# Optimal Control Algorithms and Their Analysis for Short-Term Scheduling in Manufacturing Systems
## 5 Qualitative and Quantitative Analysis of the MS Scheduling Problem  
The representation of the scheduling problem in terms of a dynamic system (1) control problem allows applications of mathematical analysis tools from modern control theory (38,43,44). For example, qualitative analysis based on control theory as applied to the dynamic system (1) provides the results listed in Table 1. This table confirms that our interdisciplinary approach to the description of MS OPC processes provides a fundamental base for MS problem decisions (or control and management) that have never been previously formalized and have high practical importance. The table also presents possible directions of practical implementation (interpretation) for these results in the real scheduling of communication with MS. For example, criteria for controllability and attainability in MS OPC problems can be used for MS control process verification for a given time interval. The attainability sets (AS) of dynamic model (1) have an important place in qualitative analysis of MS control processes. These sets allow detailed analysis of computational procedures to two-point boundary problems with fixed ends of the state trajectory \(\mathbf{x}(t)\) and a fixed time interval \((T_{0},\ T_{f}]\). An AS is a fundamental characteristic of any dynamic system. The AS approach determines a range of execution policies in the presence of disturbances over which the system can be guaranteed to meet certain goals. The union of these execution policies (i.e., feasible schedules) is called an AS in the state space. The union of possible performance outcomes from the given execution policies is called an AS in the performance space [43]. The AS in the state space depicts the possible states of a schedule subject to variations of the parameters (both planned and perturbation-driven) in the nodes and channels (e.g., different capacities, lot-sizes, etc.). Let us introduce the notation for an AS. \(\mathbf{D}_{\mathbf{x}}(t,T_{0},\mathbf{x}(T_{0}),\mathbf{U}(\mathbf{x}(T_{0} )))\) is an AS in the state space, \(\mathbf{D}_{I}(t,T_{0},\mathbf{x}(T_{0}),\mathbf{U}(\mathbf{x}(T_{0})))\) is an AS in the performance indicator's space, and \(\overline{\mathbf{D}}_{I}^{\xi}(t,T_{0},\mathbf{x}(T_{0}),\Xi,\mathbf{U}( \mathbf{x}(T_{0})))\) is an approximated AS under disturbances at moment \(t\). To interconnect schedule execution and performance analysis to the AS in the _state space_, an AS in the _performance space_ can be brought into correspondence (see Figure 4). Figure 3: Optimal schedule results. In projecting these two ASs onto each other, a certain range of the schedule execution policies and the corresponding variation of the performance indicators can be determined. A continuous time representation allows investigation of the changes in execution at each time point. Therefore, at each time point, an AS can be calculated and related to output performance. The _justification of the choice_ of the AS method is related to its dynamic nature ([43; 44]). An AS may be favourable for obtaining estimations of performance attainability and consideration of the perturbations and attainability abilities of the schedules as _time functions_. In [13; 19; 33; 34; 35; 36; 37; 38], we proposed different methods and algorithms of AS calculations. These results permit improvement of algorithm convergence in MS scheduling. The proposed model interprets MS scheduling as a response to planning goal changes, demand fluctuations, and resource availability. In this interpretation, the problem is to schedule MS in order to achieve the planned goals (e.g., MS service level). The model is scalable to other management levels of MS, i.e., orders and operations can be presented as MS configuration elements and orders, respectively. The transformation of parameters and goal criteria is also possible, i.e., the lead time can be considered as the MS cycle time. Hence, the MS strategic configuration and tactical planning can be optimized.