## 2 Materials and Methods
### Graphic-Analytical Model of FMS  
In the basic literature of the FMS field [23; 25; 33; 50; 51], manufacturing flexibility expresses the capacity of the system to produce not only a typological diversity of products but also unpredictability and instability, reduced time delivery, small batch manufacturing, rapid response to changing market conditions, change in structure-insensitive manufacturing pregnancy, rapid reconfiguration of the system to change production volumes, and geometric configuration of the piece. Flexible manufacturing systems (FMS) are oriented, open, deterministic, and complex systems. The study of such a system means to know the evolution in time of inputs/outputs and of the transfer of information. This is done using mathematical models. Developing a mathematical model consists in developing a system block diagram representation and of defining the analytical part, the graphic-analytical model developed for FMS. A system is generally defined as the set of components grouped by organizational rules which have a lot of functional relationships. The system is defined as consisting of three components: external environment (EE), composition (CS), and structure (ST). The FMS graphic processing part revolution will be designed by highlighting the physical meaning of different types of variables involved in the system:  
1. Exogenous variables are variables for which behavior is dictated by external phenomena and events, variables through which the system can communicate with the external environment: * Input variables (command) define inputs to the system (for which the values are given, known, and estimated). * Output variables define the outputs from the system (resulting from processing). * Disturbance variables external environmental influences on the system. The functional relationship (linked) between input and output variables can be expressed mathematically in the form of algebraic equations (inequalities), differential equations, integral equations, etc. 2. Endogenous variables are those in which the behavior is dictated by the interior system phenomena and events and that describe its dynamics. There are two such types of variables: state variables, defining the state of the system, and interaction variables (between subsystems), defining the links between subsystems. In the external environment (outside the system EE\({}_{0}\)--exomodium), many outlets, objective functions--performance indicators (imposed, real parameters that are available) chosen from the sensitivity indicators--are found. At the output of the manufacturing system, products with determined configurations and properties, and auxiliary materials and waste are found. The energy consumed within the system during the process is released into the environment as heat. If the output size-power is considered as the default in the study of manufacturing systems, the output size-information, instead, has a special importance for the production management by expressing the real state of the system during the process evolution on which real-time and real-environment operative manufacturing management decisions are based. Summarizing the above, it appears that the general function of a manufacturing system consists in transferring the information flow (IF) into a material flow (MF), using an energy flow (EF), so that the information creates an emerging property on input FM, transforming it into an output finished product (P\({}_{f}\)) that can be sold on the market. Knowing the overall system function, one can determine its structure, and their connecting and relating elements within the structure. For this, a time evolution analysis of the material, energy, and information flow on the transfer that is transferred is transformed in the manufacturing system from input to output.