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Mathematical Modeling, Optimal Control And Stability Analysis Mathematical Modeling, Optimal Control And Stability Analysis

For Dynamic Supply Chains For Dynamic Supply Chains

Yasser Alberto Davizon University of Texas at El Paso

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MATHEMATICAL MODELING, OPTIMAL CONTROL AND STABILITY ANALYSIS FOR

DYNAMIC SUPPLY CHAINS

YASSER ALBERTO DAVIZON CASTILLO

Master’s Program in Manufacturing Engineering

APPROVED:

Heidi A. Taboada-Jiménez, Ph.D., Chair

José F. Espiritu-Nolasco, Ph.D.

Jaime Sánchez-Leal, Ph.D.

Vladik Kreinovich, Ph.D.

Héctor Erives-Contreras, Ph.D.

José Humberto Ablanedo Rosas, Ph.D.

Stephen L. Crites, Jr., Ph.D.

Dean of the Graduate School

Copyright ©

by

Yasser Alberto Davizon Castillo

2020

Dedication

To my parents, Alberto and Beatriz for them support and guidance during my life. Also, to my

brothers: Pável, Anuar, and Alberto for their company and sponsorship during the Master

process.

I would like to express my sincere gratitude to my advisor, Doctor Heidi Taboada, and my co-

advisor Doctor José F. Espíritu, the conceiver minds of this project. Their insights and

observations have made this research experience a continuous learning process.

I also want to express my appreciation to Doctor Jaime Sánchez-Leal, Doctor Eric D. Smith, and

Doctor Oswaldo Aguirre for their advice and patience.

MATHEMATICAL MODELING, OPTIMAL CONTROL AND STABILITY ANALYSIS FOR

DYNAMIC SUPPLY CHAINS

by

YASSER ALBERTO DAVIZON CASTILLO, PhD

THESIS

Presented to the Faculty of the Graduate School of

The University of Texas at El Paso

in Partial Fulfillment

of the Requirements

for the Degree of

MASTER OF SCIENCE

Department of Industrial, Manufacturing and Systems Engineering

THE UNIVERSITY OF TEXAS AT EL PASO

December 2020

v

Acknowledgements

The author would like to thank: Industrial, Manufacturing, and Systems Engineering

Department faculty members and staff, from University of Texas at El Paso for their advising and

support as teaching assistant during the MS studies. Also, the laboratory classmates: Ana K., Juan

F., and Eduardo C.

vi

Abstract

Dynamic supply chains (SC) are important to reduce inventory, to enable the flow of

materials, and maximize profits. By this, dynamic SC require proper decision-making with

effective performance and synchronization. In operations and production planning processes,

decision-makers 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 a crucial point in the supply chain analysis. In general, SC

processes raw material, cash, and information flows, taking into account the demand profile of the

system. This research work presents the mathematical modeling, optimal control, and stability

analysis for dynamic SC. Novel mathematical models are developed to incorporate the use of

compartmental analysis in the context of SC modeling, for forward and closed-loop SC. Optimal

control (OC) formulations are developed in the context of Pontryagin maximum principle, for

energy-based OC and present-value Hamiltonian OC, with proper stability analysis for each SC

mathematical model addressed.

vii

Table of Contents

Dedication ...................................................................................................................................... iii

Acknowledgements ..........................................................................................................................v

Abstract .......................................................................................................................................... vi

Table of Contents .......................................................................................................................... vii

List of Tables ...................................................................................................................................x

List of Figures ................................................................................................................................ xi

List of Illustrations ........................................................................................................................ xii

Chapter 1: Introduction to systems theory and control for supply chains .......................................1

1.1. SUPPLY CHAIN MANAGEMENT: DEFINITIONS ..................................................1

1.2. DECISION LEVEL MAKING IN SUPPLY CHAINS .................................................2

1.2.1 Strategic level........................................................................................................2

1.2.2 Tactical level .........................................................................................................3

1.2.3 Operational level ...................................................................................................3

1.3. MATHEMATICAL MODELING IN DYNAMIC SUPPLY CHAINS ........................4

1.4. MODEL-BASED CONTROL SYSTEMS: A GENERAL TAXONOMY ...................5

1.5. CONTROL ORIENTED APPROACHES FOR SUPPLY CHAINS ............................7

1.5.1 Model-based optimal control theory .....................................................................7

1.5.2 Minimum time-optimal control.............................................................................8

1.5.3 Minimum fuel optimal control ..............................................................................9

1.5.4 Minimum energy optimal control .........................................................................9

1.6. STABILITY ANALYSIS FOR SUPPLY CHAINS ...................................................10

Chapter 2: Literature review and problem definition ....................................................................12

2.1 BACKGROUND THEORY ...........................................................................................12

2.1.1 Possibility theory ................................................................................................12

2.1.2 Knowledge-based expert systems .......................................................................14

2.2 LITERATURE REVIEW FOR CONTROL THEORY IN SUPPLY CHAINS ............15

2.2.1 Model-based optimal control ..............................................................................15

2.2.2 Robust control .....................................................................................................20

viii

2.2.3 Control approaches based on computational intelligence ...................................22

2.3 KNOWLEDGE-BASED EXPERT SYSTEM FORMULATION .................................23

2.4 PROBLEM DEFINITION ..............................................................................................25

Chapter 3: Optimal control for capacity-inventory management in serial supply chains ..............26

3.1 INTRODUCTION ..........................................................................................................26

3.2 MATHEMATICAL MODELING ..................................................................................27

3.2.1 The mixing problem and notation .......................................................................27

3.2.2 System dynamics ................................................................................................29

3.2.3 System dynamics with lead time.........................................................................30

3.2.4 Sensitivity analysis..............................................................................................31

3.2.5 Stability analysis .................................................................................................32

3.3 RESULTS .......................................................................................................................33

3.3.1 Optimal control ...................................................................................................33

3.3.2 Case study simulations ........................................................................................34

Chapter 4: Mathematical modeling and stability analysis of closed-loop supply chains ..............37

4.1 INTRODUCTION ..........................................................................................................37

4.2 MATHEMATICAL MODELING ..................................................................................40

4.2.1 Problem description and notation: Compartmental analysis ..............................40

4.2.2 CLSC mathematical modeling: System 1 ...........................................................42

4.2.3 Four-tier CLSC mathematical modeling: System 2 ............................................43

4.3 STABILITY ANALYSIS ...............................................................................................48

4.3.1 System 1: Stability analysis ................................................................................48

4.3.2 System 2: Stability analysis ................................................................................49

Chapter 5: Present value Hamiltonian-Optimal control for a production-inventory system .........50

5.1 INTRODUCTION ..........................................................................................................51

5.2 MATHEMATICAL MODELING ..................................................................................53

5.2.1 Optimality, robustness, and stability ...................................................................53

5.2.2 Systems theory and control for PI .......................................................................54

5.2.3 Systems modeling and notation: Equations of motion .......................................56

5.2.4 Linearization .......................................................................................................59

5.3 STABILITY ANALYSIS ...............................................................................................60

5.4 OPTIMAL CONTROL ...................................................................................................61

ix

5.4.1 Optimal control for PI system .............................................................................61

5.4.2 Present value Hamiltonian-Optimal control problem .........................................62

5.5 RESULTS .......................................................................................................................63

5.5.1 Case study ...........................................................................................................63

5.5.2 Simulations .........................................................................................................64

Chapter 6: Conclusions and future work .......................................................................................68

6.1 CONCLUSIONS.............................................................................................................68

6.2 FUTURE WORK ............................................................................................................68

References ......................................................................................................................................70

Glossary .........................................................................................................................................99

Appendix ......................................................................................................................................100

Vita 101

x

List of Tables

Table 1.1: Stability for supply chains ........................................................................................... 11 Table 2.1: Levels of applications conducted with PT ................................................................... 13 Table 2.2: Levels of applications conducted with KBES ............................................................. 14 Table 2.3: Classical control literature review from 2015 to 2021 ................................................ 16 Table 2.4: Optimal control literature review from 2015 to 2021 .................................................. 17

Table 2.5: Model Predictive Control literature review from 2015 to 2021 .................................. 20 Table 2.6: Robust control literature review from 2015 to 2021 ................................................... 21 Table 2.7: Control based computational intelligence literature review from 2015 to 2021 ......... 22 Table 4.1: CLSC quantitative decision-making approaches ......................................................... 38 Table 4.2: Areas of application for compartmental analysis ........................................................ 39

Table 5.1: Systems theory and control feature for PI systems ...................................................... 55

Table 5.2: System modeling parameters ....................................................................................... 58 Table 5.3: System modeling quantities ......................................................................................... 64

xi

List of Figures

Figure 1.1: Four echelons supply chain .......................................................................................... 1 Figure 1.2: Feedback system classical control ................................................................................ 5 Figure 1.3: Model-based control systems ....................................................................................... 6 Figure 3.1: Generic four echelon serial SC realization ................................................................. 29 Figure 3.2: Factory and Distributors inventory level .................................................................... 34

Figure 3.3: Wholesalers and Retailers inventory level ................................................................. 35 Figure 3.4: Demand rate ............................................................................................................... 35 Figure 4.1: Tank level representation for the compartmental analysis problem ........................... 40 Figure 4.2: System dynamics representation for the capacitated control problem ....................... 41 Figure 4.3: Two echelon representation for the capacitated control problem .............................. 41

Figure 4.4: General structure for the CLSC network .................................................................... 42

Figure 4.5: Forward CLSC dynamics ........................................................................................... 45 Figure 4.6: Reverse CLSC dynamics ............................................................................................ 46

Figure 5.1: Third-order mixed production inventory system ........................................................ 57

Figure 5.2: Inventory level for PI system ..................................................................................... 65 Figure 5.3: Inventory rate for PI system ....................................................................................... 65 Figure 5.4: Production level for PI system ................................................................................... 66

Figure 5.5: Demand level for PI system ....................................................................................... 67

xii

List of Illustrations

Illustration 5.1: Optimality, robustness, and stability for PI systems. .......................................... 54

1

Chapter 1: Introduction to systems theory and control for supply chains

This chapter presents a systems theory and control, introductory perspective, for supply

chains. In the first section, supply chain definitions are defined, considering the forward material

flows and closed-loop supply chains. At second stage, the strategic, tactical, and operational

decision-making along the supply chain are present. Mathematical modeling, control, and stability

analysis are developed in the supply chain context, are analyzed in the rest of the chapter.

1.1. SUPPLY CHAIN MANAGEMENT: DEFINITIONS

Supply chain management (SCM) refers to the cooperation process management of

materials and information flows between supply chain partners (Sucky, 2005). In a general, the

principal components of supply chains are suppliers, manufacturers, customers, logistics and

distribution networks, and retailing centers (Kamble, et al 2020). See Figure 1.1, for a four

echelons supply chain.

Figure 1.1: Four echelons supply chain

Typically, three flows are considered in supply chain dynamics: Raw material, cash and

information. Supply chains are complex dynamic systems set off by customer demands

(Marufuzzaman and Deif, 2010). As a formal definition: “A supply chain is a network of suppliers

that produce goods, both, for one another and for generic customers” (Daganzo, 2003).

2

Based on this context, this research work intends to present the mathematical modeling of

dynamic supply chains, from a model-based control theory and stability analysis context. Our

interest is to analyze the supply chains from systems and processes perspectives.

1.2. DECISION LEVEL MAKING IN SUPPLY CHAINS

There are three decision level making in the supply chain, such as strategical, tactical and

operational.

1.2.1 Strategic level

In supply chain the strategic level impacts the long term planning. In general, a taxonomy

for supply chains presents 1) Forward material flows and 2) Closed-loop supply chain (CLSC). In

general, strategic planning involves the selection of suppliers and distributors, location, and

capacity planning of manufacturing and servicing units (Chauhan et al, 2004). In (Das, 2011) a

strategic global supply chain (GSC) operates for multi-layered, multi-location-based suppliers, and

manufacturing plants by the integration quality based supply management. In (Seuring, 2009)

presents five strategic decision levels in supply chain management such as products and services,

partners and partnerships, plants and stocks, processes and planning and control. A CLSC network

consists of various conflicting decisions of forward and reverse facilities, in (Özceylan and Paksoy,

2014) the CLSC proposed model integrates the strategic and tactical decisions to avoid separated

design in both chain networks. In (Bhattacharyya, 2017) an optimization model is present for

CLSC network with a multi-echelon inventory, multi-period planning, and multi-product scenario.

In the case of green supply chains, in (Dey et al, 2018) strategic inventory on the development-

intensive and marginal-cost intensive green product types are present, while in (Kuiti, 2019)

manufacturer-retailer channel analyzes strategic decisions in a green supply chain.

3

1.2.2 Tactical level

Tactical decision level, in supply chains, presents a mid-term planning cycle. This level

determines a preliminary plan of the regular operations (Pereira, et al, 2020), usually made at an

aggregate level, it is more related demand, inventory, and supply planning. In (Ivanov, 2010)

presents planning decisions as an adaptive process to be interrelated at all the decision-making

levels, in the supply chain. In (Tian, et al 2011) the problems of tactical safety stock placement

and tactical production planning are analyzed via an iterative approach. Considering that is based

on physical parameters such as stock level, demand satisfaction, etc., in (Comelli et al, 2008)

proposes an approach to evaluate tactical production planning in supply chains, considering

financial evaluation. In (Fahimnia, et al 2014) present a tactical planning model that integrates

economic and carbon emission for a green supply chain.

1.2.3 Operational level

In supply chains, at operational level, the planning horizon is a short term (between days

and weeks). In (Schutz and Tomasgard, 2011), volume, delivery, and operational flexibility

effects, in supply chain planning under uncertain demand, are discussed. To pursue an appropriate

performance at the operational level, the supply chain design must develop on this operational

level performance (Mansoornejad, et al, 2013). The decision, at operational planning level, covers

two main areas (Monthatipkul and Kawtummacha, 2007): inventory area (related with inventory

replenishment order to satisfy demand) and transportation area (physical movement of goods

between different geographical points). In (Zhang, et al, 2020), presents operational decisions and

financing strategies in a CLSC by a manufacturer and a retailer.

4

1.3. MATHEMATICAL MODELING IN DYNAMIC SUPPLY CHAINS

Inventory management (IM) presents an important role in the operations and management

sciences. In this research work, I present a systemic analysis for IM for a class of manufacturing

systems, applying systems theory and model-based control approaches. The main goal is to explore

the dynamic nature of production systems via the development of ordinary differential equations

(linear and nonlinear), also a data-driven control approaches via system identification is addressed.

Proper mathematical models and simulations are conducted. By definition, inventory management

is part of the SCM that plans, implements, and controls the forward and reverse flow of goods and

services (as well as storage), between the point of origin and the point of consumption to meet

customer requirements. (Singh and Verma, 2017).

Inventories, in nature, are present all along the supply chain (SC) and inventory control is

a crucial activity by a company’s management (Bieniek, 2019). To present appropriate inventory

levels is crucial task for a company (Duan and Ventura, 2019), considering that quick and positive

response to customers is related to high inventory levels (which increase the cost), while low

inventory levels might cause scarcity. In general, for manufacturing firms, inventory usually

represents from 20% to 60% of the total assets. (Giannoccaro, 2003). Coordination of inventory

policies in IM for SCM between suppliers, manufactures, and distributors is a major task. As well,

to smooth material flow and minimize costs while reaches customer demand. (Giannoccaro and

Pontrandolfo, 2002).

The main considerations for IM policies along the SCM are (Ryu, et al, 2013) (1) type of

optimization (local or global); (2) control type (centralized or distributed); (3) inventory levels

nature (periodic, continuous or hybrid); (4) demand function type (stochastic or deterministic) and

(5) inventory control responsibility (self-managed or vendor-managed).

5

1.4. MODEL-BASED CONTROL SYSTEMS: A GENERAL TAXONOMY

Control theory relates the core idea to maintain equilibrium and stability state with

uncertainty and disturbance (Wiener, 1948). In general, a classical control system has its roots in

the feedback systems concept, see Figure 1.2. A general goal in control systems is to minimize the

level of error in the system, in Figure 1.2, the reference signal r(t), error signal e(t), manipulated

variable signal u(t), controlled variable yt), and output sensor signal y*(t), are the basis of a general

feedback system classical control.

Figure 1.2: Feedback system classical control

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 work, is to develop optimal controls, which are analyzed via the Pontryagin

maximum principle. Figure 1.3, presents a conceptual map for systems and control theory

approaches.

6

Figure 1.3: Model-based control systems

From Figure 1.3, a general taxonomy is present for model-based control systems, which

incorporates: Optimal control, robust control, intelligent control and adaptive control. The most

common model-based control systems strategies, nowadays are control structures which have been

applied in oil, aerospace, automotive and manufacturing industries, in general.

7

1.5. CONTROL ORIENTED APPROACHES FOR SUPPLY CHAINS

Decision-makers are facing with greater amounts of data and business decisions. There is

a growing interest in using real-time information to improve, support, and validate business

decision-making. (Sourirajan, et al 2008). Today’s enterprises work towards flexible, reliable, and

responsive business operations. Therefore, they need to implement systematic decision-making

processes. (Perea, et al 2000).

In production and logistics systems, SC and Industry 4.0 networks: Uncertainty, feedback

cycles, and system dynamics are mandatory goals (Dolgui, et al, 2018). Since 1960, control theory

(CT) has gained attention in the modeling and analysis for operational production and logistics

systems. (Ivanov, et al 2018). Considering the dynamic nature of production-inventory systems,

CT is a suitable approach to handle time-varying phenomena. (Ortega and Lin, 2004).

A supply chain is a network of facilities and distribution entities such as: suppliers,

manufacturers, distributors, retailers. (Sarimveis, et al, 2008). A supply chain is characterized by

three flows: materials and cash (forward flow) and information (backward flow).

1.5.1 Model-based optimal control theory

Nowadays, 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 criterion.

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. Applications of optimal control in management sciences and

operations research are (1) pricing; (2) scheduling; (3) logistics networks; (4) optimal transfer of

8

technology, and (5) optimal remanufacturing and recycling in closed-loop supply chains, among

others. Decision-makers in management sciences and operations research use optimal control

theory to map optimal control methods towards optimization methods based on the nature of the

complexity of the manufacturing system.

An optimal control (OC) 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 �� = 𝑓(𝑥, 𝑢, 𝑡), to find an admissible control and make the

functional objective to achieve its maximum.

A general mathematical form for an optimal control problem is:

min𝑢 𝐽(𝑢) =

1

2∫ 𝐹(𝑥, 𝑢, 𝑡)𝑡𝑓

𝑡𝑜

𝑑𝑡 + 𝑆[𝑥(𝑡𝑓)]

s.t.

�� = 𝑓(𝑥, 𝑢, 𝑡) 𝑥(𝑡0) = 𝑥0, 𝑥 ∈ 𝑋, 𝑢 ∈ 𝑈

In general, optimal control methods require:

1) A performance index

2) Dynamical systems to optimize

3) Constraints in states, inputs, and outputs

1.5.2 Minimum time-optimal control

Consider the following optimal control problem for a linear system.

9

min𝑢 𝐽(𝑢) =∫ 1

𝑡𝑓

𝑡𝑜

𝑑𝑡

s.t.

�� = 𝐴𝑥(𝑡) + 𝐵𝑢(𝑡)

𝑥(0) = 𝑥0, 𝑥(𝑇) = 𝑥𝑓 (fixed)

𝑢 ∈ 𝑈

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.

1.5.3 Minimum fuel optimal control

The general form of the minimum fuel performance index is:

min𝑢 𝐽(𝑢) =∫ |𝑢|

𝑇

0

𝑑𝑡

s.t.

�� = 𝐴𝑥(𝑡) + 𝐵𝑢(𝑡)

𝑢 ∈ 𝑈

1.5.4 Minimum energy optimal control

For a minimum energy optimal control problem, the performance index is:

min𝑢 𝐽(𝑢) =

1

2∫ 𝑢2𝑇

0

𝑑𝑡

s.t.

�� = 𝐴𝑥(𝑡) + 𝐵𝑢(𝑡)

𝑢 ∈ 𝑈

10

1.6. STABILITY ANALYSIS FOR SUPPLY CHAINS

A supply chain as a dynamical system is subject to proper stability analysis using the

following main characteristics: 1) Routh-Hurwitz stability analysis, and 2) Lyapunov direct and

indirect methods. In general, the Routh-Hurwitz stability analysis is present for single-input single-

output (SISO) dynamical systems in continuous linear time domain. While Lyapunov direct and

indirect methods provide stability results more in general for nonlinear systems.

Our aim in this section is to present, the most recent works related to stability analysis for

supply chains. A continuous time version beer game model and its stability and robust stability

properties are investigated in (Riddalls and Bennett, 2002). In (Nagatani and Helbing, 2004)

presents conceivable production strategies to stabilize supply chains, controlling the production

speed with dependence to the stock levels. A continuous time Automatic Pipeline, Inventory and

Order Based Production Control System (APIOBPCS) investigation for stability boundary via

Padé approximation and the Routh Hurwitz criteria, is present in (Warburton et al, 2004). In

(Rabelo et al, 2008) presents a methodology for manufacturing supply chains, to predict changes

due to endogenous and/or exogenous influences in the short and long-term horizons. A supply

chain can perform sudden and intense change in demand, in (Cannella and Ciancimino, 2010)

presents an analysis for collaboration and smoothing replenishment rules on supply chain using

differential equations. In (Sipahi and Delice, 2010) stability of inventory dynamics controlled by

the APIOBPCS is, with delays present in; lead time, transportation, and decision-making.

Based on this, Table 1.1 summarizes research works for stability for supply chains from

2011 to 2020.

11

Table 1.1: Stability for supply chains

Paper Contribution Stability approach

Kastsian,and

Monnigmann,

2011

This paper addresses the steady-state

optimization of a supply chain for the vendor

managed inventory, automatic pipeline,

inventory, and order based production control

systems (VMI-APIOBPCS).

Robust stability

Makuschewitz

et al,

2011

The problem to investigate input to state

stability in a proper mathematical modeling

approach for a supply chain network is

present.

Input to state stability

Wang et al,

2012

This paper analyses the stability of a

constrained production and inventory system

with a non-negative order rate via a piecewise

linear model, an eigenvalue analysis.

Lyapunov stability

Palsule-Desai

et al,

2013

A non-cooperative game-theoretic is present

to analyze the network stability in a two-tier

supply chain.

Network stability

Luo et al,

2015

A supply chain coalition stability is analyzed

considering that the retailer ally with each

other without restrains.

Coalition stability

Klug,

2016

In this paper, bullwhip and backlash effects in

supply chains are analyzed. A number of

dynamic and transient behaviours are present,

such as supply chain stability, compactness,

and bullwhip/backlash trade-offs.

Phase space

trajectory

12

Chapter 2: Literature review and problem definition

In operations and production planning processes, decision-makers require to achieve fast

and suitable decisions to validate their hypothesis about which decision incorporates more

profitable and cost down impacts in the enterprise operations. Based on this, inventory

management plays a crucial point in the supply chain analysis. Each supply chain processes raw

material, cash, and information flow, taking into account a demand forecast of the system. This

chapter presents a KBES to validate the literature review for control of production- inventory

systems. Based on, the rapid grow process to apply control and systems theory approaches for

supply chains from 2000 to 2018. A KBES is proposed considering the mathematical approach of

possibility theory (PT). Control and systems theory to explore the dynamic nature of the supply

chains. In this literature review, our goal is to analyze the evolution of control theory in the analysis

of inventory management for supply chains in a time horizon of almost twenty years.

In section 2.1, a KBES and PT-based background theory to validate the literature review

is present; in section 2.2, a literature review is conducted for control theory and supply chain

analysis is conducted. In section 2.3, the KBES methodology is presented. Finally, proper problem

definition is addressed in section 2.4.

2.1 BACKGROUND THEORY

In this section a background theory is developed for PT and KBES, taking into account the

potential opportunities as tools which enable the decision-making process.

2.1.1 Possibility theory

Decision making involves the handling of the information available about uncertainty in a

decision. (Yager, 1979). In some situations, the information on uncertainty does not have the

character of a probability. Possibility theory (PT) is a mathematical theory for dealing with certain

13

types of uncertainty. PT is an uncertainty theory to work with incomplete information. (Agarwal

and Nayal, 2015). The importance of possibility theory is that much of the information on which

human decisions are based is possibilistic, in nature, rather than probabilistic (Zadeh, 1999). PT

expresses the facility in which an event can occur, or belong to a set (Khoury, et al. 2008).

Table 2.1: Levels of applications conducted with PT

Paper Contribution Level of application

Cayrac, et al,

1996

This paper proposes allowing a

representation of the available knowledge,

through the introduction of an appropriate

representation of uncertainty and

incompleteness based on Zadeh’s

possibility theory and fuzzy sets.

Satellite fault diagnosis

Mauris, et al,

2000

This paper proposes to build a fuzzy model

of data acquired from physical sensors by

applying a truncated triangular probability-

possibility transformation.

Physical sensors

Denguir-Rekik, et

al., 2006

This paper analyzes some aspects of

uncertainty evaluations in multicriteria

decision-making in the framework of e-

commerce website recommendation.

E-commerce

website choice support

Mauris, 2007 This paper explores the possibility

expression of measurement uncertainty

applied to situations where only very

limited knowledge is available.

Measurement uncertainty

in a very limited

knowledge context

Dewalle-Vignion,

et al 2011

This paper presents a designed and

evaluated a new, nearly automatic, and

operator-independent segmentation

approach. This incorporating possibility

theory, to take into account the uncertainty

and inaccuracy inherent in the image.

Medical imaging

Guo, 2011 This paper focuses on one-shot decision

problems, which concern the situations

where a decision is experienced only once,

which are commonly encountered in

business, economics, and social systems.

Decision theory

Romero, et al,

2011

In this study a methodology for harmonic

load-flow calculation based on the

possibility theory is presented. Possibility

distributions instead of probabilities are

the input used to describe the uncertainty

in the magnitude and composition of the

loads.

Power systems

Li and Ma, 2015 This research proposes the generalized

possibilistic computation tree logic model

checking in this, which is an extension of

possibilistic computation logic model

checking.

Possibilistic computation

tree logic

14

Alsahwa, et al,

2016

This paper proposes an approach referred

as the iterative refinement of possibility

distributions by learning for pixel-based

image classification, which is based on the

use of possibilistic reasoning concepts

exploiting expert knowledge sources.

Image processing

Ren, et al., 2016 This paper, considering the insufficient

uncertainty data, proposes a possibility-

based optimal design algorithm to get a

robust and reliable optimal design of

electromagnetic devices.

Design of electromagnetic

devices

Mei, 2019 This paper intends to formulate fuzzy

control in the framework of possibility

theory by regarding the fuzzification

procedure of fuzzy control.

Fuzzy control

2.1.2 Knowledge-based expert systems

In terms of knowledge transfer, knowledge-based systems (KBS) are categorized as expert

systems and knowledge-sharing systems (Niwa, 1990). A KBES is a computer program which use

a knowledge base to solve complex problems in a particular field and achieves some reasoning

(López-Santana and Méndez-Giraldo, 2016). KBES uses human knowledge to solve problems

that commonly require human intelligence (Tripathi, 2011).

Table 2.2: Levels of applications conducted with KBES

Paper Contribution Level of application

Kusiak, 1988 In this paper a knowledge-based system

for solving the generalized group

technology which involves constraints

related to machine capacity, material

handling system capabilities, machine

cell dimensions and technological

requirements.

Cellular manufacturing

Willis, et al, 1989 This paper presents the development of

a process planning system using

knowledge-based techniques with the

derivation of component geometric

information from a three-dimensional

solid modeler.

Manufacturing process

planning

Parlar, 1989 This paper develops a knowledge-based

expert system which identify and

recommend up to 30 production-

inventory models.

Production-inventory

systems

Hsu and Chen, 1990 This research paper presents a

knowledge-based expert system for

distribution system planning.

Power systems

(distribution planning)

15

Ram and Ram, 1996 This paper provides a comprehensive

approach to design, implement, and

validate an expert system for marketing

applications such as new product

screening.

Marketing (screening

product innovations)

Mitra and Basu,

1997

This paper presents a knowledge-based

system for designing microprocessor

based systems.

Microprocessor

computer-aided design

Leo Kumar, 2019 In this paper, presents an intense

review on KBES applications in

manufacturing planning.

Manufacturing planning

2.2 LITERATURE REVIEW FOR CONTROL THEORY IN SUPPLY CHAINS

This section presents a literature review for control approaches in supply chains from 2000

to 2021, considering the control-oriented characteristics such as:

1) Model-based optimal control

2) Robust control

3) Control approaches based on computational intelligence

To provide the current state of the art in control theory in supply chains, this review

considers 70 journal papers.

2.2.1 Model-based optimal control

Supply chains present a dynamical behavior, which from model-based optimal control

perspective, the following approaches are present:

1) Classical control theory

2) Optimal control theory

3) Model predictive control

Classical control theory:

In the literature, classical control theory is used to control supply chains. In (Perea, et al

2000), present a dynamic supply chain management approach based on a dynamic framework to

model supply chains. An infinite horizon two-echelon supply chain inventory problem is present

16

in (Hosoda and Disney, 2006) and shows a sequence of the optimum ordering policies. In

(Schwaninger and Vrhovec, 2006) address a distributed control via the regulation of supply

systems from production to the customer and vice versa, applying system dynamics modeling and

simulation. The use of proportional-integral and proportional-derivative control theoretic

principles to manage the inventory replenishment process in a supply chain is present in

(Sourirajan et al, 2008). In (Yuan and Ashayeri, 2009), proposes system dynamics loops and

control theory simulations to analyze the impacts on capacity expansion strategies within a supply

chain. The Integral of the Time Absolute Error (ITAE), is present as, an appropriate control

engineering measure of resilience for inventory levels and shipment rates in (Spiegler et al, 2012).

In (Janamanchi and Burns, 2013) presents control theory concepts to develop a performance

functional that optimizes a retail supply chain, via system dynamics modeling methodology.

Table 2.3, presents a review for research papers between 2015 to 2021 with the classical

control theory developed.

Table 2.3: Classical control literature review from 2015 to 2021

Paper Contribution Control structure

Spiegler et al,

2015

A dynamic analysis of supply chain

resilience is present by the use of nonlinear

control theory.

Nonlinear control

Qing et al,

2016

This research work presents a dynamic

performance and optimization of a discrete

production control system subject supply

disruption and demand uncertainty.

Classical control

APIOBPCS

Lin et al,

2017

By the application of control engineering

approach, a feedforward forecasting in the

make to order element plays a major role

in the degree of the bullwhip effect, and

the customer order decoupling point

impacts both the bullwhip effect and the

inventory variance in the make to stock.

Classical control

IOBPCS

Zhao and Wang.

2018

In this research work, a multiproduct three-

echelon inventory control system is

analyzed for the hybrid supply chain.

Classical control

17

Optimal control theory:

In general, optimal control theory requires a mathematical model to apply Pontryagin

Maximum principle techniques. 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.

Table 2.4, presents a review for research papers between 2015 to 2021 with the optimal

control theory developed.

Table 2.4: Optimal control literature review from 2015 to 2021

Paper Contribution Control structure

Ivanov et al,

2016

This paper addresses the problem of supply

chain coordination by a robust schedule

coordination approach, applying optimal

control theory.

Optimal control

Ivanov et al,

2018

This research work extends a previous

survey in the Annual Reviews in Control

(Ivanov et al. 2012) by identifying two

new directions of control theory

applications (ripple effect analysis in the

supply chains and scheduling in Industry

4.0) and the digital technology use in

control-theoretic models.

Optimal control

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.

Optimal control

18

Zu, et al ,

2019

This paper presents a method to study the

Stackelberg differential game between a

manufacturer and a supplier.

Differential game

Wu,

2019

This paper presents a differential game for

a single manufacturer and single retailer in

a supply chain under the consignment

contract.

Optimal control

Wu and Chen,

2019

The cooperative advertising problem is

addressed, applying optimal control theory

for a supply chain under the consignment

contract in a competitive environment.

Optimal control

Yu, et al,

2020

An optimal control model is present, for a

class of low-carbon supply chain system.

A linear differential equation describes the

dynamics of emission reduction level.

Optimal control

Rarita, et al,

2021

This paper considers supply chains

modeled by partial and ordinary

differential equations.

Optimal control

Model predictive control:

In recent years, the use and application of Model Predictive Control (MPC) as an advanced

control structure has gained interest in the processes industries. In the context of supply chains, in

(Perea-López et al, 2003) presents a model predictive control strategy, for multiproduct, multi-

echelon distribution networks, to find the optimal decision variables to maximize profit in supply

chains. The presentation of the supply chain problem into a formulation to MPC application is

analyzed for a single-product, the two-node example in (Braun et al, 2003). In (Seferlis and

Giannelos, 2004), is present a two-layered optimization-based control structure for multi-product,

multi-echelon supply chain networks. Decision policies for inventory management in supply

chains are present in (Schwartz et al, 2006), via internal model control (IMC) and model predictive

control. In (Dunbar, 2007), addresses the problem of distributed control of dynamically coupled

nonlinear systems, such as supply chain systems, that are subject to decoupled constraints. The

implementation of model predictive control, to semiconductor manufacturing supply chain

management, is discussed in (Wang, et al, 2007). In (Wang and Rivera, 2008), a model predictive

19

control algorithm is addressed for tactical decision-making on semiconductor manufacturing

supply chains. Semiconductor manufacturing requires simulation modeling of discrete processes

combined with control policies such as: Discrete Event System Specification and model predictive

control, which are developed in (Huang et al, 2009). In (Li and Marlin, 2009), presents a robust

Model Predictive Control approach for supply chain optimization under uncertainties. Inventory

control for multi-item, multi-echelon distribution management in supply chains via model

predictive control is present in (Alessandri, et al, 2011). In (Subramanian, et al, 2013), proposes

the application of distributed model predictive control for inventory management in supply chain

optimization. A hybrid model predictive control formulation for inventory management in supply

chains subject to limited capacity is present in (Nandola and Rivera, 2013). In (Subramanian, et

al, 2014), an economic model predictive control with the closed-loop structure for inventory

management in supply chain optimization is discussed. A robust MPC optimization-based

decision-making for supply chain management is present in (Mastragostino et al, 2014). In (Fu, et

al, 2014), a four-echelon supply chain.is present for centralized and decentralized model predictive

control strategies, to control inventory, and to reduce the bullwhip effect. A perspective for demand

modeling in a production-inventory system is discussed in (Schwartz and Rivera, 2014), which

forecasts inventory management policies based on IMC or MPC.

Table 2.5, presents a review for research papers between 2015 to 2021 with the model

predictive control developed.

20

Table 2.5: Model Predictive Control literature review from 2015 to 2021

Paper Contribution Control structure

Fu, et al,

2016

This paper presents an inventory control

problem for a class of supply chain, which

is solved using the extended prediction

self-adaptive control approach to MPC.

MPC

Pinho, et al,

2017

This research work proposes a multilayer

model predictive control strategy to

improve the performance of the biomass

supply chain

operational level.

MPC

Dias and

Ierapetritou,

2017

This paper presents an integrated decision-

making strategies and review recent

advances from process control to supply

chain management.

Process control

Hsiao, et al,

2018

This work proposes an MPC framework

for a non-stationary semiconductor

manufacturing supply chain network from

the integrated device manufacturer

perspective.

MPC

Fu, et al,

2019

This paper presents a distributed model

predictive control approach, with minimal

information exchange and communication,

for supply chain operations and

management.

Distributed MPC

Fu, et al,

2019

This paper presents a distributed model

predictive control scheme for the inventory

management problem in supply chain

networks.

Distributed MPC

Hipólito, et al,

2020

A centralized model predictive control of

perishable goods supply chains for

logistics management is present.

MPC

2.2.2 Robust control

Considering that supply chains are dynamical systems subject to variability and

uncertainty. Several approaches have been addressed in the context of robust control such as:

1) H∞ robust control

2) Sliding mode control

3) Adaptive control

21

In (Huang et al, 2007) develops models for multi-echelon supply chain systems applying

H∞ control methods, while in (Krishnamurthy et al, 2008) consider an inventory control technique

for large-scale supply chains via adaptive control. H∞ control technique is addressed for a class of

supply chain model linearized about nominal operating conditions in (Boccadoro et al, 2008).

Considering that closed-loop supply chains (CLSC) present novel topologies, in (Huang et al,

2009) robust H∞ control strategies constrain all uncertainties for this CLSC. Another robust

control technique is sliding mode control (SMC) which in (Ignaciuk and Bartoszewicz, 2010)

SMC for inventory policy is proposed, which assurance that the demand is satisfied. Considering

that SC systems are modeled as networked systems in (Luo and Yang, 2014) a H∞ control is

present for a discrete SC.

Table 2.6, presents a review for research papers between 2015 to 2021 with the robust

control structure developed.

Table 2.6: Robust control literature review from 2015 to 2021

Paper Contribution Control structure

Ignaciuk, 2015

This paper presents the problem of

establishing a robust inventory

management policy for production-

inventory systems with time delays.

Robust control

LQ suboptimal

Li et al, 2018

This research work addresses a two-time-

scale production-inventory system as a

switching system, subjected to a

Markovian process.

H∞ control

Lesniewski and

Bartoszewicz,

2020

This article presents the case of a logistic

system with a single warehouse, in which a

model reference sliding mode controller is

developed for a perishable inventory

system.

Sliding mode control

Xu et al, 2020

This paper presents a sliding mode control

structure to manage chaotic supply chain

system this control synthesis with

dynamical analysis is useful for strategic

decision-makers in supply chain

management.

Adaptive sliding

mode

control

22

2.2.3 Control approaches based on computational intelligence

Control techniques based on computational intelligence has been developed for supply

chains, such as:

1) Fuzzy control

2) Neural networks control

3) Chaos control

4) Petri nets

In (Anne et al, 2009) investigates three-echelon supply chain dynamics subjected to

uncertainties which can exhibit saturation and chaos. Hybrid Petri nets are present in (Dotoli et al,

2009) where it proposes a modular model to describe material, financial, and information flow of

supply chains. In (Hong et al, 2010) an on-line neural network controller optimize a three-stage

supply chain is present.

Table 2.7, presents a review for research papers between 2015 to 2021 with control

techniques based on computational intelligence.

Table 2.7: Control based computational intelligence literature review from 2015 to 2021

Paper Contribution Control structure

Göksu, et al, 2015

This paper presents a mathematical

model for the synchronization and

control of a chaotic supply chain

management system.

Chaos control

Zhang and Zhao,

2015

This research work presents a dynamic

discrete switched dual-channel closed-

loop supply chain with a time delay in

remanufacturing.

Fuzzy control

Kocamaz,

et al.

2016

This paper addresses the control and

synchronization of chaotic supply chain

compensated with Artificial Neural

Network

based controllers.

Artificial

Neural Network

Zhang, et al, 2017

A Takagi-Sugeno fuzzy robust control

technique for the nonlinear supply chain

system subject to lead times is present.

Fuzzy control

Zhang, et al, 2017

This paper presents a closed-loop

supply chain subject to uncertainties,

Fuzzy control

23

with a fuzzy control model which

restrain the bullwhip effect.

Pavlov, et al, 2018

This research work is based on a hybrid

fuzzy-probabilistic approach in which

the supply chain resilience assessment

is extended by incorporating ripple

effect and structure reconfiguration.

Fuzzy systems

Zhang, et al, 2018

This paper presents a fuzzy robust

strategy for the robust operation of the

supply chain network subject to

production lead times under the

uncertain customer demand.

Fuzzy control

Zhang and Zhang,

2020

This paper addresses a fuzzy robust

control for a class of closed-loop supply

chain with lead times, to mitigate the

bullwhip effect.

Fuzzy control

Yan, et al 2021

This research work presents a

mathematical model for a supply chain

with computer-aided digital

manufacturing process for control and

synchronization.

Chaos control

2.3 KNOWLEDGE-BASED EXPERT SYSTEM FORMULATION

In this section a KBES is present as a rule-based system, from the development of nine

questions survey in a Likert scale, considering the level of application of optimal control, robust

control, and control-based computational intelligence.

The proposed methodology has been adapted from (Espino-Román, et al 2020), and

summarized such as:

1) Survey design

To measure the level of application of optimal control in SC the following questions are

part of the survey:

Q1: Considering the dynamic nature of supply chains, do you consider suitable the use via

model-based optimal control theory in the decision-making process of SC?

24

Q2: Do you consider useful the use of Pontryagin Maximum Principle in the design via

optimal controls for SC?

Q3: How often do you apply Model Predictive Control techniques in the operation of SC?

For robust control:

Q4: Assuming the measure of uncertainty, do you consider convenient apply robust control

techniques in the design of SC?

Q5: Do you consider important the design of H∞ robust controller for SC?

Q6: Sliding Mode Control, as a robust control technique, how often do you it in the design

of SC controllers?

For computer-based computational intelligence:

Q7: Do you consider convenient the development of fuzzy control techniques for SC?

Q8: In the context that SC requires data manipulation, how convenient do you consider the

application of Neural networks in SC?

Q9: Assuming SC networks are complex and dynamic nature, do you consider suitable the

application of chaos control for SC?

2) Calculate the level of uncertainty (LU) for each survey question such as:

𝐿𝑈 = ∑ 𝜋𝑖5𝑖=1 (𝑓𝑖)

𝑖 (1)

Where πi= possiblity level and fi= frequency per question.

3) Calculate the certainty factor (CF) for each question

𝐶𝐹 = 1 − 𝐿𝑈 (2)

4) Develop and calculate the CF for each category

𝑅𝑞 = 1 −∏ 𝐿𝑈𝑗𝑚𝑗=1 (3)

25

5) Construct the production rules for the AND/OR decision tree

6) For the categories: Optimal control (OC), Robust control (RC) and Computational

intelligence (CI), define the tree with higher CF.

2.4 PROBLEM DEFINITION

Considering the dynamic nature of supply chains, and the importance of inventory

management optimization in the SC operations. To the best of our knowledge, the following SC

problems are mandatory to address, in the context of:

1) Mathematical modeling: To present mathematical models using differential

equations, which in this context the application of compartmental analysis (which

has been applied in engineering and economics), becomes a novel mathematical

approach for SC inventory optimization.

2) Control oriented approaches: To develop control-oriented approaches which enable

to optimized the SC operation by the description of differential equations applying

optimal control theory via Pontryagin´s maximum principle.

3) Stability analysis: To analyze the stability of the SC by the application of proper

linear and nonlinear systems approaches, considering Lyapunov stability

approaches for the proposed mathematical models.

26

Chapter 3: Optimal control for capacity-inventory management in serial supply chains

This chapter presents capacity-inventory management modeling via system dynamics for

a serial supply chain (SC) applying model-based optimal control techniques. For mathematical

modeling purposes, a set of coupled first-order ordinary differential equations, with an analogy

from the mixing problem is addressed to relate capacity and inventory levels, taking into account

a production rate in each node of interaction. The mathematical model is present in a linear time-

invariant (LTI) state-space formulation. Stability analysis for the dynamic serial SC is present,

and also a sensitivity analysis is conducted for the capacity and production rate parameters, and

considers the effects of variations in parameters the SC. An energy-based optimal control is

developed with proper simulations.

3.1 INTRODUCTION

Traditional manufacturing converts raw material into end products, within a demand

profile. However, these manufacturing systems face problems such as long downtime, labor

inefficiency, not accurate scheduling, and weak energy consumption (Umble, 1992). Recently, SC

market forecasts in profitability are increasing. Based on this, nowadays SC has gained special

attention, within industry practitioners of automation science and control systems. Applications of

SC in industry integrates potential solutions such as Automotive, aerospace, biotechnology, etc.

Considering that manufacturing systems are dynamic and complex (Eyers, 2018), proper design

of control systems requires flexibility and should be adaptive to process disturbances. In this

context, new trends are required to present from systems theory and control theory perspective to

support accurate, agile, and flexible solutions for SC towards applications. Industrial production

systems require competitive and adaptive solutions, to ensure profitability in markets.

In manufacturing systems, decision-makers require to achieve and approximate, online

decisions. A manufacturing SC is a system composed of suppliers, manufacturers, distributors,

27

and customers, which serve customer requirements (Orji and Liu, 2018). SC integrates solutions

to decrease operations management costs such as production, logistics, and quality. In this context,

dynamic approaches for production-inventory systems are valuable to maximize profits and

minimize costs over the SC system. Recently, in SC the focus moves from the factory level

management to enterprise level (Akyuz and Erkan, 2018).

Inventory management presents a crucial role through the performance analysis of a supply

chain. By definition, inventory management is “the continuing process of planning, organizing and

controlling inventory while balancing supply and demand” (Singha and Verma, 2018). In this

research work, our main interest is to provide proper modeling, analysis, and model-based optimal

control for a serial SC, from a systems theory perspective. Designing an inventory management

system is directly related to sales, finance, production, and procurement (De Vries, 2007).

Moreover, the goal of inventory management is to minimize the average cost per unit of time

experienced for the inventory system, in the long run (Fiestras-Janeiro, et al, 2011). Considering

that decision-variables are related to inventory levels the SC, and fixed parameters are production

rate and capacity levels, our interest is to present the mixing problem from which approximate an

analogy for a four-echelon supply chain.

The rest of this chapter presents the mixing problem background theory in section 2. A

mathematical modeling description of the problem, sensitivity, and stability analysis for the system

are present in section 3; in section 4, a model-based optimal control for a composed energy

performance index is develop, finally, conclusions and future work are developed in Section 5.

3.2 MATHEMATICAL MODELING

3.2.1 The mixing problem and notation

In systems science, by definition, a system is a group of components that interacts towards

keeping a collection of relationships with the sum of the systems themselves to other entities

(Badillo, et al, 2011). A mathematical model is a mathematical description of the behavior of some

28

real-life systems. Such as physical, sociological, economics, etc. (Zill, 2017). Production and

supply chain dynamic modeling, present diverse mathematical approaches at distinct scales (Herty

and Ringhofer, 2011). Therefore, here we are interested to model the SC as mixing problems

(MPs), which is addressed with the interest to present an analogy for the capacity-inventory

management, taking into account as decision variables (the inventory level), with production rate

and capacity levels as constant parameters (which are provided), and considering the conservation

of mass from the mixing problem.

MPs, also known as “compartment analysis” in chemistry consists in creating a mixture of

two or more substances to determine some concentration of the resulting mixture (Martinez-

Luaces, 2018). MPs mathematical modeling has been used in science and engineering applications

such as: Biological systems (Bassingthwaighte, et al, 2012) and biomedical engineering

(Rahimian, et al, 2018). The MPs incorporate the rate change in concentration for a solute, which

follows the following first-order differential equation:

𝑑𝑆

𝑑𝑡= 𝑅𝑖𝑛 − 𝑅𝑜𝑢𝑡 (1)

Where: S(t) denotes the amount of solute in the tank at time t, Rin corresponds to solute

input rate and Rout is the output rate of solute.

In general, Equation 1 is expressed as:

𝑑𝑀

𝑑𝑡= 𝜙𝑖 −𝜙𝑜

𝑀

𝑉 (2)

For inventory management, a first order differential equation describes the dynamics:

𝑑𝐼

𝑑𝑡= 𝐷𝑟 − 𝜙𝑜

𝐼

𝐶 (3)

29

Where: Dr corresponds to the demand rate, ϕo (production rate); I (inventory level), C

(Capacity). In terms of inventory management, to approximate a proper analogy for the MPs

system, for two nodes interaction, the first-order differential equation is:

𝑑𝐼𝑗

𝑑𝑡= 𝜙𝑖

𝐼𝑖

𝐶𝑖− 𝜙𝑗

𝐼𝑗

𝐶𝑗 (4)

3.2.2 System dynamics

In this section, to propose an LTI system with the general form:

�� = 𝐴𝑥 + 𝐵𝑢 (5)

𝑦 = 𝐶𝑥 + 𝐷𝑢 (6)

For the set of first-order differential equations, which have been mentioned in Section 2. A

general structure of the four echelons serial SC, which is integrated by factory (F), distributors (D),

wholesalers (W), and retailers (R), is present in Figure 3.1.

Figure 3.1: Generic four echelon serial SC realization

The MPs conditions for the SC dynamics presents two cases: (1) when the inflow is equal

to outflow (φi = φo), and (2) when inflow is bigger than outflow (φi > φo).

Case 1: φi = φo In case 1, in which there is a balance between the production rate at each

node, which implies φi = φo. In general, a single-input single-output (SISO) mathematical model

for each node of the form:

𝑑𝐼0

𝑑𝑡= 𝜑𝑖 − 𝜑𝑜

𝐼𝑜

𝐶𝑜 (7)

30

Expanding Equation (7), for each echelon of the four nodes SC we have:

𝑑𝐼1

𝑑𝑡= 𝜑𝑖 − 𝜑𝑎

𝐼1

𝐶1 (8)

𝑑𝐼2

𝑑𝑡= 𝜑1

𝐼1

𝐶1− 𝜑2

𝐼2

𝐶2 (9)

𝑑𝐼3

𝑑𝑡= 𝜑2

𝐼2

𝐶2− 𝜑3

𝐼3

𝐶3 (10)

𝑑𝐼4

𝑑𝑡= 𝜑3

𝐼3

𝐶3− 𝜑4

𝐼4

𝐶4 (11)

Case 2: φi > φo. When production rates at each node follows the condition φi > φo, capacity

control presents the following relation:

𝑅𝑜𝑢𝑡 = 𝑞(𝑡)𝑟𝑜𝑢𝑡 (12)

Considering that:

𝑞(𝑡) =𝐼

𝐶𝑜+(𝜑𝑖−𝜑𝑜)𝑡 (13)

Analyzing each node, with condition of φi > φo, we have:

𝑑𝐼1

𝑑𝑡= 𝜑𝑖 − 𝜑𝑎

𝐼1

𝐶1+(𝜑𝑖−𝜑1)𝑡 (14)

𝑑𝐼2

𝑑𝑡= 𝜑1

𝐼1

𝐶1−

𝐼2

𝐶2+(𝜑1−𝜑2)𝑡 (15)

𝑑𝐼3

𝑑𝑡= 𝜑2

𝐼2

𝐶2−

𝐼3

𝐶3+(𝜑2−𝜑3)𝑡 (16)

𝑑𝐼4

𝑑𝑡= 𝜑3

𝐼3

𝐶3−

𝐼4

𝐶4+(𝜑3−𝜑4)𝑡 (17)

3.2.3 System dynamics with lead time

To incorporate, the lead time, for each echelon of the four nodes SC the set of coupled

ordinary differential equations are:

𝑑𝐼1

𝑑𝑡= 𝜑𝑖 −

𝜑1

𝐶1𝐼1(𝑡 − 𝜃1) (18)

𝑑𝐼2

𝑑𝑡= 𝜑1

𝐼1

𝐶1−𝜑2

𝐶2𝐼2(𝑡 − 𝜃2) (19)

𝑑𝐼3

𝑑𝑡= 𝜑2

𝐼2

𝐶2−𝜑3

𝐶3𝐼3(𝑡 − 𝜃3) (20)

𝑑𝐼4

𝑑𝑡= 𝜑3

𝐼3

𝐶3−𝜑4

𝐶4𝐼4(𝑡 − 𝜃4) (21)

31

Considering the fractional lead time as:

𝜃1 =𝜃𝑗

∑ 𝜃𝑗𝑛𝑖

(22)

Expanding the delayed-state inventory level 𝐼𝑗(𝑡 − 𝜃𝑗) in terms of a Taylor series:

𝐼𝑗(𝑡 − 𝜃𝑗) = 𝐼𝑗 − 𝜃𝑗𝐼�� (23)

Applying equation (23) to Eqs. (18-21):

𝑑𝐼1

𝑑𝑡= (1 −

𝜑1

𝐶1𝜃1)

−1{𝜑𝑖 − 𝜑1

𝐼1

𝐶1} (24)

𝑑𝐼2

𝑑𝑡= (1 −

𝜑2

𝐶2𝜃2)

−1{𝜑1

𝐼1

𝐶1− 𝜑2

𝐼2

𝐶2} (25)

𝑑𝐼3

𝑑𝑡= (1 −

𝜑3

𝐶3𝜃3)

−1{𝜑2

𝐼2

𝐶2− 𝜑3

𝐼3

𝐶3} (26)

𝑑𝐼4

𝑑𝑡= (1 −

𝜑4

𝐶4𝜃4)

−1{𝜑3

𝐼3

𝐶3− 𝜑4

𝐼4

𝐶4} (27)

3.2.4 Sensitivity analysis

Considering the level of variation in capacity and production rate for the SC, a sensitivity

analysis is present, which incorporates in the state space for the LTI with proper dynamics the SC.

For production rates sensitivity analysis: Considering equations (8)-(11), and derivate respect to

φ1, φ2, φ3 and, φ4, respectively it produces:

𝑑𝐼𝑘

𝑑𝜑𝑘= −

𝐼𝑘

𝐶𝑘 (28)

With k=1,2,3,4.

Analyzing the capacity control sensitivity analysis.

𝑑𝐼𝑙

𝑑𝐶𝑙= 𝜑𝑙

𝐼𝑙

𝐶𝑙2 (29)

With l=1,2,3,4.

32

To relate capacity level with production rate, to work with Equations (28) and (29), from

this we have:

𝜕𝐼1

𝜕𝐶1= −

𝜑1

𝐶1

𝜕𝐼1

𝜕𝜑1 (30)

After some mathematical analysis and integrating once, Equation (30) can be expressed as:

𝐶1𝜕𝐼1

𝜕𝐶1= −𝜑1

𝜕𝐼1

𝜕𝜑1 (31)

Equation (31), relates capacity level with production rate, based on this:

𝑑𝐶1

𝐶1= −

𝑑𝜑1

𝜑1 (32)

Integrating equation (32), and solving we have: 𝑙𝑛(𝐶1𝜑𝑎) = 𝐾.. Generalizing for each

node, and considering the LTI system:

𝐶1 =𝑒𝐾

𝜑1 (33)

Assuming K=0, Equation (33) reduces to: 𝐶1 =1

𝜑1.

3.2.5 Stability analysis

Theorem 1. An equilibrium point x∗ of the dynamical system, in equations (8)-(11), is

stable if all the eigenvalues of J∗, the Jacobian evaluated at x∗, have negative real parts such as:

𝜆1 < −𝜑1

𝐶1, 𝜆2 < −

𝜑2

𝐶2, 𝜆3 < −

𝜑3

𝐶3, 𝜆4 < −

𝜑4

𝐶4.

To prove theorem 1, the Jacobian for the dynamical system is:

33

𝐽 =

[ −𝜑1

𝐶10

𝜑1

𝐶1

−𝜑2

𝐶2

0 00 0

0𝜑2

𝐶2

0 0

−𝜑3

𝐶30

𝜑3

𝐶3

−𝜑4

𝐶4 ]

(34)

Applying the determinant to:

|𝜆𝐼 − 𝐽| = 0 (35)

From (35) the eigenvalues are:

𝜆1 < −𝜑1

𝐶1, 𝜆2 < −

𝜑2

𝐶2, 𝜆3 < −

𝜑3

𝐶3, 𝜆4 < −

𝜑4

𝐶4 (36)

Considering that production rates and capacities are positive numbers, we can conclude

that all the eigenvalues of the Jacobian have negative real parts.

3.3 RESULTS

3.3.1 Optimal control

An energy-based optimal control is solved for:

min 𝐽 =1

2∫ 𝑢2𝑇

0𝑑𝑡 (37)

s.t.

𝑑𝐼1

𝑑𝑡= 𝑢 − 𝜑1

𝐼1

𝐶1

𝑑𝐼2

𝑑𝑡= 𝜑1

𝐼1

𝐶1− 𝜑2

𝐼2

𝐶2

𝑑𝐼3

𝑑𝑡= 𝜑2

𝐼2

𝐶2− 𝜑3

𝐼3

𝐶3

𝑑𝐼4

𝑑𝑡= 𝜑3

𝐼3

𝐶3− 𝜑4

𝐼4

𝐶4}

(38)

Applying the Pontryagin maximum principle (PMP), we achieve the following associated

Hamiltonian for the problem:

34

𝐻(𝐼𝑖, 𝑢, 𝜆𝑖, 𝑡) =1

2𝑢2 + 𝜆1 (𝑢 − 𝜑1

𝐼1

𝐶1) + 𝜆2 (𝜑1

𝐼1

𝐶1− 𝜑2

𝐼2

𝐶2) + 𝜆3 (𝜑2

𝐼2

𝐶2− 𝜑3

𝐼3

𝐶3) +

𝜆4 (𝜑3𝐼3

𝐶3− 𝜑4

𝐼4

𝐶4) (39)

In order to develop the PMP, the conditions for co-states are:

��𝑖 = −𝜕𝐻

𝜕𝑥𝑖 (40)

And also, the condition:

𝜕𝐻

𝜕𝑢= 0 (41)

By the application of equations (40) and (41), to the Hamiltonian presented in equation

(39), we proceed to solve the energy-based optimal control, for the serial supply chain.

3.3.2 Case study simulations

A serial supply chain has been modeled via the application of an energy-based optimal

control (as performance index), with demand rate as the optimized variable (input for the system).

Figure 3.2: Factory and Distributors inventory level

In Figure 3.2, the inventory level for factory and distributors is present, for a time horizon

of 20 arbitrary units (au). It can be seen that factory inventory starts from a higher inventory level

compared to distributors.

35

Figure 3.3: Wholesalers and Retailers inventory level

In Figure 3.3, the analysis for wholesalers and retailers inventory levels is present,

considering a higher inventory level for wholesalers compared to retailers, in a time horizon of 20

au. In Figure 3.4, a demand rate graph for a time horizon of 20 au, is present, considering the time

evolves to the final time horizon, the demand of the serial supply chain tends to zero.

Figure 3.4: Demand rate

36

The relation inventory level at each stage of the SC versus demand presents a negative

trend with stable demand as is proposed in (Fogarty, 1991). The demand profile takes into account

that is an independent demand, which implies that demand can be affected by trends and seasonal

patterns. For an independent demand the product usage pattern is uniform and gradual.

A serial supply chain is present, which incorporates the mathematical modeling via the

mixing problem, for inventory management in the factory, distributors, wholesalers, and retailers.

Also, sensitivity and stability analysis are present. Finally, an energy-based optimal control is

developed for the system, with proper simulations. In future work, our goal is to work the mixing

problem mathematical modeling and model predictive control for dynamic divergent supply

chains.

37

Chapter 4: Mathematical modeling and stability analysis of closed-loop supply chains

Closed-loop supply chains (CLSC) integrate forward and reverse logistics, towards the

main interest to process finished goods in a re-manufacturing analysis, to achieve sustainable

solutions for enterprises. This chapter proposes a capacitated control analysis in inventory

management for a class of CLSC. To present two mathematical models for CLSC exploring the

dynamic nature of the system via compartmental analysis. Proper stability analysis for each CLSC

system is present. Considering that decision-makers require mathematical models maximize

profits and minimize costs along the CLSC. This work aim is to approximate a capacity control

for the production levels of CLSC applying the compartmental analysis with a set of coupled

ordinary differential equations. The general structure of the proposed mathematical modeling

considers production level, throughput, and capacity for each node along of each CLSC system.

4.1 INTRODUCTION

The network of forward and reverse logistics synergetic analysis integrates, closed-loop

supply chains (CLSC) (Govidan, et al, 2015), for process in which remanufactures finished goods

into the manufacturers in a closed-loop context. Most of the major work of product recovery is

developed by the manufacturer (He, et al, 2019). Recalling that, in the forward supply chain, raw

materials are manufactured into new products, while in the reverse supply chain, used products are

remanufactured into as new products, also called: remanufactured products (Yuan and Gao, 2010).

Industry practitioners, considers in general processes between 2 and 5% of scrap, for a proper

sustainable process. According to the Remanufacturing Institute, manufacturers of remanufactured

products save 85% of energy use, 86% of water use, and 85% of material use compared to

manufacturing a new product (Kumar, et al, 2017). This research work proposes a general

framework for a CLSC, which can approximate solutions for non-perishable products.

38

Decision-makers require mathematical models that maximize profits and minimize costs

along the CLSC. Based on this, to approximate a capacity control for inventory management of

CLSC applying the problem of mixtures in fluids with coupled ordinary differential equations.

A mathematical model is a mathematical description of the behavior of some real-life

system or phenomenon. Such as physical, sociological, economics, etc. (Zill, 2017). The general

structure of the proposed mathematical modeling considers throughput, inventory level,

production level, and capacity for each node along the CLSC. Compartmental analysis is present

in this research work, to develop proper mathematical modeling for the two CLSC systems. Table

4.1, presents a literature review of mathematical decision-making approaches for CLSC.

Table 4.1: CLSC quantitative decision-making approaches Article Paper scope Network

structure

Mathematical

decision making

approach Zhao, et.al, 2019 A game theory is present for benefits allocation,

in the CLSC management scheme.

Closed-loop Game theory

Wang and Shai, 2019 The practical problem describes a MNIP model

with multi-objective and multi-constraint, for

spare parts optimization, in a multi-period closed-

loop logistics.

Closed-loop Multiobjective

optimization

Su and Sun, 2019 This paper investigates a CLSC network that

includes forward-reverse logistics and the

uncertainty of demand.

Closed-loop Nondominated

sorting genetic

algorithm II

Chen, et. al, 2019 In this paper, the CLSC models consider the case

that the manufacturer can improve the product

quality.

Forward/

Reverse

Game theory

Taleizadeh, et al, 2018 This paper explores a dual-channel closed-loop

supply chain (CLSC) system consisting of a

manufacturer and a retailer.

Closed-loop Game theory

Zhang, et al, 2017 An uncertain CLSC system with hybrid recycling

channels, applies fuzzy control to restrain the

bullwhip effect.

Closed-loop Fuzzy control

Zhang and Zao, 2015 This model consists of a material recovery

subsystem with a third-party reverse logistics

provider, and a manufacturer remanufacturing

recovery subsystem, as well as a cost-based

switching signal vector

Closed-loop Fuzzy robust

control

Hosoda and Disney,

2018

A dynamic CLSC with first-order auto-regressive

demand and return processes is analyzed, with

the assumption that two processes are cross-

correlated.

Forward/

Reverse

Time series

De Giovanni, 2018

In the battery sector, retailers can offer a joint

maximization incentive to manufacturers to push

Closed-loop Optimal control

39

up green activity program efforts and use the

return rate as a marketing lever.

Kumar, et al, 2017 This paper presents green SC and of reverse

logistics programs in manufacturing

organizations.

Forward/

Reverse

Evolutionary

algorithms

Banakis, et al, 2017 This paper presents a CLSC in agri-food where

the medium used mushrooms are recovered, and

used as raw material for the production of new

growing medium.

Closed loop Multi-objective

optimization

Jindal and Sangwan,

2014

In this paper, a multi-product, multi-facility

capacitated CLSC framework is proposed in an

uncertain environment.

Closed loop Mixed integer

linear

programming

Ozceylan and Paksoy,

2013

In this paper, a new mixed integer mathematical

model for a CLSC network, which includes both

forward and reverse flows with multi-periods and

multi-parts is proposed.

Closed loop Mixed integer

linear

programming

Huang, et al, 2009 This paper focus on modeling with uncertainties,

in the CLSC for an effective, and robust control

the dynamic system.

Closed loop

Robust control

Hassanzadeh Amin

and Zhang, 2013

A general CLSC network is configured which

consists of multiple customers, parts, products,

suppliers, remanufacturing subcontractors, and

refurbishing sites.

Closed loop

Mixed-integer

nonlinear

programming

Compartmental analysis has been applied in several engineering and science areas, Table

4.2, summarizes a review of papers within areas of application.

Table 4.2: Areas of application for compartmental analysis Article Paper Scope Area

of application Martínez-Luaces,

2018

In this paper, mixing problems lead to linear ordinary

differential equation (ODE) systems, and the

corresponding qualitative analysis about the ODE solutions and their stability or asymptotical stability are included.

Chemical engineering

Bassingthwaighte, et

al, 2012

This paper presents, in pharmacokinetics,

compartmental models for describing the curves of

concentration-time of a drug following administration,

to guide a defining dosage.

Biology systems

Henley, et al, 2015 This paper analyzes the relation between a

compartmental circuit and a data-based, input-output

model of dynamic cerebral autoregulation, and dynamic

CO2-vasomotor reactivity.

Biomedical engineering

40

Rahimian, et al, 2018 A modified Pinsky–Rinzel pyramidal model is proposed

by replacing its complex nonlinear equations with

piecewise linear approximation, via compartmental

modeling.

Biomedical engineering

Griffiths, et al, 2000 This paper examines epidemic models for HTV/AIDS,

and show that the equations with a proper linearization,

achieves analytic solutions.

Medicine

Berglind, et al, 2012 In this paper, a compartmental model is examined with a

stochastic differential equation to model concentrations

of free-leucine in the plasma.

Medicine

4.2 MATHEMATICAL MODELING

4.2.1 Problem description and notation: Compartmental analysis

Compartmental analysis incorporates the rate change in concentration for a solute, which

follows the following first-order differential equation:

𝑑𝑆

𝑑𝑡= 𝜑𝑖𝑛 − 𝜑𝑜𝑢𝑡 (1)

Where:

- S(t) denotes the amount of solute in the tank at time t.

- ϕin input rate of solute

- ϕout output rate of solute

ϕi

V ϕj

Figure 4.1: Tank level representation for the compartmental analysis problem

In general, equation 1 can expressed from Figure 4.1, as:

M

41

𝑑𝑀

𝑑𝑡= 𝜙𝑖 −𝜙𝑜

𝑀

𝑉 (2)

In terms of inventory management, to approximate a proper analogy for the mixture

problem system, Figure 4.2 analyzes a single node of a supply chain echelon such as:

D

r I ϕo

Figure 4.2: System dynamics representation for the capacitated control problem

A first order differential equation describes the dynamics of Figure 4.2:

𝑑𝐼

𝑑𝑡= 𝐷𝑟 − 𝜑𝑜

𝐼

𝐶 (3)

Where: Dr corresponds to the demand rate; 𝜙𝑜 (throughput); I (inventory level); C( Capacity).

Figure 4.3 presents two-node interaction, which is described by the first-order differential

equation, in equation 4 such as:

𝑑𝐼𝑗

𝑑𝑡= 𝜑𝑖

𝐼𝑖

𝐶𝑗− 𝜑𝑗

𝐼𝑗

𝐶𝑗 (4)

Ii ϕi Ij ϕj

Figure 4.3: Two echelon representation for the capacitated control problem

Considering that a differential equation approximates the continuum, the main interest of

this project is to represent in a state-space formulation the dynamics for the capacity control for a

CLSC. Recalling, a state-space formulation is the minimal set of states, which describes the

behavior of a dynamical system.

C

i

Cj

i

Cj

i

42

4.2.2 CLSC mathematical modeling: System 1

A proposed CLSC network for non-perishable products with the general structure:

Figure 4.4: General structure for the CLSC network

The system dynamics for the general CLSC network, from Figure 4, has the following

state-space equations:

Supplier tier:

𝑑𝐼𝑠

𝑑𝑡=

𝐷𝑟

𝐶𝑠−𝜑𝑠𝑚

𝐶𝑠𝐼𝑠 (5)

Manufacturer tier.:

𝑑𝑄𝑚

𝑑𝑡=

𝜑𝑠𝑚

𝐶𝑚𝐼𝑠 +

𝜑𝑟𝑚

𝐶𝑚𝐼𝑟 −

𝜑𝑚𝑑

𝐶𝑚𝑄𝑚 (6)

Distributor tier:

𝑑𝐼𝑑

𝑑𝑡=

𝜑𝑚𝑑

𝐶𝑑𝑄𝑚 −

𝜑𝑑𝑐

𝐶𝑑𝐼𝑑 (7)

Consumer tier:

43

𝑑𝐼𝑐

𝑑𝑡=

𝜑𝑑𝑐

𝐶𝑐𝐼𝑑 −

𝜑𝑐𝑐𝑜

𝐶𝑐𝐼𝑐 −

𝑦

𝐶𝑐𝐼𝑐 (8)

Colector tier:

𝑑𝐼𝐶𝑜

𝑑𝑡=

𝜑𝑐𝑐𝑜

𝐶𝐶𝑜𝐼𝑐 −

𝜑𝑐𝑜𝑅

𝐶𝐶𝑜𝐼𝐶𝑜 (9)

Recycler tier:

𝑑𝐼𝑅

𝑑𝑡=

𝜑𝑐𝑜𝑅

𝐶𝑅𝐼𝐶𝑜 −

𝜑𝑟𝑚

𝐶𝑅𝐼𝑅 (10)

4.2.3 Four-tier CLSC mathematical modeling: System 2

Considering the dynamics of the forward CLSC, which consists of four echelons such as

Suppliers, manufacturers, distributors, and consumers. Based on Figure 4.5, the set of ordinary

differential equations, and applying equations (3) and (4), the dynamics that describes the forward

CSLC are:

For echelon 1:

Supplier 1

𝑑𝐼1

𝑑𝑡= 𝑑𝑎 − (𝜙11 + 𝜙12)

𝐼1

𝐶1 (11)

Supplier 2

𝑑𝐼2

𝑑𝑡= 𝑑𝑏 − (𝜙13 + 𝜙14)

𝐼2

𝐶2 (12)

For echelon 2:

Manufacturer 1

𝑑𝑄1

𝑑𝑡= 𝜙11

𝐼1

𝐶3− (𝜙21 + 𝜙22)

𝑄1

𝐶3 (13)

Manufacturer 2

44

𝑑𝑄2

𝑑𝑡= (𝜙12

𝐼1

𝐶4+ 𝜙13

𝐼2

𝐶4) − (𝜙23 + 𝜙24)

𝑄2

𝐶4 (14)

Manufacturer 3

𝑑𝑄3

𝑑𝑡= 𝜙14

𝐼2

𝐶5− (𝜙25 + 𝜙26)

𝑄3

𝐶5 (15)

For echelon 3:

Distributor 1

𝑑𝐼3

𝑑𝑡= 𝜙21

𝑄1

𝐶6− 𝜙34

𝐼3

𝐶6 (16)

Distributor 2

𝑑𝐼4

𝑑𝑡= 𝜙22

𝑄1

𝐶7+ 𝜙23

𝑄2

𝐶7− 𝜙35

𝐼4

𝐶7 (17)

Distributor 3

𝑑𝐼5

𝑑𝑡= 𝜙24

𝑄2

𝐶8+ 𝜙25

𝑄3

𝐶8− 𝜙36

𝐼5

𝐶8 (18)

Distributor 4

𝑑𝐼6

𝑑𝑡= 𝜙26

𝑄3

𝐶9− 𝜙37

𝐼6

𝐶9 (19)

For echelon 4:

Consumer 1

𝑑𝐼7

𝑑𝑡= 𝜙34

𝐼3

𝐶6+ 𝜙35

𝐼4

𝐶7− 𝑦1𝐼7 (20)

Consumer 2

𝑑𝐼8

𝑑𝑡= 𝜙36

𝐼5

𝐶8+ 𝜙37

𝐼6

𝐶9− 𝑦2𝐼8 (21)

45

Figure 4.5: Forward CLSC dynamics

46

Figure 4.6: Reverse CLSC dynamics

The reverse CLSC is present in Figure 4.6, with the following dynamics:

Reverse dynamics echelon 1:

Consumer 1

𝑑𝐼9

𝑑𝑡=

−(𝜑𝑅1+𝜑𝑅2)

𝐶𝐶1𝐼9 − 𝑦1𝐼9 (22)

47

Consumer 2

𝑑𝐼10

𝑑𝑡=

−(𝜑𝑅3+𝜑𝑅4)

𝐶𝐶2𝐼10 − 𝑦2𝐼10 (23)

Reverse dynamics echelon 2:

Collector 1

𝑑𝐼11

𝑑𝑡=

𝜑𝑅1

𝐶𝐶𝑜1𝐼9 −

𝜑𝑅5

𝐶𝐶𝑜1𝐼11 (24)

Collector 2

𝑑𝐼12

𝑑𝑡=

𝜑𝑅2

𝐶𝐶𝑜2𝐼9 +

𝜑𝑅3

𝐶𝐶𝑜2𝐼10 −

(𝜑𝑅6+𝜑𝑅7)

𝐶𝐶𝑜2𝐼11 (25)

Collector 3

𝑑𝐼13

𝑑𝑡=

𝜑𝑅4

𝐶𝐶𝑜3𝐼10 −

𝜑𝑅8

𝐶𝐶𝑜3𝐼13 (26)

Reverse dynamics echelon 3:

Recycler 1

𝑑𝐼14

𝑑𝑡=

𝜑𝑅5

𝐶𝑅1𝐼11 +

𝜑𝑅6

𝐶𝑅1𝐼12 −

(𝜑𝑅9+𝜑𝑅10)

𝐶𝑅1𝐼14 (27)

Recycler 2

𝑑𝐼15

𝑑𝑡=

𝜑𝑅7

𝐶𝑅2𝐼12 +

𝜑𝑅8

𝐶𝑅2𝐼13 −

(𝜑𝑅11+𝜑𝑅12)

𝐶𝑅2𝐼15 (28)

Reverse dynamics echelon 4:

Reverse Manufacturer 1

𝑑𝑄4

𝑑𝑡=

𝜑11

𝐶𝑀1𝐼1 +

𝜑𝑅9

𝐶𝑀1𝐼14 −

𝜑𝑇𝑅1

𝐶𝑀1𝑄4 (29)

Reverse Manufacturer 2

48

𝑑𝑄5

𝑑𝑡=

𝜑12

𝐶𝑀2𝐼1 +

𝜑13

𝐶𝑀2𝐼2 +

𝜑𝑅10

𝐶𝑀2𝐼14 +

𝜑𝑅11

𝐶𝑀2𝐼15 −

𝜑𝑇𝑅2

𝐶𝑀2𝑄5 (30)

Reverse Manufacturer 3

𝑑𝑄6

𝑑𝑡=

𝜑14

𝐶𝑀3𝐼2 +

𝜑𝑅12

𝐶𝑀3𝐼15 −

𝜑𝑇𝑅3

𝐶𝑀3𝑄6 (31)

4.3 STABILITY ANALYSIS

4.3.1 System 1: Stability analysis

Theorem 1. For the dynamical system in equations (5)-(10), an equilibrium point x∗ is

stable if all the eigenvalues of J∗, the Jacobian evaluated at x∗, have negative real parts such as:

𝜆1 < −𝜑𝑠𝑚

𝐶𝑠, 𝜆2 < −

𝜑𝑚𝑑

𝐶𝑚, 𝜆3 < −

𝜑𝑑𝑐

𝐶𝑑, 𝜆4 < −

𝜑𝐶𝐶𝑜+𝑦

𝐶𝑐, 𝜆5 < −

𝜑𝐶𝑜𝑅

𝐶𝐶𝑜, 𝜆6 < −

𝜑𝑟𝑚

𝐶𝑅

Proof. Developing the Jacobian for the dynamical system is:

( )

0 0 0 0 0

0 0 0

0 0 00

0 00 0

0 0 0 0

0 0 0 0

sm

s

sm md rm

m m m

md dc

d d

CCoCCo

cc

CCo CoR

o o

CoR rm

R R

C

C C C

C C

y

CC

CC CC

C C

J

−

−

−

− +

−

−

=

Calculating the eigenvalues for: |𝜆𝐼 − 𝐽| = 0, we have:

𝜆1 < −𝜑𝑠𝑚

𝐶𝑠, 𝜆2 < −

𝜑𝑚𝑑

𝐶𝑚, 𝜆3 < −

𝜑𝑑𝑐

𝐶𝑑, 𝜆4 < −

𝜑𝐶𝐶𝑜+𝑦

𝐶𝑐, 𝜆5 < −

𝜑𝐶𝑜𝑅

𝐶𝐶𝑜, 𝜆6 < −

𝜑𝑟𝑚

𝐶𝑅

We can conclude that all the eigenvalues of the Jacobian have negative real parts, by

considering that production rates and capacities are positive numbers.

49

4.3.2 System 2: Stability analysis

Theorem 2. If the forward CLSC dynamical system with equations (11)-(21) is stable, and

the reverse CLSC dynamical system with equations (22)-(31) is stable, therefore the overall CLSC

system is stable.

Proof. By calculating the eigenvalues for: |𝜆𝐼 − 𝐽| = 0 for the forward CLSC dynamical system

such as:

𝜆𝑓1 < −(𝜑11+𝜑12)

𝐶1, 𝜆𝑓2 < −

(𝜑13+𝜑14)

𝐶2 , 𝜆𝑓3 < −

(𝜑21+𝜑22)

𝐶3, 𝜆𝑓4 < −

(𝜑23+𝜑24)

𝐶4, 𝜆𝑓5 < −

(𝜑25+𝜑26)

𝐶5,

𝜆𝑓6 < −𝜑34

𝐶6, 𝜆𝑓7 < −

𝜑35

𝐶7, 𝜆𝑓8 < −

𝜑36

𝐶8, 𝜆𝑓9 < −

𝜑37

𝐶9, 𝜆𝑓10 < − y1, 𝜆𝑓11 < − y2

Also, calculating the eigenvalues for the reverse CLSC dynamical system we have:

𝜆𝑟1 < −(𝜑𝑅1+𝜑𝑅2)

𝐶𝐶1, 𝜆𝑟2 < −

(𝜑𝑅3+𝜑𝑅4)

𝐶𝐶2, 𝜆𝑟3 < −

𝜑𝑅5

𝐶𝐶01, 𝜆𝑟4 < −

(𝜑𝑅6+𝜑𝑅7)

𝐶𝐶02, 𝜆𝑟5 < −

𝜑𝑅8

𝐶𝐶03

𝜆𝑟6 < −(𝜑𝑅9+𝜑𝑅10)

𝐶𝑅1 , 𝜆𝑟7 < −

(𝜑𝑅11+𝜑𝑅12)

𝐶𝑅2, 𝜆𝑟8 < −

𝜑𝑇𝑅1

𝐶𝑀1, 𝜆𝑟9 < −

𝜑𝑇𝑅2

𝐶𝑀2, 𝜆𝑟10 < −

𝜑𝑇𝑅3

𝐶𝑀3

By considering that production rates and capacities are positive numbers. we can conclude that

all the eigenvalues of the Jacobians have negative real parts, therefore the overall CLSC is stable.

50

Chapter 5: Present value Hamiltonian-Optimal control for a production-inventory system

Enabling industrial production systems requires competitive and adaptive solutions to

ensure profitability in markets. In PI systems, decision-makers require to achieve and approximate,

online decisions. Proper manner, resilient, and sustainable PI systems present solutions to decrease

operations management costs such as production, logistics, and quality and enhances real-time or

near real-time decision making. Based on this, dynamic approaches for production-inventory

systems are valuable to maximize profits and minimize costs over the smart manufacturing system.

This chapter emphasizes the characteristics that make a production system resilient and sustainable

from the systems modeling and control theory perspectives. Optimal control presents a tradeoff

between optimality and stability conditions of the proposed dynamical system.

This chapter presents an optimal control approach for a class of resilient and sustainable

production-inventory system (PI). Modeling, analysis, and control for a PI system are present. For

systems modeling purposes, a resilient mixed capacity/third order PI is analyzed, considering an

emergy sustainability index. Resiliency factor and sustainability index are features enhance proper

PI characteristics for the system. Optimality, robustness, and stability are the top-level

characteristics which encompass a suitable control. A proper Lyapunov stability analysis is present

over a third-order nonlinear dynamical system. Optimal control formulation is conducted via

Pontryagin maximum principle, with a present value-Hamiltonian approach (PVH).

51

5.1 INTRODUCTION

Nowadays, enabling industrial production systems requires competitive and adaptive

solutions to ensure profitability in markets. Inventory management which involves the planning

and control of inventory, typically, yields significant cost savings (Chen and Chen, 2019).

Competitive companies require to produce high-quality products and respond to fast changes in

demands, within a low-cost production profile (Boukas, 2006). In PI systems, decision-makers

require to achieve and approximate, online decisions. Proper manner, PI integrates solutions to

decrease operations management costs such as production, logistics, and quality and enhances real-

time or near real-time decision making. Also, the production processes assume, all the products

are perfect and the demand rate is constant (Li, et al, 2017).

Based on this, dynamic approaches for production-inventory systems are valuable to

maximize profits and minimize costs over the manufacturing system. The modeling and analysis

of dynamic production-inventory control systems in manufacturing enterprises is essential, for

operations and planning control (Bijulal, et al, 2011). This paper presents PI from the systems

modeling and control theory perspectives. Control Theory is the area of engineering, which

analyze the time response of physical systems, by applying differential equations (Ortega and Lin,

2004). In order to solve the optimal production-inventory control problem, optimal control theory

is commonly applied to production-inventory systems (Wang and Chan, 2015). Optimal control

presents a tradeoff between optimality and stability conditions of the proposed dynamical system.

Optimal control applications have been proved in economics, management science, and industry

(Khmelnitsky and Gerchak, 2002 From control theory approaches, stochastic and deterministic

models have been proposed to analyze the production-inventory systems. In (Tan, 2002), (Zhao

and Melamed, 2007), (Ioannidis, et al, 2008), (Kogan, et al, 2010), (Kogan, et al, 2017), (Wang,

52

et al, 2018), (Li, et al, 2018), presents PI systems from a stochastic control perspective, while in

(Nandola and Rivera, 2013), (Ignaciuk, 2015), (Doganis, et al, 2008) which present deterministic

control perspective. For mathematical modeling purposes, production-inventory systems are

deterministic (Tadj, et al, 2006) and stochastic (Khoury, 2016), with linear and nonlinear dynamic

performance. Capacity level, inventory level, production rate, production level, etc. are the main

parameters to be analyzed in production-inventory systems. To present a mathematical model, for

a class of third-order nonlinear dynamical system, which approximates a PI system behavior

applying fluid and classical mechanics analogies.

This research chapter:

1) Develops a mathematical model exploring the dynamic nature of resilient and

sustainable production-inventory systems.

2) Presents the relation between optimality, robustness, and stability for PI systems.

3) Establishes the main characteristics of production-inventory system from a systems

and control theory perspective.

4) Presents the design process of a third-order nonlinear dynamical system which

captures a resilience factor and an emergy sustainability index.

5) Demonstrates stability analysis via Lyapunov direct method for the third-order

nonlinear system.

6) Develops a proper linearization of the dynamical system to proceed with a

feedback control system design.

7) Introduces the PVH-optimal control approach for energy-based formulation via the

Pontryagin maximum principle.

The rest of chapter is developed as follows. First, systems theory and then, control features

for PI systems are reviewed and mathematical modeling on PI is discussed in Section 5.2. Stability

53

analysis is presented in Section 5.3. Optimal control formulation is provided in Section 5.4 while

case study results are developed and present in Section 5.5.

5.2 MATHEMATICAL MODELING

5.2.1 Optimality, robustness, and stability

In systems and control theory, optimality, robustness, and stability are the top-level

characteristics which encompass a suitable control system. Control systems for a proper operation

in a process, require a certain level of optimality, robustness, and stability conditions. PI systems

require real-time control approaches to make real-time or near real-time online decisions. The role

of mathematical modeling is to theoretically approximate the operation of factories, production-

inventory systems, and supply chains. A good approach is to justify proper analogies with mixed

variables from fluid mechanics and electromechanical systems domain. Therefore, in this paper,

to define those characteristics that are mandatory for automation science practitioners to achieve a

PI system:

Definition 1. Principle of Optimality

An optimal policy has the property that independently of initial state and decisions, the

subsequent decisions must constitute the optimal policy regard to the state resulting from the first

decision (Bellman, 1957).

Definition 2. Robustness

In control systems, robustness is defined as the ability to maintain the performance

characteristics under the presence of system variations or perturbations (Stengel, and Ray, 1991).

Theorem 1. Lyapunov stability

The equilibrium state xe=0 of the system ��(𝑡) = 𝑓(𝑥, 𝑡), is asymptotically stable if a

Lyapunov function 𝑉(𝑥, 𝑡) can be found such that ��(𝑥, 𝑡) < 0 for all 𝑥 ≠ 0 and 𝑡 ≥ 𝑡0.

54

5.2.2 Systems theory and control for PI

Production-inventory systems present a complex dynamical, to stability and dynamic

performance (Al-Khazraji, et al, 2018). For modeling, analysis, and control of PI systems, several

approaches have been conducted: classical control theory (transfer function) and modern control

(state-space). In Illustration 5.1, our interest is to graphically, demonstrate the relation of a PI

system as a subset of SM and show that these terms are related to optimality, robustness, and

stability properties from a systems and control theory perspective. Based on definition 1, which is

related to the principle of optimality and definition 2, which is related to robustness, and Theorem

1 (Lyapunov stability).

Illustration 5.1: Optimality, robustness, and stability for PI systems.

As Illustration 5.1 shows, PI systems are at the intersection of these three concepts

mentioned above. In this work, a PI system is mathematically modeled by a third-order nonlinear

system of ordinary equations. This method is also known as a model-based systems theory and

control theory-oriented approach in the context of optimal control.

This chapter defines nine characteristics that from a systems and control theory

perspectives are necessary for PI systems such as trackability, observability, controllability,

identifiability, dissipativity, passivity, entropy, sensitivity, and profitability (refer to Table 5.1).

55

These features are required in different levels of application within the context of automation

science and control systems, while the optimality, robustness, and stability dimensions for PI

systems are presented. In Table 5.1, O refers to Optimality, S to Stability and R to Robustness.

Table 5.1: Systems theory and control feature for PI systems

Control

feature

Definition O S R

Trackability Trackability deals with the

dynamical plant behavior from

the initial to the final moment.

Refers to the plant property, it is

independent of the controller and

control. (Gruyitch, 2018)

X X X

Observability Imply that internal state variables

of the system can be externally

measured.

X X

Controllability Ability of a controller to

arbitrarily alter the functionality

of the system plant.

X X

Identifiability Refers that for a class of models,

for the identification method of

the system, a unique solution is

achieved.

X X

Dissipativity A system is dissipative if the

increase in its energy, during a

time interval, is less than that

supply rate energy.

(Bao, and Lee, 2007)

X

Passivity In passivity, if the difference

between stored energy and

supplied energy is positive the

feedback system will be stable in

an input-output sense

(Loría, and H. Nijmeijer, 2004)

X

Entropy Refer to the measure of the cost

of performing different

performance criteria, to calculate

the maximum work that can be

extracted from the repeated

operation of feedback-controlled

systems.

(Cao, and M. Feito, 2009)

X X X

Sensitivity Relates to a measure of the

system´s characteristics

dependence of a particular

element. (Shinners, 1998)

X X

Profitability Provide economical solutions

during a time horizon, which

captures tradeoffs between

revenue and costs.

X X

56

5.2.3 Systems modeling and notation: Equations of motion

Considering the dynamic nature of PI systems, it becomes of special interest to

approximate online solutions for decision-makers to generate tradeoffs between the maximization

of profits and the minimization of cost the supply chain network. The main contribution of this

paper is to present a mathematical model captures the features PI systems via optimal control

theory applying the Pontryagin maximum principle.

Based on this, a mathematical model for the mixed capacity-cyber physical production-

inventory system involves a third-order state space formulation, in which a vertical motion for

inventory level is considered, coupled by a resiliency coefficient (spring) and an emergy

sustainability index (damper), and horizontal production level flow is presented. Both, the

resiliency coefficient and the emergy sustainability index are dimensionless and real positive

parameters. The resiliency coefficient is related to the ability of the system to deal with changes in

demand, inventory, and production levels. While the emergy sustainability index measures the

contribution of the process to the economy of environmental loading; this index dissipates the

energy of the system through the damper.

Figure 5.1 presents a PI system adapted from mathematical models previously published

in (Davizon, et al, 2014), (Davizon, et al, 2015) for the spring-mass-damper dynamical system,

and from papers which analyze production-inventory systems via fluid analogies (Schwartz, et al,

2006) with applications in Semiconductor Manufacturing in (Wang, et al, 2007), (Schwartz, et al,

2009) and demand modeling oriented approaches for inventory management presented in

(Schwartz, and Rivera, 2010), ] (Schwartz, and Rivera, 2014).

57

The third order mixed PI systems refers to two subsystems: (subsystem 1) a tank-level

system and (subsystem 2) a mass-spring-damper, which coupled represents a novel PI systems

realization. This PI system approximate a manufacturing system, in continuous time.

Figure 5.1: Third-order mixed production inventory system

The tank inventory level is mathematically considered as a mass variable-mechanical

oscillator, subject to an external force, and such as it is presented in its canonical form in Equation

(1).

𝑚(𝑡)ℎ••

+ 𝛽ℎ•

+ 𝑘ℎ = 𝑑(𝑡) (1)

Systems modeling parameters are presented in Table 5.2. To propose the tank-spring-

damper dynamical system, as an analogy for a mixed capacity/production inventory system.

d(t)

q0(t)

h(t)

k β

ρ

C

58

Table 5.2: System modeling parameters

Symbol Quantity Type of parameter

d(t) Demand Manipulated variable (input)

h(t) Inventory level Controlled variable (output)

q0(t) Production level Controlled variable (output)

C Capacity Fixed parameter

Ρ

Production system

fluency (density

flow)

Fixed parameter

K Resiliency factor Fixed parameter

Β Emergy

sustainability index Fixed parameter

V Volume Fixed parameter

The horizontal production level flow is described by the following first-order equation:

𝑞0•=

1

𝛼𝐶(𝑑(𝑡) − 𝑞0) (2)

Assuming that laminar flow is present, and by the relation in which: ρ=m(t)/V, where ρ is

production-inventory system fluency (fluid density); V is the volume defined by V=C.h(t), and C

is the capacity of the system. This produces:

𝑚(𝑡) = 𝜌𝐶ℎ(𝑡) (3)

Substituting equation (3) in (1):

𝜌𝐶ℎℎ••

+ 𝛽ℎ•

+ 𝑘ℎ = 𝑢(𝑡) (4)

After mathematical manipulation, solving for ••

h :

ℎ••

+ (𝛽

𝜌𝐶)ℎ•

ℎ+ (

𝑘

𝜌𝐶) = (

1

𝜌𝐶)(𝑢

ℎ) (5)

Based on equation (5), a state-space formulation for the system, considering:

𝑥1 = ℎ and 𝑥2 = ℎ , is:

59

��1 = 𝑥2 (6)

��2 = −(𝛽

𝜌𝐶)𝑥2

𝑥1− (

𝑘

𝜌𝐶) +

1

𝜌𝐶(𝑢(𝑡)

𝑥1) (7)

To calculate the equilibrium points, we proceed with equations (6) and (7), from which:

ℎ𝑒(𝑡) = 𝑒−𝐾

𝛽𝑡 and 𝑢𝑒 = 𝐾ℎ𝑒. (See Apendix).

The full state space description for the system requires to represent equation (2), with

equations (6) and (7), as the following:

��3 = −(1

𝛼𝐶) (𝑥3 − 𝑢(𝑡)) (8)

5.2.4 Linearization

To proceed, a proper linearization of equations (6) and (7) around the equilibrium point

needs to be obtained. In the form: �� = 𝐴𝑥 + 𝐵𝑢, we have:

𝐴 = [

𝜕𝐹1

𝜕𝑥1

𝜕𝐹1

𝜕𝑥2𝜕𝐹2

𝜕𝑥1

𝜕𝐹2

𝜕𝑥2

]

{𝑈𝑒|ℎ𝑒}

(9)

where:

𝐹1 = 𝑥1 = 𝑥2 (10)

𝐹2 = 𝑥2 = −𝛽

𝜌𝐶

𝑥2

𝑥1−

𝑘

𝜌𝐶=

𝑢

𝜌𝐶𝑥1 (11)

Applying derivatives in equations (10) and (11), and evaluating for the equilibrium point:

𝜕𝐹1

𝜕𝑥1= 0 ,

𝜕𝐹1

𝜕𝑥2= 1,

𝜕𝐹2

𝜕𝑥1= −

𝛽

𝜌𝐶(1 +

𝑘

𝛽) and

𝜕𝐹2

𝜕𝑥2= −

𝛽

𝜌𝐶

For maxtrix B:

𝐵 = [

𝜕𝐹1

𝜕𝑢

𝜕𝐹2

𝜕𝑢

]

𝑈𝑒

(12)

60

𝜕𝐹1

𝜕𝑢= 0 and

𝜕𝐹2

𝜕𝑢=

1

𝜌𝐶

Finally, the linear state-space formulation for the dynamic system is:

[𝑥1𝑥2��3

] = [

0 1 0

−1

𝜌𝐶(𝛽 + 𝑘) −

𝛽

𝜌𝐶0

0 0−1

𝛼𝐶

] [𝑥1𝑥2𝑥3] + [

01

𝜌𝐶

1

𝛼𝐶

] 𝑢(𝑡) (13)

5.3 STABILITY ANALYSIS

Lyapunov stability analysis for the third-order nonlinear mixed PI systems is conducted

via the direct method. The candidate Lyapunov function is:

𝑉 =𝑞

2𝑥12 +

1

2𝑥22 +

1

2𝑥32 (14)

Applying the derivative in time to equation (14):

�� = 𝑞𝑥1��1 + 𝑥2��2 + 𝑥3��3 (15)

Substituting equations (6-8) into equation (15):

�� = 𝑞𝑥1𝑥2 + 𝑥2 (−𝑎𝑥2

𝑥1− 𝑏) + 𝑥3 (−

1

𝛼𝐶𝑥3) (16)

Where 𝑎 =𝛽

𝜌𝐶 and 𝑏 =

𝐾

𝜌𝐶. After some algebraic manipulations, equation (16) reduces to:

�� = 𝑞𝑥1𝑥2 − 𝑎𝑥22

𝑥1− 𝑏𝑥2 − (

1

𝛼𝐶) 𝑥3

2 (17)

Considering that: |𝑥1||𝑥2| = 𝑤2, equation (17) is:

�� = 𝑞𝑤2 − (𝑎 + 𝑏)|𝑤| − (1

𝛼𝐶) 𝑥3

2 (18)

Applying the following conditions: q=0 and K= -β, asymptotic stability is achieved and

61

−(1

𝛼𝐶) 𝑥3

2 ≤ 0, which implies �� ≤ 0.

5.4 OPTIMAL CONTROL

Applications of optimal control theory in engineering and sciences are broad and diverse

and can be found in aerospace, chemical, electrical, mechanical, and industrial engineering.

Economists use the term dynamic optimization to address optimal control theory problems.

Applications of optimal control in management sciences and operations research are (1)

pricing with dynamic demand and production costs; (2) scheduling and production planning

problems; (3) distribution and transportation in logistics networks; (4) optimal transfer of

technology, and (5) optimal remanufacturing and recycling in closed-loop supply chains, among

others. Decision-makers in management sciences and operations research use optimal control

theory to map optimal control methods towards optimization methods based on the nature of the

complexity of the manufacturing system.

The main contribution of this paper is to present a model-based systems theory and optimal

control theory approach based on a mathematical model captures the features of PI systems via

optimal control theory applying the Pontryagin maximum principle.

5.4.1 Optimal control for PI system

An optimal control (OC) is defined as an admissible control, minimizes a functional

objective (Naidu, 2003).

Definition 3. Given a dynamical system with initial condition x0, which evolves in time

according to the state-space equation �� = 𝑓(𝑥, 𝑢, 𝑡), to find an admissible control and make the

functional objective to achieve its maximum.

A general mathematical form for an optimal control problem is:

62

min𝑢 𝐽(𝑢) =

1

2∫ 𝐹(𝑥, 𝑢, 𝑡)𝑡𝑓

𝑡𝑜

𝑑𝑡 + 𝑆[𝑥(𝑡𝑓)]

s.t. (19)

�� = 𝑓(𝑥, 𝑢, 𝑡) 𝑥(𝑡0) = 𝑥0, 𝑥 ∈ 𝑋, 𝑢 ∈ 𝑈

5.4.2 Present value Hamiltonian-Optimal control problem

The proposed third-order nonlinear dynamical system via linearization for this research

follows the general PVH-optimal control definition:

min𝑢 𝐽(𝑢) =

1

2∫ 𝑒−𝛿𝑡𝑊(𝑥, 𝑢, 𝑡)𝑡𝑓

𝑡𝑜

𝑑𝑡

s.t. (20)

�� = 𝑓(𝑥, 𝑢, 𝑡) 𝑥(𝑡0) = 𝑥0, 𝑥 ∈ 𝑋, 𝑢 ∈ 𝑈

Where δ is a general discount factor. By applying, the Pontryagin maximum principle, for

the PVH-optimal control problem in Equation (20) as is developed and presented in (Cerda-

Tena, 2012).

𝐻(𝑥, 𝑢, 𝜌, 𝑡) = 𝑊(𝑥, 𝑢, 𝑡)𝑒−𝛿𝑡 + 𝜌𝑓(𝑥, 𝑢, 𝑡) (21)

With the following conditions:

i) �� = −𝜕𝐻

𝜕𝑥= −

𝜕𝑊

𝜕𝑥𝑒−𝛿𝑡 − 𝜌

𝜕𝑓

𝜕𝑥 with 𝜌(𝑡𝑓) = 0

ii) max H for 𝑢 ∈ Ω

iii) �� = 𝑓(𝑥, 𝑢, 𝑡) for 𝑥(𝑡0) = 𝑥0

Defining the PVH such as Π = 𝐻𝑒𝛿𝑡, this gives:

Π = 𝑊(𝑥, 𝑢, 𝑡) + 𝜌𝑒𝛿𝑡𝑓(𝑥, 𝑢, 𝑡) (22)

Let 𝑞(𝑡) = 𝜌(𝑡)𝑒𝛿𝑡, solving for 𝜌(𝑡):

63

𝜌(𝑡) = 𝑒−𝛿𝑡𝑞(𝑡) (23)

By the derivate of Equation (23) with respect to time:

��(𝑡) = −𝛿𝑒−𝛿𝑡𝑞(𝑡) + 𝑒−𝛿𝑡��(𝑡) (24)

From: Π(𝑥, 𝑢, 𝑞, 𝑡) = 𝑊(𝑥, 𝑢, 𝑡) + 𝑞(𝑡)𝑓(𝑥, 𝑢, 𝑡), we know that:

�� = −𝜕𝑊

𝜕𝑥𝑒−𝛿𝑡 − 𝜌

𝜕𝑓

𝜕𝑥 (25)

Equating equations (24) and (25), after some algebra:

�� = −𝜕𝐻

𝜕𝑥+ 𝛿𝑞 (26)

with 𝑞(𝑡𝑓) = 0

5.5 RESULTS

5.5.1 Case study

Applying the PVH for energy-based optimal control problem a general formulation such

as in equation (20) is presented. The following problem is addressed as:

min𝑢 𝐽(𝑢) =∫ 𝑒−𝛿𝑡𝑢2

𝑡𝑓

𝑡𝑜

𝑑𝑡

s.t. (27)

��1 = 𝑥2

��2 = −(𝛽

𝜌𝐶)𝑥2𝑥1− (

𝑘

𝜌𝐶) +

1

𝜌𝐶(𝑢(𝑡)

𝑥1)

��3 = −(1

𝛼𝐶) (𝑥3 − 𝑢(𝑡))

Defining the Hamiltonian similar to equation (22), and applying equation (26), for the

system described in (27):

Π(𝑥, 𝑢, 𝑡) = 𝑢2 + 𝑞1𝑥2 + 𝑞2(−𝐴𝑥1 − 𝐵𝑥2 + 𝐶0𝑢(𝑡)) +

𝑞3𝐷(−𝑥3 + 𝑢(𝑡)) (28)

64

��1 = 𝐴𝑞2 + 𝛿𝑞1 (29)

��2 = (𝐵𝑞2 − 𝑞1) + 𝛿𝑞2 (30)

��3 = 𝐷𝑞3 + 𝛿𝑞3 (31)

Finding an optimal control such as 𝜕Π

𝜕𝑢= 0, this yields:

𝑢∗ = −1

2(𝐶0𝑞2 + 𝐷𝑞3) (32)

Where: 𝐴 = 𝐶0(𝛽 + 𝐾); 𝐵 = 𝐶0𝛽;𝐶0 =1

𝜌𝐶; 𝐷 =

1

𝛼𝐶

5.5.2 Simulations

Simulations were performed in MATLAB, for the PVH based optimal control system

described in Equation 27. For mathematical modeling purposes, we worked with a linearization of

the system model which was conducted within the equilibrium points of the third-order nonlinear

PI systems.

The system modeling quantities are summarized in Table 5.3:

Table 5.3: System modeling quantities

SYMBOL C 𝜌 K 𝛽 𝛿 𝛼

QUANTITY 10 1000 500 100 0.7 1

In Figure 5.2, the inventory level for PI systems achieves a maximum overshoot

considering the dynamic nature of the system and stabilizes in a time scale of nine. Once the

maximum inventory is achieved, around the four-time scale, half of the maximum inventory level

is reached. Considering that the inventory level and inventory rates were developed from a second-

order differential equation with a time-varying approach as a function of the mass, the simulations

conducted led towards a dynamic response in which the steady-state is reached near to the time

scale of ten.

65

Figure 5.2: Inventory level for PI system

The vertical process description is achieved via the analysis in inventory level, as the first

output description of the mixed-PI systems dynamical system.

To quantify the inventory rate as a change measure in the inventory over time, Figure 5.3

presents a minimum overshoot and stabilizes a time scale of eight. The inventory rate approximates

the time derivative of the inventory level which has been presented previously in Figure 5.2.

Figure 5.3: Inventory rate for PI system

66

The horizontal flow description for the third-order mixed-PI systems achieves a second

output analysis for the production level which is shown in Figure 5.5. The maximum production

level is achieved at the time scale of nine. This result is related to the time when steady-state

conditions are reached (as shown previously in Figures 5.2 and 5.3).

A tradeoff between inventory level, inventory rate, and production levels reflects that once

the steady-state is reached, the proper inertia of the dynamical system decreases the production

goals in a specific time horizon.

Vertical and horizontal flow descriptions for the third-order PI systems are based on a

system modeling linearization around the equilibrium points.

Figure 5.4: Production level for PI system

Figure 5.5 shows the demand level as the single input variable; it presents a maximum level

at initial conditions. Once PI systems reach steady-state conditions, an inflection level in the

demand profile is achieved by the time scale of nine. At the end of the time horizon, the demand

reaches a minimum amount, which implies that low inventory level and inventory rates are present.

A tradeoff between production level and demand level is related, considering the horizontal flow

description of the system.

67

Figure 5.5: Demand level for PI system

This chapter developed a PVH-based optimal control for PI system mathematical modeling

system. Systems and control theory perspective features for PI systems were defined such as

trackability, observability, controllability, identifiability, dissipativity, passivity, entropy,

sensitivity, and profitability. A third-order nonlinear mixed-PI systems mathematical model was

developed following a model-based control system engineering and optimal control approach. A

mathematical linearization for the third-order mixed-PI systems was presented which takes into

account a vertical and horizontal flow description. In future work, our interest is to develop a four

echelon cascade system analysis for the mixed-PI systems in which more scenarios will be

addressed. Also, we expect to work with a nonlinear model predictive control approach for the

nonlinear PI system descriptions of the dynamical system. The interaction of automation science

and control systems with model-based systems engineering plays an important role in the context

of mixed-PI systems. System identification techniques will be of interest to explore, for data-driven

control-oriented approaches. Finally, fault-tolerant control analysis is important to be addressed to

consider reliability approaches over each of the parameters and elements in the mixed-PI systems.

68

Chapter 6: Conclusions and future work

This research work, based on previous chapters, presents the following conclusions and

future work, to continue improving the state of the art in mathematical modeling, optimal control,

and stability analysis for supply chains.

6.1 CONCLUSIONS

In chapter 3, a serial supply chain was present considering the application of

compartmental analysis as a mathematical model for a class of linear supply chain system, with a

proper sensitivity and stability analysis for the dynamic supply chain. An energy-based optimal

control problem for the serial supply chain was conducted, with results-oriented to inventory level

and demand rate with stable and negative decreasing profiles.

In chapter 4, closed-loop supply chains were developed via compartmental analysis for a

single-input single-output system, and two input-two output systems, in which inventory levels

and demand rate, present a stable behavior consider the linear supply chain system nature.

In chapter 5, a production-inventory system was developed with a third-order dynamical

behavior, applying Pontryagin Maximum principle via present-value Hamiltonian optimal control.

Proper Lyapunov stability analysis for the dynamical system was conducted, taking into account

the simulations for the production-inventory system.

6.2 FUTURE WORK

To extend the results achieved in this research work, to develop the following:

Mathematical modeling.

- To extend the application of compartmental analysis for closed-loop supply chains and

the circular economy taking into account the sustainability impacts in manufacturing

supply chains.

69

- To develop stochastic mathematical models to incorporate uncertainty levels within the

supply chains.

- To incorporate the time-delayed impact in the dynamic supply chains, via the use of

lead times.

Control oriented approaches.

- To develop novel control structures incorporating: Model Predictive Control, and

Economic Model Predictive Control, for the previous mathematical models developed.

- To extend robust control strategies for supply chains which incorporates uncertainty in

inventory levels and demand rates.

- To apply control based computational intelligence approaches such as chaos control,

Fuzzy control, Neural networks, etc. for novel supply chains.

- To extend results for data-driven control in supply chains.

Stability analysis.

- To incorporate novel stability analysis for control structures with extended

mathematical modeling for dynamic linear supply chains.

- To develop Lyapunov stability analysis for nonlinear supply chains, with time-delayed

state-space formulations.

70

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Glossary

Mathematical model: A mathematical model is a mathematical description of the behavior of some

real-life system or phenomenon. Such as physical, sociological, economics, etc.

System: In systems science, by definition, a system is a group of components which interacts

towards keeping a collection of relationships with the sum of the systems themselves to other

entities.

Supply chain management: Refers to the cooperation process management of materials and

information flows between supply chains.

Closed-loop supply chains: Closed-loop supply chains integrates forward and reverse logistics,

towards the main interest to process finished goods in a re-manufacturing analysis, to achieve

sustainable solutions for enterprises.

Optimal control: An optimal control is defined as an admissible control, which minimizes a

functional objective.

100

Appendix

Equations (6) and (7) are used to calculate the equilibrium points such as: ��1 = ��2 = 0. Therefore,

Equation 6 is reduced to:

−(k

ρC) +

1

ρC(u(t)

x1) = 0 (A1)

Recalling 𝑥1 = ℎ, this provide the equilibrium point:

ue = khe (A2)

With the assumption that 𝑢 = 0, in equation (5) this leads to:

−h

h− (

k

β) = 0 (A3)

Integrating over time, finally, this leads to:

he(t) = e−K

βt (A4)

101

Vita

Yasser Alberto Davizon Castillo, received a PhD in Engineering Sciences major in

Mechatronics Engineering (2014), MS in Automation Sciences (2003), and BS Physics

Engineering (2000) from Tecnológico de Monterrey, Campus Monterrey. His research work is

related to optimal control of production-inventory systems, Model Predictive Control for

Management Sciences, and control-oriented approaches for mechanical systems.

Contact Information: yadavizonca@miners.utep.edu