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

• A new method for generating max-plus equations for manufacturing flow lines.
• The method can be applied to manufacturing flow lines with any structure and can account for finite buffers as well as parallel identical stations at any stage.
• The ability to generate max-plus equations quickly is useful in both design and operation phases of manufacturing flow lines. In the system design phase, easy generation of equations enables analyzing and comparing any number of possible line configurations in an efficient manner. It can also give insights into the effects of adding buffers to the system or changing buffers sizes on various system performance measures. In the operation phase, it can be used to analyze possible line improvements and line reconfigurations due to product changes.

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.