Control of production system with production delays by means of Lambert functions

Abstract

The article presents the control of production system with production delay. The objective of the control of the production system  is the optimization of the quantity of production  by minimizing production costs in the selected time period. The production control is presented in a simple production system with delay, where the production of a single product is continuously performed. For achieving the optimal production, the Lambert functions are used in the article, which enable construction of analytical solutions of delayed differential equation governing the production process. In the article, the stability of the control of production system is investigated,  where the stability bound is derived in the analytical form. It is found that production delay causes an oscillatory stock response, which can be successfully quenched by controlling production, when the time delay and the rate of the planned production, which is connected to the information about the control error, are in the stable area.  The obtained analytical results are compared with numerical solutions, obtained by Runge-Kutta method in the programming environment Mathematica, where a perfect  matching is found.