Distribution problem is a kind of combinatorial problem.The distribution problem is formed by dividing n objects into r boxes and calculating the number of different distribution methods.The number of methods is the number of allocations.Given different conditions for objects and boxes, different distribution problems can be formed.
The distribution problem is a class of classical probability problems.That is, the problem of housing allocation in classical probability calculation.Suppose there are N boxes marked with numbers 1, 2,..., N, and n particles.The so-called assignment problem is how to assign particles to each box, where n ≤ N.Assuming that it is equally possible for each particle to be allocated to each box, the distribution method and the total number of various sub methods are as follows:
For example, calculate the probability of the following two events:
1. A={There is a particle in each of the specified n boxes}.
2. B={There are just n boxes each with a particle}.
The four distribution methods are numbered according to the sequence in the table above:
1. The first distribution method (also known as Maxwell Boltzmann statistics): each box can contain any particle and the particle can be identified
2. The second distribution method (also known as Bowser Einstein statistics): each box can contain any particle and the particle is indiscernible
3. The third distribution method: each box can accommodate at most one particle and the particle can be identified
4. The fourth distribution method (also known as Fermi Dirac statistics): each box can contain at most one particle, but the particle cannot be identified
Secondary distribution problem
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Quadratic assignment problem (QAP) is one of the most classic and challenging combinatorial optimization problems.Since Koopmans and Beckmann first proposed QAP as a combinatorial optimization problem in 1957, it has been widely used in many fields. Many problems, such as integrated circuit wiring, factory location layout, typewriter keyboard design, job scheduling, etc., can be formalized as quadratic assignment problems.In addition, the QAP problem has also been applied to statistical data analysis, archaeological data sorting and relay race team sorting.In addition, some NP hard combinatorial optimization problems, such as the traveling salesman problem, triangulation problem and the max clique problem, can also be transformed into quadratic assignment problems.Based on the theoretical and practical importance of QAP, many scholars have been inspired to study its theory, application and optimization technology in the past decades.
In 1976, Sahni and Gonzales proved that QAP is not only a NP hard problem, but also a polynomial time approximation algorithm without approximation.QAP is difficult to solveoptimization problem The main reason is the so-called "combination explosion" phenomenon, and the solution time increases exponentially with the problem size.Generally speaking, when the scale of the problem n>20, it is difficult to find its optimal solution by using classical algorithms in effective computing time, such as branch and bound method, cut plane method, etc.In order to solve the QAP problem in a practical way, people are second to none. Many heuristic algorithms are constantly proposed and applied to solve the QAP, such as simulated annealing algorithm, genetic algorithm, ant algorithmParticle Swarm Optimization、Tabu search algorithmAnd greedy random adaptive search algorithm.However, theseheuristic algorithmThere is no guarantee that the solution found must be the optimal solution. They can only give a better solution within an acceptable time.
Because of the high computational complexity and representative difficulty of solving QAP problems, many new algorithms, theories and ideas often use QAP as the standard to test their own performance after being proposed. Solving QAP problems has become one of the main manifestations of the success of optimization technology.[1]
Basic conditions
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The basic conditions of the distribution problem are:
1. Identifiable (different) or indistinguishable (same).
2. The box can be identified or not.
3. The objects assigned to the box are ordered or disordered.
4. Empty boxes are allowed or not allowed.
Both objects and boxes are indistinguishable, which is also called separation.
