Application of Graphs in the Standard Calculation of Product Cost
Abstract Generated abstract
This paper addresses the difficulty of calculating and updating standard production costs for complex machine-building products whose components and manufacturing norms are subject to frequent change. It proposes representing a product structure as a directed graph, with vertices corresponding to design elements and arcs indicating component applicability, and describes a layered graph-based recurrence for accumulating direct and embedded costs by cost item. The method organizes memory records by vertex number, processes each cost category, and produces factory cost statements for all structural elements. An implementation on the Ural-11B computer is reported, requiring about 450 commands and calculating three types of standard data for a 700-vertex network in 1.5 minutes, excluding printing.
Full Text
APPLICATION OF GRAPHS IN THE STANDARD COST CALCULATION OF PRODUCTION
Yu. N. Efimov, V. I. Kizev, G. Yu. Miroshkin, V. I. Nevraev,
P. A. Sedelnikov
(Presented by the scientific seminar of the Department of Computer Engineering)
Technical progress continuously enriches the functions of industrial products, inevitably complicating their designs. The products of a modern machine-building plant are characterized by thousands and tens of thousands of names of materials, purchased components, and parts; by hundreds and thousands of names of assemblies and instruments. The elements that make up products are characterized by complex interrelationships. The continuous process of improving product design and the technology of its manufacture is accompanied by changes in the standards that serve as the initial data for calculating production cost. A change in the standards of one design element causes changes in the cost of the product and of the entire chain of design elements through which the first is connected with the product.
The actual nature of changes in standards is multifaceted. New design elements are introduced and old ones are annulled; norms of material consumption, energy expenditures, and labor input of the production process are changed. Under these conditions, the operational tracking of changes in the cost of producing a complex product by manual methods is practically impossible. This shortcoming in determining the cost of production can be eliminated by using computers for the calculations. The problem of determining cost consists in calculating the direct costs of producing a given product and in calculating the costs introduced by the elements that make up the given product. This problem can be solved with the aid of graph theory [1].
The design of a product is represented by a directed graph reflecting the interrelationships of all the elements that make up the product.
Each vertex of the graph corresponds to a product (a design element), which is characterized by a set of direct standard cost components for each manufacturing shop per unit of the given product.
The arcs of the graph are characterized by the applicability of the given product in a unit of each of those products of which it is a component part.
In solving the problem posed, either the direct or the inverse specification of the graph may be used. The graph is represented in the form of layers with through numbering of the vertices. Sources must stand at the beginning of the topology array, and sinks at the end [2].
For the calculation of any cost item in main memory, a field is allocated whose size is a multiple of the number of vertices of the graph. To record the information corresponding to each vertex, space is allocated in this field—a cell or a group of cells.
The address of a record is a function of the number of the graph vertex:
\[ A_i=\kappa\cdot i + A_0 \]
where \(A_i\) is the address of the record;
\(\kappa\) is the number of cells in one record;
\(i\) is the number of the graph vertex;
\(A_0\) is the initial address of the field.
At the first stage, for the given cost item, at each vertex of the graph the sum of the corresponding standard cost components over all shops is found, which gives the direct production costs for each product,
\[ k_i^{\mathrm{н}}=\sum_{y=1}^{n} k_{iy}^{\mathrm{н}}, \]
where \(k_i^{\mathrm{н}}\) denotes the total direct costs for the product;
\(k_{iy}^{\mathrm{н}}\) denotes the direct costs for the product in shop \(y\);
\(n\) is the number of shops participating in the processing of the given product.
To obtain the full production cost of a product, the calculation is performed for each cost item according to the following recurrence relation:
\[ k_j^{\mathrm{п}}=\sum_{i\in \Gamma^{-1}j} k_i^{\mathrm{н}}\cdot r_{ij}+k_j^{\mathrm{н}}, \]
where \(k_j^{\mathrm{п}}\) denotes the total costs as a whole for the product; \(r_{ij}\) is the applicability of the preceding \(i\)-th product in the subsequent \(j\)-th product.
When the topology of the graph is scanned according to the indicated relation, the corresponding item of the production cost of each structural element, including the final product, is determined.
The array formed for the given cost item is written to magnetic tape, and the described process is repeated for another type of cost. After the calculation has been made for all cost items, an array will have been formed on the tape containing, for each product (for each vertex of the graph), a set of standardized cost items of the factory cost.
When the results are output to print, a statement is formed in which both the individual components of cost and the aggregate factory production cost of each product are indicated.
The algorithm described was implemented on the Ural-11B digital computer. The program contains about 450 commands. The computation time for three types of initial standard data for a network with 700 vertices and complexity coefficient 3 is 1.5 min, and the time for printing the results over the entire set of elements constituting the design is about 4 min.
REFERENCES
- C. Berge. Graph Theory and Its Applications. Moscow, Foreign Literature Publishing House, 1962.
- Yu. N. Efimov, P. A. Sedelnikov. Algorithm for the ordered recoding of graph vertices. This collection.