is one of the most widely studied combinatorial optimization problems, receiving much attentions by many scholars.
The TSP computing is emerging in the 1930's in Vienna and Harvard.
Then this problem was promoted by Whitney and Flood.
This is very interesting problem to be solved not only due to the obsession
to create various concepts and theories as well as variety of applications.
Thus, solving the TSP is one door to prove a very well-known problem, P versus NP,
which is one of the 7 millennium problems with $ 1 million prize for its solution.
One of the 7 problems, the Poincare Conjecture problem, is solved.
However, P versus NP is still upsetting to this day in the academic world whether this problem is really P = NP or P ≠ NP.
There have been many methods on offer in solving the TSP methods, ranging from exact or heuristic methods.
Prof. Anang Gani offers one method and even he gave the label "New Paradigm" in view of the TSP problem.
The method discussed today is called INTERACTION THEORY.
According to him, this theory had been developed since 1965.
This is a great effort with very very long trip,
and I think it's a great commitment beyond usual from him that we need to justly appreciate.
He received the engineering degree from ITB in 1963.
After that he went on to pursue Master of Science in industrial engineering
from the Georgia Institute of Technology USA in 1965.
He proceeded to obtain doctoral degree from Universitas Degli Study Padova, Italy, in 1972.
He had been a member of professional associations: Alpha Phi Mu, AIIE, OERSA/TIMS, GGI (France), Mastel.
He was chairman of the Graduate Program in Industrial engineering and Management ITB.
Chairman of the Department of Industrial Engineering, Director of the Institute of Engineering and Management.
And now Prof. Anang Gani will deliver a presentation entitled the Interaction Theory,
a New Paradigm to solve the Traveling Salesman Problem.
Bismillahir Rahmanir Rahim 
Assalamualaikum warahmatullahi wabarakatuh
I wish to thank the Moderator Prof. Edy Baskoro, also to Prof. Harjono and Prof. Hendra as head of Board of Professor's of Institut Teknologi Bandung (ITB) .
Traveling Salesman Problem or TSP 
is a mathematical phenomenon which is very interesting
because while it is considered a very primitive problem, 
it turns out to be a very complex matter.
There have been many methods developed, but 
there is no algorithm that is efficient and effective to solve it.
Interaction Theory is a promising method to solve
 TSP efficiently and effectively.
Initial development of the interaction theory started in 1965.
I did research on the Facilities Planning
 problem and Traveling Salesman Problem
as a special project at Georgia Tech under supervision of James Apple.
Then,1969 : PLANET ( Plant Layout and Evaluation Technique )
 was developed as one of the early concepts
for computerization in facility planning problem.
The study produced a useful method and introduced a computer application
for facilities planning and for solving the Traveling Salesman Problem.
At that time the Traveling Salesman Problem 
was not observed so much
was not observed so much
but was already recognized as an interesting problem.
Further development of the Traveling Salesman Problem is a challenging problem and much studied by experts.
Since 1966, I have been continuing my research and further developed,
a new concept which is called "The Interaction Theory" 
for solving TSP in ITB.
The basic concept of the Interaction Theory 
is to define a set of priorities
for interacting elements of a system to obtain 
the optimum solution for decision making.
I have presented the application of The Application of the Interaction Theory for Solving the Traveling Salesman Problem (TSP).
Presented at the ORSA / TIMS Joint National Meeting St. Louis, MO in 1987.
The Application for Transportation Problem in 1988 at the ORSA/TIMS Washington DC. ,
and the International Workshop on Optimal Network Topologies or " IWONT 5" in 2012.
The outline of my presentation consists 
of introduction, objectives, background,
interaction theory, example and computational experiences, and conclusion.
The TSP is known as an NP-complete problem in combinatorial optimization, which is a problem related to complexity theory.
If the TSP can be solved using an algorithm in polynomial time, this will prove that NP problem can be solved in polynomial time.
Hence P equals NP. The Interaction Theory has been developed since 1966. It is a new approach to solve TSP,
in that it is different from the existing philosophies in the field of Mathematic, Computer Science and Operations Research.
A new and simple formula is introduced, including an "interaction coefficient" of the TSP.
That is cij = xij2/(Xi . Xj) which is a breakthrough for a TSP algorithm.
The process of finding a solution does not require calculation of all paths of possible routes. It is a very huge saving in time and storage space.
Researchers in mathematics and computer science give much attention to the question of P versus NP.
They predicted that it would cause a significant impact on the advancement of research of decision problems,
search problems, and optimization problems.
"The P versus NP Problem" is considered one of the seven greatest unsolved mathematical problems.
There is a reward of $1.000.000 from The Clay Mathematics Institute of Cambridge
for someone who can prove that P equals NP or P does not equal NP.
This presentation will include seven sections.
Application of TSP in very strategic areas (robot control, road trips, mapping genomes, etc.),
then explanation of the theory of interaction, and how it opens a new paradigm and offers a new hope
for solving optimization problems.
This is considering the interaction theory gives a new concept that is simple
but capable of solving complex combinatorial problems.
The TSP is a classical problem which is actually very simple and easily stated,
but yet it is very difficult to solve.
The problem is to find the shortest possible route to visit N cities exactly once and returns to the origin city.
The difficulty for finding the shortest route is
because the amount of possible routes exponentially increases by the number of cities.
The TSP is a combinatorial problem which has a lot of alternatives,
which is 1/2 (N-1)! possible paths for N number of cities.
For example, in the case of visiting only 4 cities, there are 3 possible routes.
However, if the number of cities increases 4 times to 16 cities, the number of possible routes jumps to 653,837,184,000.
If the number of cities is 40, the amount of possible routes becomes 1,009 x 10 rank 46.
TSP with higher number of cities cannot be solved with just brute force enumeration
and distance calculation, as it is computationally intractable.
Richard M. Karp showed in 1972 that the Hamiltonian cycle problem was NP-complete,
which implies that the TSP can be classified as NP-Complete problem.
If the TSP can be solved using an algorithm in polynomial time,
this will prove that NP problem can be solved in polynomial time. Thus, P equals NP.
Alternative routes for 33 Cities are 131,565,418,466,846,756,083,609,606,080,000,000.
The Super Computer IBM ROADRUNNER CLUSTER took 28 billion years to get the first optimal solutions to keep track of all of the above alternatives,
while the age of the universe is only 14 billion years old (W J. Cook).
It's a crazy problem, indeed.
At first, there were 7 the millennium problem unsolved; but after the Poincare Conjecture was solved,
there are 6 questions left which have not been solved.
One of them is the P versus NP problem.
This is an example of the problems for the 150 and 13,509 cities.
The P versus NP problem gives an opportunity to prove P = NP. The Interaction Theory presents a new paradigm for solving the TSP.
The TSP is known as an NP-complete problem in combinatorial optimization,
which is a problem related to complexity theory. If the TSP can be solved using an algorithm in polynomial time,
this will prove that NP problem can be solved in polynomial time.
problem solved in polynomial time and NP Problem cannot be solved in polynomial time.
If the TSP can be solved using an algorithm
 in polynomial time,
this will prove that NP problem can be solved
 in polynomial time.
Up until now, no-one has been able to prove
 whether P≠NP or P=NP.
If the TSP can be solved using an algorithm in polynomial time,
this will prove that NP problem can be solved in polynomial time.
The main problem with TSP is the number of steps increases exponentially along with the increase in the size of the problem.
Therefore huge amount computer resources are required.
This new paradigm was created by eliminating the main problem.
The new paradigm in solving the TSP is indispensable.
The existing paradigm relies on Linear Programming and derivatives to obtain exact solutions,
while many heuristic methods essentially rely on probabilistic.
If we search all the elements to obtain the optimum solution,
the need for resources of time and memory are unlimited.
The level of knowledge required very high, while the case of the P versus NP is still a question mark.
Interaction theory is expected to create a new Paradigm.
It is by nature deterministic.
The procedure is so simple, resources need is limited due to the limited checking elements.
Only simple Arithmetic is needed to solve the problem.
The milestones of solving the Traveling Salesman Problem show the number of cities increased substantially after the 1990s.
Table above shows the time, the research team and the size of the instance.
G. Dantzig, R. Fulkerson, and S. Johnson solving problems 49 City in the Year 1954.
Meanwhile, since 1990, the size of the instance increases.
In 1998, D. Applegate, R. Bixby, V. Chvátal, and W. Cook solved the 13,509 cities within 4 years,
and then they broke the 15.112 cities within 22 years.
Whereas in 2009 , Yuchi Nagata solved 100,000 Cities.
Many approaches and methods are being developed nowadays.
We may separate them into two groups: exact and heuristic method.
Examples of well known exact solution approaches to solve TSP are: Linear Programming,
Integer Programming, Branch and Bound, Cutting Plane, Dynamic Programming,
Lagrange Relaxation, Ellipsoid Algorithm, and Projective Scaling Algorithm.
Since the optimal solution is unattainable, approximation or heuristic algorithms
have been developed to achieve the 'nearest-to-optimal' solution instead of having the actual optimal solution.
Examples well known heuristic approaches to solve the TSP are: Neural Network, Genetic Algorithm,
Simulated Annealing, Artificial Intelligent, Expert System, Fractal, Tabu Search,
the Threshold Algorithm, Nearest Neighbor, and Ant Colony Optimization
The general formula for the TSP consists of Objective function,
constraints, and constraint elimination sub tour.
It is very interesting to watch the amount of constraint which increases so rapidly
with an increase in the number of cities.
Imagine for n = 50 Cities, then the number of constraints is 10 rank 60
and for n = 120 the number of constraints is 2 x 10 rank 179.
This then affects in a very high need for computer resources.
We have explained the P versus NP Problem, TSP and existing methods.
Next we will discuss the interaction Theory, a method to solve TSP efficiently and effectively.
In 1965 I did a research on the Facilities Planning problem as a special project
at Georgia Tech under supervision of Prof. James Apple.The research
was entitled "Evaluation of Alternative Materials Handling Pattern".
Later, J. M. Davis and K. M. Klein further continued my original work to develop the model
and the computer program for Facilities Design and Layout.
Then M. P. Deisenroth improved and finalized it in 1971 under the name "PLANET",
or Plant Layout Analysis and Evaluation Technique, under direction of James Apple.
Since 1966, I had been continuing my research and further developed a new concept
which is called "The Interaction Theory" in ITB.
Early research of the Interaction Theory resulted in the development of the model representing the flow of materials.
The model is in the 'From -- To' chart which provides quantitative information
of the movement between departments.
This chart is similar to the common mileage chart on the road map.
In the case of the TSP, it is known as the distance matrix which comes from a set of point coordinates.
Interaction is the relationship of physical or non-physical elements in a system.
The basic concept of the Interaction theory is to define a set of priorities
for the interacting elements of a system to obtain the optimum solution for decision making.
Interaction occurs not only between the two elements.
Interaction occurs not only in the form of action and reaction,
but also in the form of distance, finance, cost, frequency, communications,
chemical and physical connection and others.
This relationship is modeled in the form of a matrix that has a value matrix that is essentially the absolute value.
The absolute value cannot be used to determine priority. Instead, it is the environment that creates new value.
I called it interaction coefficient. That is a relative value of the elements of a system.
A description of the relative value of the elements of a system is analogous to the perception of two parallel lines.
Our perception of two parallel horizontal lines changes when lines that form a radiating pattern are added.
It seems that the two lines are not parallel anymore.
This optical illusion occurred because of the interaction of the two parallel horizontal lines
with the radiating pattern lines.
Hence, there are two perceptions; the perfectly parallel horizontal lines,
and the distorted horizontal lines which is influenced by adding radiating lines.
Similarly, two small circles have a "different" size because each is influenced by the environment.
They look to be in different size within groups of small and large circles,
while in fact both circles are of equal size.
Another example is two lines with arrows in the opposite direction.
This table shows an example of the asymmetric TSP for 5 cities. The cost, not the distance, to travel between the cities is defined in a matrix.
In this example, the travel cost in cell 13 is 10, in cell 34 is 10 and in cell 41 is also 10.
They are the same absolute value. However, in the Interaction Theory, these values of 10 are not the same,
because they are affected by the other values in their respective column and row.
In the case of cell 13, there is interaction with other values in the same row, which are 700 in cell 12 and 20 in cell 14,and also interacts
with other values in the same column, 800 in cell 23 and 30 in cell 43.
Same case for cell 34 and 41, which are affected by the values of their respective rows and column.
In this case, the relative value of X13 is lower than the relative values of X34 and X41 even though all have the same absolute value of 10,
because of the interaction with its column and row that have higher values.
The very important problem here is to find the relative values of a system,
the interaction coefficients, which will be used to solve the TSP.
The core of the interaction Theory is the interaction coefficient formula, which after further steps provides the route selection priority.
that significantly reduces the time and resources needed to solve the TSP.
The formula for the interaction coefficient cij=xij2/(Xi. . X.j.)
It has been developed in order to have new relative value. xi.j Represents the relationship between the row i and column j.
Xi is the sum of all elements in row i and X.j is the sum of all elements in column j.
Interaction Theory not only solves the TSP but other optimization problem: Traveling Salesman Problem,
Vehicle Routing Problem, Transportation Problem, Logistic, Assignment Problem, Network Problem,
Set Covering Problem, Minimum Spanning Tree (MST), Decision Making, Layout Problem,
Location Problem, Financial Analysis, Clustering, and Data Mining.
Formulation of the TSP with Interaction Theory is very simple.
The main activity of the existing algoritms of TSP is searching the optimal solution
from so many alternatives. The selection is related to the priority.
The algorithm is divided into two general phases: Preparation phase and Processing phase.
Preparation phase consist of 5 main steps.
First step,defining distance between cities or the interaction matrix or IMAT.
Second,calculating Normalization of IMAT or NIMAT. Third,calculating the interaction coefficient matrix or ICOM.
Fourth,sorting the interaction coefficient as the sorted ICOM or SICOM Fifth step,
prioritizing interaction between cities using the delta interaction matrix or DIM.
The interaction matrix or IMAT is a matrix that represents the absolute value between two elements.
For the specific case of solving the TSP, this matrix represents the distance, time, or travel cost between 2 cities.
Normalization of IMAT or NIMAT is necessary step to normalize the matrix elements,
with each element is added in front of the numbers with the number 1 plus zero,
taken from the digits of the largest element.This new matrix is called NIMAT.
The interaction coefficient represents the relative value of interaction between elements to other elements.
In this step, the interaction coefficients in the same row are sorted.
In each row, the interaction coefficients are ordered either in ascending or descending order depending on the objective.
In the case of the TSP, the interaction coefficients are sorted in ascending order in order with the purpose of finding the shortest route.
The searching is related to the priority. The role of the DIM is very strategic.
The DIM is a critical tool in decision making to choose optimal tour. Because the columns in DIM are arranged in order of priority, not all the matrix elements
must be examined, but only very small matrix elements, specifically elements which are in the first 1-20 column.
This will affect in the reduction of the number of iterations, thus saving tremendous amount of time and memory space.
Processing phase is the searching process to choose the shortest path
that are closest to the optimum tour and discard alternative routes that do not make complete tour.
An application example of the interaction theory algorithm is as follows The problem
is to find a shortest route of the asymmetric TSP consists of 7 cities.
Note that the symmetric TSP is much easier to solve that then the asymmetric TSP.
The interaction matrix of 7 cities which represents the distance of the asymmetric TSP is shown in this table.
The Normalization of Interaction Matrix or NIMAT is a modification
of IMAT where all elements have been added by 100.000, except diagonal cells are added by 110.000.
The reason is to avoid selecting diagonal elements. Interaction coefficient is now "normal".
The Interaction Coefficient Matrix or ICOM is calculated using the formula ci,j=  xi,j2/(Xi. .X.j), and is shown in this table.
The Sorted Interaction Coefficient Matrix or SICOM shows ascending sequence of cells in each row, as shown in this table.
The Delta Interaction Matrix or DIM shows a zero column in the first column and each row.
It shows the incremental cell values, which are used for considering cells selection.
For example, d21 with incremental value 95 is better than d15 with value 135.
Next we search the optimal solution. First column of DIM contains all city numbers,except 7.
Then the next process is to check the second column.
There are three cells in column 2 which contain city number 7.
Their values are d57 which equals 4, d47 which is 76 and d37 that is 92.
In this case d57 was chosen as the smallest value and eligibility limit.
Then the TSP solution is obtained by the order of 1-3-2-4-5-7-6-1 with a value of 2758.
Please note in this case that no further processing phase is needed as the data
in the DIM instantly provide the optimum tour without the need of any further calculation.
Experiment in solving instances from TSPLIB shows that the algorithm is very efficient and effective.
One major advantage of The Interaction Theory is the process of finding a solution
does not require investigation of all DIM columns. Maximum requirement is about 20 columns.
Then the search for the optimal solution is very fast, reducing the search space significantly.
For The Mona Lisa Instance with 100,000 cities or 100.000 columns,only 8 columns are required.
This means great saving in time and memory space.
The application of Interaction Theory in solving various TSP instances yields optimum results.
What is more interesting is that sometime it even generates more than one optimal solution.
For example application for 101 cities instance generate 31 solutions, while for 657 cities instance gives 4 solutions.
For famous The Mona Lisa Instance with 100.000 cities, 7 solutions are found with the distance of 5,757,191,
which is the same with Yuichi Nagata's record, the smallest result at this time.
Good news for us is a challenge to solve 115.475 cities.
In the final results of the leader board, my staff Mohammad Syarwani is included in The Top 10
Capacity and application of interaction theory
Monalisa
The conclusion is that the Interaction Theory creates a new paradigm to the new efficient and effective algorithm for solving the TSP.
Overall, the Interaction Theory shows a new concept which has potential
for development in mathematics, computer science and Operations Research and their applications.
The philosophy behind the Interaction Theory method is quite different from the existing methods
in the field of mathematics, Computer Science and Operations Research.
This method might create greater change in the practical applications due to the fact that the basic philosophy is very common, easy to understand, and easy to use,
The formulation of the problem is accurate, easy and flexible,
The procedure is very simple. Simple knowledge of arithmetic is sufficient in performing the algorithm,
The method gives the exact result,The operating time is very short,
The method is robust and very effective for solving problems manually, not only simpler but also complicated and large scale problems,
Applicable for all types of TSP: symmetric, asymmetric, minimize and maximize problem.
