Generating max-plus equations for efficient analysis of manufacturing flow lines

Abstract

An elegant mathematical tool, max-plus algebra, can be used to model manufacturing flow lines in the form of linear state-space-like equations for application in manufacturing systems analysis and control. A method for quick and efficient generation of the max-plus equations for manufacturing flow lines of any size or structure has been developed. The generated equations are used to model flow lines with finite buffers between stations as well as parallel identical stations.A flow line is initially assumed to have infinite buffers for all stations and no parallel identical stations. An adjacency matrix is then used to encode the layout of the flow line and determine the relationships between different stations in the line. The max-plus equations for the line are then generated for the simplified line while capturing the line dynamics in two matrices that are function of the processing times of the different stations in the line. Additional terms are used to model finite buffers and the parallel identical stations.The developed manufacturing systems modelling method using max-plus algebra is intuitive and easy to understand and code in software. It is a decision support tool which facilitates quick analysis of different configurations of manufacturing flow lines and assessment of several what-if scenarios. A case study is presented where the max-plus equations are generated for three possible assembly line configurations for a back flushing control valve. The effect of buffer sizes and changes in the assembly time of one of the stations on the total line idle time is analyzed using the generated model.