The topic of this thesis is Lean management with the application of Lean management principles for the purpose of improving a production or service process. In the first part of the paper, there is a theoretical part in which the basics of Lean management, basic principles and some important lean tools and methods are elaborated. In the following, the five postulates of Lean management are explained in detail and each principle that makes up the popular "House of lean production" is elaborated. This includes visual management, standardization of work, jidoka principle, Just-in-time principle and involvement. The mentioned principles are explained to achieve the basic understanding of the philosophy of Lean management, so that it is possible to apply them in practical situations from production. In the second part of the paper, the theoretical background of the case study and ways of visual presentation of the study results are explained. The case study was used as a tool in the observation and analysis of one of the production processes in the St. Juraj of Cemex Hrvatska. With the help of a case study, opportunities for process improvement were presented, and with the help of the Lean management methodology, proposals were presented for the improvement of the maintenance process of the raw material mill for cement production. At the same time, the greatest attention was directed towards eliminating the muda of transport and waiting time, general improvement of process capabilities and improvement of work safety. Safety at work is particularly emphasized because the working environment of a cement plant can be dangerous for human health and life. As a conclusion of this work, a table of the effect assessment of the proposed improvements is presented, where a positive effect is expected in the direction of reducing the frequency and duration of transportation of storage elements and improving communication and involvement of workers using sign boards and practicing kaizen activities.