## 1 Introduction  
One of the high-performance methods of finishing parts is grinding them with fine-grained wheels. Circular external grinding with fine-grained wheels can provide parts with a processing accuracy of five accuracy standards, with a surface roughness of \(Ra=0.02-0.01\) microns [1]. The productivity of the process is higher than the productivity of such widespread finishing methods as superfinishing and semi-mechanical finishing [2; 3; 4; 5; 6; 7].  
Ginding of parts with fine-grained wheels is used in the ball bearing, machine tool, and instrument-making industries, as well as in the processing of cold rolling rolls in the metallurgical industry [8]. In recent years, models of high- and especially high-precision circular grinding machines have been developed and mastered, and manufacturing technology and formulations of fine-grained wheels have also been developed. All of these are prerequisites for the wider introduction of grinding with fine-grained wheels into the industry as a progressive finishing method.  
The main issues in the development of technological processes for grinding parts to obtain a surface roughness of \(Ra=0.02\) and surface accuracy according to the fifth quality are: determining the number of grinding operations of parts, choosing the characteristics of grinding wheels, and assigning grinding modes [9; 10; 11; 12].  
Recommendations on these issues are diverse [12]. For example, for the case of grinding the rolls of cold rolling machines to obtain \(Ra=0.02-0.01\) microns, various recommendations suggest carrying out finishing in one, two, three, four, or more operations [10]. The variety of recommendations for grinding with fine-grained wheels is explained not only by the multiplicity of factors affecting the process, but also by the lack of methods for optimizing the process.  
The use of fine-grained wheels makes it possible to provide a given roughness of \(Ra=0.04-0.02\) microns for the surface of the rolls; however, the complexity of their processing increases sharply, and intensive wear of the abrasive tool is observed [13]. Attempts to intensify the process lead to a deterioration in the output quality indicators; that is, it is not possible to obtain a part with a given surface roughness. It is possible to solve this problem by optimizing the number of technological operations. Similarly to other process parameters, the optimal number of operations when grinding parts with fine-grained wheels should be determined on the basis of a technical and economic analysis. The complexity of processing the part can be taken as a parameter reflecting a variable share of the cost of the process. With an increase in the number of finishing operations, the auxiliary time increases, due to the additional costs of rearranging the part and the wheel, and the machine processing time of the part decreases, since larger-grained wheels are used for intermediate operations, providing greater productivity [12]. When constructing technological processes of grinding in one, two, three, or more operations, each option must ensure that the part is obtained with a given surface roughness, a given accuracy, and the required quality of surface layers. The optimal one can be taken as the one that provides the least complexity in processing the part [13].  
Thus, the object of research is the process of the fine grinding of cold rolling rolls.  
The subjects of the study are the regularities of determining the optimal number of operations that ensures the roughness of the surface of cold rolling rolls is \(Ra=0.04-0.02\) microns with the least complexity of processing the part.  
The present work aims, based on theoretical and experimental study of the grinding process with fine-grained wheels, to develop a methodology for calculating the number of operations for the fine grinding of cold rolling rolls in order to obtain a surface roughness of \(Ra=0.04-0.02\) microns, which allows for the least labor-intensive processing of the part.