Analysis and Solution of The Model Production System with Time Delays Using Max-Plus Algebra

Max-Plus algebra has been used to model and analyze network problems, such as railway networks, project scheduling, queuing systems, and production systems. In this network problem, Max-Plus Algebra is used for scheduling analysis. The assumption used in this analysis is a normal situation, there are no changes in the system that affect scheduling. One factor that is important and needs to be considered is the occurrence of delays in the system which affect scheduling. The objectives of this research are 1) to analyze the effect of system delays on scheduling, and 2) to determine interventions in the form of switching on the server to overcome delay propagation due to schedule delays at certain stages in the system. The simple production system in this research involves 3 servers. The arrangement of the three servers is a combination of parallel and series arrangements. The simulation in this research uses the Scilab. The switching carried out in this research is by speeding up the process on the server by 1 unit of time. The results of this research are 1) direct delay propagation occurs if the delay occurs on the server that contains the eigenvalues. If a delay occurs on a server that does not contain eigen values then there will be no delay propagation provided that the amount of delay on that server does not exceed (t – 1) where t is the time required for the entire process from that server to the next server. If the delay exceeds (t – 1) then delay propagation will occur. 2) Based on simulations using Scilab, delay propagation can be overcome by switching one or more servers according to the delay point. Delays on the server containing the eigenvalues can only be resolved by switching the server. Meanwhile, on servers that do not contain eigenvalues, switching can be done using a combination of these servers.