Caroline Thierry

Gérard Bel

André Thomas

1 Supply

Simulation for supply Chain management: An Overview

come

-the system under study is the supply chain of a given enterprise, and we can consider:

-the internal supply chain of an enterprise which focus on functional activities and processes and on material and information flows within the enterprise. In this case supply chain management may be viewed as the integration of previously separate operations within a business enterprise.

-the external supply chain of the enterprise which includes the enterprise, the suppliers of the company and the suppliers' suppliers, the customers of the company and the customers' customers (SCOR). In this case supply chain management mainly focuses on integration of operation and cooperation between the enterprise and the other actors of the supply chain.

-the supply chain under study is a network of enterprises (without a focus on one particular enterprise of the supply chain): a supply chain is a "network of organisations that are involved, through upstream and downstream linkages, in the different processes and activities that produce value in the form of products and services in the hands of the ultimate consumer." ([CHR 92]). In this viewpoint, the focus is on the virtual and global nature of business relationships between enterprises. In this case supply chain management mainly focuses on cooperation between the supply chain actors.

Supply chain management

Supply chain processes: the integrated supply chain point of view

To describe supply chains with a process point of view, we refer to the Supply Chain Operations Reference (SCOR) model. SCOR is a cross-industry standard for supply chain management and has been developed and endorsed by the Supply-Chain Council (SCC). SCOR focuses on a given company and is based on five distinct management processes: Plan, Source, Make, Deliver, and Return. Supply chain management addresses different types of problems according to the concerned decision horizon. Long range (strategic) decisions are concerned with supply chain configuration: number and location of suppliers, production facilities, distribution centres, warehouses and customers, etc. Medium and short ranges (tactical and operational) are concerned with decisions material management: inventory management, planning processes, forecasting processes, etc.

On the other hand information management is also a key parameter of supply chain management: integrating systems and processes through the supply chain to share valuable information, including demand signals, forecasts, inventory and transportation etc. In this collaborative supply chain, as far as a supplier-buyer partnership is established, several problems arise:

-how to exchange/share information? -is it possible to perform mutual problem solving? -how to set up global supply chains indicators?…

Thus the problem of the centralisation or distribution of the information and decision processes within the supply chain becomes a main challenge for the supply chain managers.

Material flow

Simulation for Supply Chain Management

Why to use simulation for Supply Chain Management?

As far as simulation is concerned the objective is to evaluate the supply chain performances. We distinguish tree ways to conduct Supply Chain performance measurement:

• analytical methods, such as queuing theory,

• Monte-Carlo methods, such as simulation or emulation,

• physical experimentations, such as lab platforms or industrial pilot implementations. In this context of Supply Chain, analytical methods are impractical because the mathematical model corresponding to a realistic case is often too complex to be solved. Obviously, physical experimentations suffer from technical-and cost-related limitations. Simulation seems the only recourse to model and analyse performances for such large-scale cases. Simulation enables, on the one hand, the design of the supply chain and on the other hand, the evaluation of supply chain management prior to implementation of the system to perform what-if analysis leading to the "best" decision. This simulation includes supply chain flows simulation and decision process dynamics. In the field of supply chain management, simulation can be used to support supply chain design decisions or evaluation of supply chain policies. As far as supply chain design decisions are concerned, the following decisions can be considered:

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How to use simulation for Supply Chain Management?

To attempt to specify the different ways to use simulation for SCM it is important to differentiate, on the one hand, the real system (the "real world") and on the other, its simulation model.

In fact, the simulation model must be built according to its usage and/or the SCM function that we want to model or to evaluate. Different classes of models can be highlighted to understand the variety of SC simulation models according to:

-the systemic decomposition of the SCM system -Decision system -Information system -Physical system -the level of distribution of the system: -simulation model for centralized SCM system evaluation A centralised SCM system consists in a single information and decision system for the different entities of the supply chain under study.

-simulation model for distributed SCM system evaluation A distributed SCM system consists in a distribution of the decision system over different entities of the supply chain under study.

As a matter of fact, the execution of the simulation can be performed: -In a centralised way on a single computer

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In a decentralised way: -on multiprocessor computing platforms : parallel simulation -or on geographically distributed computers interconnected via a network, local or wide: distributed simulation.

As far as decentralisation of the simulation is concerned, "the execution of a single main simulation model, made up by several sub-simulation models, which are executed, in a distributed manner, over multiple computing stations". [TER 04] The need of a distributed execution of a simulation across multiple computers derives from four main reasons [TER 04]:

-to reduce execution simulation time […] -to reproduce a system geographic distribution […] -to integrate different simulation models that already exist and to integrate different simulation tools and languages […] -to increase tolerance to simulation failures […]" -to test independently different control models -to progressively deploy a control system, -to prepare at SC control changes" More over it is important to stress that simulation mostly focuses on the dynamics of the supply chain processes concerning both physical and decision systems (i.e. production management systems see §1.3.1.).

Supply chain management simulation types

This section is dedicated to the presentation of the different types of models and approaches mainly used for supply chain management simulation.

As seen before, an important part of the model is the decision system model (hierarchical planning and control processes). So the first subsection presents the main production management models which are used in supply chain management.

Then, the different kinds of well known simulation models will be quickly presented. For each of them we will highlight how the different production management models can be linked with the simulation model.

Production management models focus

The objective of this part is to focus and to present a very synthetic and simplified description of production management models in order to introduce, in a following part, how they can be integrated in a supply chain simulation model. Here we focus only on the production processes even the approach could be extended to supply and distribution processes.

There are two main categories of production management models.

Time bucket models

In production planning and control, and mainly for long and middle term, we are concerned with the determination of quantities to be produced per time period for a given horizon in order to satisfy demand or/and forecast. To perform these decision processes, time bucket models are needed. They are characterized by:

-Decision variables: produced, stocked or transported quantities -Data: capacities of the resources (in number of parts per period, for example) -Constraints: conservation of flow, bill of materials, limited capacities, demand satisfaction… Example: For a production line composed of two production resources (see figure 1.6.). The demand is d t and the production resources capacities are C R1,t , C R2,t . Each item is produced from one single component.

The planning model variables are: -x Ri ,t = quantity of items to be produced on resource Ri during time period t -y Ri ,t = quantity of items to be transported from resource Ri during time period t -Ii Ri,t = input inventory level on resource Ri at the beginning of time period t.

-Io Ri,t = output inventory level on resource Ri at the beginning of time period t. The planning model constraints are:

-Ii R1,t+1 = Ii

-x R1,t ≤ C R1,t -x R2,t ≤ C R2,t -Ii R1, t0 = ∞ -Ii R, t ≥ O ∀R∈{R1, R2}, ∀t -Io R, t ≥ O ∀R∈{R1, R2}, ∀t -x R,t ≥ 0 ∀R∈{R1, R2}, ∀t -y R, t ≥ 0 ∀R∈{R1, R2}, ∀t
Joint to these models, methods are used to perform the plan: MRP like methods, mathematical programming, constraints programming, meta-heuristics.

Starting time models

In production planning and control, and mainly for short term, we are also concerned with the determination of the starting time of tasks on different resources. For that we use starting time models (sequence of timed events). These models are characterised by: -Decision variables: starting time of tasks (t i ) -Data: ready dates (r i ,) due dates (d i ) -Constraints: precedence, resource sharing, due dates Example: -t i ≥ r i -t i ≥ t j + p j OR t j ≥ t i + p i -t i + p i ≤ d i Joint to these models, methods are used to perform the schedule: mathematical programming, constraints programming, meta-heuristics…

Simulation types

Due to the special characteristics of supply chains, the supply chain simulation model building is difficult. The two main difficulties are highlighted, and then the different types of models for supply chain management simulation are quickly presented.

Size of the system

One characteristics of supply chain simulation is the huge number of "objects" to be modelled. A supply chain is composed of a set of companies, composed of a set of factories and warehouses, composed of a set of production resources and stocks. Between all these production resources circulate a set of components, parts, assembled parts, sub-assemblies and final products. So the number of "objects" of the model can be very large.

Complexity of the production management system

To simulate a system it is necessary to simulate the behaviour of the "physical" system and the behaviour of the "control" system. For a supply chain this implicates that it is necessary to model the behaviour of the supply chain management system of each company and the relation between these production management systems (cooperation).

As this supply chain management system is very complex, it can be difficult to model it in details. However it is absolutely necessary to model it, as it is this system which controls the product flow in the supply chain. So, according to objective of the simulation study and the kind of chosen model, various aggregated or simplified models of the production management system must be designed. Following sections present different examples of these models.

Different kinds of models for supply chain management simulation

Simulation model

A simulation model is composed of a set of "objects" and relations between these objects; for example: in a supply chain main objects are items (or set of items) and resources (or set of resources).

Each object is characterised by a set of "attributes". Some attributes have a fixed value (example: name) others have a value which varies during the time (example: position of an item in a factory).

The state of an object at a given time is the value of all its attributes. The state of a system at a given time is the set of the attributes of the objects included in the system.

The purpose of a simulation model is to represent the dynamic behaviour of the system.

There are various modelling approaches according to how state variations are considered:

-States vary continuously: continuous approach -States vary at a special time (event): discrete event approach.

The following parts of this section will introduce the chapters 2 to 4 which will go into details on the viewpoint and present related works (State of art and recent works).

Simulation of supply chain management using continuous simulation approach

In this section we will introduce system dynamics, a continuous simulation approach where states vary continuously. Chapter 2 will go into details and present recent works related to this viewpoint of simulation for supply chain management.

System dynamics

This new paradigm has been first proposed by Forester for studying "Industrial Dynamics".

Companies are seen as complex systems with [KLE 05] :

-different types of flows: manpower, technology, money, and market flows.

-stocks or levels which are integrated into time according to the flow variations System dynamics is centred on the dynamics behaviour. It is a flow model where it is not possible to differentiate individual entities (like transport resources).

Management control is performed by making variations on rates (production rates, sale rates,…). Control of rates can be viewed as a strong abstraction of common production management rules.

The model takes into account the "closed loop effect": the manager is supposed to compare continuously the value of performance indicator to a target value. In case of deviation he implements corrective actions.

Example:

-I t2 = I t1 + p(xr t1,t2 -dr t1,t2 ) -xr t1,t2 = production rate between two dates t1 and t2 -dr t1,t2 = sale rate between two dates t1 and t2 -p = time duration between t1 and t2

Production management models/ simulation models

The two models do not consider the same objects states.

-In systems dynamics objects are continuous flows. The behaviour of these flows are represented by a differential equation (with derivate) which is integrated using a time sampling approach.

-In planning models the objects are resources and their activities. It is considered that the attributes of these activities change only at a special periodic dates. There is no notion of derivate. This kind of model seems well adapted to supply chain simulation as it has been design by Forester for "Industrial Dynamics" studies which used same concepts as those used lately in supply chain studies.

Simulation of supply chain management using discrete event approach

In this section we will detail the discrete event approach. We will distinguish between time bucket driven approach and even driven approach. This differentiation is based on the time advance procedures which characterized these two approaches. Chapter 3 will go into details and present recent works related to this viewpoint of simulation for supply chain management.

For the "discrete event approach" they are:

-different ways to "see the world": activities, event and process The main practices to "mix" various types of models and time advance procedures are listed below:

Time bucket driven approach

Discrete events simulation using time bucket driven approach is rarely used for job shop simulation but it well fits for simulation of supply chain management (see the specific characteristics of this simulation in §1.3.2.1 and §1.3.2.2).

Time bucket driven discrete events models

In such a model: -time is divided in periods of a given length: time bucket -time is incremented step by step with a given time bucket. At the end of each step a new state is calculated using the model equations. So in this approach it can be considered that events (corresponding to a change of the state) occur at each beginning of a period -the lead time for an item on a production resource is considered small compared to the size of the time bucket -the main states are the states of resources (or set of resources) during a given period: they describes the activities in which resources are implicated in a given time period. They are characterised by the quantities of items processed in this activity in a given time period: for example the number of items of a given type manufactured, stocked or transported by a given resource in a given period.

-the simulation has to determine all the states of all the resource at each period of a simulation run. This kind of model is also called "Spreadsheet simulation" [KLE 05]. We don't adopt this designation because spreadsheet is a tool with which it is possible to use all the modelling approach.

Simulation models

It must be noticed that the planning models presented in § 1.3.1 are also time bucket models which are well known and used in production management domain. We will see hereafter that they are very similar to time bucket driven discrete events simulation models but that they are used in a different way in simulation.

In order to illustrate that, we consider a very simple example of a production line composed of two production resources with no specific production management. Shop floor control is a first-in first-out strategy, k is the number of parts from M1 to be used to produce one part on M2. The simulation model uses the following state variables:

-Ii Ri,t is the input inventory level of resource Ri at the beginning of time period t. -Io Ri,t is the output inventory level of resource Ri at the beginning of time period t. It can be noticed immediately that this model is very similar to the production management model presented in § 1.3.1.1.

In order to illustrate that, let's consider a simulation with this model corresponding to the following hypothesis: resource R1 sends parts to resource R2 according to a production and transportation plan determined outside of the system. So Ii R1,t0 , Ii R2,t0 , x R1,t , , x R2,t , y R1,t , y R2,t are known at the beginning of the simulation.

In this case, the true state variables of the model are Ii R1,t , Ii R2,t , Io R1,t , and Io R2,t .

The simulation must determine the variation over time of this variables taking into account the values of the exogenous variables (x R1,t , , x R2,t , y R1,t , y R2,t ) . Thus simulation allows evaluation of the proposed production and transportation plan. It is also possible to introduce hazard in the behaviour of the model. This shows that the same model can be used in a: -simulation decision process: taking into account x R1,t x M2,t , y R1,t and y R2,t . . the problem is to determine Ii R1,t , Ii R2,t , Io R1,t , and Io R2, .

-production planning decision process: in a centralised planning (APS or SCM like) the problem is to determine x Ri,t and y Ri,t which satisfy the constraints of the planning model (stock capacity, supplier demand ). Notice: it is possible to use a "what if" approach with the planning model testing different demands or different production management policies. In this "what if" approach the problem is solved several times, each time with this different data. Then it is possible to see the influence of these data on the generated plan. This approach is not considered in this book: we refer to simulation only when the dynamics of the system is considered. 1.3.4.1.3. Production management models/ simulation models Now the question is: how the different production management models can be linked to a discrete events simulation model with time bucket approach? Time bucket production planning model can be easily linked to the global simulation model as the modelling approach is the same. In this case the two models will be joined up: the simulation model focuses on the circulation of the flow of parts, the planning model determines the quantities to be produced. The Chapter 3 provides a study of both discrete events and time bucket simulation used for supply chain management and proposes case studies to illustrate the pivotal role that simulation can play as a technique for decision aid.

If we consider now the other category of production management models that we call in § 1.3.1.2 "starting time models" (scheduling,…) we can state that:

-"time bucket driven discrete events simulation models" do not use the same "objects states" than "starting time production management models" (which use "start time of an activity").

-between two periods the bucket driven activities simulation model does not represent the state of the system. So the start time of an activity is not known and cannot be used as data in a "starting time" scheduling model. The only way to obtain a good approximation of this date can be to use a very small time period. But this is often not possible because this will contradict the fundamental hypothesis for this kind of model: the production duration for an item on a production resource is very less than the time bucket of the model.

Event driven approach

In this sub-section the main characteristics of the discrete events models for supply chain management simulation using event driven approach is presented. Remember that this approach is intensively used for job shop simulation. So it can be consider as convenient to use this kind of model for supply chain simulation.

However using the specific characteristics of supply chain management simulation (see §1.3.2.1 and §1.3.2.2) can induce some difficulties for this kind of simulation. The main difficulty comes from the size of the model induced by this context. It can be not efficient to model the circulation of each individual part in each production resource of the different companies of the supply chain: the number of events can become prohibitive and slow down considerably the simulation which can become unworkable. It is why it is often necessary to use model reduction techniques introduce here after §1.5.2. As seen before, chapter 3 provides a study of both discrete events and time bucket simulation used for supply chain management and proposes case studies to illustrate the pivotal role that simulation can play as a technique for decision aid. We remind hereafter the main characteristics of this approach.

1.3.4.2.1. Event driven approach for discrete events simulation In a event driven discrete events model: -the main states are the states of items (or set of items) -the simulation must determine the dates of all the events (state variation) which occurs during a simulation run -each state is characterised by the resource utilised by a given item at a given time and correlatively the "occupation state" of the resources. Example: position of a given item ("on a given production resource" , "in a given stock", or "being transported by a given cart")

-each state variation is represented by a "state variation logic" -time advance event to event. A "simulation engine" using " a "time table" determine the date of the "next event" (ex: the delivery date of a job).

1.3.4.2.2. Production management models/ simulation models Consider again the question which is to know how the different planning models can be connected in a event driven discrete events simulation?

The time buckets planning models cannot be directly connected to an event driven discrete events simulation because the modelling approach is not the same. We will see chapter 3 how different adaptations can be realized in order to allow connections.

The "starting time models" presented in §1.3.1.2 (scheduling level) can be directly connected to an event driven discrete events simulation because they use the same modelling approach.

In summary for this event driven approach:

-Simulation models and planning models does not use the same "objects states". -Simulation models and scheduling or shop floor control models use the same "objects states"

Simulation of supply chain management using games

In this section we will introduce the business games then chapter 4 will go into details and present recent works related to this viewpoint of simulation for supply chain management.

Games and simulation

Different games can be used to perform simulation. Games enable simulating real conditions off-line, and exploring new ideas or strategies in a safe, interactive and also fun environment. Basically, the complexity of their model allows splitting games into two classes:

-board games have a model simple enough to be played with tokens or pieces that are placed on, removed from, or moved across a "board" (a premarked surface, usually specific to that game).

-sophisticated games have a more realistic model requiring, for example, to be run on computerized devices.

Production management models/ simulation models

In this kind of simulation model (board games), the simulation of time can be made using either a clock which synchronises the players, or the time of the simulation is the real time (each player evolves independently). [KLE 05] distinguishes:

-Strategic games: in these games every player represents a company competing or collaborating through other companies by interacting with the simulation model during several rounds. The well-known beer game belongs to this category.

-Operational games: in these games every player represents an actor (e.g. a worker of a workshop) interacting with the simulation model either during several rounds or in real time. Examples include games for training in production scheduling.

Decision systems and simulation models (systems)

The preceding sections have presented the main concepts which are used in supply chain management simulation and introduce the first part of the book (chapter 2 to 4).

The second part of the book is dedicated to the problem of distribution of the supply chain management simulation. This concept of simulation distribution is extremely important in the case of supply chain simulation because of the naturally distributed aspect of the supply chain by its elf. This section introduces this part of the book (chapter 5 to 10).

Models and system distribution

There is a consensus on the architecture of simulation environments putting the emphasis on modularity between the control system CS and the shop-floor system SF. This separation principle enables to introduce the concept of emulation. Emulation is not new, it is used in automation to test computer aided manufacturing software, for example [COR 89]. Fusaoka proposed a theoretical formulation and an experimental run consisting in verifying the assertion SF∧CS⊃G [FUS 83], where G is the required performance level of the shop floor. The real shop floor system (SFr) may be replaced by a model (SFm) , that we call an emulated shop floor. Likewise, a model of the control system (CSm) can be used instead of the real one (CSr). Therefore, four experimental situations can be defined, using either models or real systems [PFE 03]:

1. (SFr, CSr) Experimentation consists in deployment of the real control system at the Shop floor. It is the more classical case. 2. (SFr, CSm) A Control system model is applied for the real shop floor. This configuration could be used to test a new control system. 3. (SFm, CSr) The real control system is used with a shop floor model (emulation with the real control system) 4. (SFm, CSm) Both shop floor and control system are modelled.

Let's first focus on a single company of the supply chain or on a centralized SCM system. The real system is made up of the physical system, the information system and the control (or decision) system (Cases 1 and 4 on Figure 1.13). These three systems compose the SCM system. Basically, building a simulation model leads to design a virtual model representation of these (or at least one of these) three preceding systems implemented on a computer S1, as seen in §1.3. Let's now consider different cases where this model can be distributed.

If we want to evaluate the effect of different control rules, on a specific physical system, it could be interesting for example to build an emulation system corresponding to this physical system. This emulation model being controlled by the real decision System (Case 3 on Figure 1.14.) connected to the actual information system. Actually, emulation aims at mimic the behaviour of the physical system only. It can be seen as a virtual shop floor which can be connected to an external control system. Like simulation, emulation can be used to model complex cases, but emulation remove the additional task of modelling decision processes (this task being often one of most difficult as stated by [ VAN

Figure 1.14. Emulation system connected to real control system

Using emulation provide modularity between tests cases and control systems to be tested. This modularity is useful to try a control system in various situations, or to try various control systems on a same test case. It can also be useful to validate the real control system before actually deploying it.

Obviously, the same concept can be use for a Supply Chain System (Figure 1.15. and 1.16.). But, Supply Chain being networks of companies often independents (i.e. § 1.1.1.) , simulation models can be built in a centralised way (Figure 1.15.) or in a distributed way (Figure 1.16.). In a distributed context, different simulation models can be implemented on different computers, each one representing companies' behaviour.

Figure 1.16. Distributed simulation models of a Supply Chain

As underlined in the preceding sections, SC management systems are traditionally organised in a hierarchical way. Different decision functions exist: Planning, Master Scheduling, detail scheduling and Control according to the classical MRP² System describe by [VOL 05]. This kind of architecture exists in each company belonging to the SC. In internal SC, the same ERP software could be used, but in external one, often different information and decision systems ERP must be connected, leading to interoperability problems and/or synchronization problems.

The following parts of this section will introduce the chapters 5 to 10. These different chapters will describe respectively simulation problems relative to centralized architectures, simulation synchronization problems and distributed simulation architectures.

Centralized simulation

The decisions that are usually taken before planning on the implementation of any supply chain can be classified into two categories: structural (for long term objectives) and operational (for short term goals). Simulation can be used as a tool for carrying out the decision making process for both structural and operational decisions thanks to dynamic simulation of material flow and taking into account of all random phenomena. In opposite of distributed simulation, in a centralized approach, one single simulation model reproduces all the supply chain structures (entities and links).

The chapter 5 is dedicated to this kind of approach. It presents a brief literature review on supply chain centralized simulation and discusses on two developed centralized simulation approaches. Effectively, for most simulation evaluation approaches, supply chain processes are modeled to perform "what-if" analysis. Firstly, a discrete-event simulation-based optimization is used to estimate the operational performances of the solutions suggested by the optimizer. The optimizer was developed based on the NSGA-II, which is considered one of the best multiobjective optimization using a genetic algorithm [START_REF] Pratap | A fast and elitist multiobjective genetic algorithm: NSGA-II[END_REF].

Moreover this chapter 5 illustrates the applicability and efficiency of the two preceding approaches using three industrial applications. In the first one, a case study from automotive industry will be presented. The objective is to improve the profitability and responsiveness of the company's supply chain by redesigning its production-distribution network. A centralized simulation-based optimization approach is used for the optimization of facility open/close decisions, production order assignment and inventory control policies. In the second case, the authors applied the centralized simulation-based multi-objective genetic algorithm approach to a real-life case study of a multi-national textile supply chain, which consists of several suppliers, a single distribution center and an aggregated customer. The modeling and simulation details are discussed and numerical results are presented analyzed. In the third case, another automotive industry case, a generic model is proposed, which can be used by different automotive industries. The developed model is limited to only the interactions between the assembly line and its direct suppliers. Taking into account that the model is generic, it is able to help supply chain decision-makers in their choices.

Multi Agents Systems decision simulation

New forms of organizations have emerged from Supply Chain concept in which partners have to collaborate and have strong collaboration. Producing enterprises operate as nodes in a partner network and share activities to produce and deliver theirs goods. In such context, integration of planning of all the nodes is needed, that is to say, that partners have to be able to distribute and synchronize their activities.

To obtain the optimal performance level in such dynamic environment, multiagents Systems (MAS) can be used. Effectively, MAS are composed (as a Supply Chain network is) of a group of agents that can take specific roles within the organizational structure. Different agents may represent different objects belonging to the studied network. This idea is not new; Parunak used agents for manufacturing control or collaborative design ([PAR 96] or [START_REF] Parunak | What Can Agents Do in Industry, and Why? An Overview of Industrially-Oriented R&D at CEC[END_REF]) nevertheless these approaches are particularly well adapted when studying supply chain management.

Chapter 6 and 7 are dedicated to MAS usage for SCM. The first one highlights the interest to use MAS for Supply Chain simulation and the second one, considers MAS decisional system simulation for enterprise network.

Just like a Supply Chain in which distributed activities and decisions are made in order to obtain a global optimal performance, MAS simulation leads to a distributed system, within there is generally no centralized control, to have a global point of view; where agents act in an autonomous way and do not have locally the global knowledge, to obtain nevertheless a global optimum. Effectively, several analogies between Supply Chains and MAS can be highlighted:

• The multiplicity of acting entities •

The entities properties, abilities or decision-making capabilities… • Information sharing and task distribution…

A review of research works on agent-based supply chains modelling and simulation is also made in chapter 6. On the other hand, chapter 7 presents specific contributions in agent-based Supply Chains modelling and simulation for decision system development. The first part of this chapter will concern the Supply Chain control, and the second one, is related to the design of decision system based on simulation.

Simulation for Product Driven Systems

As said before, in the distributed Supply Chain and manufacturing control context, Multi agent Systems (MAS) are often used according to the fact that each company could act, in some circumstances, in an autonomous way. Consequently, it is possible to implement agents to describe their behaviour. Thus, the SCM System could be composed by planning and scheduling agents and by agents representing physical elements as products, for example. Moreover, it is also possible to built emulation model for distributed supply chains. This kind of model can be made in a centralized or distributed way (using several models and computers for physical, control and decision systems). This last possibility is interesting for all contexts where products and/or physical entities are able to take some autonomous decisions. The main idea, is to focus the decisionmaking processes as near as possible to the shop-floor or physical system, where events (disturbing or not) actually occurs. Current researches focusing on autonomy are for instance Holonic Manufacturing Systems (HMS), multi-agents based control, or more generally intelligent manufacturing. These take their roots both in fundamental research such as distributed artificial intelligence, artificial life or cooperative control, and also in practical experiences such as kanban-controlled systems, or empowered operators.

Centralised control systems showed their limits to efficiently respond to frequent changes, which led researchers on the way of distributed manufacturing systems ([DIL 91]). But advances in this domain show limitations with systems stability's and global optimisation.

The qualities and complementarities of both centralised and distributed approaches (hybrid architectures) let foreseen considerable benefits of coupling them together, adding the global optimisation abilities from centralised control systems with the reactivity and the possible robustness of decentralised ones. Both hierarchical and heterarchical approaches share benefits and drawbacks. As a consequence has emerged the idea of coupling both systems, with the aim of ensuring a global optima while keeping the heterarchical systems reactivity. This concept would be realistic by using such technologies as RFID. This technology enabled to postulate that embedding intelligence into the product could lead to some types of product-driven systems. ([WON02])([MOR 03]). This concept need to use in a different way simulation, especially, the simulation tool must reproduce the "communicant product" behaviour. Consequently, the simulation tool is built in two parts, the first one, is an "emulation model" where the entities represent items and don't have any attributes (no information and decision are implemented in the model), and the second one, is a "control model" where the entities represent an information flow activated by events occurring in the emulation model. As shown previously, assessing the impact of synchronising physical and informational flows needs to model them as distinct flows (figure 1.17.). In that way, it could be interesting to represent them in two distinct models, that works simultaneously and have to be synchronised. Moreover, the interface standardisation enables to exchange different control models with the same physical emulation one. To represent the implementation of RFID technologies, with fixed readers, the notion of synchronisation points is implemented in the model, which are points where the physical system emits events to update the information system. This event update could launch a decision process that will react by acting on the emulation model. Chapter 8 will be dedicated to this concept.

Models synchronisation = HLA distributed simulation approaches

To face flexibility and reactivity SC problems, recent researches consist in developing adapted simulation environments, allowing the analysis and the evaluation before considering an operational deployment. In a Supply Chain context, we can imagine that building a unique SC model could be a very difficult task, that could lead to simplifying, to strong hypotheses and finally to unrealistic results.

Moreover, running such a model could lead to data problems. As a result, there is a real need for Distributed Simulation (DS) in SCM. That is to say, that a unique control system could manage several physical systems simulation models (Figure 1.18.) or that is to say that such an implementation needs a communication protocol allowing the exchange of information between the various components.

The appearance of standards of distributed simulation specification allows facilitating the implementation of such simulations. A treatment in distributed simulation must be ensured to respect the existing causality relations. Moreover, it is important to take into account all events arising as the time go on, that is to say, it is important to manage the time. Indeed, the problems of messages coordination between the partners of the supply chains and of synchronization of these partners must be managed. The chapter 9 will presents various techniques of existing distributed modelling and simulation (DEVS (Discrete Event system Specification), SIMBA (Simulation Based Applications), HLA (High Level Structures)) by exposing the characteristics.

To have a no ambiguous description of the system and to have a definition of discrete events simulation algorithms whose validity is founded and verifiable, DEVS and SIMBA formalisms can be useful to obtain models formal specification. The American defence has developed the HLA (High Level Architecture) protocol in order to synchronize within a great simulation, simulators being carried out on different computers.

Figure 1.18. Multi-lines synchronization problems

This chapter 9 relates to the study of the multi-lines synchronization problems in internal logistic. Emulation and control models will be presented. Following this study is illustrated by an industrial case. This application takes into account two production sites containing several lines of assembly. HLA ensures interoperability between these various models.

Simulation software

To evaluate decision impact or to choose a management production or Supply Chain organisations, it is today natural to use simulation. Law and Kelton [LAW 91] summarise several reasons for the spectacular increase in the use of simulation in the field of manufacturing and Supply Chain systems. Consequently, we can find today a lot of relevant simulation software, more and more used according to the complexity inherent of the Supply Chain problems.

The main goal of the chapter 10 is to highlight simulation software functionalities. Firstly, a software typology will be proposed. This typology is establish, for discrete-events simulation, according to some literature criteria as event, activity or process approaches, etc… Secondly, supply chain test games will be presented. They are described by a knowledge model and particular formalism that will be explained. These test games are useful to choose or to analyse different simulation software. Finally, a special methodology will be proposed to help modeller to specify his needs and to choose his simulation tool.

Simulation methodology

Evaluation of simulation models

The simulation model quality evaluation is a hard problem. It isn't possible to do that in a formal way (especially for discrete events simulation). At least, we want to have a model behaviour leading to obtain simulation measure indicators values, as close as possible, than the same indicators measures on the real system. As we said previously, the model contains always approximations due to necessary simplifications. Thus, to evaluate this quality, we have to focus on the simulation system architecture, on the one hand, and on its proposed results (indicators), on the other.

Two criteria concern the study of reference architecture quality:

-the first one concerns the architecture structural proprieties (nature of information, use and implementation easiness, reusability, etc.. )

-the other one concerns the operational performances of the simulation model.

It's possible to study the structural aspects through a theorical approach, without any application. But on the other hand, operational performances must be evaluated by simulation runs.

Reduction of simulation models

In the simulation model, the number of "objects" of the model and the number of events occurrences can be very big. In consequence, the simulation duration on a computer can be not acceptable for an operational use as stressed before in § 1.3.4.2. So it is necessary to reduce the model size of a supply chain.

To reduce the model of a supply chain various approaches exists:

-Abstraction which is a« method for reducing the complexity of a simulation model while maintaining the validity of the simulation results with respect to the question that the simulation is being used to address » ([FRA 95] -Figure 1 Brooks and Tobias [BRO 00] suggest a "simplification of models" approach for those cases where the indicators to be followed are the average throughput rates. They suggest an eight stage procedure. The reduced model can be very simple and then an analytical solution becomes feasible and the dynamic simulation redundant. Their work is valid in cases where the required results are averages and where the aim is to measure throughput.

Hung and Leachman [HUN 99] propose a technique for model reduction applied to large wafer fabrication facilities. They use "total cycle time" and "equipment utilization" as decision-making indicators to do away with the Work Centre (WC). In their case, these WC have a low utilization rate and a fixed service level (they use standard deviation of batch waiting time as a decision-making criterion).

Tseng [TSE 99] compares the regression techniques applied to an "aggregate model" (macro) by using the "flow time" indicator. Indeed, he suggests reducing the model by mixing "macro" and "micro" approaches so as to minimise errors in the case of complex models. Here again, for the "macro" view, he only deals with the estimation of flow time as a whole. For the "micro" approach, he constructs an individual regression model for each stage of the operation to estimate its individual flow time. The cumulative order of flow time estimates is then the sum of the individual operation flow time estimates. He tries then to mix the macro and micro approaches.

The reducing model problem

Within the framework of control decision making scenario evaluation such model reductions could be useful. Moreover, concerning Supply Chain planning, the more interesting decision making level is the Master Planning. At this level of planning, load/capacity equilibrium is obtained via the "management of critical capacity" function or Rough-Cut Capacity Planning. Consequently it could be interesting to put forward a reduced model (figure 1.20. explains its principle) in which we find the bottlenecks and the "blocks" which are "aggregates" of the work centers required by released manufacturing orders (MO) [THO 05].

The Work Centers (WC) remaining in the model are either conjectural and structural bottlenecks or WC which are vital to the synchronization of the MO. All other WC are "aggregated blocks" upstream or downstream of the bottlenecks.

By "conjunctural bottleneck" we mean a WC which, for the MPS and predictive scheduling in question, is saturated. This is to say that it uses all available capacity. By "structural bottleneck" we mean a WC which (in the past) has often been in such a condition. Effectively, for one specific portfolio (one specific MPS) there is only one bottleneck -the most loaded WC -but this WC can be another WC than the traditional bottlenecks.

We call a "synchronization work center" one or several resources enabling the planning of MO with bottlenecks and those without to be synchronized. To minimize the number of these "synchronization work centers", we need to find WC having the most in common amongst all this MO portfolio not using bottlenecks and which figure in the routing of at least one MO using them.

EB1 XT4 XTS

3 WC are aggregated in one « Bloc »

Figure 1.20. Reduced model -Principle

A reduction algorithm highlights these so-called "synchronization" work centers. Indeed, the MO using structural or conjunctural bottlenecks may be synchronized and scheduled in comparison with one another thanks to the scheduling of these bottlenecks. But for certain MO that do not use them, synchronization WC will need to be used.

Another states reduction using the notion of bottleneck

In this sub section we show on examples of model reduction using the notion of bottleneck.

With this modelling approach, the "physical part" of the factory is modelled as a network of interconnected flow shops with the following hypothesis:

-in each flow shop items cannot overtake each others -in a given flow shop there can be identical machine in parallel -an item is launch in a given flow shop only when all its components are available (assembly) -these hypotheses are consistent with "product line" organisation of enterprise tendency -Detailed model -Reduced model (states reduction using the notion of bottleneck)

-Industrial application This reduction method has been applied (i.e. [TEL 03]) to a factory included in an aeronautic supply chain. The model of the factory is shown on the right size of the following figure. The reduced model using this kind of method is presented on the left side of this figure. There is a strong reduction of the number of resources modelled nevertheless in a validation phase; the results of the simulation with the reduced model have been compared successfully with the real case.

Figure 1 . 1 .

 11 Figure 1.1. The SCOR processes ([SCO 05])

Figure

  Figure 1.2. which is adapted from the SSCP-Matrix [STA 00] synthesises the different supply chain decision processes:

  evaluation of supply chain control policies are concerned, the following decisions can be considered:

Figure 1 . 5 .

 15 Figure 1.5. Systemic decomposition of the SCM system

Figure 1 . 6 .

 16 Figure 1.6. Time bucket model (example)

Figure 1 . 7 .Figure 1 . 9 .

 1719 Figure 1.7. Events, activities, processes

Figure 1 . 10 .

 110 Figure 1.10. Discrete events simulation

Figure 1 .

 1 Figure 1.11. Production management models/ simulation models (example)

Figure 1 .

 1 Figure 1.12. Simulation process

Figure 1 . 17 .

 117 Figure 1.17. Product Driven System Simulation

Figure 1 . 21 .

 121 Figure 1.21. Model reduction -a case study

  1.2. which is adapted from the SSCP-Matrix [STA 00] synthesises the different supply chain decision processes:

	1.1.2.3. Supply chain processes: the collaborative supply chain point of view
	Let's now consider (figure 1.4) at least two independent organizational units
	(legal entities).
			Source	Make	Deliver	Sale	Source	Make	Deliver	Sale
	SUPPLIERS	Long term Medium term	Lot size Material requirement Purchasing quantities Inventory level Supply network design Supply network planning Suppliers selection	Production network design Sales forecasting and demand planning Distribution structure Supply network modes design Lot size Transportation quantities and Master scheduling Distribution Planning Capacity booking Inventory level Sales & Operation Planning Capacity booking Inventory level Distribution network design Master distribution planning Plant location Production network design	Distribution network design Lot size Material requirement scheduling Lot size Master Capacity booking Inventory level Sales & Operation Planning Purchasing quantities Inventory level Supply network Production network Supply network planning Plant location Suppliers selection design design	modes quantities and Transportation Distribution Planning Distribution structure Capacity booking Inventory level Distribution network Master distribution planning design	Sales forecasting and demand	planning	CUSTOMERS
			Purchase	Production	Transport		Purchase	Production	Transport	Demand
		Short term	scheduling Start and finish delivery dates	Supply scheduling scheduling Suppliers selection Start and finish dates for each operation Start and finish transportation dates	Sales & Plant location Demand fulfillment and ATP	Master scheduling Start and finish delivery dates	scheduling Start and finish dates for each operation	scheduling Start and finish transportation dates	fulfillment and ATP
			Ordering materials	Purchasing quantities Inventory level network planning Shop floor Warehouse control replenishment	Operation Planning	Transportation Capacity booking Inventory level Ordering distribution planning materials	Shop floor control	Warehouse replenishment	CUSTOMERS
							quantities and modes			Inventory level
										Scheduled receipts
										(End of planning
										period)
	End of processing Time on a resource			Start and finish delivery dates	Start and finish dates for each operation	Start and finish transportation dates		Inventory level Scheduled receipts
									(End of planning
		Parts location					period)
						Material flow
			Figure 1.3. Dynamic behaviour of supply chain management system

Figure 1.4. Different supply chain decision processes (2 independent units))

Simulation Ii Perturbations Ri,t Ri,t Io Ii y R1,t x R1,t x R1,t0

		R2,t
	y	R2,t	Simulation
			Aléas

  06] and presented here before § 1.3.2.2).

		Control System			Control System
	On computer R1 in real time	On computer R1 in real time
		connected to Physical		connected to emulation
		and Information Systems	and Information Systems
		with			with
	Hierarchical Decision making process	Hierarchical Decision making process
	Decisions	Production execution actions	Production events reports	Decisions	Production actions execution	reports events Production
		Information System		Information System
		On computer R1			On computer R1
		in real time connected		in real time connected
	to physical and control Systems	to physical and control Systems
		Production execution	Production events	Production execution	Real System	Production events
		actions	reports	actions		reports
		Physical System			Physical System Model
		(items, resources, …)		(items, resources, …)
		Real System		Real System emulation Model On computer S1

of a Supply Chain on two computers S1 and S2

		Control System A			Control System B
		On computer R1 in real time			On computer R1 in real time
		connected to Physical			connected to Physical
		and Information Systems	Information		and Information Systems
		with			with
		Hierarchical Decision making process		Hierarchical Decision making process
	Decisions	Production execution actions	Production events reports	Decisions	Production execution actions	Production reports events
		Information System A			Information System B
		On computer R1			On computer R1
		in real time connected			in real time connected
		to physical and control Systems		to physical and control Systems
		Production execution actions	Production events reports		Production execution actions	Production events reports
		Physical System A (items, resources, …)	Parts		Physical System B (items, resources, …)
		Real System -Company A			Real System -Company B
			Real Supply Chain
		Control System Model A			Control System Model B
		On computer R1 in real time			On computer R1 in real time
		connected to Physical			connected to Physical
		and Information Systems			and Information Systems
		with			with
		Hierarchical Decision making process		Hierarchical Decision making process
	Decisions	Production execution actions	Production events reports	Decisions	Production execution actions	Production reports events
		Information System Model A		Information System Model B
		On computer R1			On computer R1
		in real time connected			in real time connected
		to physical and control Systems		to physical and control Systems
		Production execution actions	Production events reports		Production execution actions	Production events reports
		Physical System Model A			Physical System Model B
		(items, resources, …)			(items, resources, …)
		Real System -Company A			Real System -Company B
		Control System A			Control System B
		On computer R1 in real time			On computer R1 in real time
		connected to Physical			connected to Physical
		and Information Systems	Information		and Information Systems
		with			with
		Hierarchical Decision making process		Hierarchical Decision making process
	Decisions	Production execution actions	Production events reports	Decisions	Production execution actions	Production reports events
		Information System A			Information System B
		On computer R1			On computer R1
		in real time connected			in real time connected
		to physical and control Systems		to physical and control Systems
		Production execution actions	Production events reports		Production execution actions	Production events reports
		Physical System A (items, resources, …)	Parts		Physical System B (items, resources, …)
		Real System -Company A			Real System -Company B
			Real Supply Chain
		Control System Model A			Control System Model B
		On computer R1 in real time			On computer R1 in real time
		connected to Physical			connected to Physical
		and Information Systems	Information		and Information Systems
		with			with
		Hierarchical Decision making process		Hierarchical Decision making process
	Decisions	Production execution actions	Production events reports	Decisions	Production execution actions	Production reports events
		Information System Model A		Information System Model B
		On computer R1			On computer R1
		in real time connected			in real time connected
		to physical and control Systems		to physical and control Systems
		Production execution actions	Production events reports		Production execution actions	Production events reports
		Physical System Model A (items, resources, …)	Parts		Physical System Model B (items, resources, …)
		Real System -Company A			Real System -Company B
		Centralized simulation Model

Parts Information Centralized simulation Model of a Supply Chain on a computer S1 Figure 1.15. Centralized Supply Chain Simulation Model

  .19.). Its objective is to reduce the calculus combinatory -Aggregation which is a « form of abstraction by which a set of data or variables with common characteristics can replaced by an aggregated data or variable » [MER 87] -Number of events reduction which consists in replacing "part of a discrete event model by a variable or a formula" ([ZEI 76]) Even though the most researches concerning model reduction are relative to manufacturing flows, it could be useful to analyze their results, especially concerning reduction problem, to highlight similarities between Manufacturing process simulation models and Supply Chain simulation models. Amongst various authors, Zeigler was the first to deal with the reduction simulation model problem [ZEI 76]. In his view, the complexity of a model is relative to the number of elements, connections and model calculations. He distinguished four ways of simplifying a discrete simulation model in replacing part of the model by a random variable, coarsening the range of values taken by a variable and grouping parts of a model together. Innis and al [INN 99] first listed 17 simplification techniques for general modelling. Their approach was comprised of four steps: hypotheses (identifying the important parts of the system), formulation (specifying the model), coding (building the model) and experiments.

	1.6.2.1. Reducing model literature review
		Model abstraction techniques Model abstraction techniques
		Reduce complexity of a simulation model Reduce complexity of a simulation model
	Model Boundary modification (real word considered factors) Model Boundary modification (real word considered factors)	Modification of behaviours Modification of behaviours	Model form modification (simplify input/ouput relationships : tables, probabilities, …) Model form modification (simplify input/ouput relationships : tables, probabilities, …)
	Hierarchies of Hierarchies of	Parameters Parameters
	models models	elimination elimination
	State Aggregation State Aggregation	Temporal aggregation Temporal aggregation	Entity aggregation Entity aggregation	Function aggregation Function aggregation
	(states distinctions of which (states distinctions of which	(reduce granularity, (reduce granularity,	(change entity hierarchy: (change entity hierarchy:	(agregate a set of functions (agregate a set of functions
	has no interest for the has no interest for the	Merge quasi Merge quasi	by function, by by function, by	that the entities perform that the entities perform
	simulation results) simulation results)	simultaneous events) simultaneous events)	structyure) structyure)
	Figure 1.19. A Taxonomy of Model. Abstraction Techniques [FRA 95]

Conclusion

In this introduction chapter we have presented the main concepts which are used in supply chain management simulation. The specificities of this kind of simulation and the modelling problem difficulties in this context have been highlighted. Different types of approaches and model as been presented to solve this problem. At last, the links between the distribution level of both the system and the model have been characterised.

The following of the book includes three mains parts.

The first part takes the viewpoint of the simulation model types:

-continuous simulation (chapter 2) -event system -event driven or time bucket driven (chapter 3) -simulation games (chapter 4) The second part takes the viewpoint of the distribution level of the system and the model:

-Centralized approaches (chapter 5).

-Of the interest of agents for supply chain simulation (chapter 6).

-Decisional system simulation of enterprise network with MAS (chapter 7).

-Simulation for Product Driven Systems (chapter 8).

-HLA distributed simulation approaches for supply chain (chapter 9). Then a third part is dedicated to the simulation products (chapter 10). Even if we are convinced of the importance of the simulation methodology, no part of this book is explicitly dedicated to this aspect. Nevertheless the simulation methodologies (reduction simulation models, the simulation model validation and the simulation analysis) will be evocated throughout the different chapters:

-a presentation of such simulation concepts and techniques highlighted in the chapter 1, -applications of these concepts and techniques in case studies that illustrate the pivotal role of simulation in decision-making process.


Anna Carla Araujo

Jose Luis Silveira

Recife-PE THRILLING: AN APPLICATION OF THREAD MILLING PROCESS

Keywords: Thrilling, Thread Milling, Drilling, Mechanistic Model, Force Prediction

This paper presents an application of thread milling and drilling models for prediction of forces generated by thrilling tools. The chip thickness and cutting force models are developed considering the unique geometry of the tool. The mechanics of cutting for drilling is applied for the first stage of cut and the thread milling model is applied on a helical path.

INTRODUCTION

The increased use of CNC technology has created the conditions for a future-oriented technology for the production of internal and external threads. Thread milling is a very useful alternative to thread cutting or thread forming in a large number of work cases. It offers some advantages as: high cutting speeds and feeds, multi-functional designs reduce cycle times by combining drilling, chamfering and threading operations into one tool (the same tool for right-hand and left-hand threads), threads with excellent form, finish, and dimensional accuracy, easy machining of difficult materials, Eliminating the possibilities and consequences of tap breakage and reversal of the spindle is not required.

The thrilling tool (Fig. 1) is used for threads in the size range from M4 to 3/4". The tool makes drilling into solid material, chamfering of the thread hole and thread milling in one work process without tool change producing a recess on the bottom of the hole. Mostly used in short-chipping materials.

Mechanistic approach is based on the geometry of the process and relates the forces to the chip load through the specific forces. The specific forces are empirically related to the fundamental machining parameters and the coefficients for a given tool and work piece material combination determined from calibration experiments. [START_REF] Chandrasekharan | A calibration procedure for fundamental oblique-cutting model coefficients based on a three-dimensional mechanistic drilling force model[END_REF] used drilling force signal to determine the coefficients to predict end milling forces. Using the same tool and work piece materials, the specific cutting force is calculated based on the cutting velocities, geometry of cut and the mechanistic coefficients as the mechanistic coefficients are universal for all machining processes. The thrilling tool is used for both operations: drilling and thread milling. Using the same theory from [START_REF] Chandrasekharan | A calibration procedure for fundamental oblique-cutting model coefficients based on a three-dimensional mechanistic drilling force model[END_REF] and as drilling process it is well known, the method confirms the modeling of thread milling operation and also can be used to define the velocities before the machining via drilling measurement and thread milling prediction. As the tool is unique for the process, with the same material and same geometry is important to compare the mechanics of cutting during the different phases comparing parameters found in different operations.

TOOL GEOMETRY

The tool geometry is presented in Fig. 3 and4. Compound of nf continuous flutes (normally two), each flute has a chisel edge, a drilling cutting edge and a milling cutting edge. The mechanism of thrilling process must be analyzed separately for those three regions.

Drilling point

The Chisel edge does not cut but only spreads the material sideways by an indentation mechanism. Instead of using the laws of cutting, the mechanism of indentation must be used. It has a width of 2ω and an edge angle of ψ c . The drilling cutting edge goes from the external diameter of the chisel edge until the diameter d d . In the tool front view (Fig. 3) the point angle that locates the chisel edge and the cutting lips is κ. The vertical distance from the bottom of the tool and the end of the drilling cutting edge is h d . Between the cutting lips and the thread cutting edge there is a vertical edge that separate the drilling part from the thread milling part. This edge will cut as an end milling tool. Its height is h e

Thread milling tool part

The upper part of the milling cutting edge is a flute that makes the desired thread form (Fig. 4). The thread geometry is characterized by the thread angle ξ and thread pitch p. The helix angle is λ and the rake angle for the thread milling cutting edge is α h . The maximum diameter of the tool thread is denoted as d e and the minimum as d i . Both parts cut together and the programmer should be careful not to machine the previous thread surface when making more then one pitch. (1)

(2)

The variation in the disk diameter along the tool causes radial depth of cut to vary for each disk. The angle between the flutes depends on the numbers of flutes Nf and is defined by the tool geometry as

(3)

MODEL DEVELOPMENT

A mechanistic oblique-cutting force model is used to develop a three-dimensional force model for the process. The calibration algorithm from [START_REF] Chandrasekharan | A calibration procedure for fundamental oblique-cutting model coefficients based on a three-dimensional mechanistic drilling force model[END_REF] is used to extract the oblique butting model coefficients from the drilling stage and it is applied to predict the thread milling forces and prevent from tool breakage and chip welding. This procedure is possible because for the same work piece and tool materials the specific cutting force is the same for any cutting process.

The force signal has a different pattern from the entry of the tool to in the permanent drilling stage and from the end milling part. The interest is to predict the thread milling part based on the drilling forces. As can be seeing in the Figure 5, the first step, when tool is already in the work piece with a pilot hole, is very stable. As the chisel edge it is not acting, the cutting lip is the only part that it is cutting and then the model is very simple to compare with the experimental forces to calculate the specific cutting force.

Figure 5. Drilling Force [START_REF] Chandrasekharan | A calibration procedure for fundamental oblique-cutting model coefficients based on a three-dimensional mechanistic drilling force model[END_REF]

Extracting Specific Coefficients from Drilling Stage.

The approach used in simulating other processes, in which forces are calculated using the uncut chip area is not completely applied to drilling. Drilling forces are more effectively simulated by dividing the cutting edge into two parts: the part inside an indentation zone, that yield a plastic region of forces produced by the chisel edge due to the material removal by extrusion, and the part that follows the mechanics of cutting.

Thread milling prediction.

In Araujo (2005), the process has been simplified as a simple linear movement in order to develop and validate the model. In this work, the model is applied in the helical path movement, where the tool goes in x, y and z directions at the same time. Although it is important to consider that the z velocity is very small compared to the cutting velocity and the radius of the curvature of the hole is large compared to the chip thickness. The depth of cut b(Ω) changes, where Ω is the position of the tool in relation to the center of the hole.

The force calculated by Araujo (2005), F tC , F rC and F zC , are described in the tool axis referential. In x, y and z directions the forces can be written as:

(4) Figure 6. Tool Trajectory

Chip Modeling.

Tlusty concept of the contact interface between the tool and the work piece is used (Araujo, 2003). It is considered two different types of cutting Type I and Type II. Type III [START_REF] Yang | Analysis of force shape characteristics and detection of depth-of-cut variations in end milling[END_REF] for low axial and high radial depth of cuts it is not used as in thread milling the associated radial depth of cut is small compared to the tool diameter. In the thread milling geometry e is the radial depth of cut and b(Ω) is the axial depth of cut. The difference between the initial and final axial depth of cut is normally one pitch.

(5)

The action of one flute is projected in a plane and this plane is unfolded (unrolled). The cutting edge is then a thready line inclined by α h and it moves from left to right. The entire cutting action can be divided in three phases depending on the length of the cutting edge in contact with the work piece. The limits of each phase are indicated by ψ n , where the subscript n goes from 1 to 4 depending on the phase. For Type I cutting, the length of the cutting edge increases from 0 to the maximum depth b in phase A, and the rotational angle µ goes from the entry angle, θ en (ψ 1 ), to δ (ψ 2 ), which is defined as (6)

In phase B, the cutting edge length in contact with the work piece remains constant and θ goes from δ to the exit angle θ ex (ψ 3 ). As the tool is going out of the work piece, the cutting edge length gradually decreases (phase C), and the tool is completely out of the work piece when θ is greater than ψ 4 = θ ex + δ.

The angular position of a point P on the cutting edge is denoted as φ(θ) and the points on the cutting edge span from φ i to φ f ,. which values depend on the phase and the type of cutting. (Araujo, 2005) The total cutting force can be calculated summing the elemental forces on each disk along the tool in contact with the work piece.

For the thread milling process θ en = 0. Because of the thread, the diameter of the cutting edge varies as given in Eq. 1. This variation in the diameter causes changes in the radial depth of cut and consequently the exit angle, which can be written a , (7) where r o =(d E -d e )/2. In the presence of run out, each flute rotates at a different radius of rotation R c . The runout is characterized by the parameters parallel axis offset run out ρ and its locating angle λ. ρ is the distance from the spindle axis to the tool axis and λ is the angle between an arbitrary flute and the direction of the run out. Since each disk element has different diameter, R c for the cutting edge of n th flute and i th disk element is given as [START_REF] References | A model for thread milling cutting forces[END_REF] The chip thickness ate time t can be written as:

(9) where f t is the feed per tooth and the rotation angle of the cutting edge can be written as:

(10)

Forces and Specific Pressure.

The mechanistic modeling approach is taken for the cutting force model similar to that given in [START_REF] Kline | The effects of run-out on cutting geometry and forces in end milling[END_REF]. The elemental forces on each disk are expressed as (Araujo, 2005) (11) where K t , K r and K z are the specific cutting energies in the tangential, radial and axial directions, respectively, calculated as a function of the spindle speed, feed rate and the tool-piece material coefficients a 0 , a 1 , a 2 , b 0 , b 1 and b 2 .

(12)

These directions are located on the cutting edge. The tool referential is located on the tool-piece contact moving with Ω in relation. ( 13)

RESULTS

The results shown in Figures 7, 8 and 9 were calculated for an Aluminum work piece using the coefficients taken from previous experiments from [START_REF] References | A model for thread milling cutting forces[END_REF]. Although, the experimental coefficients should be taken from the drilling stage to be applied into the thread milling stage. In the future work, experiments will be run to validate the procedure.

CONCLUSIONS

The model for thrilling operations was developed based on the thread milling and drilling model. As the thread milling velocities secure range are not well known, the specific pressures taken from the drilling stage can be used to predict forces on the thread milling stage.

Figure

  Figure 1. Thrilling Tool

Figure 3 .

 3 Figure 3. Tool Point Geometry

Figure 4 .

 4 Figure 4. Thread Milling Part The thread milling part can be analyzed as a stack of disks. Each disk has ∆z thickness and a variable diameter d(z) as shown in Fig.4. The variable diameter d(z) can be written as a function of the disk height z measured from the bottom of the tool.

Figure 7 .

 7 Figure 7. Drilling Force.

Figure 8 .

 8 Figure 8. Thread milling force

ACKNOWLEDGEMENTS

The lead author would like to gratefully acknowledge the fellowship received from CAPES, a Brazilian council for scientific and technological development. All of the authors are grateful for the council and the University of Illinois at Urbana-Champaign in the name of Professor Shiv Kapoor for the support and directions provided during the course of this research.


Anna Carla Araujo -Annaaraujo

Marcelo Amorim

NONLINEAR DYNAMICS OF MILLING PROCESS

Keywords: Milling, Force Prediction, non-linear dynamics, non smooth systems, chatter

The cutting processes include different manufacturing procedures as milling, turning and drilling. In all of these, the machine tool vibration plays an important role concerning the cutting characteristics. Therefore, its correct understanding is essential in order to improve the workpiece surface quality and avoid tool breakage. Chatter and squeal are some undesirable phenomena related to an improper functioning. This article discusses the nonlinear dynamics associated with the machine tool vibration relative to the workpiece considering a nonsmooth single-degree of freedom system. Milling process is of concern assuming different machining characteristics. The cutting force depends not only on the current position of the tool and the workpiece but also on a delayed value of the displacement as it is a function of the chip thickness. Therefore, the equation of motion is represented by a delay-differential equation that is solved employing the Runge-Kutta method together with a Taylor's series approximation employed to estimate the difference terms. Numerical simulations are carried out showing some situations related to proper and improper functioning during the cutting process.

Introduction

Machining is one of the oldest process for shaping components and due to its versatility and precision achieved through continual innovation, research and development, has become an indispensable process used in manufacturing industry.

The machine tool vibration during the cutting process plays an important role concerning the workpiece surface quality and also the tool durability. Chatter and squeal are some undesirable phenomena related to an improper functioning. During certain cutting conditions, the motions of the workpiece-tool system are characterized by large amplitudes, which are not desirable for obtaining a good surface finish. The undesirable motions, which are often referred to as chatter, can result in wavy surfaces on the workpiece, inaccurate dimensions, and excessive tool wear.

Machining has been modeled by dynamical approach in order to help a proper comprehension of the process [START_REF] Moon | Dynamics and Chaos in Manufacturing Processes[END_REF]. For turning operations, [START_REF] Chandiramani | Dynamics of 2 dof regenerative chatter during turning[END_REF] analyzed the dynamics of cutting considering a two-degree of freedom (2-dof) and orthogonal cutting modeling to predict chatter. [START_REF] Pratt | Design and modelling for chatter control[END_REF] studied the boring bars for turning, which commonly presents chatter problems and experimentally determined modal properties of the tool. Also dedicated to turning, [START_REF] Kalmar-Nagy | Subcritical hoft bifurcation in the delay equation model for machine tool vibrations[END_REF] showed the existence of a subcritical Hopf bifurcation in the delay-differential equation used to describe the machining equations of motion. [START_REF] Zhao | Dynamics and stability of milling process[END_REF], [START_REF] Insperger | Stability of up-milling and downmilling, part 1: alternative analytical methods[END_REF], [START_REF] Gradisek | On stability prediction for milling[END_REF] and [START_REF] Mann | Milling bifurcation from structural asymmetry and nonlinear regeneration[END_REF] investigated the stability of milling process with single and twodof systems for different experimental conditions. The mathematical model is represented by delay-differential equations.

Milling process has the peculiarity to have a contact/non-contact behavior due to its geometry and due to tool vibration. This kind of behavior is related to nonsmooth systems that are usually associated with either the friction phenomenon or the discontinuous characteristics as intermittent contacts [START_REF] Savi | Numerical and experimental investigations of the nonlinear dynamics and chaos in non-smooth systems[END_REF]. This article represents the milling process by a nonsmooth one-degree of freedom oscillator that is related to cutting and non-cutting behaviors, respectively related to contact and non-contact situations. The force description is done by mechanical approach and is used as an excitation to the system dynamics. Under these assumptions, the equation of motion is represented by a delay-differential equation that is solved employing the Runge-Kutta method together with a Taylor's series approximation employed to estimate the difference terms. Numerical simulations are carried out showing some situations related to proper and improper functioning during the cutting process.

Machining Model

Machining process is very complex because it involves different related phenomena. During cutting processes, for example, it is common to have temperature variations that induce dramatic changes in expected behaviors. Although all these complexity, some simple models may be used in order to obtain useful information concerning tool and workpiece behaviors as dynamic tool prediction.

The milling process is of concern in this article and its modeling is done by considering that it is a full immersed milling in the y direction and, therefore, the tool vibration is neglected in this direction. Hence, a single-degree of freedom system is assumed as shown in Figure 1, a schematic picture of the cutting process presenting the tool and the workpiece. Basically, the model considers a mass-spring-dashpot system representing the tool. Linear behavior is assumed for these elements and, therefore, they are respectively represented by parameters m, k and c. The workpiece, on the other hand, is represented by a spring-dashpot, represented by parameters k s and c s . The distance between the tool and the workpiece is represented by a negative gap g as the tool is engaged into the workpiece. This oscillator represents the end milling tool and its center has a displacement x that is associated with the sum of the predicted position plus the distance around the center due to the tool vibration [START_REF] Moon | Dynamics and Chaos in Manufacturing Processes[END_REF]. Under these assumptions, the milling process can be represented by the following equation:

mẍ + c ẋ + kx = 0, without contact (1) mẍ + (c + c s ) ẋ + (k + k s )x = -F x , with contact (2)
where F x represents the cutting force that may be properly described, as discussed in the next section.

Cutting Force Prediction

The prediction of the cutting force in machining has attracted the attention of many researchers that proposed different forms of modeling: analytical, numerical, experimental and hybrid approaches [START_REF] Ehmann | Machining process modeling: a review[END_REF]. Mechanical [START_REF] Kline | The prediction of cutting forces in end milling with application to cornering cuts[END_REF] and empirical approaches are certainly the most usual modeling employed in order to estimate cutting forces during milling. In this paper, we consider the prediction of forces in the end milling that involves instantaneous cutting forces modeled by mechanical approach with a dynamical deflection included in the force model.

The instantaneous differential cutting force for a single tooth f i with no helix angle is written based on Martelotti equation [START_REF] Tlusty | Dynamics of cutting in end milling[END_REF]:

f i (t) = K c t c b (3)
where t c is the uncut chip thickness, b is the depth of cut and K c is a vector with the specific cutting force coefficients.

The uncut chip thickness t c for the end milling can be written as a function of the angle φ i (t), the angle of each cutting flute i with respect to the normal direction of the feed per tooth s t [mm/rev] ( [START_REF] Kline | The prediction of cutting forces in end milling with application to cornering cuts[END_REF] as:

t ci (t) = s t sin φ i (t) (4)
The feed per tooth is related to the feed v f [mm/min], the spindle speed ω[rpm] and the number of teeth N t by:

s t = v f ω N t (5)
By considering φ 1 (t) the angle for a first flute, defined as the first flute that touches the piece and ξ the angle between the teeth:

φ i (t) = φ 1 (t) + ξ(i -1) = 2πω 60 t + ξ(i -1) (6)
If the flute is in contact with the piece, φ i (t) is inside an angle range defined by the initial and final angles of the contact, θ i and θ f , and there is a contribution of this flute for computing the total force. If φ i (t) is not inside that range, the flute force is zero as there is no material to cut [START_REF] Araujo | Force prediction in thread milling[END_REF].

θ i < 2πω 60 t + ξ(i -1) < θ f =⇒ f i (t) = 0 (7)
By assuming a full immersed milling, where the initial and the final angles are, respectively, θ i = 0 and θ f = π, the force are given by:

f i (t) = K c b s t sin φ i (t) if sin φ i (t) ≥ 0 f i (t) = 0 if sin φ i (t) < 0 (8)
The total force is calculated by the sum:

F(t) = N t i=1 f i (t) = b s t N t i=1 K c (t) sin φ i (t) whenever sin φ i (t) ≥ 0 (9)
The specific cutting force coefficients can be written in tangential and radial components, K t and K r . These components, for mechanistic models, are supposed to be constant for each tool-workpiece pair [START_REF] Araujo | A model for thread milling cutting forces[END_REF].

K c (t) = K x (t) K y (t) = cos φ i (t) sin φ i (t) sin φ i (t) -cos φ i (t) K t K r ( 10
)
The total force F(t) can be written as a function of F x (t) and F y (t) components:

F x (t) = b s t N t i=1 K r sin 2 φ i (t) + K t sin φ i (t) cos φ t (t) whenever sin φ i (t) ≥ 0 (11) F y (t) = b s t N t i=1 s t K t sin 2 φ i (t) -K r sin φ i (t) cos φ t (t) whenever sin φ i (t) ≥ 0 (12)
Notice that this approach does not considers that the vibration changes the cutting force. At this point, a dynamical approach is of concern in order to introduce these forces as excitations of the dynamical system that represents the milling process.

Equations of Motion

The end milling tool position x = x(t) of the center is represented by the sum of the predicted position plus the distance around the center due to the tool vibration [START_REF] Moon | Dynamics and Chaos in Manufacturing Processes[END_REF].

x c = x o + x = s t ωt 60 + x (13)
By considering a dynamical point of view, a new parcel should be added (or removed) from the cutting force from Equation 11 as the tool position is changing and there is more (or less) material to cut. Since the tool is vibrating from the beginning of the cutting process, the material machined over (or below) the predicted trajectory changes the instantaneous cutting area for the acting flute. The additional material is computed as ∆x = x(t) -x(t -τ ), where τ is the time between the passage of two flutes.

τ = 60 ξ 2πω (14)
Under this assumption, the force can be rewritten adding the terms x = x(t) and x τ = x(t -τ ) to the feed per tooth parcel:

F x (t) = b Nt i=1 s t + ∆x K r sin φ i (t) + K t cos φ i (t) (15)
Moreover, by considering the idea of the nonsmooth oscillator, the force is zero when there is no flute in contact with the workpiece and also, while in vibration, when the tool position is less then feed per tooth, the negative gap between the piece and the workpiece.

F x (t) = b Nt i=1 (s t + ∆x)(K r sin φ i (t) + K t cos φ i (t)) if x ≥ -s t F x (t) = 0 if x < -s t (16)
It should be highlighted that the cutting process is related to a maximum force, F max , and if this level is achieved its value is suddenly reduced and instantaneously released.

Therefore, employing this forces as the excitation of the nonsmooth oscillator that represents the cutting process, the system dynamics may be represented as follows:

mẍ + c ẋ + kx = 0 if x < -s t (17) mẍ + (c + c s ) ẋ + (k + k s )x = -F x = -K(s t + ∆x) if x > -s t (18)
These equations represents different behavior for the system with and without contact. When the tool has contact with the workpiece, the damping and the stiffness are a sum of the workpiece and tool. On the other hand, when the tool lost the contact with the workpiece, the damping and stiffness are only calculated by considering the tool vibration as a bending beam.

Note that K is a constant for this rotation angle φ i , and it is not dependent from x:

K = n i=1 (K t b sin 2 (φ i ) + K r b sin(φ i ) cos(φ i )) whenever sin(φ i ) ≥ 0 (19)

Numerical Procedure

Numerical simulations are carried out considering the fourth order Runge-Kutta method, however, in order to deal with the difference term in the right hand side, it is considered two different situations. The first one is when t > τ and therefore, the x τ = x(t -τ ) has been already computed and the equation of motion may be rewritten as follows:

ẍ + (c + c s ) m ẋ + (k + k s + K) m x = - K(s t -x τ ) m ( 20
)
This system may be written in terms of a first order system as follows,

ẋ v = 0 1 -p 1 -p 2 x v + 0 p 3 ( 21
)
where:

p 1 = (c + c s ) m ; p 2 = (k + k s + K) m ; p 3 = -K(s t -x τ ) m ; (22)
For time instants where t < τ , it is necessary to estimate the values of x τ that is done by employing a Taylors series expansion [START_REF] Cunningham | A non-linear differential-difference equation of growth[END_REF]:

x -x τ = τ ẋ - τ 2 2 ẍ ( 23
)
Under this assumption, the equation of motion become:

mẍ + (c + c s ) ẋ + (k + k s )x = -K(s t + τ ẋ - τ 2 2 ẍ) (24)
And this equation can be written as a first order system as:

ẋ v = 0 1 -p 5 /p 4 -p 6 /p 4 x v + 0 p 7 /p 4 (25)
where

p 4 = 1 - Kτ 2 2m ; p 5 = (c + c s + Kτ ) m ; p 6 = (k + k s ) m ; p 7 = -Ks t m (26)

Numerical Simulations

In order to show the capabilities of the proposed model, numerical simulations are carried out considering an end milling tool with diameter d = 19.05mm, four flutes and no helix angle. The tool characteristics are represented by the mass, the stiffness and the damping coefficients: m = 0.6kg, k = 2000N/mm and c = 0.004Ns/mm, respectively. The workpiece properties are represented by the stiffness (k s = 2 • 10 7 N/mm) and the damping (c s = 0.005Ns/mm), values taken from [START_REF] Mann | Milling bifurcation from structural asymmetry and nonlinear regeneration[END_REF]. The maximum force that the chip does not brake is taken as

F max = 2000N.
Our analysis starts by considering the influence of depth of cut b on the system dynamics. Therefore, it is assumed different values of this parameter while the others are fixed. Besides the other presented parameters, it is assumed that s t = 0.15mm and ω = 200rpm.

Figure 2 shows the maximum displacement in situation with and without contact. Notice that, for small values of s t , system is always in contact, presenting a proper functioning. Nevertheless, the increase in this value tend to produce situations where the system starts to work presenting both behaviors. This curve may be useful to identify regions of stable, regular motions, associated with desirable functioning. At this point, the same analysis is done considering b = 0.5 mm and ω = 200 rpm fixed, varying the feed per tooth s t . As expected, the same qualitative results are obtained and, the increase in feed values tends to cause improper functioning.

Figure 4 shows the maximum displacement in situation with and without contact. Once again, this curve may be useful to identify regions of stable, regular motions, associated with desirable functioning. Figure 5 shows steady-state displacement time history, phase space and force F x when s t = 0.016mm, b = 0.5mm and ω = 200 rpm. Notice that non-contact behavior is always related to large displacements and therefore, are undesirable for machining purposes. In order to observe a critical machining situation, it is considered a simulation with s t = 0.05mm, b = 0.15mm and ω = 1500 rpm (Figure 5). This Figure presents the steady-state tool displacement, phase space plots and force F x , showing a contact/non-contact response with great amplitudes variations.

Conclusions

The analysis of milling process is proposed by a dynamical approach. A nonsmooth singledegree of freedom oscillator is proposed in order to model this cutting process. The forces involved during cutting is modeled by considering a mechanical approach and are used as an excitation of the dynamical system. The mathematical model is represented by a nonsmooth delay-differential equations that is numerically solved by the Runge-Kutta method together with a method based in Taylors series in order to estimated delayed displacements. Numerical simulations are carried out considering the influence of depth of cut and feed per tooth in the system dynamics. Results show that non-contact vibrations are related to improper functioning and can be used as a criterion to define functioning conditions.

Figure 1 :

 1 Figure 1: Dynamic of an End Milling Tool

Figure 2 :

 2 Figure 2: Maximum tool displacement in steady-state when s t = 0.15mm, ω = 200rpm and b ∈ [0.01, 0.30] mm

  Figure3shows steady-state displacement time history and phase space plots for s t = 0.15mm, b = 0.01mm and ω = 200 rpm. Notice that non-contact behavior is always related to large displacements and therefore, are undesirable for machining purposes.

Figure 3 :

 3 Figure 3: Numerical response for s t = 0.15mm, b = 0.01mm and ω = 200 rpm

Figure 4 :

 4 Figure 4: Maximum displacement in steady-state when ω = 200rpm, b = 0.5mm and s t ∈ [0.001, 0.030] mm

Figure 5 :

 5 Figure 5: Numerical response for s t = 0.016mm, b = 0.5mm and ω = 200 rpm

Figure 6 :

 6 Figure 6: Numerical response for s t = 0.05mm, b = 0.15mm and ω = 1500 rpm

Acknowledgements

The authors acknowledge the support of Brazilian Research Councils CNPq and FAPERJ.