Definition:

In linear programming problems, some optimal solutions may be fractions or decimals, but for some specific problems, it is often required that the solutions of some variables must be integers. For example, when the variable represents the number of machines, the number of workers or the number of cars loaded, etc. In order to meet the requirements of integers, at first glance, it seems that all you have to do is round off the obtained non-integers. In fact, integers are not necessarily feasible optimal solutions, so there should be a special method to solve integer programming. In integer programming, if all variables are limited to integers, it is called pure integer programming; If only some variables are limited to integers, it is called mixed integer programming. A special case of integer programming is 0 1 programming whose variables are limited to 0 or 1. Unlike linear programming problems, integer and 0 1 programming problems have not yet found a general polynomial solution.

combinatorial optimization

Combinatorial optimization can usually be expressed as an integer programming problem. Both of them are looking for the best solution that satisfies certain constraints in limited options. There are many typical problems that reflect the broad background of integer programming. Such as knapsack (or loading) problem, fixed cost problem, harmonious expedition problem (combination dual problem), effective expedition problem (combination coverage problem), traveling salesman problem, vehicle routing problem and so on. Therefore, the application scope of integer programming is also extremely extensive. It has many applications not only in industrial engineering design and scientific research, but also in computer design, system reliability, coding and economic analysis.

integer programming

Since R.E. Gomory put forward the cutting plane method in 1958, integer programming has become an independent branch. In the past 30 years, people have developed many methods to solve various problems. The most typical method to solve integer programming is to generate a related problem step by step, which is called the derivative of the original problem. Every derivative problem is accompanied by a relaxation problem that is easier to solve than it (derivative problem is called the source problem of relaxation problem). Determine the destination of its source problem by solving the relaxation problem, that is, whether to abandon the source problem or generate one or more of its own derivative problems to replace it. Then, select a derivative problem of the original problem that has not been abandoned or replaced, and repeat the above steps until there are no unresolved derivative problems. At present, the more successful and popular methods are branch and bound method and cutting plane method, both of which are formed under the above framework.

0- 1 programming

0- 1 programming plays an important role in integer programming. On the one hand, many practical problems, such as distribution, land selection and delivery, can be attributed to this kind of planning. On the other hand, integer programming with arbitrary bounded variables is equivalent to 0- 1 programming, and many nonlinear programming problems can be expressed as integer programming problems by 0- 1 programming method, so many people are committed to this direction. The common method for solving 0- 1 programming is branch and bound method, and there are some special methods for various special problems, such as Hungarian method for solving assignment problem.