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# An Artificial Neural Hopfield Network

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As an optimization method for searching for a global optimum, it is proposed to use an artificial neural Hopfield network to determine the optimal route from the list of subcritical paths.

Optimization of the inter-shop technological route using the modular principle of the technological process, permutational models and the neural network of Hopfield. Isolines or isosurfaces with the same values of optimization parameters, for example, costs (C), capital investments (К), turnover (G) are shown inside the framework of each workshop, and each point on this line represents a variant of the technological process obtained with the help of graph (graph-tree, network graph, Euler graph, etc.) illustrating the structure of technological operations of the production (technological) process or the module of technological operations of this workshop or production site.

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The bypass sequence of the workshops can be determined either by means of a standard permutation model of the inter-shop technological routes, which is determined by analysis:

• inter-shop technological routes for the manufacture of parts-representatives of groups of parts, or analogues from the number previously mastered in production;
• routes obtained by the traveling salesman method, known from system theory

It is also possible to use other methods of combinatorial optimization. An example of constructing a graph of a permutation model for inter-shop technological routes of products: 1-casting; 2-blacksmithing; 3- mechanical; 4-thermal; 5-assembly; 6-paint and varnish shops; 7-testing station. In combinatorial optimization problems, it is required to find the best of the finite, but usually a very large number of possible solutions. If the problem is characterized by a characteristic number of elements (the dimension of the problem), then the typical number of possible solutions from which to make a choice grows exponentially – or more quickly – ! (according to the well-known Stirling formula for sufficiently large ). This property makes a simple method of enumerating all the options, in principle guaranteeing a solution for a finite number of alternatives, is extremely inefficient, because such a solution requires an exponentially long time. Effective decisions are recognized that guarantee the answer for polynomial time, which grows as a polynomial with increasing dimension of the problem, i.e. as N.

The difference between polynomial and exponential algorithms goes back to von Neumann. Problems that allow a guaranteed finding of the optimum of the objective function in polynomial time form a class P. This class is a subclass of a broader class of problems NP in which, in polynomial time, we can only estimate the value of the objective function for a particular configuration, which is naturally much easier than choosing the best of all configurations. Until now, it is not known exactly whether these two classes coincide or not. This problem, of which many mathematical copies have already been broken. If these classes were the same, for any combinatorial optimization problem, the exact solution could be guaranteed to be found in polynomial time. Practically solvable are considered problems that admit a polynomial solution, at least for typical (and not worst) cases.

For more difficult problems, it would be sufficient to have a weaker condition-finding suboptimal solutions, local minima of the objective function, which do not differ too much from the absolute minimum. Neural network solutions are just parallel algorithms that quickly find suboptimal solutions to optimization tasks, minimizing the objective function in the course of its functioning or training.

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