Calculus of Variations, Maximum Principle, and Dynamic Programming for Supply Chain Optimal Control

Carlos E. López-Landeros (), Ricardo Valenzuela-González (), Elias Olivares-Benitez () and Sergio Frausto-Hernández ()
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Carlos E. López-Landeros: Universidad Panamericana
Ricardo Valenzuela-González: Universidad Panamericana
Elias Olivares-Benitez: Universidad Panamericana
Sergio Frausto-Hernández: Tecnológico Nacional de México/IT Aguascalientes

A chapter in Handbook of Visual, Experimental and Computational Mathematics, 2026, pp 845-875 from Springer

Abstract: Abstract This chapter explains how supply chain dynamic optimization is an optimal control problem and presents the general mathematical model. Then, it describes the calculus of variations, the maximum principle, and dynamic programming. The necessary and sufficient optimization conditions for each of them and the transversality conditions for the free horizon are deduced. Furthermore, the theory of stochastic processes is reviewed to formulate the stochastic optimal control problem and its optimality conditions through Itô’s lemma and dynamic programming. Finally, an illustrative production planning example in free horizon using the three optimal control tools is solved for both the deterministic and stochastic cases. It should be noted that this chapter does not constitute a comprehensive review of optimal control theory applied to supply chain issues. It focuses solely on their mathematical foundations and their direct application to a basic production-inventory problem. However, references to more advanced studies on the subject are provided for the interested reader.

Keywords: Calculus of variations; Maximum principle; Dynamic programming; Itô’s lemma; Stochastic optimal control (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/978-3-032-16368-4_83

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