Proactive Scheduling and Routing of MRP-Based Production with Constrained Resources

Abstract

This research addresses the challenges of proactive scheduling and routing in manufacturing systems governed by the Material Requirement Planning (MRP) method. Such systems often face capacity constraints, difficulties in resource balancing, and limited traceability of component requirements. The lack of seamless integration between customer orders and production tasks, combined with the manual and time-consuming nature of schedule adjustments, highlights the need for an automated and optimized scheduling method. We propose a novel optimization-based approach that leverages mixed-integer linear programming (MILP) combined with a proprietary procedure for reducing the size of the modeled problem to generate feasible and/or optimal production schedules. The model incorporates dynamic routing, partial resource utilization, limited additional resources (e.g., tools, workers), technological breaks, and time quantization. Key results include determining order feasibility, identifying unfulfilled order components, minimizing costs, shortening deadlines, and assessing feasibility in the absence of available resources. By automating the generation of data from MRP/ERP systems, constructing an optimization model, and exporting the results back to the MRP/ERP structure, this method improves decision-making and competes with expensive Advanced Planning and Scheduling (APS) systems. The proposed innovation solution—the integration of MILP-based optimization with the proprietary PT (data transformation) and PR (model-size reduction) procedures—not only increases operational efficiency but also enables demand source tracking and offers a scalable and economical alternative for modern production environments. Experimental results demonstrate significant reductions in production costs (up to 25%) and lead times (more than 50%).

1. Introduction

In modern manufacturing environments, effective planning and scheduling remain critical challenges, especially in systems based on traditional Material Requirements Planning (MRP) or its successors such as Manufacturing Resource Planning (MRP II) and Enterprise Resource Planning (ERP). Although widely used, these systems often fall short when confronted with dynamic production demands, limited resources, and the need for rapid adaptation to order changes. Key limitations include the lack of real-time integration with actual production execution, the inability to account for machine and labor availability, and limited support for cost or time optimization. Moreover, the rigidity and high implementation costs of conventional ERP/MRP II solutions often make them unsuitable for agile production contexts.

To address these shortcomings, we introduce a novel approach based on mathematical modeling—specifically, mixed-integer linear programming (MILP)—designed to enhance production planning by integrating proactive routing and scheduling capabilities. Unlike traditional MRP systems, our method incorporates additional constraints and decision variables that reflect the partial and time-dependent use of diverse resources, including tools, software, and human labor. It also dynamically traces the origin of demand for each component, minimizes idle time between operations, and supports adaptive routing under real-time conditions.

Our contribution further includes an innovative model-size reduction technique that enables the computationally efficient generation of feasible and near-optimal production schedules. This framework facilitates both cost and time minimization and supports comprehensive feasibility assessments under various operational scenarios, such as resource downtimes or machine failures. By automating data exchange between MRP/ERP systems and the optimization engine, the proposed method offers a scalable and cost-effective alternative to expensive Advanced Planning and Scheduling (APS) systems, ensuring traceability and strategic flexibility in production.

This paper proceeds as follows: Section 2 reviews the relevant literature to contextualize our contribution. Section 3 presents the optimization model and methodological details. Section 4 discusses implementation and computational results. Section 5 concludes with implications and future research directions.

2. Related Works

Traditional Material Requirements Planning (MRP), along with its successors such as MRP II (Manufacturing Resource Planning) and Enterprise Resource Planning (ERP), has long served as the foundation of production planning in manufacturing environments. However, these systems have often been criticized for their lack of responsiveness to real-time operational constraints and limited optimization capabilities. As Vollmann [1] notes, classic MRP struggles to handle capacity constraints and frequently generates infeasible production schedules when resource availability fluctuates. To address these limitations, researchers have proposed various enhancements to MRP-based systems. For instance, Ptak and Smith [2] introduced the concept of Demand-Driven MRP (DDMRP), which improves adaptability in conventional planning models through buffer-based scheduling. Nevertheless, DDMRP still lacks integrated optimization and routing features when resources are constrained. In recent years, attention has shifted towards the integration of mathematical optimization into production planning. Pinedo [3] provides a comprehensive overview of scheduling models and emphasizes the growing use of mixed-integer linear programming (MILP) to model complex production systems under resource and precedence constraints. Graves [4] and Herrmann [5] propose hybrid approaches that link MRP data with MILP models for batch-level scheduling. While these hybrid MRP-MILP approaches [4,5] integrate optimization with MRP data, they typically lack three key capabilities: (a) full traceability of demand sources at the component level down to customer orders; (b) dynamic routing under real-time resource constraints; and (c) comprehensive resource coverage (tools, labor, software). Moreover, Shukla et al. [6] explored dynamic scheduling using real-time shop-floor data, highlighting the importance of responsiveness but without offering an integrated solution for MRP-driven contexts. Similarly, Holsapple and Sena [7] examined the role of decision support systems in manufacturing but did not extend their work to the mathematical optimization of scheduling.

Recent advancements in production planning leverage Industry 4.0 technologies, such as digital twins [8] and reinforcement learning [9,10], to enhance real-time decision-making [11]. However, these approaches often address isolated subproblems (e.g., single-machine scheduling) or lack integration with MRP’s demand-driven logic. Our work bridges this gap by combining MILP optimization with MRP traceability while accounting for dynamic resource constraints and addressing them holistically (tools, labor, software, etc.), a challenge acknowledged but not fully addressed in contemporary research [12,13].

3. Proactive Scheduling and Routing in MRP-Based Production Planning

In MRP-based planning, the system calculates the required dates and quantities of components and raw materials needed to fulfill customer orders using data on inventory levels, product structure, lead times, and other relevant information. It aggregates these requirements by component and date, and then transforms them into production and purchase orders. At this point, the source/customer order/that generated the requirement typically becomes untraceable. The system then transfers component orders for execution. Using technological data, including routes, it allocates the necessary resources/machines/employees/to these orders at defined times, thereby generating a schedule for their execution. In MRP II/ERP systems, these allocations are not always permissible. To address this, the Capacity Resource Planning (CRP) module verifies and adjusts them before passing the schedule to the Shop-Floor Control (SFC) module (see Figure 1). None of these stages, i.e., MRP, CRP, or SFC, incorporates optimization. Building on this, we propose our own approach (Section 4).

Our approach departs from the above-mentioned methods by integrating a comprehensive MILP model with features tailored to the operational constraints of MRP-driven production. In contrast to earlier solutions, our model considers not only machine use but also a wide array of additional resources (e.g., labor, tools, software), allowing for partial and time-specific utilization. It also introduces dynamic routing and scheduling, precise traceability of component-level demand origins, and the ability to evaluate order feasibility under various failure or shortage scenarios. Furthermore, it supports proactive and optimized decision-making. In addition, a proprietary procedure reduces the size of the modeled problem, enabling efficient computation without sacrificing model fidelity. This makes our approach both practical and cost-effective when compared to expensive APS systems.

Table 1. Comparison of the proposed approach with alternatives.

Approach	Optimization Guarantee	MRP Integration	Traceability	Computational Cost
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

provides a general comparison of the proposed approach with existing alternatives.

4. Mathematical Model for Proactive Scheduling and Routing

To address the limitations of MRP-based production planning systems (Section 1 and Section 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)?

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 B_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. Second, the model’s structure, including its set of constraints and multi-objective function, allows us to answer individual questions Q1–Q4 by appropriately selecting constraints and elements of the objective function, thereby enabling dynamic model construction for each specific question.

Table 2. Decision variables, indices, and parameters of the mathematical model.

Symbol	Description
Indices
T	Set of execution periods (t–execution period index t ∈ T)
W	Set of machines/workstations (w–machine/workstation index w ∈ W)
R	Set of additional resources (r–resource index r ∈ R)
F	Set of variants of equipping machines/workstations with additional resources (f–variant index f ∈ F)
O	Set of customer orders (o–order index o ∈ O)
E	Set of tasks to be performed (e–task index e ∈ E)
U	Set of predicted resource unavailability states (u–state index u ∈ U)
Parameters
pw_ro	Agreed end date of the customer order o. If pw_r0 = 0, the date is arbitrary.
pw_zo,e	If task e is part of the customer order o, then pw_zo,e = 1; otherwise, pw_zo,e = 0.
pw_go,e	If task e is the last task in the customer order o, then pw_go,e = 1; otherwise, pw_go,e = 0.
pt_wf,e,w	If in a variant of equipping f task e can be executed on machine w, then pt_wf,e,w = 1; otherwise, pt_wf,e,w = 0.
pw_tf,e,w	Number of quanta/time segments required to execute task e on machine w in equipment configuration f.
pt_gf,e,w,r	Quantity of additional resource r required to execute task e on machine w in equipment configuration f. If pt_gf,e,w,r = 0, then the resource is not needed.
pw_ke1,e2	If pw_ke1,e2 = 1, then task e2 must be performed before task e1. If pw_ke1,e2 = 0, then there is no direct sequence relation between tasks e1 and e2.
pw_me1,e2	Minimum technological break between the start of task e1 and the end of e2. Valid only if pw_ke1,e2 = 1.
pw_ce1,e2	Maximum technological break between the start of task e1 and the end of e2. Valid only if pw_ke1,e2 = 1.
pw_de1,e2	If task e1 and e2 must be executed on the same machine, then pw_re1,e2 = 1; otherwise, pw_re1,e2 = 0.
pt_mr	Available quantity of additional resource r.
pd_au,w	If in the state of resource unavailability, u machine w is available, then pd_au,w = 1; otherwise, pd_au,w = 0.
pd_bu,r	Amount by which the availability of additional resource r is reduced in the state of unavailability u?
Decision variables
Xu,f,e,w,t	If in the unavailability state u in period t task e is executed on machine w in equipment configuration f, then Xu,f,e,w,t = 1; otherwise, Xu,f,e,w,t = 0.
Bu,e	If in the unavailability state u we are unable to perform task e, then Bu,e = 1; otherwise, Bu,e = 0.
Yu,f,e,w	If in the unavailability state u machine w is in equipment configuration f and executes task e, then Yu,f,e,w = 1; otherwise, Yu,f,ew = 0.
Su,e	Planning period number in the unavailability state u in which the execution of task e begins.
Zu,f,e,w,t	If in the unavailability state u the execution of task e on machine w in the equipment configuration f begins in period t, then Zu,f,e,w,t = 1; otherwise, Zu,f,e,w,t = 0

presents the model’s parameters and decision variables.

Table 3. Parameters and components of the objective function.

Symbol	Description
Parameters
cp_ww	Cost of operating machine w per execution period
cp_rr	Cost of using additional resource r per execution period
Determined/calculated quantities
Cost_1u	Cost of machine use under unavailability state u
Cost_2u	Cost of additional resource use under unavailability state u
Cou_1u	Number of periods of work under unavailability state u
Decision variables
Cu,t	If work is carried out under unavailability state u in period t, then Cu,t = 1; otherwise, Cu,t = 0

outlines the individual components of the objective function, while

Table 4. Description of constraints.

No.	Constraint	Description and Explanation
(1)	Task duration constraint	Ensures full execution of each task over the required number of time units on the assigned machine under a given resource availability scenario. Prevents partial or incomplete execution.
(2)	Task start time definition	Defines the exact start period of a task based on when execution begins. Links the binary variables Z and X.
(3)	Unique assignment constraint	Ensures each task is assigned to only one machine and one configuration variant. Prevents resource conflicts and maintains task integrity.
(4)	Feasibility or failure flag	Schedules each (via Z variables) or marks it unfulfilled (B = 1). Allows the model to proceed even if full feasibility is not possible—crucial for what-if and failure analyses.
(5)	All-or-none task execution per order	Marks an entire customer order as unfulfilled if any of its tasks remain unfulfilled. Reflects real-world dependencies (e.g., missing component = undeliverable product).
(6)	Start time computation	Calculates the planned start time of each task by linking the binary Z variables with the numeric start period S.
(7)	Precedence and minimum delay	Enforces technological precedence between tasks with a minimum delay (e.g., curing, cooling). Prevents premature scheduling.
(8)	Maximum delay constraint	Limits the delay between sequential tasks to avoid issues like spoilage or degradation. Ensures tightly connected operations are not overly delayed.
(9)	Deadline enforcement	Ensures customer orders (final tasks within them) are completed by specified due dates unless explicitly marked as infeasible (B = 1).
(10)	Resource capacity constraint	Prevents overuse of additional resources (e.g., labor, tools) based on their availability. Respects realistic capacity ceilings even under failure/unavailability states.
(11)	Same machine assignment for task groups	Forces selected tasks (e.g., sequential or setup-sensitive ones) to be scheduled on the same machine. Supports process consistency and reduces changeovers.
(12)	Binary constraints	Forces variables like X, Y, Z, and B to be binary (0 or 1), reflecting yes/no decisions (task scheduled or not, resource used or not, etc.).
(13)	Machine cost calculation	Computes part of the objective function: total cost of using machines over time. Each active machine unit contributes to the cost.
(14)	Resource cost calculation	Calculates the cost of using additional resources over time, accounting for their duration and intensity of use.
(15)	Activation of time periods (lower bound)	Flags a period as active if any task is scheduled in it. Supports total period usage tracking.
(16)	Activation of time periods (upper bound)	Opposite of (15); ensures that only periods with scheduled tasks are marked as active. Works with (15) to enforce consistent tracking of working time.
(17)	Total work period count	Tallies all active periods (C variables). Used in time-based objective functions (e.g., makespan minimization).
(18)	Binary flag for work periods	Forces the C variable to be binary, indicating whether a given period includes any activity. Part of time minimization logic.

provides a brief description of constraints (1)–(18). The objective function primarily relates to cost (two components) and also includes one time-related component.

$${\displaystyle \sum _{\mathrm{t}\in \mathrm{T}}(\mathrm{pd}\text{\_}{\mathrm{a}}_{\mathrm{u},\mathrm{w}}\cdot \mathrm{pt}\text{\_}{\mathrm{w}}_{\mathrm{f},\mathrm{e},\mathrm{w}}\cdot {\mathrm{X}}_{\mathrm{u},\mathrm{f},\mathrm{e},\mathrm{w},\mathrm{t}})}=\mathrm{p}\mathrm{w}\text{\_}{\mathrm{t}}_{\mathrm{f},\mathrm{e},\mathrm{w}}\cdot {\mathrm{Y}}_{\mathrm{u},\mathrm{f},\mathrm{e},\mathrm{w}}\forall \mathrm{u}\in \mathrm{U},\mathrm{f}\in \mathrm{F},\mathrm{e}\in \mathrm{E},\mathrm{w}\in \mathrm{W}$$

(1)

$${\mathrm{X}}_{\mathrm{u},\mathrm{f},\mathrm{e},\mathrm{w},\mathrm{t}}={\displaystyle \sum _{(\mathrm{t}1\in \mathrm{T}\wedge \mathrm{t}1<\mathrm{t},\mathrm{t}1\ge \mathrm{t}-\mathrm{p}\mathrm{w}\text{\_}{\mathrm{t}}_{\mathrm{f},\mathrm{e},\mathrm{w}})}{\mathrm{Z}}_{\mathrm{u},\mathrm{f},\mathrm{e},\mathrm{w},\mathrm{t}1+1}}\forall \mathrm{u}\in \mathrm{U},\mathrm{f}\in \mathrm{F},\mathrm{e}\in \mathrm{E},\mathrm{w}\in \mathrm{W},\mathrm{t}\in \mathrm{T}$$

(2)

$${\displaystyle \sum _{\mathrm{f}\in \mathrm{F}}{\displaystyle \sum _{\mathrm{e}\in \mathrm{E}}{\mathrm{X}}_{\mathrm{u},\mathrm{f},\mathrm{e},\mathrm{w},\mathrm{t}}}}<=1\forall \mathrm{u}\in \mathrm{U},\mathrm{w}\in \mathrm{W},\mathrm{t}\in \mathrm{T}$$

(3)

$${\displaystyle \sum _{\mathrm{f}\in \mathrm{F}}{\displaystyle \sum _{\mathrm{w}\in \mathrm{W}}{\displaystyle \sum _{\mathrm{t}\in \mathrm{T}}{\mathrm{Z}}_{\mathrm{u},\mathrm{f},\mathrm{e},\mathrm{w},\mathrm{t}}}}}=1-{\mathrm{B}}_{\mathrm{u},\mathrm{e}}\forall \mathrm{u}\in \mathrm{U},\mathrm{e}\in \mathrm{E}$$

(4)

$${\mathrm{B}}_{\mathrm{u},\mathrm{e}1}={\mathrm{B}}_{\mathrm{u},\mathrm{e}2}\forall \mathrm{o}\in \mathrm{o},\mathrm{e}1,\mathrm{e}2\in \mathrm{E}\wedge \mathrm{p}\mathrm{w}\text{\_}{\mathrm{z}}_{\mathrm{o},\mathrm{e}1}=1\wedge \mathrm{p}\mathrm{w}\text{\_}{\mathrm{z}}_{\mathrm{o},\mathrm{e}2}=1$$

(5)

$${\mathrm{S}}_{\mathrm{u},\mathrm{e}}={\displaystyle \sum _{\mathrm{f}\in \mathrm{F}}{\displaystyle \sum _{\mathrm{w}\in \mathrm{W}}{\displaystyle \sum _{\mathrm{t}\in \mathrm{T}}({\mathrm{Z}}_{\mathrm{u},\mathrm{f},\mathrm{e},\mathrm{w},\mathrm{t}}\cdot \mathrm{t})}}}\forall \mathrm{u}\in \mathrm{U},\mathrm{e}\in \mathrm{E}$$

(6)

$${\mathrm{S}}_{\mathrm{u},\mathrm{e}1}\ge {\mathrm{S}}_{\mathrm{u},\mathrm{e}2}+{\displaystyle \sum _{\mathrm{f}\in \mathrm{F}}{\displaystyle \sum _{\mathrm{w}\in \mathrm{W}}(\mathrm{p}\mathrm{w}\text{\_}{\mathrm{t}}_{\mathrm{f},\mathrm{e}2,\mathrm{w}}\cdot {\mathrm{Y}}_{\mathrm{u},\mathrm{f},\mathrm{e}2,\mathrm{w}})}}+\mathrm{p}\mathrm{w}\text{\_}{\mathrm{m}}_{\mathrm{e}1,\mathrm{e}2}\forall \mathrm{u}\in \mathrm{U},\mathrm{e}1,\mathrm{e}2\in \mathrm{E}\wedge \mathrm{p}\mathrm{w}\text{\_}{\mathrm{k}}_{\mathrm{e}1,\mathrm{e}2}=1$$

(7)

$${\mathrm{S}}_{\mathrm{u},\mathrm{e}1}-{\displaystyle \sum _{\mathrm{f}\in \mathrm{F}}{\displaystyle \sum _{\mathrm{w}\in \mathrm{W}}(\mathrm{p}\mathrm{w}\text{\_}{\mathrm{t}}_{\mathrm{f},\mathrm{e}2,\mathrm{w}}\cdot {\mathrm{Y}}_{\mathrm{u},\mathrm{f},\mathrm{e}2,\mathrm{w}})}}-{\mathrm{S}}_{\mathrm{u},\mathrm{e}1}\le \mathrm{p}\mathrm{w}\text{\_}{\mathrm{c}}_{\mathrm{e}1,\mathrm{e}2}\forall \mathrm{u}\in \mathrm{U},\mathrm{e}1,\mathrm{e}2\in \mathrm{E}\wedge \mathrm{p}\mathrm{w}\text{\_}{\mathrm{k}}_{\mathrm{e}1,\mathrm{e}2}=1$$

(8)

$$\begin{array}{l}{\mathrm{S}}_{\mathrm{u},\mathrm{e}}\le (1-{\mathrm{B}}_{\mathrm{u},\mathrm{o}})\cdot (\mathrm{p}\mathrm{w}\text{\_}{\mathrm{r}}_{\mathrm{o}})-{\displaystyle \sum _{\mathrm{f}\in \mathrm{F}}{\displaystyle \sum _{\mathrm{w}\in \mathrm{W}}\mathrm{p}\mathrm{w}\text{\_}{\mathrm{t}}_{\mathrm{f},\mathrm{e},\mathrm{w}}\cdot {\mathrm{Y}}_{\mathrm{u},\mathrm{f},\mathrm{e},\mathrm{w}}}}\forall \mathrm{u}\in \mathrm{U},\mathrm{p}\in \mathrm{P},\mathrm{o}\in \mathrm{O}\wedge \mathrm{p}\mathrm{w}\text{\_}{\mathrm{g}}_{\mathrm{o},\mathrm{e}}=1\\ \wedge \mathrm{p}\mathrm{w}\text{\_}{\mathrm{r}}_0\ne 0\end{array}$$

(9)

$${\displaystyle \sum _{\mathrm{f}\in \mathrm{F}}{\displaystyle \sum _{\mathrm{e}\in \mathrm{E}}{\displaystyle \sum _{\mathrm{w}\in \mathrm{W}}(\mathrm{p}\mathrm{t}\text{\_}{\mathrm{g}}_{\mathrm{f},\mathrm{e},\mathrm{w},\mathrm{r}}\cdot {\mathrm{X}}_{\mathrm{u},\mathrm{f},\mathrm{e},\mathrm{w},\mathrm{t}})}}}\le \mathrm{p}\mathrm{t}\text{\_}{\mathrm{m}}_{\mathrm{r}}-\mathrm{p}\mathrm{d}\text{\_}{\mathrm{b}}_{\mathrm{u},\mathrm{r}}\forall \mathrm{u}\in \mathrm{U},\mathrm{r}\in \mathrm{R},\mathrm{t}\in \mathrm{T}$$

(10)

$${\mathrm{Y}}_{\mathrm{u},\mathrm{f},\mathrm{e}1,\mathrm{w}}={\mathrm{Y}}_{\mathrm{u},\mathrm{f},\mathrm{e}1,\mathrm{w}}\forall \mathrm{u}\in \mathrm{U},\mathrm{f}\in \mathrm{F},\mathrm{w}\in \mathrm{W},\mathrm{e}1,\mathrm{e}2\in \mathrm{E}\wedge \mathrm{p}\mathrm{w}\text{\_}{\mathrm{d}}_{\mathrm{e}1,\mathrm{e}2}=1$$

(11)

$$\begin{array}{l}{\mathrm{X}}_{\mathrm{u},\mathrm{f},\mathrm{e},\mathrm{w},\mathrm{t}}\in \{0,1\}\forall \mathrm{u}\in \mathrm{U},\mathrm{f}\in \mathrm{F},\mathrm{e}\in \mathrm{E},\mathrm{w}\in \mathrm{W},\mathrm{t}\in \mathrm{T}\\ {\mathrm{Z}}_{\mathrm{u},\mathrm{f},\mathrm{e},\mathrm{w},\mathrm{t}}\in \{0,1\}\forall \mathrm{u}\in \mathrm{U},\mathrm{f}\in \mathrm{F},\mathrm{e}\in \mathrm{E},\mathrm{w}\in \mathrm{W},\mathrm{t}\in \mathrm{T}\\ {\mathrm{Y}}_{\mathrm{u},\mathrm{f},\mathrm{e},\mathrm{w}}\in \{0,1\}\forall \mathrm{u}\in \mathrm{U},\mathrm{f}\in \mathrm{F},\mathrm{e}\in \mathrm{E},\mathrm{w}\in \mathrm{W}\\ {\mathrm{B}}_{\mathrm{u},\mathrm{e}}\in \{0,1\}\forall \mathrm{u}\in \mathrm{U},\mathrm{e}\in \mathrm{E}\end{array}$$

(12)

$$\mathrm{Cos}\mathrm{t}\text{\_}{1}_{\mathrm{u}}={\displaystyle \sum _{\mathrm{f}\in \mathrm{F}}{\displaystyle \sum _{\mathrm{e}\in \mathrm{E}}{\displaystyle \sum _{\mathrm{w}\in \mathrm{W}}{\displaystyle \sum _{\mathrm{t}\in \mathrm{T}}(\mathrm{c}\mathrm{p}\text{\_}{\mathrm{w}}_{\mathrm{w}}\cdot {\mathrm{X}}_{\mathrm{u},\mathrm{f},\mathrm{e},\mathrm{w},\mathrm{t}})}}}}$$

(13)

$$\mathrm{Cos}\mathrm{t}\text{\_}{2}_{\mathrm{u}}={\displaystyle \sum _{\mathrm{f}\in \mathrm{F}}{\displaystyle \sum _{\mathrm{e}\in \mathrm{E}}{\displaystyle \sum _{\mathrm{w}\in \mathrm{W}}{\displaystyle \sum _{\mathrm{r}\in \mathrm{R}}{\displaystyle \sum _{\mathrm{t}\in \mathrm{T}}(\mathrm{c}\mathrm{p}\text{\_}{\mathrm{r}}_{\mathrm{r}}\cdot \mathrm{p}\mathrm{t}\text{\_}{\mathrm{g}}_{\mathrm{f},\mathrm{e},\mathrm{w},\mathrm{r}}\cdot {\mathrm{X}}_{\mathrm{u},\mathrm{f},\mathrm{e},\mathrm{w},\mathrm{t}})}}}}}$$

(14)

$${\mathrm{C}}_{\mathrm{u},\mathrm{t}}\le {\displaystyle \sum _{\mathrm{f}\in \mathrm{F}}{\displaystyle \sum _{\mathrm{e}\in \mathrm{E}}{\displaystyle \sum _{\mathrm{w}\in \mathrm{W}}{\mathrm{X}}_{\mathrm{u},\mathrm{f},\mathrm{e},\mathrm{w},\mathrm{t}}}}}\forall \mathrm{u}\in \mathrm{U},\mathrm{t}\in \mathrm{T}$$

(15)

$${\mathrm{C}}_{\mathrm{u},\mathrm{t}}\ge {\mathrm{X}}_{\mathrm{u},\mathrm{f},\mathrm{e},\mathrm{w},\mathrm{t}}\forall \mathrm{u}\in \mathrm{U},\mathrm{f}\in \mathrm{F},\mathrm{e}\in \mathrm{e},\mathrm{w}\in \mathrm{w},\mathrm{t}\in \mathrm{T}$$

(16)

$$\mathrm{C}\mathrm{o}\mathrm{u}\text{\_}{1}_{\mathrm{u}}={\displaystyle \sum _{\mathrm{t}\in \mathrm{T}}{\mathrm{C}}_{\mathrm{u},\mathrm{t}}}\forall \mathrm{u}\in \mathrm{U}$$

(17)

$${\mathrm{C}}_{\mathrm{u},\mathrm{t}}\in \{0,1\}\forall \mathrm{u}\in \mathrm{U},\mathrm{t}\in \mathrm{T}$$

(18)

5. Implementation

We implemented the proposed mathematical model (1)–(18) using the AMPL language [14] and the Gurobi mathematical programming environment [15]. AMPL enables solving the model using various solvers such as Gurobi, LINGO, CPLEX, and SCIP. The implementation process, illustrated in Figure 2, involves several stages. In the first stage, we generate mathematical models using constraints (1)–(12) and elements of the objective function (13)–(18). Solving these models provides answers to questions Q1–Q4. Next, we use the proprietary PT procedure (Section 5.1) to extract and transform parameters and coefficients from the MRPII/ERP system database. Based on these, we construct the implementation model in AMPL. Due to the high computational complexity of MILP models, we apply the proprietary PR procedure (Section 5.2) to reduce model size. This procedure decreases the number of decision variables and simplifies certain constraints, i.e., reduces the size of the model. We then solve the reduced model using the Gurobi solver. Each solution not only answers one of the questions Q1–Q4, but also provides the corresponding schedule and resource allocation for the given set of orders being executed.

5.1. Data Transformation and Preparation Procedure (PT)

The PT procedure enables the collection and transformation of data required to compute model parameters (1)–(18). Some parameters—such as a set of machines (W), a set of additional resources (R), and a set of customer orders (O)—are retrieved directly from the database (Figure 3) without the need for their transformation. Others—such as a set of execution periods (T), a set of machine equipment variants (F), and a set of tasks to be performed (E)—require conversions and/or data transformations. These transformations and conversions yield direct values for model parameters in the pw_*, pd_*, and pt_* groups (Table 2). Then, using the product structure (MRP conversion), we define production tasks for each component and generate a graph representing their execution (Figure 4) using the PT procedure (Appendix B, Figure A2).

Example of a generation graph: an active order for product A triggers demand for components B, C, D, and E (product structure). Regardless of the production route, producing A requires three tasks (

Table A5. Production routes (Table allocation–Figure 3).

Item (i)	Route (f)	Operation (o)	Workstation (w)	Worker (r)	tpi	ti	tzi
A	1	1	W1	R1	10 (const)	1	10 (const)
A	2	1	W2	R1	10 (const)	1	10 (const)
F	1	1	W1	R1	10 (const)	1	10 (const)
F	2	1	W3	R1	10 (const)	1	10 (const)
C	1	1	W1	R3	10 (const)	1	10 (const)
C	2	1	W2	R4	10 (const)	1	10 (const)
G	1	1	W1	R1	10 (const)	1	10 (const)
G	2	1	W2	R3	10 (const)	1	10 (const)
G	3	1	W1	R3	10 (const)	1	10 (const)
B	1	1	W4	R5	10 (const)
D	1	1	W4	R5	10 (const)
E	1	1	W4	R5	10 (const)
H	1	1	W4	R5	10 (const)

Note—const means that the time does not depend on the batch size.

), resulting in tasks E1, E2, and E3—performed in sequence: preparation, production, and completion. Producing component B requires one operation, represented by task E4, which must be completed before task E3 can begin, and so on.

At this stage, we define machine configuration variants based on the feasible technological execution of individual tasks on machines and the additional resources required. These configurations serve as the foundation for generating dynamic routing during model solving. Figure 5 presents examples of machines/workstation equipment configurations.

5.2. Procedure for Reducing the Size of the Modeled Problem (PR)

The PR procedure, illustrated by the pseudocode of which is shown in Figure 6, applies two approaches to data representation: one used in relational databases (Figure 3), and the other in mathematical programming (matrix model). In the matrix model, each element, whether a variable or a parameter, is written in its appropriate position based on matrix indices (dimensions). Even if an element equals 0, the model still records it in the corresponding cell. This not only significantly increases the matrix size but also leads to the generation of sparse matrices. In contrast, relational models store only the rows corresponding to admissible combinations of key attribute values (model indices). If a particular combination is inadmissible, the row simply does not exist. Based on this fundamental difference in data representation formats, we developed the PR procedure using both matrix and relational models, allowing for a significant reduction in size by leveraging the data of the modeled problem. For the model described in Section 4, the PR procedure proceeds as follows. Using the data from the MRP II/ERP system tables (Figure 3), it retrieves all tasks (e) and machine (w) indices, which are the key values of the tasks and machines tables. Then, for each combination of these indices, it checks whether a corresponding row exists in the allocation_tasks table. If no row is found for a given (e,w) pair, the procedure removes the associated decision variables X, Y, and Z, and deletes constraints (1), (2), (11), and (16) as they are indexed by e,w. For example, if no matching row is found for e = 2, w = 3, and e = 5 and w = 3 in the allocation_tasks table, the model removes the variables X_u,f,2,3,t, X_u,f,5,3,t, Y_u,_f,2,3, Y_u,f,5,3, Z_u,f,2,3,t, Z_u,f,5,3,t, along with all corresponding constraints (1), (2), (11), (16) concerning these index pairs. The absence of such indicates that the corresponding tasks cannot be allocated to the selected machines. The same logic applies to the index pairs (e,u) and (r,u).

6. Computational Experiments

We evaluated the proposed proactive scheduling and routing approach and conducted experiments using real-world data from a kitchen furniture manufacturing facility. Experiments were conducted in several stages (Section 6.1, Section 6.2 and Section 6.3) on a computer with the following specifications: Intel(R) Core(TM) i5-2430M CPU @ 2.40GHz, 8.0 GB, 64-bit operating system, x64 processor (Windows 10).

6.1. Initial Case Study—Order Fulfillment and Baseline Comparison

In the first phase, we analyzed two customer orders: one for 100 units of product A and another for 150 units of product F. The primary objective was to verify feasibility (Q1): Can the system fulfill customer orders with the available resources?

We compared the following two methods:

• Baseline Approach—using standard MRP II/ERP with manual scheduling.
• Proposed MILP Approach—using the model (1)–(18) with integrated PT/PR procedures.

As shown in Table 5, the MILP-based method yielded the following results:

• A cost reduction of approximately 15% (from 6100 to 5600 units).
• A shorter execution time by eight periods (23% reduction).

Table 5. Results of the first and second stage experiments.

Q	Constraints	Objective Function	Cost_1	Cost_2	Cou_1
Q1	MRPII/ERP	---	3800	3900	80
Q1	1–12	----	2840	2760	72
Q2	1–14	Min (Cost_1 + Cost_2)	2310	2435	64
Q2A	1.12,15–18	Min(Cost_1 + Cost_2), Cou_1 ≤ 54	2480	2435	50
Q3	1.12,15–18	Min Cou_1	2480	2435	48
Q3A	1.12,15–18	MinCou_1, Cost_1 + Cost_2 ≤ 4800	2310	2435	55
Q4A	1.12,15–18 W2 unavailable	Min (Cost_1 + Cost_2)	2840	2675	73
Q4B	1.12,15–18 W2 unavailable	Min Cou_1	3010	2760	55
Q4C	1.12,15–18 R2 unavailable	Min Cou_1	2310	2435	55
Q4D	1.12,15–18 R1 unavailable	Min (Cost_1 + Cost_2)	---	---	---
Q4E	1.12,15–18 R2 unavailable	Min (Cost_1 + Cost_2)	2840	2675	73

Table 5. Results of the first and second stage experiments.

Q	Constraints	Objective Function	Cost_1	Cost_2	Cou_1
Q1	MRPII/ERP	---	3800	3900	80
Q1	1–12	----	2840	2760	72
Q2	1–14	Min (Cost_1 + Cost_2)	2310	2435	64
Q2A	1.12,15–18	Min(Cost_1 + Cost_2), Cou_1 ≤ 54	2480	2435	50
Q3	1.12,15–18	Min Cou_1	2480	2435	48
Q3A	1.12,15–18	MinCou_1, Cost_1 + Cost_2 ≤ 4800	2310	2435	55
Q4A	1.12,15–18 W2 unavailable	Min (Cost_1 + Cost_2)	2840	2675	73
Q4B	1.12,15–18 W2 unavailable	Min Cou_1	3010	2760	55
Q4C	1.12,15–18 R2 unavailable	Min Cou_1	2310	2435	55
Q4D	1.12,15–18 R1 unavailable	Min (Cost_1 + Cost_2)	---	---	---
Q4E	1.12,15–18 R2 unavailable	Min (Cost_1 + Cost_2)	2840	2675	73

This demonstrates that the model not only ensures feasibility but also improves schedule efficiency compared to traditional approaches.

6.2. Optimization Under Different Objectives (Q2–Q4)

In the second phase, we applied the model to extended scheduling scenarios aligned with questions Q2–Q4:

• Q2: Minimize total production cost (Cost_1 + Cost_2).
• Q3: Minimize the number of execution periods (Cou_1).
• Q4: Assess feasibility and optimize performance under resource unavailability.

Several scenario variants were introduced:

• Q2A: Minimize cost under a time constraint (Cou_1 ≤ 54).
• Q3A: Minimize time under a cost constraint (Cost_1 + Cost_2 ≤ 4800).
• Q4A: Minimize cost under the unavailability of resource W1.
• Q4B: Minimize time under the unavailability of resource W1.
• Q4C: Minimize time under the unavailability of additional resource R2.
• Q4D: Minimize cost under the unavailability of additional resource R1.
• Q4E: Minimize cost under the unavailability of additional resource R2.

Key insights (Table 5) include the following:

• Cost optimization (Q2) yielded the lowest cost (4745 units), while time optimization (Q3) shortened execution to 48 periods—nearly half the baseline duration.
• Q3A satisfied both constraints with only a modest cost increase, demonstrating the model’s flexibility.
• Q4D was the only infeasible case, revealing the unavailability of resource R1. This resource turned out to be critical for fulfilling product A, which requires R1 for the critical “polishing” step (

  Table A3. Workstations/machines parameters (Table machines–Figure 3).

  Workstation (w)	Production Capacity (Capa)	Cost of Work (Cost)
  W1	200	40
  W2	200	30
  W3	200	50
  W4	200	5

  ). Since no alternative resources or routing options exist (

  Table A6. Customer orders (Table orders–Figure 3).

  Item (i)	Planning Period (tm)	Customer Order Size (qu)
  A	4	100
  F	4	150

  ), the model correctly flagged infeasibility, highlighting its ability to identify critical resource dependencies.

These results confirm that the model can enhance the feasibility and cost-efficiency of MRP-driven systems, as well as improve responsiveness. Furthermore, they confirm the model’s ability to explore trade-offs between cost and time and its capacity to proactively support a “what-if” analysis for risk-aware planning.

6.3. Scalability and Performance Evaluation

In the final phase, we tested the computational performance of the model on larger datasets by varying the number of orders and the planning horizon. We recorded:

• Time to first feasible solution (T-FIRST);
• Time to optimal solution (T-OPTIM).

This was for both the full MILP model and the MILP model enhanced with the PR model-size reduction procedure (see

Table 6. Results of the last stage of experiments (performance experiments).

Q	O	No.	T-FIRST (MILP)	T-OPTIM (MILP)	T-FIRST (MILP + PR)	T-OPTIM (MILP + PR)
Q2	2	85	1.2	4.06	0.30	1.77
Q3	2	85	15.87	46.67	2.30	28.93
Q2	5	85	34.67	67.56	4.56	15.24
Q3	5	85	234.67	765.56	8.56	43.24
Q2	10	85	74.26	349.45	5.63	62.63
Q3	10	85	1587.34	3287.11	67.34	456.34
Q2	20	150	142.34	1245.76	12.45	234.47
Q3	20	150	403,426	>5000.00	76.45	1884.45
Q2	50	150	434.21	7834.56	15.34	345.67
Q3	50	150	6372.45	>7500.00	143.45	2186.56

O—number of orders, No.—number of execution periods, T-FIRST—time to obtain the first feasible solution, T-OPTIM—time to obtain the optimal solution. MILP—a mathematical programming solver was used to solve the problem, MILP + PR—a mathematical programming solver with a size reduction PR procedure was used for the solution.

).

Observations:

• PR consistently accelerated solution times—by more than 20× in some cases.
• For instance, when solving Q3 with 50 orders over 150 time periods:

  ○

  Without PR: T-OPTIM exceeded 7500 s.

  ○

  With PR: Reduced to ~2187 s.

• For moderate-sized instances (10–20 orders), PR reduced solution time from hours to minutes without sacrificing solution quality.

These findings show that the PR algorithm significantly improves scalability, making the model computationally feasible for real-world-sized problems.

7. Conclusions

This study presents an innovative, mathematically grounded approach that addresses critical limitations of traditional MRP/ERP systems in handling proactive production scheduling and routing under resource constraints. By formulating and implementing a mixed-integer linear programming (MILP) model enriched with dynamic routing capabilities, partial resource utilization, and traceability of demand sources, the proposed method significantly enhances the responsiveness and efficiency of MRP-based production environments. The model assumes discrete, deterministic production conditions and is less suited for continuous processes (e.g., chemical plants) without modifications to account for flow dynamics.

The integrated PT and PR procedures automate data transformation from MRP/ERP systems and optimize model size, enabling real-time and computationally efficient decision-making. Computational experiments based on real-world production data confirm substantial performance gains: cost reductions of up to 25% and execution time reductions exceeding 50%, depending on the optimization objective. The model also demonstrates high adaptability to operational disruptions such as resource unavailability or equipment failure.

In contrast to costly and rigid APS systems, the proposed solution provides a flexible, scalable, and economically viable alternative for manufacturers seeking to modernize production planning while maintaining compatibility with existing MRP infrastructures. Future research will extend the framework to stochastic environments [16], multi-facility coordination, and AI-based decision support—areas where the current deterministic, single-facility model requires adaptation [17,18].

In addition, we intend to apply the proposed framework in our other research domains, such as the VRP/CVRP [19] and timetabling problems [20]. We plan in the implementation area that the results will be visualized using Gantt charts (machine/task timelines) and network diagrams (dynamic routing paths). For what-if scenarios (Q1–Q4), dashboards highlight with colors which decisions are acceptable, unacceptable, or optimal.