This research communication aims to present a curricular proposal to relate the importance on the application of optimal control theory for dynamic supply chains, for industrial engineering students and practitioners. This by considering the role of model based optimal control theory in management sciences, specifically for dynamic supply chains (DSC) at operational level with emphasis in high volume production systems.In general, control theory relates the core idea to maintain equilibrium and stability state with uncertainty and disturbance (Wiener, 1948). Moreover, a classical control system has its roots in the feedback systems concept. A general goal in control systems is to minimize the level of error in the system under some performance criteria. Mathematical modeling and analysis of control systems present the following taxonomy: Model-based control and data-driven control. In general, model-based control has the following sub-classification: Optimal control, robust control, adaptive control, and intelligent control. Our aim, in this research communication, is to present the importance of optimal control theory, which are analyzed via the Pontryagin maximum principle, and evaluate its impact to improve the curricular proposal for DSC for industrial engineering practitioners.In the past, the interest to investigate the dynamic behavior of management control system, has been growing (Connors, 1967). The mathematical description of dynamical system is often in terms of difference or differential equations. A general theory is present that permits us to determine the optimal control of such a system according to some suitable performance criteria. Necessary and sufficient conditions of optimality are derived for a class of optimal control problems. Our interest is to present the conventional optimal control problems, for linear time-invariant dynamical systems in the context of industrial engineering education. Applications of optimal control in management sciences and operations research are (1) pricing; (2) scheduling; (3) logistics networks; (4) optimal transfer of technology, and (5) optimal remanufacturing and recycling in closed-loop supply chains, among others.Therefore, decision-makers in management sciences and operations research use optimal control theory to map dynamic optimization methods towards optimization methods based on the nature of the complexity of the DSC system. In the following section an optimal control theory for DSC is addressed, considering the role of mathematical modeling, calculus of variations and the optimal control theory applied to DSC.Nowadays, supply chain management (SCM) refers to the cooperation process management of materials and information flows between supply chain partners (Sucky, 2005). From here, the principal components, in generic, manufacturing supply chains networks are suppliers, manufacturers, customers, logistics and distribution networks, and retailing centers (Kamble, et al 2020). As a general definition: "A supply chain is a network of suppliers that produce goods, both, for one another and for generic customers" (Daganzo, 2003).Dynamic supply chains are important to minimize inventory, to allow the flow of raw material, cash, information, labor force and energy, while maximize profits along the entire operation (Taboada et al, 2022). By this DSC operation requires decision-making with synchronization. Mathematical modeling for optimal control theory, from engineering education perspective justify proper skills in ordinary differential equations (ODE) which mainly are based for linear systems of first and second order in the DSC context (Zill, 2016), this considering that the roots of the applicability of OCT is the state-space formulation, for the description of the evolution for the DSC system, by considering the definition of inputs, states and outputs for the dynamical system mathematical description. However, DSC can be model via partial differential equations (Davizon, et al 2023) this type of mathematical modeling is out of scope of our research communication.Besides, while the main goal along the DSC is to regulate the inventory levels, which in nature, are present all along the DSC, while inventory control is a crucial activity by a company's management (Bieniek, 2019). To present appropriate inventory levels is crucial task for an enterprise (Duan and Ventura, 2019), considering that fast customers response is related to high inventory levels (which increase the cost), while low inventory levels might cause scarcity.Calculus of variations (CV) as its roots in the seventeenth century from classical optimization problems such as the brachistochrone, it was formalized by Euler and Lagrange, which developed the Euler-Lagrange equation. The problem of calculus of variations emerges from mathematical analysis of functions (Komzsik, 2009). In general, CV is a mathematical discipline related to determine the function, curve, or surface that optimizes a given functional-typically expressed as an integral dependent on a function with proper derivatives. In practice, CV expand classical calculus by finding entire functions, paths, or shapes that minimize or maximize a given physical or performance criterion.Furthermore, connects mathematical modeling with physical intuition, preparing future engineers to approach interdisciplinary problems with a rigorous and systematic methodology. . In essence, it offers a unified framework for understanding how natural and artificial systems evolve toward optimal configurations.In general, optimal control theory requires a mathematical model to apply the Pontryagin Maximum principle techniques. An optimal control is defined as an admissible control, which minimizes a functional objective. Based on this, given a dynamical system with initial condition x0, which evolves in time according to the state space equation □□̇= □□(□□, □□, □□), to find an admissible control and make the functional objective to achieve its maximum. A general mathematical form for an optimal control problem is:min □□ □□(□□) = 1 2 ∫ □□(□□, □□, □□) □□□□ □□□□ □□□□ + □□□□(□□ □□ ) s.t.(1)□□̇= □□(□□, □□, □□) □□(□□ 0 ) = □□ 0 , □□ ∈ □□, □□ ∈ □□ Therefore, optimal control methods requires: 1) A performance index; 2) Dynamical systems to optimize; 3) Constraints in states, inputs, and outputs.Consider the following optimal control problem for a linear system.min □□ □□(□□) = ∫ 1 □□□□ □□□□ □□□□ s.t.(2)□□̇= □□□□(□□) + □□□□(□□) □□(0) = □□ 0 , □□(□□) = □□ □□ (fixed)This is called, a minimum time optimal control problem based on the context that the objective is to transfer the state x from a fixed initial point xi to a fixed final point xf in a minimum time, which implies a time-optimal way. In supply chain systems, minimum-time optimal control improves coordinated, and efficient transitions between inputs, states and outputs, from production setups to logistics reconfiguration.The minimum-fuel optimal control problem enables sustainable, energy-efficient, and low-emission supply chains by optimizing transportation, production, and network operations to balance economic and environmental tradeoffs, such as positive and negative externalities in the minimum fuel optimal control approach.The general form of the minimum fuel performance index is:min □□ □□(□□) = ∫ |□□| □□ 0 □□□□ s.t.(3)□□̇= □□□□(□□) + □□□□(□□)□□(0) = □□ 0 , □□(□□) = □□ □□ (fixed) □□ ∈ □□ 2.The minimum-energy optimal control problem improves supply chain performance, by minimizing energy consumption across transportation, production, and logistics processes, promoting the equilibrium between economic efficiency and environmental goals. For a minimum energy optimal control problem, the performance index is:min □□ □□(□□) = 1 2 ∫ □□ 2 □□ 0 □□□□ s.t. (4)□□̇= □□□□(□□) + □□□□(□□) □□(0) = □□ 0 , □□(□□) = □□ □□ (fixed) □□ ∈ □□Optimal control theory justifies its application in DSC by enabling time-dependent decisions that optimize inventory, production, and distribution. Additionally, it supports practices with real-time, automated systems, making DSC more optimal, adaptive, and robust, from a systems theory perspective.In relation to optimal integrated ordering and production control in a supply chain for finite capacitated warehouses is analyzed in (Song, 2009). In (Kogan et al, 2010), address a differential game where the effect of information asymmetry is present under stochastic demand. Considering the dynamic nature of goods flows process, efficient inventory management in production-inventory systems is present in (Ignaciuk and Bartoszewicz, 2010). In (Ignaciuk and Bartoszewicz, 2012), a control theory approach, is present, for the problem of inventory control in systems with perishable goods. The applicability of optimal control theory for supply chain management is analyzed in (Ivanov et al, 2012), this is based on the fundamentals of control and systems theory. Also, the problem of supply chain coordination by a robust schedule coordination approach, applying optimal control theory is addressed in (Ivanov et al, 2016). In (Dolgui et al, 2018) a survey on optimal control applications to scheduling in production, supply chain, and Industry 4.0 systems via deterministic maximum principle is present. An optimal control model is present, for a class of low-carbon supply chain system is depicted in (Yu et al, 2020).In (Rarita, et al 2021) considers supply chains modeled by partial and ordinary differential equations.In industrial engineering, real-world problems such as: to allocate resources, schedule production, or manage inventories so that costs are minimized or revenue is maximized, involve decision-making in a certain time horizon. Optimal control theory develops mathematical modeling, optimization, and systems engineering, as an essential part of industrial engineering education. It trains future engineers to think dynamically, make informed decisions, skills that are mandatory in modern industry and automation-driven, at operational level, for high volume production systems. Within industrial engineering, these principles are applied to problems such as production scheduling, inventory management, and process automation. By integrating mathematical modeling, systems analysis, and decision-making, optimal control theory advance the development of critical competencies for designing and managing industrial systems, at operational level.Based on this, a suitable curricular proposal is mandatory to address the application of optimal control theory in dynamic supply chains, this by means of the context of feedback systems proper analogy applied in a Gantt diagram, refer to Optimal control theory plays an important role in industrial engineering education as it provides students with analytical and computational tools to model, optimize, and regulate DSC. By the principles of the calculus of variations, it provides a systematic platform for determining timedependent control strategies that achieve optimal performance under given constraints, as we previously presented.In dynamic supply chains, operations and production planning processes, enables decision-makers to develop convenient and suitable decisions, to validate their hypothesis about which decisions incorporate more profit while reducing costs on enterprise operations. Based on this, inventory management plays an important role in the DSC analysis. In future work, we will incorporate optimal control theory into industrial engineering curricula to enhance students, capacity to address real-world challenges through rigorous, optimization-based approaches. Engineering education faces difficulties due to misunderstandings that obstruct students to comprehend core scientific concepts (Guerra and Meneses, 2025). Moreover, optimal control for DSC application aligns with emerging paradigms such as Industry 4.0 and smart manufacturing, where adaptive and data-driven control strategies are of interests to explore in practical case studies.Finally, the analysis and optimization of DSC are made possible by the mathematical modeling, CV, and optimal control, all of which are critical components of engineering education. and successfully integrated mathematical theory with engineering practice for future industrial engineering practitioners, strengthening analytical thinking and the capacity to build effective systems and technologically sophisticated solutions for processes. Table 1.
Davizón et al. (Thu,) studied this question.