A mathematical model of the distribution of production flows

Savenkova N.P., Artem'eva L.A., Dryazhenkov A.A., Laponin V.S., Mokin A.Yu.

Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics

In the report a mathematical model is proposed. The model allows us to obtain a quantitative estimate of matherial flows within a large-scale factory. A formalization of the verbal statement of this problem is proposed that includes a direct graph construction. The graph corresponds to phased production process. A two-dimensional grid [2] is defined, and in the nodes of the grid the production flows are evaluated. The space of grid functions and corresponding inner product [1] are also defined. Evaluation of production flows is reduced to the conditional minimization problem for the quadratic functional. The novelty of the problem is that matherial balance is obtained due to a minimization problem solving that takes into account tight equality-type constraints and inequality constraints. The minimizing functional incorporates both known measurements of some of the sought variables and recipes of the production processes. Among the features of the problem are gully-type functional and large scale of the considered production which contains thousands of objects that deals with dozens of production names. The last circumstance requires using special optimization methods developed in [3], [4].

Existence and uniqueness problems are discussed together with possible ways to speed up the algorithm.


1. Il'in V.A., Kim G.D. Linear algebra and analytical geometry. Moscow. MSU Publ. 1998.

2. Samarskiy A.A. The theory of difference schemes. Moscow. Nauka. 1989.

3. Antipin A.S. Extrapolational methods for calculation of a saddle point of Lagrange function and their application to problems with separable block structure. Zh. Vychisl. Mat. Mat. Fiz. (1986), Volume 26, Number 1. P. 150 - 151.

4. Antipin A.S. Iterative methods of predictive type for computing fixed points of extremal mappings. Izv. Vyssh. Uchebn. Zaved. Mat. (1995), Number 11. P. 17 - 27.

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