The mass of tasks of an economic nature, planning problems, and even solving problems from other spheres of human life is associated with variables related to integers. As a result of their analysis and the search for optimal solutions, the concept of an extreme problem appeared. Its features are the above feature to take an integer value, and the task itself is considered in mathematics as integer programming.
The main direction of using tasks with variables taking integer values is optimization. A method that uses integer linear programming is also called a clipping method.
The Gomori method got its name from the mathematician who first developed the algorithm in 1957-1958, which is still widely used to solve integer linear programming problems. The canonical form of the integer programming problem allows us to fully and fully reveal the advantages of this method.
The Gomori method as applied to linear programming greatly complicates the task of finding the optimal values. Indeed, integerity is the main condition, in addition to all parameters of the problem. There are frequent cases when a problem, having admissible (integer) plans, if the objective functions have restrictions on an admissible set, does not reach the maximum in the solution. This is due to the absence of integer solutions. Without this condition, as a rule, a suitable vector is found in the form of a solution.
To substantiate numerical algorithms in solving problems, it becomes necessary to impose various additional conditions.
Using the Gomori method, one usually considers many problem plans to be limited by the so-called polyhedron of solutions. Based on this it follows that the set of all integer plans for the task has a finite value.
Also, to guarantee the integrity of the function, it is assumed that the coefficients of the values are also integers. Despite the severity of such conditions, it is possible to relax them a little.
The Gomori method, in essence, involves constructing constraints that cut off solutions that are not integer-valued. At the same time, no solutions of the integer plan are cut off.
The algorithm for solving the problem includes finding suitable options by the simplex method, without taking into account the integer conditions. If in all components of the optimal plan there are solutions related to integers, then we can assume that the goal of integer programming is achieved. It is possible that an unsolvability of the problem is discovered, so we get proof that the integer-programming problem has no solution.
It is possible that non-integer numbers are present in the components of the optimal solution. In this case, a new constraint is added to all the constraints of the task. A new limitation is characterized by the presence of a number of properties. First of all, it must be linear, must cut off an integer-free plan from the found optimal set. No integer solution should be lost, truncated.
When constructing constraints, one should choose the component of the optimal plan with the largest fractional part. This restriction will be added to the existing simplex table.
We find a solution to the problem using ordinary simplex transformations. We check the solution of the problem for the existence of an integer optimal plan, if the condition is satisfied, then the problem is solved. If again a result was obtained with the presence of non-integer solutions, then we introduce an additional restriction and repeat the calculation process.
Having completed a finite number of iterations, we obtain the optimal plan for the problem posed to integer programming, or we prove the unsolvability of the problem.