Method of Homori. Solving integer programming problems

The mass of problems of an economic nature, planning problems and even the solution of questions from other spheres of human life activity is connected with variables that refer to whole numbers. As a result of their analysis and the search for optimal methods of solution, the concept of an extremal problem appeared. Its features are the above feature to take an integer value, and the problem itself is treated in mathematics as integer programming.

The main direction of using tasks with variables that take integer values is optimization. A method that uses integer linear programming is also called the clipping method.

The method of Homori got its name by the name of mathematician, who first developed in 1957-1958 the algorithm, which is still widely used for solving integer linear programming problems. The canonical form of the integer programming problem makes it possible to fully discover the advantages of this method.

The Gomori method for linear programming significantly complicates the problem of finding optimal values. After all, integer is the main condition, in addition to all the parameters of the problem. It is not uncommon for a problem, when having a feasible (integer) plan, if the objective function has constraints on an admissible set, the solution does not reach the maximum. This is due to the absence of integer solutions. Without this same condition, as a rule, a suitable vector is in the form of a solution.

To substantiate numerical algorithms in solving problems, it becomes necessary to superimpose various additional conditions.

Using the Gomori method, the set of problem plans is usually considered to be a bounded so-called polytope of solutions. Proceeding from this it follows that the set of all integral plans for the problem in question has a finite value.

Also, to ensure the integerity of a function, it is assumed that the coefficient values are also integers. Despite the severity of such conditions, they can be sent to a bit.

The method of Homori, in fact, involves the construction of constraints that cut off decisions that are not non-integer. In this case, there is no cutoff of any solution to the integer-valued plan.

The algorithm for solving the problem involves finding the appropriate variants 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 undecidability of the problem will be revealed, so we get a proof that the integer programming problem has no solution.

A variant is possible when there are non-integer numbers in the components of the optimal solution. In this case, a new restriction 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, it must cut off the non-integer plan from the optimal set found. No single integer solution should be lost, cut off.

When constructing the constraint, you should choose the component of the optimal plan with the largest fractional part. It is this restriction that will be added to the existing simplex table.

We find the solution of the obtained problem using ordinary simplex transformations. We check the solution of the problem for the presence of an integer optimal plan, if the condition is satisfied, then the problem is solved. If again the result was obtained with the presence of non-integer solutions, then we introduce an additional restriction, and we repeat the process of calculations.

Having carried out a finite number of iterations, we obtain an optimal plan for the problem posed before integer programming, or prove the unsolvability of the problem.