Stochastic optimal control of an automotive supply chain using dynamic programming
Publisher
Carlos Eduardo López Landeros
Date Issued
2025
Author(s)
López Landeros, Carlos Eduardo
Advisor(s)
Valenzuela González, Ricardo
Olivares Benítez, Elías
Type
text::thesis::doctoral thesis
Abstract
La cadena de suministro automotriz es una de las más dinámicas y grandes a nivel
mundial. Ante los diversos desafíos de suministro, producción y transporte que enfrenta, los
socios de esta cadena necesitan coordinar sus operaciones para lograr eficiencia, robustez y
confiabilidad. Esta coordinación es posible en estado transitorio y en ambientes con
incertidumbre usando teoría de control. En tal virtud, en este trabajo se propuso un modelo
de control óptimo estocástico para una cadena de suministro automotriz multi-producto de
cinco eslabones operando bajo una estrategia push-pull, con el objetivo de minimizar el
costo total de operación en un horizonte de tiempo definido. El modelo consideró
restricciones de capacidad, operaciones de mezcla de partes, requerimientos de ensamblaje,
tasas variables de suministro y de producción-inventario. Se asumió que la demanda de
automóviles sigue una distribución normal, con costos unitarios de materias prima
fluctuando como movimiento geométrico Browniano y niveles de disponibilidad de piezas
variables comportándose como procesos de Itô. La dinámica se modeló como un balance de
entradas y salidas en los nodos del sistema mediante ecuaciones diferenciales ordinarias. La
solución del modelo propuesto se obtuvo con la adaptación y mejora de un algoritmo
numérico de control óptimo basado en Programación Dinámica con el cual se determinaron
los niveles óptimos de producción-inventario para cada socio de la cadena, impulsados por
las tasas de adquisición de materia prima entrante. Se compararon diferentes
configuraciones deterministas y estocásticas del modelo y los hallazgos proporcionaron
información útil para los gerentes en las áreas de producción, inventario y transporte. Estos
resultados son evidencia de los beneficios que representa modelar la cadena de suministro
desde una perspectiva de control óptimo estocástico para la operación coordinada de
sistemas industriales.
mundial. Ante los diversos desafíos de suministro, producción y transporte que enfrenta, los
socios de esta cadena necesitan coordinar sus operaciones para lograr eficiencia, robustez y
confiabilidad. Esta coordinación es posible en estado transitorio y en ambientes con
incertidumbre usando teoría de control. En tal virtud, en este trabajo se propuso un modelo
de control óptimo estocástico para una cadena de suministro automotriz multi-producto de
cinco eslabones operando bajo una estrategia push-pull, con el objetivo de minimizar el
costo total de operación en un horizonte de tiempo definido. El modelo consideró
restricciones de capacidad, operaciones de mezcla de partes, requerimientos de ensamblaje,
tasas variables de suministro y de producción-inventario. Se asumió que la demanda de
automóviles sigue una distribución normal, con costos unitarios de materias prima
fluctuando como movimiento geométrico Browniano y niveles de disponibilidad de piezas
variables comportándose como procesos de Itô. La dinámica se modeló como un balance de
entradas y salidas en los nodos del sistema mediante ecuaciones diferenciales ordinarias. La
solución del modelo propuesto se obtuvo con la adaptación y mejora de un algoritmo
numérico de control óptimo basado en Programación Dinámica con el cual se determinaron
los niveles óptimos de producción-inventario para cada socio de la cadena, impulsados por
las tasas de adquisición de materia prima entrante. Se compararon diferentes
configuraciones deterministas y estocásticas del modelo y los hallazgos proporcionaron
información útil para los gerentes en las áreas de producción, inventario y transporte. Estos
resultados son evidencia de los beneficios que representa modelar la cadena de suministro
desde una perspectiva de control óptimo estocástico para la operación coordinada de
sistemas industriales.
License
Acceso Abierto
How to cite
López Landeros, C. E. (2025). Stochastic optimal control of an automotive supply chain using dynamic programming. (Tesis doctoral). Universidad Panamericana.
Table of contents
Table of Contents
LIST OF FIGURES...............................................................................................................................................III
LIST OF TABLES.................................................................................................................................................IV
LIST OF ALGORITHMS ....................................................................................................................................IV
LIST OF ACRONYMS...........................................................................................................................................V
MATHEMATICAL NOMENCLATURE...........................................................................................................VI
A. General..................................................................................................................................................VI
B. Model....................................................................................................................................................VII
C. IDP Algorithm...................................................................................................................................VIII
ABSTRACT ...........................................................................................................................................................IX
CHAPTER 1 INTRODUCTION ................................................................................................................... 1
1.1. Study Problem.....................................................................................................................................2
1.2. Justification .........................................................................................................................................4
1.3. Objectives............................................................................................................................................5
1.3.1. General ....................................................................................................................................................5
1.3.2. Specifics..................................................................................................................................................5
1.4. Outline .................................................................................................................................................6
CHAPTER 2 FUNDAMENTALS.................................................................................................................. 7
2.1. SC dynamic optimization as an optimal control problem...............................................................8
2.2. Optimal control methods ...................................................................................................................9
2.2.1. Maximum Principle.................................................................................................................................9
2.2.2. Dynamic Programming .........................................................................................................................10
2.3. Stochastic processes theory..............................................................................................................12
2.3.1. Itô processes..........................................................................................................................................12
2.3.2. Solution of stochastic differential equations .........................................................................................14
2.4. The stochastic optimal control problem .........................................................................................14
2.4.1. General formulation ..............................................................................................................................15
2.4.2. Optimality conditions............................................................................................................................15
CHAPTER 3 LITERATURE REVIEW....................................................................................................... 17
3.1. Early dynamic modeling in production and logistics ....................................................................18
3.2. Automotive SCs stochastic and dynamic modeling .......................................................................18
3.2.1. Optimization-based approach................................................................................................................19
3.2.2. Simulation-based approach ...................................................................................................................20
3.3. Optimal control in general SCs .......................................................................................................21
3.4. Research gaps....................................................................................................................................26
CHAPTER 4 PROBLEM FORMULATION ................................................................................................. 28
4.1. Statement...........................................................................................................................................29
4.1.1. The automotive supply chain network ..................................................................................................29
4.1.2. Assumptions..........................................................................................................................................30
4.2. Dynamic modeling ............................................................................................................................31
4.3. Uncertainty modeling .......................................................................................................................32
4.3.1. Car demand ...........................................................................................................................................32
4.3.2. Unit raw material costs..........................................................................................................................32
4.3.3. Transportation capacities.......................................................................................................................33
4.4. The stochastic optimal control model .............................................................................................34
4.4.1. Objective function .................................................................................................................................35
4.4.2. Dynamic constraints..............................................................................................................................35
4.4.3. Path constraints, control bounds, and end-time conditions ...................................................................36
CHAPTER 5 METHODOLOGY................................................................................................................ 37
5.1. Numerical Dynamic Programming .................................................................................................38
5.1.1. Iterative Dynamic Programming ...........................................................................................................39
5.1.2. vIDP computational implementation.....................................................................................................40
5.1.3. Refinement of the vIDP algorithm solution ..........................................................................................42
5.2. The uncertainty characterization....................................................................................................43
5.2.1. Normal car demand ...............................................................................................................................43
5.2.2. GBm unit raw material costs.................................................................................................................44
5.2.3. Generalized Itô processes for parts availability levels ..........................................................................45
5.3. vIDP-AR for the stochastic optimal control problem....................................................................45
5.4. Computational experiments.............................................................................................................46
5.4.1. Deterministic approach..........................................................................................................................46
5.4.2. Stochastic approach...............................................................................................................................47
CHAPTER 6 RESULTS............................................................................................................................ 48
6.1. General case of study........................................................................................................................49
6.2. Deterministic approach....................................................................................................................50
6.2.1. Configuration 1: equal raw material procurement rates........................................................................51
6.2.2. Configuration 2: non-equal raw material procurement rates.................................................................52
6.2.3. Comparison of configurations...............................................................................................................53
6.2.4. Sensitivity analysis................................................................................................................................54
6.3. Stochastic approach..........................................................................................................................55
6.3.1. GBm unit raw material costs under normal demand .............................................................................55
6.3.2. Availability of raw material as generalized Itô processes.....................................................................61
6.3.3. Threefold stochastic configuration........................................................................................................64
6.3.4. Comparison of stochastic configurations ..............................................................................................67
6.4. Main findings ....................................................................................................................................68
6.5. Managerial insights ..........................................................................................................................71
CONCLUSIONS....................................................................................................................................... 72
APPENDIX A.......................................................................................................................................... 75
Journal papers ...............................................................................................................................................75
Copyrights......................................................................................................................................................76
Congresses......................................................................................................................................................77
APPENDIX B.......................................................................................................................................... 79
Automatic refinement subroutine ................................................................................................................79
vIDP-AR for the ASC threefold configuration ...........................................................................................81
REFERENCES....................................................................................................................................... 104
LIST OF FIGURES...............................................................................................................................................III
LIST OF TABLES.................................................................................................................................................IV
LIST OF ALGORITHMS ....................................................................................................................................IV
LIST OF ACRONYMS...........................................................................................................................................V
MATHEMATICAL NOMENCLATURE...........................................................................................................VI
A. General..................................................................................................................................................VI
B. Model....................................................................................................................................................VII
C. IDP Algorithm...................................................................................................................................VIII
ABSTRACT ...........................................................................................................................................................IX
CHAPTER 1 INTRODUCTION ................................................................................................................... 1
1.1. Study Problem.....................................................................................................................................2
1.2. Justification .........................................................................................................................................4
1.3. Objectives............................................................................................................................................5
1.3.1. General ....................................................................................................................................................5
1.3.2. Specifics..................................................................................................................................................5
1.4. Outline .................................................................................................................................................6
CHAPTER 2 FUNDAMENTALS.................................................................................................................. 7
2.1. SC dynamic optimization as an optimal control problem...............................................................8
2.2. Optimal control methods ...................................................................................................................9
2.2.1. Maximum Principle.................................................................................................................................9
2.2.2. Dynamic Programming .........................................................................................................................10
2.3. Stochastic processes theory..............................................................................................................12
2.3.1. Itô processes..........................................................................................................................................12
2.3.2. Solution of stochastic differential equations .........................................................................................14
2.4. The stochastic optimal control problem .........................................................................................14
2.4.1. General formulation ..............................................................................................................................15
2.4.2. Optimality conditions............................................................................................................................15
CHAPTER 3 LITERATURE REVIEW....................................................................................................... 17
3.1. Early dynamic modeling in production and logistics ....................................................................18
3.2. Automotive SCs stochastic and dynamic modeling .......................................................................18
3.2.1. Optimization-based approach................................................................................................................19
3.2.2. Simulation-based approach ...................................................................................................................20
3.3. Optimal control in general SCs .......................................................................................................21
3.4. Research gaps....................................................................................................................................26
CHAPTER 4 PROBLEM FORMULATION ................................................................................................. 28
4.1. Statement...........................................................................................................................................29
4.1.1. The automotive supply chain network ..................................................................................................29
4.1.2. Assumptions..........................................................................................................................................30
4.2. Dynamic modeling ............................................................................................................................31
4.3. Uncertainty modeling .......................................................................................................................32
4.3.1. Car demand ...........................................................................................................................................32
4.3.2. Unit raw material costs..........................................................................................................................32
4.3.3. Transportation capacities.......................................................................................................................33
4.4. The stochastic optimal control model .............................................................................................34
4.4.1. Objective function .................................................................................................................................35
4.4.2. Dynamic constraints..............................................................................................................................35
4.4.3. Path constraints, control bounds, and end-time conditions ...................................................................36
CHAPTER 5 METHODOLOGY................................................................................................................ 37
5.1. Numerical Dynamic Programming .................................................................................................38
5.1.1. Iterative Dynamic Programming ...........................................................................................................39
5.1.2. vIDP computational implementation.....................................................................................................40
5.1.3. Refinement of the vIDP algorithm solution ..........................................................................................42
5.2. The uncertainty characterization....................................................................................................43
5.2.1. Normal car demand ...............................................................................................................................43
5.2.2. GBm unit raw material costs.................................................................................................................44
5.2.3. Generalized Itô processes for parts availability levels ..........................................................................45
5.3. vIDP-AR for the stochastic optimal control problem....................................................................45
5.4. Computational experiments.............................................................................................................46
5.4.1. Deterministic approach..........................................................................................................................46
5.4.2. Stochastic approach...............................................................................................................................47
CHAPTER 6 RESULTS............................................................................................................................ 48
6.1. General case of study........................................................................................................................49
6.2. Deterministic approach....................................................................................................................50
6.2.1. Configuration 1: equal raw material procurement rates........................................................................51
6.2.2. Configuration 2: non-equal raw material procurement rates.................................................................52
6.2.3. Comparison of configurations...............................................................................................................53
6.2.4. Sensitivity analysis................................................................................................................................54
6.3. Stochastic approach..........................................................................................................................55
6.3.1. GBm unit raw material costs under normal demand .............................................................................55
6.3.2. Availability of raw material as generalized Itô processes.....................................................................61
6.3.3. Threefold stochastic configuration........................................................................................................64
6.3.4. Comparison of stochastic configurations ..............................................................................................67
6.4. Main findings ....................................................................................................................................68
6.5. Managerial insights ..........................................................................................................................71
CONCLUSIONS....................................................................................................................................... 72
APPENDIX A.......................................................................................................................................... 75
Journal papers ...............................................................................................................................................75
Copyrights......................................................................................................................................................76
Congresses......................................................................................................................................................77
APPENDIX B.......................................................................................................................................... 79
Automatic refinement subroutine ................................................................................................................79
vIDP-AR for the ASC threefold configuration ...........................................................................................81
REFERENCES....................................................................................................................................... 104