# Standardization and unification, mathematical problems in

Problems in which optimal series of products and their component parts have to be determined.

An optimal series of products is a set of various types of products selected from an initial series that enables one to meet all forms of demands on the required quantity with minimum total expenditure in the course of the development, production and use of all the products. An optimal series of products exists when, as the number of distinct types of products increases, the expenditure incurred during their development increases monotonically, while the serial and operating costs diminish.

The terminological difference between problems of standardization and problems of unification is to a certain extent questionable. They reflect different ways of looking at the question of the delineation between standardization and unification. For example, problems of choosing optimal series for simple individual and mass-produced products come under the heading of standardization, while problems of choosing optimal series for complex, expensive products and their components come under the heading of problems of unification. Another approach to the delineation of problems of standardization and unification is based on the degree of detail with which the structure of the products in the initial series is studied. If the products of different types in the initial series are completely different from each other and do not have identical, i.e. unified, components, then one speaks of a single-level standardization problem, or simply of a standardization problem. Taking into account the structure of the products and the fact that different products may have the same components, one speaks of a two-level standardization problem. By studying the structure of the components of the products in greater detail, it is possible to obtain an $n$-level standardization problem. Unification problems are $n$-level standardization problems, where $n>1$. If it is supposed that in defining an optimal series of complex products one must also, as a rule, define an optimal series of their most important components, then the two approaches to the delineation between standardization and unification described above coincide.

The simplest quantitative method for solving problems of standardization by the establishment of reasonable parameters and dimensions of the machinery and equipment is the use of a system of preference numbers, based on geometric progressions. The established series of preference numbers $R5,R10,R20,R40$ are series of geometric progressions with respective ratios

$$10^{1/5}\approx1.6,\quad10^{1/10}\approx1.25,$$

$$10^{1/20}\approx1.12,\quad10^{1/40}\approx1.06.$$

If for the class of products in question the optimality of one of these series is proved, and the minimum value of the main parameter $a_0$ is chosen, then, subsequently, the values of the main parameter of all the other products in the series can be obtained by rounding off, where necessary, the values $a_0q^n$, $n=1,2,\ldots,$ where $q$ is the ratio of the chosen series.

An approach based on the system of preference numbers gives a very approximate solution to standardization problems. Moreover, the domain of feasibility of this approach is confined to the narrow class of comparatively simple one-dimensional standardization problems in which the products in the series are characterized by one main parameter. In most cases, particularly where complex and expensive products (which cannot be characterized by one main parameter) are concerned, the optimal solution of problems of standardization and unification has to be defined using very strong mathematical methods.

Mathematical models designed for use in solving problems of standardization and unification generally reduce to fairly complex multi-extremal problems of non-linear programming, the solution of which requires modern computing methods and computers with high operating speed and a large memory.

For certain special classes of problems in which characteristics can be of essential use, simpler effective solution methods are possible.

#### References

[1] | A.A. Koktev, "Foundations of standardization in engineering" , Moscow (1973) (In Russian) |

[2] | Yu.V. Chuev, G.P. Spekhova, "Technical problems of operations research" , Moscow (1971) (In Russian) |

[3] | V.L. Beresnev, E.K. Gimadi, V.T. Dement'ev, "Extremal problems of standardization" , Novosibirsk (1978) (In Russian) |

[4] | I.B. Vapnyarskii, "Numerical methods of solving problems of the mathematical theory of standardization" USSR Comp. Math. Math. Phys. , 18 : 2 (1978) pp. 169–171 Zh. Vychisl. Mat. i Mat. Fiz. , 18 : 2 (1978) pp. 484–487 |

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Standardization and unification, mathematical problems in.

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