An application of distributed model predictive control to supply chain management

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FEDERAL UNIVERSITY OF SANTA CATARINA GRADUATE PROGRAM IN PRODUCTION ENGINEERING

Éder Vasco Pinheiro

AN APPLICATION OF DISTRIBUTED MODEL PREDICTIVE CONTROL TO SUPPLY CHAIN MANAGEMENT

Florianópolis 2017

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Éder Vasco Pinheiro

AN APPLICATION OF DISTRIBUTED MODEL PREDICTIVE CONTROL TO SUPPLY CHAIN MANAGEMENT

Dissertation presented to the Grad-uate Program in Production Engi-neering in partial fulfillment of the requirements for the degree of Mas-ter in Production Engineering, area Logistic and Transport.

Advisor: Prof. Enzo Morosini Frazzon, Dr.

Florianópolis 2017

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Ficha de identificação da obra elaborada pelo autor,

através do Programa de Geração Automática da Biblioteca Universitária da UFSC.

Pinheiro, Eder Vasco

An Application of Distributed Model Predictive Control to Supply Chain Management / Eder Vasco Pinheiro ; orientador, Enzo Morosini Frazzon, 2017. 122 p.

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós Graduação em Engenharia de Produção, Florianópolis, 2017.

Inclui referências.

1. Engenharia de Produção. 2. Cadeias de Suprimento. 3. Controle Preditivo Distribuido. 4. Planejamento Logístico Operacional. I. Frazzon, Enzo Morosini. II. Universidade Federal de Santa

Catarina. Programa de Pós-Graduação em Engenharia de Produção. III. Título.

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ACKNOWLEDGEMENTS

I am grateful to the Federal University of Santa Catarina, its Graduate Program in Production Engineering and my adviser Enzo Morosini Fraz-zon, for letting me achieving a Master degree. Through the course of these years, many people contributed somehow to this final dissertation, I thank them all, but especially Dr. Jürgen Pannek.

This work should be dedicated to my family which deserves my thanks for all the support they have given to me. I am especially grateful to my wife Vanessa who gave me diary support.

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Time delays between taking a decision and its effects on the state of the system are com-mon and particularly troublesome.

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RESUMO ESTENDIDO

Os produtos e servi¸cos não apresentam valor até que eles literal-mente estejam no lugar e no instante que os consumidores precisam. O conjunto de princípios e técnicas que busca garantir esses valores baseados no tempo e na localiza¸cão é atribuído a defini¸cão de logística. Portanto, sempre que houver um processo incluem o planejamento, a implementa¸cão e o controle de fluxos de informa¸cões e de bens ou servi¸cos entre origem ofertantes e destinos consumidores, então trata-se de um processo logístico. A batrata-se para este trabalho é o planejamento operacional, parte do processo logístico em uma cadeia de suprimentos.

Uma Cadeia de Suprimento é um conjunto de estruturas e pro-cessos que uma ou várias organiza¸cões usam para atender a demanda de um grupo de clientes. Entre as várias formas de visualizá-la, pode-se descrevê-la através de uma rede ou grafo cujos nós são os elos da cadeia e os arcos as conexões entre eles. Entre o conjunto de processos da cadeia de suprimentos, a logística tem seu papel de destaque ampli-ado ao longo da evolu¸cão do seu conceito. Com isso, nas cadeias de suprimento, muitas vezes, a defini¸cão do processo logístico ultrapassa as barreiras de uma única organiza¸cão. Assim surge a Gestão da Cadeia de Suprimentos, que é composta por todas as atividades envolvidas na integra¸cão da cadeia de suprimentos e na coordena¸cão dos fluxos de materiais, os fluxos de informa¸cões e os fluxos financeiros. A integra¸cão da cadeia de suprimentos visa promover a vantagem competitiva à ca-deia como um todo em detrimento de apenas um dos seus elos.

Os desafios da gestão da cadeia de suprimentos surgem de pro-blemas relacionados às mudan¸cas no meio ambiente da cadeia, que normalmente contêm várias incertezas, especialmente as relacionadas a demanda e tempos dos processos. O planejamento, em todos os seus horizontes, constituiu uma a¸cão básica para este processo de busca por melhoria da eficiência, a performance logística. Entre as etapas de pla-nejamento, estratégico, tática e operacional, o Planejamento Operacio-nal em logística é uma atividade que desempenha um papel importante na supera¸cão de problemas de curto prazo na gestão da cadeia. Um dos problemas nesta atividade é a estabiliza¸cão das variáveis de estado, especialmente a variável nível de estoque.

O problema de interesse da proposta deste trabalho de disserta¸cão é a estabiliza¸cão e otimiza¸cão do nível de estoque contra a varia¸cão da demanda, de forma a buscar a maior redu¸cão possível do Efeito Chicote. Como consequência, este trabalho trata da liga¸cão entre a teoria de con-trole e gestão da cadeia de suprimentos através de Modelos de Concon-trole Preditivo. A motiva¸cão para apresentar um trabalho com essa

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gem surgiu a partir da leitura do trabalho Supply Chain Optimization

via Distributed Model Predictive Control (PANNEK; FRAZZON, 2014). Os

benefícios de aplicar a metodologia de controle preditivo à problemas logísticos provêm do fato de que tal metodologia facilita o uso de vários tipos de modelos preditivos do sistema e, também, por ela utilizar valo-res atuais e passados do processo enquanto prevê o comportamento do sistema em um horizonte de planejamento. Usando essa capacidade de previsão, os modelos de controle preditivo minimizam uma fun¸cão ob-jetivo e calculam uma sequência de controles satisfazendo as restri¸cões e o modelo do sistema.

A metodologia de controle preditivo para um modelo discreto do sistema segue uma idéia simples: a cada instante de tempo, usando o estado atual do sistema como estado inicial da sequência, a a¸cão de con-trole é obtida através da solu¸cão de um problema de concon-trole ótimo em um horizonte finito de tempo. Com isso se obtém sequência finita de valores de controle e dela somente o primeiro elemento é aplicado ao sistema. A simplicidade e qualidade desta metodologia mostra-se pelo amplo uso que ela tem em processos industriais. Além disso, as refe-rência bibliográficas indicam a prevalência dela no controle de sistemas complexos quando a solu¸cão dadas por equa¸cões de programa¸cão dinâ-mica são computacionalmente intratáveis em problemas com grande di-mensionalidade. Mais ainda, os controles distribuídos preditivos podem viabilizar a solu¸cão de problemas ainda maiores, dos quais a sociedade depende fortemente, como redes de tráfego rodoviário, redes de água, redes eléctricas e redes de transporte intermodal.

Aliado a adequabilidade de um framework para otimiza¸cão den-tro do planejamento operacional da logística de uma cadeia de supri-mentos, o cenário atual da introdu¸cão da tecnologia da informa¸cão e comunica¸cão como solu¸cões à logística motiva ainda mais este tipo de estudo, o qual tem forte tendência a se desdobrar como ferramenta para solu¸cão em tempo real às fortes mudan¸cas das premissas usadas para o planejamento operacional. De fato, as tecnologias da informa¸cão tendem a facilitar a coleta de dados e a atua¸cão remota, fazendo com que os dados estejam disponíveis online e em tempo real. Portanto, os aperfei¸coamentos dos modelos matemáticos e das metodologias que proporcionam apoio a tomada de decisão rápida são fundamentais.

As Cadeias de Suprimento têm a propriedade de serem sistemas distribuídos que podem ou não ter uma gestão centralizada. Por isso, apesar deste trabalho destacar um modelo preditivo distribuído para a cadeia logística de três estágios, o modelo com o controle centralizado também foi descrito e implementado. Assim, além deste trabalho apre-sentar um modelo conceitual baseado em Controle Preditivo Distribuído

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para a Gestão da Cadeia de Suprimento, para efeitos de compara¸cão do resultado, também apresenta um modelo com gestão centralizada. Por meio de experimentos desenvolvidos a partir da implementa¸cão dos modelos em Matlab, analisa-se algumas propriedades tais como estabi-lidade dos procedimentos, a influência do horizonte de planejamento no controle preditivo e o comportamento do Efeito-Chicote quando há varia¸cões da demanda final.

Como ferramenta de apoio a tomada de decisão, a abordagem apresentada busca dar maior entendimento da dinâmica do sistema, que se torna necessário aos gestores que participam de algum elo da cadeia e precisam considerar a dinâmica das rápidas mudan¸cas da de-manda. No contexto de uso desta abordagem em um cenário real deve-se considerar possíveis restri¸cões de acesso a informa¸cão entre elos da cadeia de suprimentos. Além disso, pelo problema em questão possuir o fator humano da tomada de decisão pelos gestores do processo logís-tico a aplica¸cão das variáveis de controle não acontece como acontece nos processos industriais, em que a a¸cão é executada instantaneamente. Todas essas questões teóricas ou experimentas são tratadas ao longo da disserta¸cão.

O primeiro capítulo apresenta detalhadamente o problema de interesse que gerou os desenvolvimentos realizados nesta disserta¸cão. Além de tratar o contexto do estudo, esse capítulo destaca os objetivos, as contribui¸cões e as delimita¸cões do trabalho. O segundo capítulo con-tém toda a revisão de literatura estruturada, assim como a metodologia utilizada para reunir as bibliografias e as expor de forma organizada. Portanto, esse capítulo dedicado à exposi¸cão das referências bibliográfi-cas foi organizado em quatro se¸cões, uma para descrever a metodologia, duas para apresentar o entendimento dos conteúdos básicos do tema de pesquisa e uma para apresentar o estado da arte do assunto específico do trabalho. No terceiro capítulo é feita toda a descri¸cão e dedu¸cão do modelo de controle preditivo formulado para o planejamento operaci-onal da cadeia de suprimentos com três elos. Como são apresentados o modelo de gestores distribuídos e com gestor centralizado, cada um deles possui sua própria se¸cão. Ademais as duas se¸cões, a terceira se¸cão apresenta uma breve visão sobre as restri¸cões sobre usos práticos dos modelos. O quarto capítulo apresenta o resultado dos experimentos re-alizados em Matlab. Esse capítulo contém uma se¸cão para análise dos resultados do modelo distribuído e outra para uma análise comparativa dele com o modelo centralizado. As conclusões são apresentadas no quinta capítulo assim com as observa¸cões sobre pesquisas que podem surgir como sequência ao que foi desenvolvido nesta disserta¸cão. Por fim, os códigos produzidos na implementa¸cão dos modelos estão

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pre-sentes após a lista das referências bibliográficas que foram citadas em todo o texto.

Palavras-chave: Cadeias de Suprimento. Controle Preditivo Distri-buído. Planejamento Logístico Operacional.

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ABSTRACT

A Supply Chain is a set of structures and processes an organiza-tion uses to deliver an output to group of customers. Among this set of processes, the subset, defined as the logistics, is dedicated to providing values to customers making products or services available to them at the appropriate location and time. The more quickly and completely the demand can be met, the better the logistics process performance. Broadening the concept of logistic, the Supply Chain Management en-compasses all activities involved in integrating a Supply Chain and co-ordinating materials, information and financial flows in order to fulfill the ultimate customer demands. This integration functionality aims to improve the competitive advantage of the Supply Chain as a whole.

The challenges in Supply Chain Management arise from prob-lems related to changes in the Supply Chain environment, which nor-mally contain some degree of uncertainty. The Planning is the first action to evolve this process forward improving efficiency. This way, the Operational Planning plays a great role in overcoming Supply Chain Management short-time problems. A problem in this step is stabilization of states variables, especially the stock level. The problem of interest is the implementation of a controller for a set of dynamically coupled linear subsystems called Three-Stage Supply Chain. Hence, the main purpose of this work is optimization and stabilization of stocks levels against demand variation considering this entire Supply Chain with a model predictive controller formulation.

The Model Predictive Control is one of the techniques which can be applied to systems adaptation. Its advantages are particularly the benefits relating the use of a system model, and both current and his-torical measurements of the process to predict the systems’ behaviour for a planning horizon. Using this prediction ability, it minimizes an objective function and calculates a control sequence satisfying the con-straints of the system. The Supply Chain (SC) has the property to be a distributed system, which can have centralized or even distributed managers. Therefore, this work presents a Distributed Model Predic-tive Control procedure to stabilize the stocks levels considering the dis-tributed view of the SC, and also a centralized Model Predictive Control for the purpose of comparability. Furthermore, it presents the results from a computational analysis of the application of both models.

Keywords: Supply Chain. Distributed Model Predictive Control. Op-erational Logistic Planning.

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LIST OF FIGURES

1.1 Flows representation of a three-stage SC . . . 28

1.2 Venn diagram representing the two axes of research and their intersection, which is the objective proposal research 32 2.1 Number of articles over time . . . 36

2.2 Pareto Criterion or Rule 80-20 . . . 37

2.3 Supply Chain interconnected elements . . . 41

2.4 Illustration of the Bullwhip Effect . . . 46

2.5 Model predictive control - rolling horizon illustration . . 49

2.6 Ilustration for network of the Beer Game . . . 57

3.1 Information and material flow for the three stage SC . . 65

3.2 Model predictive control - rolling horizon illustration . . 68

3.3 Procedure Flow: Decentralized Approach . . . 69

3.4 Information and material flow for the three stage SC . . 70

3.5 Procedure Flow: Centralized approach . . . 72

4.1 E1-S1: Graphics of Stock, Unfulfilled order and Order . . 75

4.5 E3-S1: Stock, Unfulfilled order and Order for distributed model . . . 79

4.6 E3-S1: Stock, Unfulfilled order and Order for centralized model . . . 79

4.7 E3-S2: Stock, Unfulfilled ordera and Order for distributed model . . . 80

4.8 E3-S2: Stock, Unfulfilled order and Order for centralized model . . . 80

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LIST OF TABLES

2.1 Number of papers by source . . . 38 3.1 Summary of variables’ notations for stage p and time tn. 65 4.1 Values for each experiments . . . 74

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LIST OF ABBREVIATIONS AND ACRONYMS

DMPC Distributed Model Predictive Control . 17, 30–33, 51, 52, 56–59, 63, 72, 78, 81, 83, 84

ICT Information and Communication Technology . . . . 30 M Manufacturer . . . . 27–29, 63, 64, 66, 75, 77 MPC Model Predictive Control 17, 29–33, 35, 38, 47–56, 59–61,

63, 66, 67, 69–73, 78, 81, 83

QP Quadratic Programming . . . . 52 R Retailer . . . . 27–29, 63, 64, 66, 69, 75 RH Receding Horizon . . . . 43 RHC Receding horizon control . . . . 43 S Supplier . . . . 27–29, 63, 64, 66, 69, 75, 77 SC Supply Chain 17, 19, 27–33, 39–45, 53, 54, 56–60, 63, 65, 66,

68, 70–72, 81, 83, 84

SCM Supply Chain Management 17, 27, 28, 30–33, 35, 38–42, 44, 47, 53, 54, 56, 59, 72, 83, 84

SQP Sequential Quadratic Programming . . . . 52 SS Supply Chain Stages . . . 63, 68, 70, 71

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CONTENTS

1 Introduction 27

1.1 Problem Context . . . 27 1.2 Problem Statement . . . 30 1.3 Objective, Contributions and Delimitations . . . 31 1.4 Research methodology . . . 32 1.5 Organization of the dissertation . . . 33

2 Literature Review 35

2.1 Search Methodology . . . 35 2.2 Supply Chain Management . . . 39 2.2.1 Logistics Management Concepts . . . 39 2.2.2 Oscillation and the Bullwhip Effect . . . 45 2.3 Model Predictive Control . . . 47 2.4 Applications of MPC in SCM . . . 53 2.4.1 Centralized models . . . 53 2.4.2 Decentralised or distributed models . . . 56 3 A Model Predictive Control Framework for Supply Chain

Operational Planning 63

3.1 A Distributed Managers Model . . . 63 3.2 A Centralized Manager Model . . . 69 3.3 Some restraints about using the model in a real system 72

4 Experiments 73

4.1 Distributed Model Results . . . 75 4.2 The Centralized against the Distributed Model . . . . 78

5 Conclusion 83

Bibliography 87

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1 INTRODUCTION

This document describes the work required to get a Master’s Degree in Production Engineering. In this chapter there is context, purpose, focus and significance for this study. Furthermore, it presents a description of the methodology used to perform the work in a scientific context. The last chapter subject is the overview of the following chapters.

1.1 PROBLEM CONTEXT

Products and services would not have value unless they are with customers when and where they wish to consume them. There is a set of principles and techniques which seeks to achieve this value and it is called the logistical process. The Logistic is the pro-cesses of "planning, implementation, and controls the efficiency, of effective forwards and reverses flows and storage of goods, services and related information between the origin and consumption points in order to meet customers’ requirements" (CSCMP, 2013). This way, the logistic process creates value by timing and positioning in-ventory. Then, the more quickly and completely the demand can be met, the better the logistics process performance. This process is the combination of a firm’s order management, inventory, transporta-tion, warehousing, materials handling, and packaging as integrated throughout a facility network (BOWERSOX et al., 2002).

The SC, also called the value chain or demand chain, are the network of organizations engaged in provide a product or service from supplier to customer, or in the words of Chopra and Meindl (2010), it consists of all parties involved in fulfilling costumers’ re-quirements. That logistic definition given by (CSCMP, 2013), and reinforced by (BALLOU, 2006), implies that logistic is part of a larger process which consists in the management of SC. That is the understanding considered for this dissertation, which also means to agree with Bowersox et al. (2002) who wrote that logistics, in contrast to Supply Chain Management (SCM), is the work required to move and position inventory throughout a supply chain. As such, logistics is a subset of and occurs within the broader framework of a SC.

The Figure 1.1 shows a simplified representation of flows in a three stage SC consisting of a Supplier (S), a Manufacturer (M), and a Retailer (R). More specifically, this SC is a logistic chain, but throughout this work it is treated by its broad meaning, the

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28 Chapter 1. Introduction

chain of suppliers. Over these stages there are flows of information and materials. The SCM has the task of integrating organizational units along a SC and coordinating all the flows (STADTLER; KIL-GER, 2007; CSCMP, 2013). Then, the great interest is to manage appropriately these flows to reduce costs while maintaining a good service level, i.e, there is no shortage of what customers demand to the Retailer.

Figure 1.1: Flows representation of a three-stage SC.

Source: Own elaboration

These information flows are essentially the order of goods to the upstream stage, while the material flows are its downstream displacements. In the Figure 1.1, Supplier and Manufacturer are upstream from Manufacturer and Retailer, respectively, and on the other side, R and M are respectively downstream from M and S. Therefore, for example, the Manufacturer asks the Supplier what it needs (information flow), and the Supplier gives the Manufacturer its requests (material flow).

Along the three-stage SC, dynamics are driven by the cus-tomer demand, which is the quantities of goods cuscus-tomers want per day. Then, demand triggers all the information and material flows. Therefore, when Retailer attends the demands for sale, it is impacted by reducing its stock level, that is the buffer of goods avail-able for customers purchase. Every day, the Manufacturer receives an order from the Retailer. An order is a quantity that a stage be-lieve will be needed to fulfill its present end future demands. Fur-ther, the order from the Retailer is the demand that the Manufac-turer seeks to fulfill. This demand information triggers a process inside this stage. This process consists of checking stock and dis-patching goods to satisfy the quantity demanded until the present day. Checking stock is to verify the quantity just available for selling

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1.1. Problem Context 29

and the dispatching consists of delivering that quantity as soon as possible. While delivering the products to the downstream stage, it is natural to exists a delay into that operation. Such delay should be taken into account because it can reduce performance in the logistic process.

The foregoing flows between Retailer and Manufacturer simi-larly exists for Manufacturer and Supplier. Then, every day the Man-ufacturer orders the quantity it needs for the Supplier, which should check stock and prepare for the delivery of goods downstream. This context assumes the Supplier does not order at any other stage and it is self-sufficient, however there is a delay between the Supplier identifying its needs and having goods available.

The context of the Three-Stage SC is the same as the "Beer Game", which Sterman (1989) created to explain many phenomenon occurring when flowing information and products between differ-ent agdiffer-ents. Although the name refers to beer, the product can be in-terpreted as any kind. For example, when used amongst high school students, this game is recast as the "apple juice game". When this game is applied in a company to explain what happens in SC, then it is customized to represent the product of its industry. Here, for the illustration purpose, it can be thought of as the Supplier offer-ing bottles to the Manufacturer, who brews and "bottles" the beer, and then ships it to the Retailer for sale to customers. Despite the simplicity of the beer game, it is an example of a situation where cyclical instability arises from the failure of decision makers to ac-count for time delays (STERMAN, 2000).

The dynamics of SC exists under an optimization process which depends on the demand over time. Then, its continuous necessity for adaptation makes the Model Predictive Control (MPC) a natural framework to deal with this class of problem (IVANOV et al., 2012). The MPC, or receding horizon control, has now become a standard control methodology for industrial and process systems. Its wide adoption from the industry is largely based on the inherent ability of the method to efficiently handle constraints and the non-linearity of multi-variable dynamic systems (SARIMVEIS et al., 2008).

The MPC methodology follows a simple idea: at each discrete-time instance, using the current state of the system as the initial state, the control action is obtained by on-line solving a finite-horizon open-loop optimal control problem. In this way, a finite-optimal con-trol sequence is obtained, from which only the first element is kept and applied to the system, that here is the SC illustrated by Fig-ure 1.1. The procedFig-ure is repeated after each state transition. The

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30 Chapter 1. Introduction

MPC is prevalent in the control of complex systems where the off-line solution of the dynamic programming equations is computa-tionally intractable due to the high problem dimensionality (CAMA-CHO; BORDONS, 2007).

The planning is part of logistic process, and managing the in-formation and material flows between suppliers and customers into the SC consider decisions, or planning, levels classified as strategic (long term), tactical (medium term), and operational (short term) (CHOPRA; MEINDL, 2010). These levels are defined by the time period in which decisions are to be made and operational planning works with the decisions which need more frequent to be taken. The daily decisions is the interest of this work besides the MPC which is suitable for models this problem, giving a solution to support oper-ational planning.

In addition to the suitability of the MPC framework for opti-mization on SC planning, the current logistic scenario which Infor-mation and Communication Technology (ICT) rapidly being intro-duced has also motivated this research. Indeed, this ICT facilitates data collection and remote actualization, making data more avail-able online. Then, improvements of math models and frameworks are even more important to better explain the dynamics of many distributed systems. Special attention is given to the feasibility of the application of this rolling horizon framework, and also the im-pact of service level for the Bullwhip Effect, also called Whiplash Effect, or Forrest Effect (FORRESTER, 1961).

1.2 PROBLEM STATEMENT

The problem of interest is to implement a controller for this set of dynamically coupled linear subsystems called Three-Stage SC in order support operational planning decision-making of stock level, ordering rate and unfulfilled ordering. These subsystem could have a central or a distributed manager. Then a centralized and a distributed MPC are implemented. These model description are presented in Chapter 3. The choice for both approaches is because, although decision centralization or a more available of customer de-mand information are better solutions proposed to reduce Bullwhip-Effect (CHEN; LEE, 2012), some industries cannot have it.

What first motivated this work was the propose to investigate the suitability of the Distributed Model Predictive Control (DMPC) to the optimization in Supply Chain Management, as pointed by

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1.3. Objective, Contributions and Delimitations 31

Pannek and Frazzon (2014). Furthermore, this paper has induced this work which intends to verified some of the theoretical points about an implementation of a consolidated conceptual model and, then to study more about the application of the Distributed Model Predictive Control (DMPC) to optimize the Supply Chain opera-tional planning. Stabilization and feasibility are seeking to provide that each subsystem does not deviate too far from the previous policy, consistent with traditional MPC move suppression penalties. This infeasibility could exist because, even though some degree of coordination is desired, the stockholders cannot divulge all the in-formation about their local states and objectives. For that reason, a centralized approach has also been implemented as a way to com-pare the distributed model against the centralization.

1.3 OBJECTIVE, CONTRIBUTIONS AND DELIMITATIONS

The general objective of this Master’s Dissertation is to ana-lyze the applicability of DMPC to model the SC dynamic and provide optimization to operational planning management of the Three-Stage logistic chain.

This objective lies in the intersection of two research axes, which are DMPC and SCM planning. Therefore, the scope of this work considers these axes and, especially, their intersection, as Fig-ure 1.2 presents.

Aiming this general objective, this work outputs four specific contributions which are its specific objectives:

• Describe a conceptual model for application of Distributed Model Predictive Control to optimization of operational plan-ning of manufacturing SC;

• Develop a simulation tool in Matlab for the problem of a Three-Stage SC;

• In addition to technical programming, implementation effects have to be analyzed and documented with respect to input disturbances on the chain a well as to the influence of the horizon of the controller;

• Describe a conceptual centralized MPC, coding it in Matlab, and compare with the distributed approach.

What delimit the boundary of this work is the conceptual plication. Although the models were implemented, there was no

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ap-32 Chapter 1. Introduction

Figure 1.2: Venn diagram representing the two axes of research and their intersection, which is the objective proposal research.

plications to real case application. Then, the system model was the engine processing for processing the MPC proposed. Besides, a for-mal prove o convergence and stabilization of the procedures which were implemented were not presented.

1.4 RESEARCH METHODOLOGY

The research methodology is deductive-experimental contain-ing three steps with asynchronous executions. The first step is the bibliographical revision that results in a reference model intending to present the state-of-art around applications of MPC to SCM. The second step contains the statement and deduction of the models for the Three-Stage SC in a MPC framework. Finally, the third step con-cerns the experiments and analysis regarding the results from the model which was implemented in Matlab.

The bibliographical study of the two axes which Figure 1.2 displays, i.e, axis MPC and axis SCM, was based mainly on clas-sic literature. However, when searching references for the subject into the axes intersection, some other interesting texts were studied and considered amongst the collection of references. The searching for documents of applications of MPC to SCM, or more specifically, DMPC to SCM, the Scopus and Science Direct are the main research

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1.5. Organization of the dissertation 33

databases. The keywords used in this searching were "Model pre-dictive control” and “Supply chain management”, both locally con-nected and restricted to the articles titles, keywords and abstract. Besides, the progress of the research also used some references of the bibliographies which resulted from the searching.

The construction of the general models, also called concep-tual model, starts from literature results and follows with a visual design to clear the steps. Finally, the conceptual model was imple-mented in Matlab, which were the tool to generate the data and graphics for analyzing the results.

1.5 ORGANIZATION OF THE DISSERTATION

This dissertation contains five chapters. They were organized in a sequence intending to present a constructive explanation of the dissertation subjects.

• Chapter 1 gives context, purpose, focus, significance and de-limitation for this study.

• Chapter 2 contains the literature review organized into two parts: fundamental topics (Section 2.2 and Section 2.3, which are important for understanding the developments described in the following chapters; and the state of art related applica-tions of MPC and DMPC to SCM are in Section 2.4.

• Chapter 3 has the explanations and deduction of the equations and procedures for the models which are objectives of this work. This chapter has two main section, each one describing one of the models, the distributed model (Section 3.1) and the centralized model (Section 3.2). Following these two sec-tions, the last one delineate some issues concerning practical implementation of the methodology into a logistic system. • Chapter 4 present the experiments which were performed for

the problem of a three-stage SC run in the Matlab environ-ment. There are two sections in this chapter, one section is ded-icated to the experiments concerning the distributed model and another to a comparative analysis between the the dis-tributed and a centralized model.

• Chapter 5 has the conclusion of the dissertation. It comments about the results achieved and a perspective for further devel-opments relating this subject.

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2 LITERATURE REVIEW

The review of literature concerning the subject of this disser-tation follows the axis presented at Figure 1.2 of Section 1.3 which are components parts for this work. Therefore, the review starts from general to specific, i.e, the fundamental topics are presented first, and then the applications, which were found in the literature. The theory behind both general axes, Model Predictive Con-trol (MPC) and Supply Chain Management (SCM), are fundamen-tal for this work. They provide the theoretical grounding needed in understanding what following, applications of MPC to solve SCM some planning problems. Furthermore the search for works already dealing with that kind of approach to SCM planning problems is very important, because its a way to present what has been written on this topic of work.

2.1 SEARCH METHODOLOGY

The methodology of literature review is organized in two parts. The first one is about the fundamental topics and second con-cerning the specific theme of the dissertation. Fundamental topics review were based only on books that could be accessible.

Aiming to capture what has already been done into the inter-section of the research’s axes showed by Figure 1.2, a more com-plete methodology is proposed. It follows four steps:

i. identifying the suitable set of keywords;

ii. searching for the collection of articles in the databases; iii. selecting the Portfolio of articles to be analyzed;

iv. studying the articles.

The step i required a previous overview to got the keywords which best describe the subject. In order to get the keywords, the study of some works which could give a general vision about the theme was necessary. This first group of articles for a general vision were those which have simultaneously the expressions "Control The-ory" and "Supply Chain Management" Chain Management" in their titles, keywords or abstract. The chosen databases as sources for searching the bibliographies in this study was Scopus and Science Direct. The reasons for this choice are flexibility for logical com-binations of keywords during the search and the great amount of available articles.

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36 Chapter 2. Literature Review

These articles has made possible identifying the suitable set of keywords necessary for the step ii. Hereby,the selected words for the first subject were "model predictive control" and "receding horizon control", and for the second one were "supply chain" and "supply chain management". The step ii follows the rule: if at least one of those keywords about the first subject and at least one of those about the second subject are in the article’s title, or abstract, or keywords, then the article is selected, i.e, at least one keyword of each subject should be mentioned. This search resulted in 104 articles sourced just from journals and proceedings.

Although the searching keywords were searched among the words from titles or keywords or abstract of the articles, some of the 104 articles are not related to the subject of this work. Because of that, after reading all hundred and four abstracts, five articles were excluded from the set. Therefore, the step ii results in a set of 94 ar-ticles that is the input for step iii. The graphic of Figure 2.1 presents the distribution of these articles along the years, and this series in-dicates growth in the bibliographical production along these years.

Figure 2.1: Number of articles over time.

The step iii has the purpose to select articles considering their relevance. Thus, a filter was applied over the set of 94 articles. This filter is based in the Pareto Rule by considering the numbers of

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ci-2.1. Search Methodology 37

tations. The number of citations for each article were taken from Google Academics. These quantities are in the graphic of Figure 2.2 which shows that 85% of the citations are from 20% articles. There-fore, eighty articles (or 20% of 94) was selected. Nevertheless, all of the eighty articles were published between the years 2003 and 2013. Considering that the articles which were published after 2013 could not have enough time have so many quantity of citations as the articles before 2013, then all the articles sourced from journals and published after 2013 were added to the portfolio. Then, at the end of step iii the resulted portfolio amount is thirty one, that is the sum of thirteen articles published after 2013 into the initial set of works and the remainder eight articles after applying the Pareto Rule filter.

Figure 2.2: Pareto Criterion or Rule 80-20.

The Table 2.1 presents the number of articles by source for the selected portfolio. This table also shows the source type and its qualification from Capes 2014. The source Computers & Chemical Engineering, which is Qualis Capes 2014 A1 has the most quantity of papers published about this subject.

Step iv allowed classifying the thirty-one articles in two main groups related to the centralization feature of the models. Then, first group contains the papers with centralized focus on the con-trol technique, while second one contains the a decentralized or distributed one.

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Table 2.1: Number of papers by source.

Source Papers Type Capes

Computers & Chemical Engineering 7 Jour. A1 Computers & Operations Research 3 Jour. A2 Proceedings of the American Control 3 Conf. Inter. Journal of Produc. Economics 2 Jour. C Computers & Industrial Engineering 2 Jour. B1

Automatica (Oxford) 2 Jour. A1

IEEE Trans. on Auto. Control (Print) 1 Jour. A1 Euro. Journal of Operational Research 1 Jour. A1

Annual Reviews in Control 1 Jour. A2

Int.Workshop Ass. and Fut. Dir. NMPC 1 Conf. IEEE Trans. on Automatic Control 1 Jour. A1 Proc. in Applied Math. and Mechanics 1 Jour. B4 Indu. & Eng. Chemistry Research 1 Jour. A2 Inter. Journal Advan. Manufac. Tech. 1 Jour. B1 Trans. of the Inst. Meas. and Control 1 Jour. NF

Journal of Forest Research 1 Jour. B2

Logistics Research 1 Jour. NF

Chemical Engineering Transactions 1 Jour. B5 Source: Own Elaboration with data from QUALIS CAPES 2017 and Google Academics

The following sections present the literature review organized into fundamentals (Section 2.2 and Section 2.3), and specific appli-cations of MPC to solve SCM some planning problems (Section 2.4). The fundamentals topics for this dissertation are both general axes: Model Predictive Control and Supply Chain Management and its not an extensive exposition, although it indicates the literature which have more details concerning the subjects. This axes were illus-trated by the scheme displayed in Figure 1.2 of Section 1.3. By the way, this axes provide the theoretical grounding needed in under-standing what following in next sections and chapters. The specific applications, presented in Section 2.4, contains some main points about the collection of articles with is the state of art on the main

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2.2. Supply Chain Management 39

subject of this dissertation.

2.2 SUPPLY CHAIN MANAGEMENT

Several authors tried to put the essence of SCM into a con-cise definition. During the nineties until present days many of them have done it very well, but each definition has small differences compared to other. Nevertheless, their mindset consists of a target group to be managed, the objectives and the means for achieving these objectives (STADTLER, 2005; CHRISTOPHER, 1992). In this context, this chapter concern logistics concepts and issues, its evo-lution to SCM, and the values it produces to customers into a SC, which then is also called as value chain.

2.2.1 Logistics Management Concepts

The concept of logistics evolves from the beginning of civi-lization. Previously it was basically related to the transport of goods among productions and consumers centers (BALLOU, 2006). The techniques and worldwide markets evolution brought to light poten-tial gains that logistic alternatives could allow. Today the concept of logistic is concerned with the effective and efficient availability of goods or services through a network of suppliers and customers.

The CSCMP (2013) defines logistics as the processes of "plan-ning, implementation, and controls the efficient, of effective for-ward and reverses flow and storage of goods, services and related information between the point of origin and the point of consump-tion in order to meet customers’ requirements". This way, the lo-gistic process creates value by timing and positioning inventory, it is the combination of a firm’s order management, inventory, trans-portation, warehousing, materials handling, and packaging as inte-grated throughout a facility network (BOWERSOX et al., 2002). Lo-gistics is important because it more the available goods or services it creates value for all stakeholders. Value in logistics is expressed in terms of time and place. Products and services would not have value unless they are in the with customers when and where they wish to consume them. To many firms throughout the world, logis-tics has become an increasingly important value-adding process for time, space, and others consequent reasons (BALLOU, 1997).

The Supply Chain, also called the value chain or demand chain, are the network of organizations engaged in provide a prod-uct or service from supplier to customer, or in the words of Chopra

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and Meindl (2010), it consists of all parties involved in fulfilling cos-tumers’ requests. That logistic definition given by (CSCMP, 2013) and accepted by (BALLOU, 2006) implies that logistic is part of a larger process which consists in the management of SC. That is the understanding considered for this dissertation, which also means to agree with Bowersox et al. (2002) who wrote that logistics, in contrast to supply chain management, is the work required to move and position inventory throughout a supply chain. As such, logistics is a subset of and occurs within the broader framework of a SC.

Supply Chain Management can be defined as the manage-ment of material, information and financial flows through a net-work of organizations that aim to produce and deliver products or services to consumers (TANG, 2006). This broad concept has many variants and they rely on two main notions: a logistic framework, and the objective to achieve linkage and coordination between the processes of all involved entities, i.e, suppliers, customers and in-vestors (CHRISTOPHER, 1992). Then, a good understand of what is the SCM must contemplate the relation between the work of agents in the chain and the framework of logistics. Further, SCM consists of firms collaborating to leverage strategic positioning and to improve operating efficiency. The evolution of logistics as integrated logistics serves to link and synchronize the overall supply chain as a continu-ous process and it is essential for effective supply chain connectivity (BOWERSOX et al., 2002). While the purpose of logistical work has remained essentially the same over the decades, the way the work is performed continues to radically change.

The object of SCM is obviously the SC (STADTLER, 2005). Each organization has a set of processes that makes it works. As written by Sterman (2000), a Supply Chain is the set of structures and processes an organization uses to deliver an output to a cus-tomer. Further, these organizations compound a system organized in a network (JINGSHUANG et al., 2008). As discussed by Aitken (1998), this network consist of nodes, representing organizations, and links between them, manifesting the interactions within them.

The macro view of SC while creating values to all stakehold-ers consists of the stock and flow structures for acquisition, storage, conversion of inputs into outputs, and the management policies gov-erning the various flows. Due to the not clear barrier between logis-tic and SC, this work call SC as synonymous to logislogis-tic chain. Aitken (1998) has adapted the theoretical understanding of Supply Chain to the context of networks. Then, he considers the SC as a network of connected and interdependent organizations mutually and

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co-2.2. Supply Chain Management 41

operatively working together to control, manage and improve the flow of materials and information from suppliers to end users. For a better understanding of SC and its environment, its considered necessary to expand their theoretical understanding to the context of networks consisting as a type of coupled system. This type of coupled system has four highly interconnected elements such as: suppliers, manufacturers, distribution networks and customers, as shown in Figure 2.3.

Figure 2.3: Supply Chain interconnected elements.

Whilst the term Supply Chain Management (SCM) is now widely used (NOVAES, 2007), it could be argued that it should re-ally be termed demand chain management to reflect the fact that the chain should be driven by the market, not by suppliers (CHRISTO-PHER, 1992). Equally, the word chain should be replaced by net-work since there will normally be multiple suppliers and, indeed, suppliers to suppliers as well as multiple customers to be included in the total system.

The SCM concept is easier understood with the knowledge that businesses boundaryless are exceed, meaning that internal func-tional barriers are eroded in favor of horizontal process

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manage-42 Chapter 2. Literature Review

ment; externally, the gap between vendors, distributors, customers and the firm close gradually (CHRISTOFIDES et al., 2013). Today’s turbulent business environment has caused a greater awareness among managers of the financial dimension of decision making, with business managers progressively becoming more driven by the goal of enhancing shareholder value.

The suppliers in a chain are involved simultaneously in sev-eral other chains (CHRISTOPHER, 1992). For discussion purposes it is useful to outline the supply chain as a single and independent entity but, in reality, it is contained within a network of organiza-tions. The aim of many SC studies has been to attempt to isolate and analyse individual phenomena, instead of relating the issues to the broader and more general processes and structures, of the networks in which they are embedded. The law of reductionism has been ap-plied to understanding supply chain integration and management. However, as observed, supply chains exist within the context of net-works, which must be able to compete in the market as the supply chains they contain. Perceiving the supply chain as a single line en-tity disguises the complexity in which it exists (STERMAN, 2000).

The decision for producing a product begins from the tomers. A restrict concept to SCM system could be viewed as cus-tomers requesting a favourable commodity by visiting retailers in a distribution network, and this motive is transferred to manufactur-ers to meet the opportunity through a network of supplimanufactur-ers, manu-facturers, distributors, and retailers. This network is named a pro-duction/distribution/inventory system because it consists of an in-ventory management part, a logistic network, and production pro-gramming.

In a SC, a set of decision-maker facilities (suppliers, manufac-turers, warehouses, distributors, and retailers) cooperate for getting the demands or forecasting them, ordering, procuring materials or outsourcing parts of the production procedure, manufacturing or assembling and finishing final products, stocking inventory, trans-porting, and finally, delivering the final products to customers. Sup-ply chains should be programmed by an efficient manager (or set of managers) who applies previous experiences with modern meth-ods. This combination is named a supply chain management system and runs two processes of decisions and actions: as orders and ship-ments.

In some of the industries, such as industries that have a defi-nite customer or contract, changing in demand patterns across time rarely happen (CHRISTOPHER, 1992). In the supply chain of these

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products, in a specific time, such as at the beginning time of con-tract, or at the beginning time of production, a correct pattern of customer demand is forecasted by famous methods, and production planning goes ahead for having suitable customer demand planning. The problem is solved offline in this time, as zero time, and its out-put is used for programming and scheduling shipments between supply chain entities, manufacturing planning, and holding or man-aging inventory volumes (AITKEN, 1998). Therefore, in this situ-ation, the optimization problem of the supply chain management system, considering the required constraints, is just solved for zero time and then production, warehousing, and distribution plans are given to related persons, managers, and employers of each division.

The programming and control of SC can be made by differ-ent methods, such as deterministic analytical models, stochastic an-alytical models, and simulation models coupled with desired opti-mization objectives and network performance measures. To employ control methods with prediction abilities is very suitable for supply chain management systems, because decision prediction by looking to future demand exists inherently (IVANOV et al., 2012). Reced-ing horizon control (RHC) is one of them that uses the RecedReced-ing Horizon (RH) concept.

In general, supply chains operate as pull systems driven by the orders that customers place to the retailers, and their general operation is as follows: retailers accumulate orders from customers and commit to satisfy them as long as they arrive before a certain deadline. Orders arriving after the deadline will be logged for the next period (STERMAN, 1989). At the end of the a some period, retailers start satisfying the accumulated orders upon product avail-ability, and if the products are in stock, the retailers pack and ship them to the customer, otherwise, either those orders remain in the file of orders to be fulfilled (if the company follows the policy of backorders), or the orders are lost.

Since product availability is the key factor to keep a good level of customer satisfaction, retailers need to estimate their fu-ture demands and place the respective replacement orders to their supplying nodes to get the product. However, if they place more orders than required they would pay extra storage and inventory holding costs (CHRISTOFIDES et al., 2013). If they place fewer or-ders than needed, then their customer satisfaction level would drop. This process repeats itself throughout the distribution network until the orders reach the manufacturing site, where the plant manager has to define a production and a raw material acquisition plan to

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satisfy them.

The challenges in supply chain management arise from prob-lems related to changes in the Supply Chain environment. Because the management of the upstream and downstream relationship, from suppliers to customers, aim to deliver superior customer value at less cost, this optimization problem tends to be multiobjective. Fur-ther challenges arise as the Supply Chain Management focuses on achieving a more profitable outcome for all parties within the chain (CSCMP, 2013). Then to achieve that focus, in SCM it is necessary to plan and run the chain, to manage and review the results, and to replan the process.

Solving the logistic problems, regarding their difficulties are activities happening every day around the world, within each type of supplying process. The difficulty is overcome by rationalizing the Supply Chain Management planning processes. Planning is some-times associated with optimization in SCM, i.e, it seek efficiency as benefit. The planning process normally is organized into strate-gic, tactical and operational levels (STADTLER; KILGER, 2007). The strategic level deals with decisions that have a long-lasting effects on the firm. This includes decisions regarding the number, location and capacities of warehouses and manufacturing plants, or the flow of materials through the logistics network. The tactical level typ-ically includes decisions that are updated anywhere between once every quarter and once every year. This includes purchasing and pro-duction decisions, inventory policies and transportation strategies including the frequency with which customers are visited. Finally, operational level refers to day-to-day decisions such as scheduling, routing and loading trucks (CHOPRA; MEINDL, 2010).

All SCM complexities turn to challenges faced by SC man-agers who are seeking to enhance shareholder value is to identify strategies that improve free cash flow generation (STADTLER; KIL-GER, 2007). Today, for competitive customer service to be main-tained the market requires environmentally friendly products, a good portfolio mix, rapid development of new products, high quality and reliability, after-sales services, etc. Furthermore, SC managers need to consider the dynamics of a rapidly changing market environment, such as variability in demand, cancellations and returns, as well as the dynamics of internal SC operations, such as processing times, production capacity pitfalls and the availability of materials. These operational SC risks and disruptions can have severe long-term ef-fects on a firm’s financial performance.

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2.2.2 Oscillation and the Bullwhip Effect

The SC are distributed systems with a strong coupling be-tween the actions of the stockholders. Consequently the performance of the whole system depends on the correct coordination of all the SC echelons. The normally fails for this coordination make appear issues of stream fluctuations in the chain (STERMAN, 2000). This persistent instability and oscillation is known as the phenomena of the Bullwhip Effect.

This phenomenon consists of the increment of the variabil-ity of the order rates towards the producers, as the illustration Fig-ure 2.4 shows. Sterman (2000) has pointed that it is the result of a bad coordination between these agents of the SC. The economic effects of this are translated into unnecessary costs to the compa-nies, that must invest into extra capacity or subcontract to cope with the high variations in the demand (CHOPRA; MEINDL, 2010; DISNEY, 2007). In this context, support tools take into account the restrictions imposed by the different causes of the bullwhip effect, such as: batching, shortage gaming, lead-times and demand signal processing, and are very useful to help in the decision making tasks.

Field and experimental studies show that people often ignore the time delays in a wide range of systems (STERMAN, 1989; PIN-HEIRO et al., 2016). The management policies are designed to keep the stock at their target levels, compensating for usage or loss and for unanticipated disturbances in the environment. Often there are important delays between the initiation of a control action and the result, creating a supply line of unfilled orders. These patterns of behavior are fundamental to the basic physical structure of stock management systems and supply chains.

Oscillation arises from the combination of time delays in neg-ative feedbacks and failure of the decision maker to take the time delays into account. Field and experimental studies show that peo-ple often ignore the time delays in a wide range of systems. The beer game are but one example of situations where cyclical insta-bility arises from the failure of decision makers to account for time delays. There is not one single cause for the failure to account for time delays and the supply line (STERMAN, 1989). Ignoring time delays is one of the fundamental misperceptions of feedback that leads to poor performance in systems with high dynamic complex-ity. Accordingly to Sterman (2000), the failure to understand the role of time delays worsens the faced instability and leads to more surprises usually unpleasant reinforcing of the belief that the world

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is inherently capricious and unpredictable, and strengthening the short term focus still more.

Figure 2.4: Illustration of the Bullwhip Effect.

A range of factors, from information availability to individual incentives, all contribute. But behind these apparent causes lies a deeper problem (MOYAUX et al., 2007). The supply line is often in-adequately measured, but if people understood the importance of the supply line they would invest in data collection and measure-ment systems to provide the needed information. Alongside of this, compensation incentives often encourage people to ignore the de-layed consequences of today’s actions, but if investors understood the structure and dynamics of the market they could redesign com-pensation incentives for their agents to focus on long-term perfor-mance.

The bullwhip effect has been recognized in SCM literature as one of the main barriers in improving supply chain performance (FU et al., 2015). Whereas much has been done on understanding and reducing the bullwhip effect in two echelon or multi-echelon sup-ply chain, prior efforts for limiting order variation largely focused on information sharing (MOYAUX et al., 2007). In fact information exchange has been regarded as one of the main ways for taming the bullwhip effect.

Because the fierce competition in present global markets, the introduction of products with short life cycles and the heightened expectation of customers have forced manufacturing enterprises to invest in and focus attention on their logistics systems. This, to-gether with changes in communications and transportation tech-nologies, for example, mobile communication and overnight

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deliv-2.3. Model Predictive Control 47

ery, has motivated continuous evolution of the management of lo-gistics systems. In these systems, items are produced at one or more factories, shipped to warehouses for intermediate storage and then shipped to retailers or customers. Consequently, to reduce cost and improve service levels, logistics strategies must take into account the interactions of these various levels in this logistics network. This network consists of suppliers,manufacturing centers,warehouses, dis-tribution centers and retailer outlets, as well as raw materials, work-in-process inventory and finished products that flow between the facilities (BRAMEL; SIMCHI-LEVI, 1997).

The treatment of logistic as a strategic business lead it to an integrating scenery. Novaes (2007) considers this scenery as an up-per stage of logistic evolution. Moreover, the supply chain manage-ment consists of the effective integration of the main entities of the supply chain. This view, therefore, indicates the SCM as an evolu-tion of logistics pattern. Likewise, Aitken (1998) indicates supply chain management as "an integrative approach to dealing with the planning and control of the material flow from suppliers to end users".

2.3 MODEL PREDICTIVE CONTROL

The MPC is a process control methodology that is being in-creasingly employed across several industrial sectors (CAMACHO; BORDONS, 2007). The popularity of MPC in industry stems in part from its ability to tackle multivariable processes and handle pro-cess constraints. Furthermore, accordingly to Camacho and Bordons (2007) it perhaps is the most general way of posing the process con-trol problem in the time domain. The MPC is not a specific concon-trol strategy, but rather an ample range of control methods, a method-ology, to generate controllers. The main difference of MPC from an-other control strategy, such as stochastic dynamic programming and optimal control, is that the control input is not computed a priori, as an explicit function of the state vector, but it is computed on-line the rolling-horizon. Thus, MPC is prevalent in the control of com-plex systems where the off-line solution of the dynamic program-ming equations is computationally intractable due to high problem dimensionality (VENKAT, 2006).

The MPC methodology uses the model of the system and the concept of open-loop optimal feedback. The system model is used to predict and optimize the future system behavior (PANNEK; GRüNE,

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2011). Moreover, past and current state measurements are the in-puts used to estimate the current state of the system at each time step. Then, with the system model as constraint, an optimization problem is solved to determine an optimal open-loop policy from the present (estimated) state (MAYNE et al., 2000). The MPC trick is to inject only the first output move into the plant (the system). At the subsequent time step, the system state is re-estimated using new measurements. The optimization problem is resolved and the optimal open-loop policy is recomputed.

A system is a set of things together, or parts of a mechanism, or an interconnecting network (OXFORD, 2016). It constitutes a part of the universe of interest to some study and it is characterized by a relationship between its inputs and outputs. Specifically for the context of this dissertation, the system is the Three-Stage logistic chain presented in Figure 1.1. The model of a system mimics its behavior, but it is not the real world but merely a human construct to help us better understand the systems. In general all models have an information input, an information processor, and an output of expected results. The type of a model is the way in which the system model is described mathematically: transfer function or state-space (NEGENBORN; MAESTRE, 2014).

The basic MPC elements are the prediction system model, the objective function, and obtaining the control law (CAMACHO; BORDONS, 2007). Although practically every possible form of mod-elling a process appears in a given MPC formulation, this chapter will describe the issues related to the methodology in the context of discrete time systems. The restricting of the exposure due to the focus of the subject of this dissertation. Moreover, the exposition is restricted to quadratic objective function, and to only a few opti-mization methods to solve nonlinear models with that kind of objec-tive. Optimization is an important issue to MPC because to obtain the control law it is necessary solving a minimization problem, and this is going to be described in the following.

A discretization of the time is enumerated of time periods, that is, the time is counted in digital values tn in \BbbR whose the val-ues of n are from \BbbN . When a system or its model exists in the dis-crete time, it is called a disdis-crete time system. This kind of system is controlled when from each time instant tn, its process state x(tn), taking values in \BbbR d_{, has is future behavior x(t}

n+1)influenced by a control input u(tn)from \BbbR mwith the rule of a function

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2.3. Model Predictive Control 49

This function is the law or dynamics of the system. Then, for all n in \BbbN and state value x(tn)in \BbbR d, the next state is

x(tn+1) = f (x(tn), u(tn)). (2.1) For any system system such as Eq. 2.1, the MPC methodology steps is based on the following simple idea:

1. At each discrete time instance tn, as present in Figure 3.2, the control action is obtained by solving on-line a finite-horizon open-loop optimal control problem, using the current state of the system as the initial state.

2. A finite-optimal control sequence is obtained, from which only the first element is kept and applied to the system. The proce-dure is repeated after each state transition.

Figure 2.5: Model predictive control - rolling horizon illustration.

This procedure of two steps can lead to a tracking control, which is the task to determine the control inputs u(tn) such that

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x(tn) follows a given reference xref(tn) as good as possible. This means that if the current state is far away from the reference, then the controller should want track the system towards the reference and if the current state is already close to the reference, then it should to keep it there. The stabilization problem exists when the reference is constant, that is, xref(tn) = xfor all n \in \BbbN , and x is given a real number. Then, besides the full generality the tracking problem, with such a constant reference it is reduced to a stabiliza-tion problem (PANNEK; GRüNE, 2011). Since it is necessary to be able to react to the current deviation of x(tn) from the reference value, the controller should have u(tn)in feedback form, i.e., in the form u(tn) = \mu (x(tn)) for some map between the state space and the control space

\mu : \BbbR d\rightarrow \BbbR m_.

In MPC the function \mu is not generate a priori, but it is build by the process of optimization in the rolling-horizon procedure.

The future outputs for a determined horizon N , a fixed inte-ger number which is called the prediction horizon, are predicted at each instant tn using the process’ model from Eq. 2.1. These pre-dicted outputs x(tn+1)for n varying from 0 until N depend on the known values up to instant tn and on the future control signals u(tn). The set of future control signals is calculated by optimizing a determined criterion to keep the process as close as possible to the reference trajectory, which can be the set-point itself or a close approximation of it. This criterion is a function

L : \BbbR d\times \BbbR m_{\rightarrow \BbbR}

whose domain is Cartesian product of state and control spaces. This function usually takes the form of a quadratic function of the errors between the predicted output signal and the predicted reference trajectory, that is, for n natural number equal or less than N

L(x(tn), u(x(tn)) = d \sum i=1 x2 i(tn) + m \sum i=1 u2 i(tn). (2.2) Furthermore, for some systems the state and control variables can only take values into intervals. These are called resource constraints and have form

lx \leq x(tn) \leq ux lu\leq u(tn) \leq uu,

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2.3. Model Predictive Control 51

where the lower bounds lxand lu, and the upper bounds uxand uu are constant real numbers.

In order to generate the control values at each time instant the optimization problem with objective function Eq. 2.5 and con-straints by Eq. 2.3, bounds for the stages, and by the system model, the equation Eq. 2.1. Therefore, the short notation for this optimiza-tion problem is SPn_: _{\mathrm{m}\mathrm{i}\mathrm{n}} _{J (x(t} 0), u) u subject to (2.1) and (2.3) (2.4) where J (x0, o) = N +n \sum k=n L(x(tk), u(x(tk)). (2.5) The control signal u(tn)is sent to the process whilst the next control signals calculated are rejected, because at the next sampling instant x(tn+1)is already known. The process is repeated with this new value and all the sequences are brought up to date. Thus the u(tn+1)is calculated (which in principle will be different from the u(tn)because of the new information available) using the receding horizon concept.

In this formulation there is just one controller, and then it is called the a centralized MPC. When a series of static optimization problems the standard MPC formulation Eq. 2.4 then it is called a descentralized controller. Depending on the communication reules among the descentralized systems it is called a DMPC. The model whose fully description is presented in Section 3.1 is one example of DMPC formulation.

In general, the controllers can be classified depending on how many of them participate in the solution of the control problem and the relative importance between them (NEGENBORN; MAESTRE, 2014). The control system is centralized if there is a single controller that solves the plant-wide problem. The control is decentralized when there are local controllers in charge of the local subsystems of the plant that require no communication among them. When there are different control layers coordinated to take care of the process the control system is hierarchical. In this case, upper layers manage the global objectives of the process and provide references for the lower layers, which control directly the plant. Finally, if the local

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controllers communicate in order to find a solution for the overall control problem the control system is distributed.

The MPC is usually implemented in a centralized controller which has the full knowledge about the process and calculates the whole control sequence for the system (SCHERER et al., 2015). Although, the size of the problems faced today by control engi-neers has grown enormously as the limitations imposed by the com-munication and computational capabilities decrease. In this sense, there are strong incentives to have decentralized or distributed con-trol schemes, such as DMPC. Indeed, accordingly to Negenborn and Maestre (2014), the society heavily depends on infrastructure systems, such as power grids, water distribution networks, traffic systems, road-traffic networks and intermodal transport networks. Nevertheless, these systems have also several drawbacks that have to be taken into account, being the main one the loss of performance in comparison with a centralized controller. This loss depends on the degree of interaction between the local subsystems and the coor-dination mechanisms between the agents (NEGENBORN; MAESTRE, 2014).

The type of coordination mechanism that can be realized rely upon the information structure, i.e, the connectivity and capacity of the interagent communication MPC network (CAMPONOGARA et al., 2002). Therefore, the DMPC frameworks can be divided into cooperative and non-cooperative strategies. A cooperative strategy exists when the agents controlling the subsystems optimize over a common overall objective considering only local variables. In a non-cooperative strategy, the agents exchange information but optimize their own objectives (SCHERER et al., 2015). Under some condition about coordination mechanism, it is possible that a DMPC has the same performance as the centralized MPC, such as the case proved in Camponogara and Scherer (2011) for cooperative strategies.

This kind the optimization problem Eq. 2.4 is a quadratic program for which efficient algorithms exist for its solution such as Trust Region Algorithms, Interior Point Algorithms, Quadratic Programming (QP) and Sequential Quadratic Programming (SQP) (MARTINEZ; SANTS, 1995). Those are optimization methods and explicit solutions can be obtained because the criterion is quadratic, the model is linear, and there are no constraints. In constrained opti-mization, the general aim is to transform the problem into an easier subproblem that can then be solved and used as the basis of an it-erative process. A characteristic of a large class of early methods is the translation of the constrained problem to a basic unconstrained