Mathematical Model for Determining the Time of Preventive Replacements in the Agricultural Machinery Service System with Minimal Repair

Abstract

In this paper, a semi-Markov model for determining the optimal time for preventive replacements according to the age of technical objects is presented. In the analyzed system of transportation, due to its specific characteristics, the basic type of renewal process carried out is minimal repair. Minimal repairs of technical objects in semi-Markov models have been analyzed in the literature to date. In the system studied, the technical objects (sets of agricultural tractors with trailers), due to the continuous operation of combine harvesters, should carry out the assigned tasks of transporting agricultural crops without interruption. The damage to agricultural tractors that arises during the implementation of transport tasks should be repaired in the shortest possible time. The repairs to damaged tractors are carried out primarily by the Technical Emergency Service and, due to their purpose and scope, may be considered minimal repairs. The effectiveness of the function of the tested technical objects is analyzed by two criteria functions, which are very important for the system managers. These are profit per unit of time and availability. In the analyzed case, it is the availability to carry out the assigned transport tasks. The conditions for the existence of a maximum of criterion functions have been written for the assumptions. The analyzes carried out, which are presented in the work, are illustrated with sample calculations. It has been proven that, under general assumptions, the criterion functions considered in the paper have exactly one maximum. On the basis of the conducted analysis, sufficient conditions for the existence of a maximum of these functions were formulated. In the analyzed transport system, it is possible to increase the efficiency of the function of the technical facilities in use as a result of planning additional preventive replacements (increasing the frequency of these replacements). This is especially important for a system where transport units must have high availability.