# Proactive Scheduling and Routing of MRP-Based Production with Constrained Resources
## 4 Mathematical Model for Proactive Scheduling and Routing  
To address the limitations of MRP-based production planning systems (Sections 1 and 2), we propose a mathematical model that supports and optimizes decision-making in scheduling and routing. Figure 1 illustrates the functional scope of our approach in relation to MRP II/ERP modules. The model enables automated and optimized decisions that are typically made manually or semi-automatically using the MRP/CRP/SFC modules. These modules often fail to provide optimal--or even feasible--decisions. The model relies on the following assumptions:  
* Production planning is carried out using the MRP method;
* In addition to basic resources (machines, workstations, etc. ), the production processes also involve additional resources (personnel, software, tools, etc. );
* The system tracks demand sources in the production process (i.e., identifies which customer orders generate which component demands);
* Resource and additional resource availability do not need to be blocked for the entire duration of each task/operation;
* Dynamic routing is permitted, i.e., routes are not fixed before production begins but can be selected and created dynamically during production, based on pre-defined equipment configurations for machines/workstations;
* The model supports proactive decision-making (answers to \"what-if\" scenarios);
* Some decisions are optimized with respect to cost and time. The model assumes a deterministic environment (fixed demand/resource availability) and focuses on single-site planning. Future work will address stochastic demand, multi-facility coordination, and real-time disruptions (e.g., machine breakdowns). Based on these assumptions, we developed a mathematical model for optimizing scheduling and routing decisions. The optimization model (1)-(18) provides answers to the following questions:  
* Q1: Can the system fulfill customer orders with the available resources (what is the resulting schedule and execution method)? * Q2: How can the system fulfill customer orders at the lowest cost using the available resources (what is the optimal schedule and execution method)? * Q3: How can the system fulfill customer orders in the shortest possible time using the available resources (what is the optimal schedule and execution method)? * Q4: Can the system fulfill customer orders under specific resource unavailability scenarios (e.g., machine failure, reduced availability of some resource or additional resources, etc.--what is the schedule and execution method)? \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Approach** & **Optimization Guarantee** & **MRP Integration** & **Traceability** & **Computational Cost** \\ \hline Proposed MILP & Optimal/near-optimal & Native & Full & Moderate (with PR) \\ Reinforcement Learning & Heuristic & Partial & Limited & High (training) \\ Genetic Algorithms & Heuristic & Challenging & Limited & Variable \\ Hybrid MILP/Metaheuristics & Heuristic & Partial & Limited & High \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the proposed approach with alternatives. If the answers to Q1 and Q4 are negative, the proposed model identifies the specific tasks that cannot be completed, allowing users to determine which customer orders remain unfulfilled. We formulated the model as a MILP problem. It incorporates two key innovations that distinguish it from existing mathematical programming approaches in production planning and control. First, we selected the set of decision variables to ensure that the model always yields a solution. This is achieved by introducing an additional decision variable \(\text{B}_{\text{u,e}}\), used in constraints (4), (5), and (9). As a result, the model avoids NSF/No Solution Found outcomes, which provide no feedback on which tasks are infeasible or why.