The interaction theory is a new promising mathematical theory which has been developed at institut Teknologi Bandung (ITB) Indonesia by Gani. Its applications provide a powerful method for solving the various combinatorial optimization problem.
The philosophy  behind this method is quite different  from the  existing methods in the field of Operations Research, Management Science and Computer Science. The method gives the exact solution and very  effective for solving manually not only simple but also complicated problems. The procedure is so simple that a knowledge of simple arithmetic is
sufficient.

The interaction theory is a new mathematical theory which Has been developed at institut Teknologi Bandung (ITB) Indonesia.
Its applications provide a powerful method for solving the various combinatorial problems such as the Traveling Salesman Problem (TSP), transportation Problem, transshipment problem, assignment problem, Minimum spanning tree (MST), network, set covering, matching layout, location and clustering. The Traveling Salesman Problem (TSP) known as the NP-complete problem, has become something of an obsession with two generations of mathematicians for its notoriety as a very difficult problem, since it takes excessive time in arriving at the solution. The TSP is widely unsolvable by polynomial algorithm. A large number of solution techniques has been developed, but most of them are limited by  the problem size.  The existing mathematical programming techniques can not afford to solve this problem.  It is the same situation that classical optimization technique failed to attack the mathematical Programming problems.  We come to the conclusion that we call for a new  mathematical philosophy and a new approach in order to have a new atmosphere. Therefore, we need breakthrough. Khachian proposed a new method with his ellipsoid algorithm, but still there are many obstacles for precision required.  Karmarkar  brought an unusually technique to linear programming which is known as projective geometry.  However, we still need to have the "formal formula".
The interaction theory tries to offer a new breakthrough for solving  the combinatorial optimization problems.

The interaction theory is a new mathematical theory for solving combinatorial optimization problems.  The basic concept of the
interaction theory is to define a set of priorities for the interacting elements of a system to attain a certain objectives.  Interaction between these two elements implies mobility of things, idea, cost, time or people. The interaction between two elements comes from the idea that :

a) Every element or component   its own special Characteristic (for example moon and world, male and female).
b) For   further    development and grow of these elements interaction, is needed.
c) Interaction does not only exist between two elements, but also among several elements.
d) Interaction could be expressed in the form of :

• Action – reaction Distance
• Cost (debit - credit)
• Social interaction
• Communication
• Chemical or physical process.

The interaction is a relationship between elements or vertices of a non-empty set which could be set forth either in a graph or matrix.  However, it is easier to express these interactions in matrix form, since increasing the number of element and the interaction, will make it difficult to figure out the interaction by a graph.  The interaction matrix can have the form of a form-to-chart which provides information on the interaction between elements or vertices.  The number of elements Usually represent, some measure of interaction between the two specified elements.  This measure might be the amount of cost, time,
distance, people, communication etc.

Cij represents the interaction coefficient from element i to element j. The interaction coefficient (Cij) is used to determine the priority in row i or in column j.  The interaction theory is employed to determine the degree of interrelationship coeficient between these two elements.

The interaction theory has also developed to solve the various problems in combinatorial optimization problems such as transportation problem, assignment problems, MST, network etc. with only one iteration. The interaction theory is a promising mathematical theory of which applications provide a very powerful method which enables one to solve
the various combinatorial problems such as :

• Traveling Salesman Problem (Symmetric and Asymmetric).
• Transportation Problem.
• Transshipment Problem.
• Assignment problem.
• Network problem
• Set Covering Problem.
• Minimum Spanning Tree (MST)
• matching Problem.
• Decision Making.
• Layout Problem.
• Location Problem
• Financial Analysis.
• Clustering.
• Forecasting.
• Multivariate.

1. The philosophy is not difficult to comprehend.
2. The formulation of the problem is easy and flexible.
3. The procedure is so simple.
4. The knowledge of simple arithmetic is sufficient.
5. The method gives the exact result.
6. The operating time is very short.
7. The method is very effective for solving manually not only the simple but also complicated and large scale problems.

Its simple implementation would create the broad subject or area of applications such as :

• Computer System.
• Industrial System.
• Economics.
• Psychology.
• Social Sciences.
• Military.
• Physics.
• Chemistry.
• Electronic System.
• Communication Network System.
• Transportation System.
• Architectural Design.
• Strategy and Policy Studies.

We see that the interaction theory is still a new philosophic and mathematical concept and we feel that the further development might create the various new efficient methods and practical applications for solving the various practical problems solving the various practical problems.