Damien Riquet

Gilles Grimaud

Michaël Hauspie

DISCUS: A massively distributed IDS architecture using a DSL-based configuration

Nowadays, cloud computing becomes quite popular and a lot of research is done on services it provides. Most of security challenges induced by this new architecture are not yet tackled. In this work, we propose a new security architecture, based on a massively distributed network of security solutions, to address these challenges. Current solutions, like IDS or firewalls, were not formerly designed to detect attacks that draw profit from the cloud structure. Our solution DISCUS is based on a distributed architecture using both physical and virtual probes, along with former security solutions (IDS and firewalls). This paper describes DISCUS SCRIPT, a dedicated language that provides an easy way to configure the components of our solution.

I. INTRODUCTION

Cloud computing is a popular model that provides large resources. A commonly admitted definition by the NIST [START_REF] Mell | The NIST Definition of Cloud Computing[END_REF] describes cloud computing as "a model enabling ubiquitous, convenient, on-demand network access to a shared pool of configurable computing resources". With the increasing popularity of cloud computing, its security importance is rising and could delay cloud computing adoption for hesitant companies.

In 2010, a study [START_REF] Spiceworks | New study sees rise in cloud services adoption among small and medium businesses in first half of 2010[END_REF] reported that 62 % of small and medium businesses had no plan to move to cloud computing due to security concerns. Academia researchers [START_REF] Armbrust | Above the clouds: A berkeley view of cloud computing[END_REF], industry actors [START_REF] Shankland | Hp's hurd dings cloud computing[END_REF], and government organizations [START_REF]Cloud computing risk assessment[END_REF] share the same perspective. That is why cloud providers need to propose a turnkey solution that deals with security concerns like intrusion detection or large-scale coordinated attacks.

Even though the "cloud computing" designation is quite recent, a lot of security concerns have already been studied. In [START_REF] Chen | What's new about cloud computing security?[END_REF], Chen et al. list what are the new challenges of cloud computing. One of these challenges is facing black hats who maintain the appearance of regular users but in fact perpetrate cybercrime or cyber attacks [START_REF] Bitweasil | Cryptohaze cloud cracking[END_REF][START_REF] Bryan | Cloud computing : A weapon of mass destruction?[END_REF]. Also, they mention that shared resource environment could introduce unexpected side or covert channels. Eventually, they point out that cloud providers need to isolate their structure at different granularities (virtual or physical machines, LANs, clouds or datacenters).

Isolation of the cloud components prevents some attacks that draw profit from its structure such as internal attacks (attacks led from the inside of the cloud targeting internal hosts).

Figure 1 pictures a cloud structure with several entry points, several levels of network devices and physical hosts. This basic structure could be either targeted from outside (horizontally dashed) or used to target external hosts (vertically dashed). Our work focuses on detecting these attacks as well as internal attacks (plain), even inside the same cluster or physical machine. The later attacks are yet seldom studied in the literature and, Figure 1: Attacks on the cloud to the best of our knowledge, no solution has been proposed to deal with them.

Current network security solutions in the cloud are Intrusion Detection Systems (IDS), Intrusion Prevention Systems (IPS) and firewalls. For example, Amazon Web Services (AWS) cloud [START_REF]Amazon web services: Overview of security processes[END_REF] uses instance isolation and firewalls. This type of device is designed to detect intrusion or attacks from outside to inside of the structure. This is why traditional security solutions are not able to detect internal intrusions, because they are located at the border of the network or simply not aware of the global security state.

In [START_REF] Riquet | Large-scale coordinated attacks : Impact on the cloud security[END_REF], we showed that distributed attacks could easily evade these security solutions, especially in structure with several entry points, like cloud computing.

Though cloud computing relies on datacenter structures, a new challenge arising is to protect the cloud network as well as the end users. Cloud computing security must be adaptable to dynamic architecture (end users may change during an attack). Moreover, cloud users may be malicious or an attacker could have the control of one or several internal hosts. Cloud security must ensure applications availability and data integrity. Finally, it must prevent distributed denial of service (DDoS), worms spreading and such large-scale coordinated attacks (even inside the cloud infrastructure).

Our proposal introduces a new architecture designed for large scale infrastructure, located at multiple places and subject to attacks (either internals or not). This solution, named DISCUS, is based on a massively distributed structure using heterogeneous probes. Physical and virtual probes are scattered across the network and collaboratively detect intrusions and attacks. Physical probes could be a cheap and reconfigurable device (typically a FPGA) placed behind an host or directly integrated into the switches. Virtual probes are integrated into physical hosts and could be based on existing security solutions like Snort [START_REF] Roesch | Snort-lightweight intrusion detection for networks[END_REF] or Bro [START_REF] Paxson | Bro: a system for detecting network intruders in real-time[END_REF].

The main feature of our solution is to provide a common framework for distributed security based on heterogeneous security devices. Any security solution could be integrated into DISCUS, providing that a back-end exists for this device. One of the supported back-end is Snort and we intend to develop back-ends for firewalls (for example pfSense) or operating systems (through a kernel module).

One of the challenge induced by this architecture is the administration of all the probes. The administrator uses DISCUS SCRIPT to write global security rules once, then the compiler is responsible for instantiating every device with relevant rules.

Our solution tackles new challenges arising with cloud computing. Also, in such structures, customers change constantly and a security solution must be aware of it. Goals of DISCUS are to provide a reconfigurable security solution that is fitted to large-scale infrastructures and also identify or isolate precisely an host among thousands.

The paper is structured as follows. First, Section II introduces related work. Section III describes the architecture of DISCUS, its components and its deployment process. Section IV presents the dedicated language used to configure devices of our solution and Section V proposes an analysis of several security rules written with this language. Finally, Section VI concludes this paper.

II. RELATED WORK

There are three main areas of prior work that are related to this research: (a) security devices used in large scale networks, (b) techniques to distribute these devices and (c) higher-level domain-specific language for security rules description.

Rimal et al. present cloud computing in [START_REF] Rimal | A taxonomy and survey of cloud computing systems[END_REF] as the concept that addresses the next evolutionary step of distributed computing. They add that there is a growing concern about its security. Users store confidential information in these architectures, and in wrong hands, it could create civil liability.

Main components of network security structures are firewalls and intrusion detection systems. Bellovin et al. describe firewalls in [START_REF] Bellovin | Network firewalls[END_REF] as components placed between two networks. All the traffic between these networks must pass through the firewall. Also, firewalls filter authorized traffic, defined by local security policies. Debar et al. outline in [START_REF] Debar | Towards a taxonomy of intrusion-detection systems[END_REF] that the main task of an Intrusion Detection System (IDS) is to monitor the usage of systems and to detect insecure states. IDS detect attempts and active misuses by legitimate users or external parties to abuse their privileges or exploit security vulnerabilities. Popular IDS, like Bro [START_REF] Paxson | Bro: a system for detecting network intruders in real-time[END_REF] or Snort [START_REF] Roesch | Snort-lightweight intrusion detection for networks[END_REF], are based on signatures rules that can match undesired behavior or network traffic.

Early motivation to distribute security systems was to tackle modern network topologies. In [START_REF] Snapp | Dids (distributed intrusion detection system) -motivation, architecture, and an early prototype[END_REF], Snapp et al. introduce DIDS, a distributed IDS monitoring an heterogeneous network of computers. In the late 90s, Bellovin et al. also proposed in [START_REF] Bellovin | Distributed firewalls[END_REF] to distribute firewalls because conventional firewalls rely on the notions of restricted topology and controlled entry points. While first solutions were centralized (several probes on the network and a central correlation unit), recent solutions are distributed (each component detects attacks and tries to collaborate with the other components) [START_REF] Zhou | A survey of coordinated attacks and collaborative intrusion detection[END_REF]. The latter are based on agents that analyze network traffic in order to collaboratively detect intrusions using alert correlation.

What we propose through DISCUS is not an agent-based infrastructure that correlates alerts coming from every probes. We want these agents to be able to share data so that all components can be aware at anytime of the state of the network. Our solution uses a content-based Peer-to-Peer system where each agent collaborates and disseminates local analysis. Previous works, like [START_REF] Li | Towards scalable and robust distributed intrusion alert fusion with good load balancing[END_REF][START_REF] Marchetti | Peer-topeer architecture for collaborative intrusion and malware detection on a large scale[END_REF][START_REF] Yegneswaran | Global intrusion detection in the domino overlay system[END_REF], use Distributed Hash Tables to distribute IDS or to detect large-scale attacks as DDoS, worm spreading or distributed portscan. Dedicated languages, also known as Domain-Specific Languages, are tailored for a specific application domain. A network security DSL can describe an exploited vulnerability, how an attack is performed, what are the consequences of such attack, or how to react.

Exploit languages [START_REF] Salgueiro | Using constraints for intrusion detection: the NeMODe system[END_REF][START_REF] Eckmann | Statl: An attack language for state-based intrusion detection[END_REF] are used to describe how an attack is performed and the stages of the attack. Knowledge languages [START_REF] Michel | Adele: an attack description language for knowledge-based intrusion detection[END_REF][START_REF] Cuppens | Lambda: A language to model a database for detection of attacks[END_REF] have a global expertise of described attacks and know the implications of a single event. They use pre and post conditions to trigger security rules. Detection languages [START_REF] Paxson | Bro: a system for detecting network intruders in real-time[END_REF][START_REF] Roesch | Snort-lightweight intrusion detection for networks[END_REF] describe the detection of an attack according to a number of occurrences of events. Response languages [START_REF] Salgueiro | Using constraints for intrusion detection: the NeMODe system[END_REF][START_REF] Kanoun | Towards dynamic risk management: Success likelihood of ongoing attacks[END_REF] describe how to react when an attack is detected.

Our solution consists in describing security rules as events, structures and functions. Following languages use the same concepts. One of the first network security language proposed was the Bro language [START_REF] Paxson | Bro: a system for detecting network intruders in real-time[END_REF] that describes security rules using variables, structures, functions and statements. STATL [START_REF] Eckmann | Statl: An attack language for state-based intrusion detection[END_REF] is a state-based language that use security rule to describe transition from an automaton.

III. DISCUS

The main goal of DISCUS is to detect and stop attacks targeting or using large network architecture, like cloud computing. Also, we want this solution to be easily (re)configurable because new exploits are discovered every day. People who are responsible for the security of a network should be able to load the new security rule easily, that is why we provide a simple and expressive dedicated language.

This section introduces DISCUS. First, it describes the proposed solution, then the security devices integrated into the solution, and eventually how to configure the whole structure.

A. Architecture overview

Commonly used network security solutions are firewalls and Intrusion Detection Systems. These solutions usually consider that attacks are led from an external network targeting the protected network (or vice versa). That is not true anymore for large networks like in cloud computing. Because these devices are not adequate for this type of network, we propose another security structure based on common security solutions, combined with a massively distributed structure composed of security probes.

Figure 2 represents the proposed structure. In this basic structure, firewalls remain the entry points from the outside of the network and filter undesired traffic. New elements are security probes, either physical or virtual. We want these security probes to be as close to the hosts as possible. Thus, we can finely analyze network traffic that goes through these at their source. This can also let security administrator isolate one or several hosts which have been detected as malicious. Physical probes are placed behind physical hosts or plugged to the switch or directly integrated into it. Virtual probes are necessary when virtual machines are running on a physical machine (typical scenario in cloud computing). In this case, we need to analyze traffic that comes from every virtual machine, in order to prevent attacks inside a physical host. These probes are configured by an administrator responsible for the security of the structure. He describes security rules thanks to DISCUS SCRIPT, a dedicated language, and uses the provided compiler to generate a unique binary for every security probe. After being configured, security components cooperate to detect intrusion occurring on the network. If a decision has to be made (for example, how to react when an intrusion has been detected), decision can be delegated to powerful components of the structure like firewalls.

B. Security probes

In our solution proposal, we introduce new security probes. The goal of these security probes is to be placed in cutthrough configuration between hosts and network elements. Physical probes are physically plugged directly on the host or the closest switch. Virtual probes are integrated into physical hosts running virtual machines. The latter could be existing security solutions such as Snort or Bro, customized in order to be able to collaborate into DISCUS. They process network data from these virtual hosts in order to prevent attacks occuring inside a same physical host. Also, they can isolate a virtual machine which has been detected as being a malicious host.

C. Configuration of the structure

Configuration of security devices of our solution is done in few steps. Figure 3 depicts these steps. At first, (1) the security administrator write security rules using the dedicated language. Then, thanks to a configuration file describing the network structure, (2) he compiles these rules using DISCUS SCRIPT's compiler. Eventually, (3) the compiler generates a binary program for every probe and components of the security structure. Each binary is produced by selecting and adapting a subset of the rules matching the context of the target. For example, a rule specifying that network traffic from a specific internal host is forbidden will only be consistent for the security probes monitoring this specific host. In this section, we introduce the dedicated language proposed along with our solution DISCUS. In our case, we wanted to configure easily probes and security elements of the structure. We use a static typing and declarative language. Besides, security rules expressed in this language are described as events, because it is natural to describe a sequence of network packets as a sequence of events. Following subsections introduce main syntax elements of the language. We will not discuss in this paper some syntax elements (like structures or enumerations) because they are very similar to C instructions, as well as types (because of the length of the paper). The only types used in this paper are enums, netaddr and intX, where X is the number of bits of the integer.

A. Table declaration

A security device usually keeps contextual data in order to run analysis. It includes hosts on the network, state of the connections, number of connections, and so on. Tables are aggregate types that store contextual data. For example, it can be used to store the number of failed TCP connections in order to detect portscan. Such tables are local or shared between components of the security solution.

Listing 1: Example of a local table declaration

t a b l e attack_attempts { i n t 3 2 attacker; i n t attempt_counter; t i m e last_attempt; };

Listing 1 declares a table storing the number of attack attempts for a specific IPv4 source. When this number reaches a fixed threshold, the security device reacts or raises an alert.

One of the goal of DISCUS is to provide a distributed system able to detect large-scale attacks. This type of detection is possible through tables which can be global or local. A local table could be assimilated to a database stored directly on the security device (like common security solutions). Shared tables are spread over the network and each device is able to get or update entries of an table. When the compiler is processing the security rules, it has the full knowledge of which security devices can store data (thanks to the topology configuration), which security devices insert or update entries in which table and so on. It is the role of the compiler whether or not to distribute a table and where to distribute it.

Such distributed architecture could be based on existing systems like Distributed Hash Table . A network overlay such as Peer-to-Peer systems deals with joining and leaving device and provides fault-tolerance and scalability.

B. Purge and deletion of tables

Our solution is based on a multitude of security probes storing tables. Unfortunately these devices have limited resources and cannot store unlimited data. That is why table entries need to be deleted. There are two scenarios for deletion of entries: no more resources on the device (need to purge the system) or obsolete entries (deletion of obsolete entries).

Listing 2: Example of a deletion statement remove attack_attempts when (now -last_transmission) > 3600;

In Listing 2, we specify that we do not want to keep entries that are too old (in this case, last attempt has been made at least one hour ago). Purge statements are declared in a similar manner, that is why we won't discuss it here.

C. Event-based security rules

Major statements of this dedicated language are security rules. We chose to adopt an event-based approach because network attacks could be naturally separated into events spread over the time. Moreover, event based systems can be easily and very efficiently implemented on FPGA. Listing 3 presents the syntax used to define a new event. A rule is triggered when an event (on A) occurs and conditions are satisfied. An event is defined using a name and a list of parameters. When a rule is triggered, it can execute several statements and/or raise one or several events (raise D), immediately or after a delay. There are several statements: create or delete a table entry, update a field in a table or run a command.

D. Characteristics of the language

In this subsection, we point out several characteristics of the dedicated language. First, we want the compiler to be minimalist, in other words, we want that almost all elements are defined using the language (for example, ip or tcp structure will be defined using the language and not hard-coded in it). Also, the compiler is able to detect inconsistent or conflicting rules and alert the security administrator so that he could correct corresponding rules. Finally, the compiler decide where the data is stored across the network of security devices.

Our language has several properties as well. It is a statically typed language, mainly because we want to avoid programming mistakes or semantic misinterpretations. Also, the language detects cycles, thus it can guarantee the detection process will terminate. Finally, the compiler deletes inconsistent or unused rules : orphan event, sequences of events that lead to a dead leaf (an terminal rule that does nothing for example).

V. SECURITY RULES EXAMPLES

In this section, we describe several security rules using DISCUS SCRIPT. In the following example, we want to prevent a SYN flood attack, a simple but effective denial of service. A basic idea to detect this type of attack is to pay attention to the number of opened connections between two hosts that are not fully established. When this number exceeds a fixed threshold, the security solution can react (for example, blacklist the establishing host, close all connections and so on).

We consider that the table attack_attempts from Listing 1 is defined for the remainder of this example. We also consider that an event tcp_packet, triggered when a TCP packet is received, is implemented into the library of the solution.

At first, we need to create a contextual structure that stores the state of TCP connection. When such entry table is created and the remote host answers back a TCP SYN-ACK, we try to detect SYN flood. This attack consists in partially opening a lot of TCP connections in order to exhaust the target's resources. So, a basic detection method is to look for TCP connections not fully opened after a short delay (here 250 ms). We consider for the following listings that the arguments are the same. When the number of attempted attacks reaches a fixed threshold, the security solution reacts. In this case, we decide to blacklist the attacker for one hour. where e x i s t s a i n attack_attempts w i t h a.src == src, a.attempt_counter >= THRESHOLD run blacklist(src, 3600); VI. CONCLUSION Cloud computing is a powerful architecture. It can provide several layers of services according to the needs of end users. Currently, only security concerns delay its massive adoption.

In this paper, we propose DISCUS, a new architecture that could solve security issues arising with cloud computing. Our proposal is based on a massively distributed structure that relies on commonly used security solutions but also probes scattered across the cloud structure. These probes could be physical or virtual and are located close to rented hosts. They collaboratively detect intrusions and attacks that target or use the cloud structure. To configure all these probes, we propose in this article DISCUS SCRIPT, a dedicated language, that describes security rules.

At the moment, the compiler is implemented, as well as the Snort backend. We are currently experimenting this backend and working on the configuration and deployment of security rules. Future work includes distribution of data across the probes and probes reconfiguration on the fly.

Figure 2 :

 2 Figure 2: DISCUS architecture

Figure 3 :

 3 Figure 3: DISCUS deployment

Listing 3 :

 3 Security rule syntax on A(args) where ... / * C o n d i t i o n s * / i n s e r t ... / * T a b l e s * / update ... / * T a b l e s f i e l d s * / run ... / * S c r i p t o r a c t i o n f r o m t h e l i b r a r y * / r a i s e D [ i n ...]; Conditions are optional and a rule must have at least a statement. More examples of security rules are given in Section V.

  on syn_flood_check( n e t a d d r src)

  When a new TCP connection is being established, we create an entry and store it locally. Because of the length of the paper, we will omit parameters not used in this example.

	enum tcp_state {...};
	t a b l e tcp_table {
	n e t a d d r src, dst;
	i n t 1 6 p_src, p_dst;
	enum tcp_state state;
	t i m e last_trans;
	};
	where flags == SYN
	i n s e r t i n t o tcp_table {
	src = src; dst = dst;
	p_src = p_src; p_dst = p_dst;
	tcp_state = TCP_HANDSHAKE_SYN;
	last_transmission = now;
	};

on tcp_packet(..., n e t a d d r src, n e t a d d r dst, i n t 1 6 p_src, i n t 1 6 p_dst, i n t 9 flags, ...)

  If the TCP connection is still not fully established after that short delay, we use a local table entry that store the number of attempted attacks. When the structure exists, we only need to update attempt_counter and the time of the last attempt. on syn_flood_attempt(...) where e x i s t s t i n tcp_table w i t h t.src == src, t.dst == dst, t.p_src == p_src, t.p_dst == p_dst, t.state == TCP_HANDSHAKE_SYN and e x i s t s a i n attack_attempt w i t h a.attacker == dst update a.attempt_counter += 1 update a.last_attempt = now r a i s e syn_flood_check(src)

	attacker = dst;
	attempt_counter = 1;
	};
	on tcp_packet(...)
	where flags == SYN | ACK
	and e x i s t s t i n tcp_table
	w i t h t.src == src, t.dst == dst,
	t.p_src == p_src, t.p_dst == p_dst,
	t.state == TCP_HANDSHAKE_SYN
	r a i s e syn_flood_attempt(src, dst, p_src, p_dst)
	i n 250 ms;

on syn_flood_attempt(...) where e x i s t s t i n tcp_table w i t h t.src == src, t.dst == dst, t.p_src == p_src, t.p_dst == p_dst, t.state == TCP_HANDSHAKE_SYN and not e x i s t s a i n attack_attempt w i t h a.attacker == dst i n s e r t i n t o attack_attempt {


Sarah Deveaud

David Barbosa Da Silva

Charles Gumiaux

Eric Gloaguen

Yannick Branquet

Alexandre Lima

Arnaud Villaros

Laurent Guillou-Frottier

Jérémie Melleton

Which parameters control the Variscan pegmatite field-scale organization ?

The emplacement of LCT-type pegmatite field is recently admitted to be controlled by regional tectonic structures rather than by the proximity to a granitic body. In particular, the emplacement of Monts d'Ambazac pegmatites (France), which is controlled by an oriented faults-system. In order to strenghten these findings, we investigated two pegmatite fields which present similar mineralizations, but different spatial organizations: the Forcarei pegmatite field (FPF,Spain) and The Barroso-Avão pegmatite field (BAPF, Portugal). The FPF is limited in its southern edge by the Celanovamigmatitic dome (CMD), where some pegmatites occurred. These observations favor a priori the model of direct crustal anatexis (DCA). Moreover, geostatistics and synkinematic criteria as shear-bands that are observed in pegmatites seem to suggest that the leftlateral Serra do Suido shear-zone near to the FPF has a major role during its emplacement. Indeed, pegmatites are clustered and oriented in the same broad N-S direction as the shear-zone. The formation of BAPF also matches the DCA model since it is located near to the southern edge of the CMD. The distribution of pegmatites shows no preferred orientation and cross-cutting of pegmatite-subtypes. The lack of shear-zone in BAPF indicates the key role of such structures on the field zonation. These preliminary results favor the influence of shear-zones as flow channels on i) the variscan pegmatite fields emplacement and ii) pegmatite subtypes field-scale organization. These results have been enhanced by premilinary numerical model which investigates the ascent of pegmatitc melts through crustal-scale permeable zones.


Rafik Hajrya
email: rafik.hajrya@ensam.eu

Nazih Mechbal
email: nazih.mechbal@ensam.eu

Principal Component Analysis and Perturbation Theory Based Robust Damage Detection of Multifunctional Aircraft Structure

Keywords: Damage detection, blind source separation, principal component analysis, information theory, angle between subspaces, matrix perturbation theory, analytic bound, robustness, Conformal Load-Bearing Antenna Structure, temperature effect

A fundamental problem in structural damage detection is to define an efficient feature to calculate a damage index (DI). Furthermore, due to perturbations from various sources, we need also to define a rigorous threshold whose overtaking indicates the presence of damages.

In this paper, we develop a robust damage detection methodology based on Principal Component Analysis (PCA). We present first an original DI based on the projection on the separation matrix, and then we drive a novel adaptive threshold that doesn't rely on statistical assumptions. This threshold is analytic and is based on matrix perturbation theory (MPT). The efficiency of the method is illustrated using simulations on a composite smart structure with PZT and experimental results performed on a Conformal Load-Bearing Antenna Structure (CLAS) laboratory test.

Nomenclature

Introduction

The aerospace industry is aiming, more than ever, to reduce their energy consumptions and emissions. Despite recent advances in aircraft engines, the determinant energy-saving factors are reducing weight and optimizing the aerodynamic performances. Moreover, the aerospace companies have also to satisfy safety constraints, structural integrity (mechanical stiffness and strength) and some customer requests (integration new features as for example, multimedia and private communications). As shown in figure 1, a present commercial aircraft is composed of many antennas located on the structure and used for communication, navigation and radar activity. Furthermore, in a near future, the number of these antennas is expected to increase in order to improve comfort customers by allowing multimedia private communications and internet accesses. Taken into account of these customer requests will result into to an increase of weight, and a decrease in the aerodynamic performances due to the huge number of antennas that will protrude from the outer mould line (OML) of the aircraft.

To respond to these antagonist criteria, ongoing research and development programs have come up with the concept of multifunctional aircraft structure (MAS). The principle is to take the advantage of composite materials to integrate airframe structure with a functional system.

The objective is to perform a number of tasks such as transmit/receive (T/R) function, structural health monitoring (SHM), active control [START_REF] Mechbal | Damage Tolerant Active Control: Concept and State of the Art[END_REF], shape control and Conformal Load-Bearing Antenna Structure (CLAS) [2;3]. The concept of CLAS is to replace existing antennas, particularly blades and wires that protrude from the OML of aircraft with airframe structure, that: (i) supports primary structural loads, (ii) conforms to the OML and (iii) can perform the transmit/receive function of the existing antennas [START_REF] Callus | Conformal Load-Bearing Antenna Structure for Australian Defence Force Aircraft[END_REF]. CLAS will then reduce drag and have the potential to reduce weight and enhance the electromagnetic performance. However, the electromagnetic performances of this CLAS depend of their structural health, which can be clobbered by mechanical stress, impact damage or by delamination of a part of the antenna network from the host structure. Therefore, monitoring the health of the CLAS is a mandatory necessity.

In this paper, a methodology is proposed to achieve SHM on a CLAS composite structure.

Our study is a part of a more global project (Smart Materials and Structures for Electromagnetics, MSIE) supported by the Aerosapce Paris Région [START_REF] Msie | State of the Art for the Projet MSIE: Smart Materials and Structures for Electromagnetics[END_REF]. The MSIE project aims to demonstrate the practical use of metamaterials in CLAS and to monitor their performances by performing in the case of damage, SHM for restorative maintenance tasks and shape control compensation.

SHM is the most widely known form of MAS. It has been driven by recent developments in sensor technology [START_REF] Giurgiutiu | Development and Testing of High-Temperature Piezoelectric Wafer Active Sensors for Extreme Environments[END_REF], manufacturing, signal processing [START_REF] Staszewski | Time-Frequency and Time-Scale Analyses for Structural Health Monitoring[END_REF], applied mechanics and material sciences. SHM has led to a variety of efficient methods to detect, locate, classify, quantifying varying degrees of damage and estimating the remaining useful life of the structure [START_REF] Chang | A Potential Link from Damage Diagnostics to Health Prognostics of Composites Through Built-in Sensors[END_REF].

Several techniques have been employed to perform structural monitoring [START_REF] Boller | Encyclopedia of Structural Health Monitoring[END_REF], and various damage indices have been proposed to characterize the change in dynamic characteristics caused by damage. Some of these DIs are based on extracting features from the observed output vector , using numerous tools such as short Fourier transform [START_REF] Ihn | Detection and Monitoring of Hidden Fatigue Crack Growth Using a Built-In Pizoelectric Sensor/Actuator Network[END_REF], wavelet [START_REF] Sohn | Wavelet-Based Active Sensing for Delamination Detection in Composite structures[END_REF] and Hilbert-Huang transform [START_REF] Yang | Hilbert-Huang Based Approach for Structural Damage Detection[END_REF].

To perform SHM, the structure has to be smart. Smartness supposes spatial information which is obtained by network of sensors/actuators built in/on the host structure. This spatial diversity are achieved through multivariate and magavariate analysis to reduce data and to extract some simplified and hidden pattern [START_REF] Friswell | Sensor Validation for Smart Structures[END_REF][START_REF] De Boe | Principal Component Analysis of a Piezosensor Array for Damage Localization[END_REF][START_REF] Hajrya | Proper Orthogonal Decomposition Applied to Structural Health Monitoring[END_REF]. As the name implies, multivariate analysis is concerned with the analysis of multiple variables or measurements, but treats them as a single entity.

Blind separation source (BSS) is a family of techniques, which belong to multivariate statistics. BSS attempts to reveal hidden variables, named sources and mixing/separating matrices, only from their dynamical measured vectors . Among the methods of the BSS family, one can cite: Principal Component Analysis (PCA) [START_REF] Jolliffe | Principal Component Analysis[END_REF], Independent Component Analysis (ICA) [START_REF] Hyvärinen | Independent Component Analysis: Algorithms and Applications[END_REF], Second-Order Blind Identification (SOBI) [START_REF] Comon | Handbook of Blind Source Separation[END_REF]. In this paper, we focus on PCA.

PCA has been used to extract damage features. Friswell and Inman [START_REF] Friswell | Sensor Validation for Smart Structures[END_REF] proposed a damage index based on the angle between subspaces. [START_REF] De Boe | Principal Component Analysis of a Piezosensor Array for Damage Localization[END_REF] have used the same approaches to detect a piezoelectric faulty sensor. Recently, [START_REF] Mujica | Q-Statistic and T2-Statistic PCA-Based Measures for Damage Assessment in Structures[END_REF] have explored the use of PCA and T 2 and Qstatistics to detect damage.

The key issue in damage detection is to ascertain with confidence if damage is present even if the structure is subject to environmental perturbations. In the above references, the threshold associated to the proposed damage indices are defined manually [START_REF] Mujica | Q-Statistic and T2-Statistic PCA-Based Measures for Damage Assessment in Structures[END_REF], or following the assumption of Gaussian distribution of the damage index [START_REF] Kerschen | Sensor Validation Using Principal Component Analysis[END_REF].

In this study, we propose to develop a new robust analytical threshold (bound) that handle the effect of noise measurements, temperature changes and ensure the presence of damage. Our approach uses singular value decomposition (SVD) and matrix perturbation theory (MPT).

MPT considers how the features of a matrix namely singular values, singular vectors and subspaces change, when the matrix is subject to perturbations [START_REF] Stewart | Matrix Perturbation Theory[END_REF]. In the proposed approach, we study perturbations that could rise in the subspaces of the separation matrix to define an analytic threshold to the damage index proposed in this paper.

The remainder of this paper is organized as follows. In preliminary, we introduce the BSS problem and the principal component analysis technique. In section 3, the damage detection procedure is described, and it is divided into four subsections; the first and second subsections are devoted respectively toward the development of feature extraction and our proposed damage index; in subsection 3.3, we present our new approach to define the threshold. We outline in subsection 3.4 a framework. In section 4, finite element simulation's results on a composite plate are presented. In section 5, we present the experimental study of a conformal load-bearing antenna structure (CLAS), and the damage detection results are presented in section 6. Finally, conclusions and further directions will be drawn in section 7.

BSS problem and PCA algorithm

BSS problem

In blind source separation, the goal is to recover unknown source signals from their linear measured mixtures without any information on the mixing coefficients [START_REF] Comon | Handbook of Blind Source Separation[END_REF]. We suppose in the sequel a linear mixture. Hence, let be a zero mean measurement vector from sensors at time index :

[ ] (1)
The measurement matrix gathering samples is defined as follows:

[ ] (2)
The linear BSS problem is defined as [START_REF] Comon | Handbook of Blind Source Separation[END_REF]:

(

where is the mixing matrix, [ ] is the sources vector and [ ] represents all the uncertainties and perturbation effects (modeling uncertainty and noises). In our study, we will consider the noise-free model (4), dealing with uncertainties and perturbations will be done by generating a robust bound for detection decision:

BSS is an estimation problem, and it is accomplished by finding simultaneously an estimated sources vector ̂ and a separation matrix noted only from the observed data :

̂ (5)
where [ ] and are respectively the estimated sources vector at instant and the separation matrix. Equation ( 5) is called the separation model.

Clearly, without additional assumptions, the BSS problem (equation ( 5)) is ill-posed. Cardoso [START_REF] Cardoso | The Three Easy Routes to Independent Component Analysis, Contrasts And Geometry[END_REF] has shown that the BSS problem can be resolved using assumptions done on the nature of sources. In this paper, we assume that the sources are temporally identically and independently distributed (iid) and Gaussian, which leads to the principal component analysis method.

Principal component analysis

The separation of sources relies on the basic knowledge of the mutual independence of source components. It is then natural to resolve the BSS problem by minimizing a dependence criterion between these components. In our study, we adopt the mutual information as the separation criterion [START_REF] Cover | Elements of Information Theory[END_REF]. From equation ( 5), the solution is to find the matrix , so that the mutual information is zero. In the case where the sources are assumed to follow Gaussian distribution, the mutual information ( ) is given by [START_REF] Hyvärinen | Independent Component Analysis Using Mutual Information[END_REF]:

( ) ∏ ( ) (6)
If the components of the vector have to be statistically independent, mutual information defined in ( 6) is equivalent to:

( ) ∏ ( ) (7)
Knowing that covariance matrix is symmetric, equality ( 7) is satisfied if the covariance matrix is diagonal: the sources are uncorrelated and statistically independent [START_REF] Jolliffe | Principal Component Analysis[END_REF].

Principal component analysis of a measurement vector is defined as a pair of matrices { } such that the covariance matrix is factorized:

(8)
where [ ] is the matrix of eigenvectors and is a diagonal matrix of eigenvalues.

The separation matrix, noted now , is given by:

In order to reduce the number of principal components, we use the cumulative percent variance CPV [START_REF] Valle | Selection of the Number of Principal Components: The Variance of the Reconstruction Error Criterion with a Comparison to Other Methods[END_REF]. It is a measure of the percent variance captured by the first PCs:

∑ ∑ (10)

Damage detection procedure

Feature extraction

We seek to detect changes in the structure by monitoring specific features estimated from baseline observation set (referred by the exponent "s") and observation set of a current or unknown state (referred by the exponent "u"). We note , , respectively, the measurement matrix of the structure in a healthy and unknown state.

To define the feature, we use the separation matrices and . From ( 9), we have:

(11) (12)
If the reduction using a fixed CPV is possible (see subsection 2.2), the separating matrix and are rewritten as follows:

[ ( )] [ ] [ ] (13)
where: In the same way, we define the matrix using the following equation:

[ ( )] [ ] [ ] (14)
The presence of a damage will modify the measurement matrix , and consequently the separation matrix , compared to the matrix . Therefore, the subspaces generated by are deflected to those of . The idea here is to monitor changes of specific subspaces.

Let, { }, { } be respectively the range subspace of a matrix and its transpose. { } , { } correspond to the orthogonal projection of the range subspace { } and { }. One way to define orthogonal projections is to use SVD, i.e.

(

) 15
The orthogonal projections { } , { } are respectively defined as follow [START_REF] Golub | Matrix Computation[END_REF]:

{ } { } (16)
From the definition of the separating matrices and , the matrix of left singular vectors is equal to the identity matrix:

Therefore, no information can be derived from the matrix of left singular vectors, thus we will use the matrix of right singular vectors as a feature. To build our proposed damage index, we will use projections on the transpose of matrices and as a damage feature. This is the subject of the following subsection.

Damage index

Following the work of [START_REF] De Boe | Principal Component Analysis of a Piezosensor Array for Damage Localization[END_REF], the proposed damage index is constructed using the calculation of the principal angle vectors between the range subspace of the matrix and .

We note [ { } { }], the principal angle vectors between the range subspace { } and { }. Using the SVD tool, the Euclidean norm of the sinus angle of these vectors is defined as follow [START_REF] Davis | The Rotation of Eigenvectors by a Perturbation[END_REF]:

‖ [ { } { }]‖ ‖ {( ) } {( ) } ‖ ‖( {( ) } ) {( ) } ‖ (18)
In [START_REF] De Boe | Principal Component Analysis of a Piezosensor Array for Damage Localization[END_REF], the principal angle vectors are calculated using the QR decomposition and the cosines of the principal angles. Their damage index is given by the largest angle. Here, we propose the following damage index:

Proposal 1: Damage index

Let be a smart structure with sensors, and define equations (13), ( 14) and [START_REF] Comon | Handbook of Blind Source Separation[END_REF].

Damage could be detected by monitoring the following damage index:

‖ [ { } { }]‖ (19)
where is the number of principal components retained.

In the absence of disturbances and when the structure at an unknown state is healthy, the damage index is zero. Otherwise, the structure is damaged. However, as the measurements are corrupted by noises and unknown perturbations, the detection decision could be biased. Moreover, these perturbations might also affect the distinguishability between damage and environmental disturbances and consequently the isolationability of damages. To deal with this problem, we propose using matrix perturbation theory, to calculate an analytical upper bound of these small perturbations. This is the aim of the next subsection.

Analytic threshold

As stated previously, the proposed damage index is based on subspace projections and decompositions. Due to various perturbation sources such as finite data effect, measurement noises and temperature change, perturbations arise in subspaces. Therefore, false alarms could appear.

To handle these perturbations, we propose to use matrix perturbation analysis that can result in defining an analytical bound with no rely of statistical assumptions usually performed for damage detection. Matrix perturbation theory considers how matrix functions such as subspaces change when the matrix is subject to perturbations.

In this work, we consider that separation matrix obtained from the baseline, is slightly perturbed, i.e.

̃ ( 20
)
where ̃ is a perturbed version of with perturbation .

Then, we propose to define a robustness measure that estimates how much the damage index (defined in [START_REF] Mujica | Q-Statistic and T2-Statistic PCA-Based Measures for Damage Assessment in Structures[END_REF]) is affected. The overtaking of this rigorous threshold will indicate the presence of damage. It will also minimize false-positive and false-negative alarms.

To drive the proposed analytical bound, we use early work of Wedin [START_REF] Wedin | Perturbation Bounds in Connection with Singular value Decomposition[END_REF] on perturbed matrices in connection with SVD. The idea is to estimate ‖ ‖ by performing several tests or simulations on the healthy structure and to evaluate the gap between specific singular values in order to find an upper bound for . Background materials on matrix perturbation theory and Wedin framework results can be provided in [21;26]. It is to be noted that our approach is an unsupervised learning mode (ULM), which implies that data from a damaged state are not used to build this bound. This ULM is mandatory necessary for realworld application.

The proposed analytic bound is derived following three major steps:

First step: testing the healthy structure Consider the variation that the separation matrix is subject due to the environmental disturbances. To describe this variation , we perform a second test on the healthy structure, we apply the PCA method and we determine the new separation matrix noted ̃ . The matrix is defined by the following relationship:

̃ ( 21
)
The SVD of the matrix ̃ is given by:

̃ [ ( )] [ ̃ ̃ ] [ ̃ ̃ ] ̃ ̃ (22)
We first define the following residual matrices and norms:

̃ ̃ ̃ ̃ ̃ [( ̃ ) ̃ ̃ ] ̃ (23)
̃ ̃ ( ̃ ) ̃ (24)
And then evaluate their two norm, i.e.:

‖ ‖ ‖ ̃ ‖ ‖ {( ̃ ) } ‖ (25)
‖ ‖ ‖ ‖ ‖ { ̃ } ‖ (26)
where

{ ̃ } and {( ̃
) } are respectively the orthogonal projection of the matrix ̃ and its transpose.

From these relations, we introduce the number :

‖ ‖ ‖ ‖ (27)
This number quantifies the magnitude of the environmental disturbances.

We are now going to matrix perturbation theory through the work of Wedin and its principal results.

Second step: Matrix perturbation theory [START_REF] Wedin | Perturbation Bounds in Connection with Singular value Decomposition[END_REF] Our aim is to find an analytic upper bound to (see [START_REF] Mujica | Q-Statistic and T2-Statistic PCA-Based Measures for Damage Assessment in Structures[END_REF]). Consequently, when we have estimates ‖ ‖ and the gap between the least singular value of ̃ and the largest singular value of .

Assume now, that , such that :

‖ ‖ ( 28
) ( ̃ ) ‖ ̃ ‖ (29)
So:

̃ ‖ [ { ̃ } { }]‖ ̃ (30)
̃ ‖ [ {( ̃ ) } { }]‖ ̃ (31)

Third step: elaboration of the bound

Recall that the damage index (see [START_REF] Mujica | Q-Statistic and T2-Statistic PCA-Based Measures for Damage Assessment in Structures[END_REF] ) is calculated from the separation matrix of the healthy state and of the structure in an unknown state, while the term ̃ is calculated from the separation matrix and ̃ from a second healthy test.

As the matrix of singular vectors of separation matrices equals the identity matrix, the term ̃ . Consequently, the term ̃ satisfies:

̃ ( 32
)

Proposal 2: Analytic bound

The proposed bound is defined as follow:

(33)

where:

quantifies the magnitude of the environmental disturbances and it is defined in [START_REF] Davis | The Rotation of Eigenvectors by a Perturbation[END_REF], and are defined respectively in [START_REF] Wedin | Perturbation Bounds in Connection with Singular value Decomposition[END_REF] and [START_REF] Hajrya | Contrôle Santé des Structures Composites : Approche Expérimentale et Statistique[END_REF],

is the number of principal components retained, is a tuning parameter adjusting the bound.

The proof of this proposal is sketched in [START_REF] Hajrya | Contrôle Santé des Structures Composites : Approche Expérimentale et Statistique[END_REF] and is quite straightforward, from results in [START_REF] Wedin | Perturbation Bounds in Connection with Singular value Decomposition[END_REF].

Our proposed damage index and its associated bound are quantified numerically, so we can calculate a detection rate , which is defined as follow:

(34) if then the structure is damaged, otherwise it is healthy.

Framework of the damage detection procedure

For practical implementation, we outline in this subsection the following framework. We assume that the measurements have been scaled (zero mean and unity variance).

Step 1: Tests of the healthy structure state

Example through FE simulations

Before applying our damage detection methodology on a conformal load-bearing antenna structure (CLAS), we have first applied it through a finite element (FE) model of a composite plate.

Consider a rectangular carbone epoxy plate with dimensions and made up of 16 layers. The layer sequences are: (0°2, 45°2, -45°2, 90°2, 90°2, -45°2, 45°2, 0°2)

and the mechanical properties of the composite material in nominal conditions are illustrated in table 1. In our case, nominal condition means that effects of temperature change are not taken into account. Using the controllability and observability gramians, we have performed in a previous work [START_REF] Hajrya | Active Damage Detection and Localization Applied to a Composite Structure Using Piezoceramic Patches[END_REF], an optimal placement of piezoceramic patches with dimensions . These piezoceramic patches belong to the PZ 29 family, table 2 depicts the mechanical and electrical properties of these PZT. The FE model of the composite plate equipped with its piezoceramics was developed using Structural Dynamic toolbox [START_REF] Balmes | Structural Dynamics Toolbox User's Guide[END_REF].

We have chosen a model with 195 elements, each element has dimension of .

The plate is under free-free boundary conditions. Figure 2 shows the composite plate model with the optimal placement of the PZT.

To apply the damage detection methodology described in subsection 3.4, we have first built the baseline of the healthy plate model in nominal conditions (see table 1). For this purpose, we have used PZT 7 as an actuator, while the others PZT are sensors ( , then the continuous state space of the model was calculated with modes. The damping was assumed to be equal to 0.05% for all modes. Once the continuous state space calculated, we have discretized it with sampling frequency . The excitation of actuator PZT 7 consists in a signal pulse with 1ms width, signal sensors were recorded with time samples.

In section 2, we have stated that PCA can allow use to reduce the dimension of the measurement matrix. According to a fixed CPV of 98%, we have retained seven principal components ( ).

Perturbation by measurement noises

We have first simulated the effect of environmental perturbations as a noise disturbance acting on the dynamic response data: 56 simulations were achieved using the composite plate model in different conditions of measurement noises. Table 3 summarizes these simulations. The first reflects the composite model in a healthy state with 0.25% variance noise of the energy contained in the simulated data sensors, simulations n°2 to 50 correspond to the healthy state, but with different values of noise variance (from 0.26% to 0.75% with a step of 0.01%). These simulations will allow us to build the analytic bound . It is to be noted, that in the nominal conditions, the value of coefficient defined in proposal 2 (see relation (33)) is equal to 1.

Simulations 51-55 correspond to damaged states of the composite plate model. These simulations are associated respectively to a reduced stiffness of 5% for element 17, 56, 140, 37 and 184 (see figure 2), with a variance noise of 0.5%. Figure 3 depicts the time response of sensor PZT 1 for simulations n°1 (healthy state) and n°56 (damaged state: reduced stiffness of 5% for the element 17), only the 512 first samples are displayed.

Table 4 depicts the results of the damage detection through these simulations. One can see that for the damaged states, the associated damage index is upper than the proposed bound , so the damaged states are well detected. Figure 4 is a visualization manner for the decision making.

Perturbation by temperature change

The main of this subsection is to study through the FE model, the effect of temperature in the decision making regarding alarm false.

With the inherent anisotropy of composite materials, any attempt to simulate the effect of environmental factors like temperature requires relevant experimental data from the structure in an enclosed heated space. These data will feed the FE model to yield reasonably accurate prediction. However, it is expected that first order thermal effects in an instrumented composite will be associated with the difference between the evolution of constitutive behavior of the composite matrix, its fibers and the piezoelectric. As a rough approximation, one can consider varying moduli for the composite plate with fixed mechanical properties for the piezoelectric material.

In order to mimic the temperature effects, we have modified the moduli by a percentage [ ] . Table 5 presents the basis of the modification under certain temperature change. According to this rough approximation, simulations were achieved. It is to be noted that for each simulation, 0.25% variance noise measurement was added to each dynamic response. The damage indices associated to these simulations are depicted in table 6. The results depicted in this above table show that the damage index is greater than the analytic bound. Knowing that these simulations concern a healthy model, false positive alarms (FPA) were detected. So, we have to deal with these FPA.

Handling the effect of the temperature

Our strategy to handle the effect of temperature was to adapt the value of the bound by a judicious choice of the coefficient , defined in (33).

Let, ( ̃ ) , , be the perturbed separation matrices, obtained respectively from simulation 56 to 60. Each of these matrices captures the temperature effect through their singular values. According to this reasoning, we define as:

( ̃ ) , (35)
In order to test this strategy, we have simulated the temperature effect for the healthy and damaged models. Table 7 summarizes the studying model states in this subsection.

Table 8 shows the value of the updated bound, and the damage index associated to the different conditions of the composite model (see table 7). One can see that for the damaged states, the associated damage index is upper than updated bound . Moreover, in the case of temperature effect of the healthy state model, the damage index is less than the bound, so the false positive alarms presented in table 6 were entirely removed. In order to have visualization for the decision making, we have plotted in figure 5, the damage rate versus the simulation number (according to table 7); we can obverse that the healthy and damaged model states are well separated, even if the temperature is present in our modeling manner. According to these results, we can assure that if the temperature change belongs to a certain interval, no false positive alarms will be detected.

Experimental test bench: Conformal load-bearing antenna structure

The CLAS employed in this study consists of a piece of a composite fuselage. The dimension of this structure is and it is made up with the same layers as the composite plate model. The mechanical proprieties of the composite material are presented in table 1. Figure 6, 7 and 8 show respectively the antenna network of the CLAS, the seven PZT bonded on it (dimensions , PZ29 family) and their locations.

Before applying the damage detection methodology, we have done simultaneously a radiation pattern and a measure of strain using the PZT sensors (figure 9) in order to check if there is coupling between the electromagnetic phenomena of the antenna network and the electrical information transmitted by the piezoelectric sensors. This study has shown that the two phenomena can coexist in the same time without any interaction.

The CLAS depicted in figure 8 was used first to build a baseline set. For that, we have used PZT 2 as an actuator, while the six others PZT are sensors ( ). The input excitation, and the data acquisition were done using a commercial system dSPACE®. This excitation consists in a signal pulse with 1ms width; signals were acquired with sampling frequency , and time samples were recorded for each channel: one corresponding to the excitation applied to the PZT actuator, and the others concern the measurements collected by the PZT sensors. Several baseline tests with enough time lag between them were done on the CLAS ( ) in order to calculate the analytic bound . It is to be noted that these tests were done under temperature of T=18°C and T=24°C. In these conditions, the coefficient was set to one. Using a buckling device (figure 10), we have provoked in a second time a delamination of the antenna part from the host structure (figure 11).

Application of the damage detection to the CLAS

In this section, we proceed to the damage detection method within the framework of the CLAS. This method is governed by the damage index (see proposal 1), and by the analytic bound (see proposal 2).

The damage detection methodology illustrated in subsection 3.4 was applied to the CLAS to detect two kinds of damage. The first concerns the detection of a hole (damage 1 with diameter of ) located at the middle of the CLAS (see figure 10), the second damage concerns a delamination of an antenna network part, and it is separated from the host structure by a distance of (see figure 11). Let's begin by presenting the results of damage 1.

Damage detection result: damage 1

Table 9 depicts the result of our methodology to detect the hole of the CLAS. In this case, the damage index , and it is upper than the bound . Moreover the damage rate . This result allows us to assert that the first damage was well detected. Now, we present the result of damage 2.

Delamination of an antenna part

Conclusion

The important objective of a damage detection algorithm is to ascertain with efficiency the presence of damage. This decision can be done using a statistical model associated with the damage index. However, this model is an approximation due to the finite number of available baseline data. In this paper, we have explored the use of matrix perturbation theory (MPL) in the context of SHM.

Through this paper, we have proposed first a damage index based upon the change of the principal angle vectors between subspaces. This damage index does not require the knowledge of the mechanical model of the structure. So, it can be applied to more complex structures. To enhance the decision on the presence or not of damage, we have associated to this DI a bound. This later is entirely analytical, and it is suitable for adaptive damage diagnosis. This adaptability was tested to tackle the effect of temperature on a FE composite plate. Several changes of stiffness for the simulation setup and real delamination for the CLAS were well detected, with no false-positive alarms. Experiments on temperature effect (using enclosed heated space) and changes in boundary conditions are underway.

  the structure in a healthy state ̃ = Measurements matrix of a second experiment = Measurements matrix of the structure at unknown state = Matrix which reflect the effect of disturbances = Matrix of left singular vectors = Matrix of right singular vectors Variance = Matrix of right singular vectors Variance = Matrix of right singular vectors Variance = Eigenvalue and = Numbers associated to the matrix perturbation theory (subsection 3.3)

Figure 1 :

 1 Figure 1: Antenna of the AIRBUS A318 [5]

  values, the matrix of right singular vectors and the separation matrix associated to the principal subspace of the healthy structure. respectively the matrix of singular values, the matrix of right singular vectors and the separation matrix associated to the residual subspace of the healthy structure.

Figure 2 :Figure 3 : 4 :Figure 4 :

 2344 Figure 2: Finite element model of a composite plate bounded with piezoceramic patches

Figure 5 :

 5 Figure 5: Results of handling the effect of temperature in a FE composite model

Figure 6 :

 6 Figure 6: Antenna network of the CLAS Conformal load-bearing antenna structure Dimensions: 800 x 150 x2 mm3 PZT: 30 x20 x0.2 mm3

Figure 7 :Figure 8 :Figure 9 :Figure 10 :Figure 11 :

 7891011 Figure 7: Location of the piezoceramic patches on the CLAS

Step 2: Test of the structure at unknown state 2

		4.1.1. Calculate the separating matrix ( ̃	) ,	, following (11),
		4.1.2. Apply SVD to the matrix ̃	following equation (13),
		4.1.3. Calculate the variation	using the following equation:
				̃
		4.1.4. Calculate the analytic bound	following (33) (see proposal 2),
	4.2. Calculate the mean of the bound	:
			∑
	Step 5: Decision-making
	Check if	. If it is the case, then the structure is damaged, otherwise it is healthy.
	Step 3: Calculate the damage index
	3.1. Apply PCA to the measurement matrix ,
	3.2. Calculate the separating matrix	following equation (11),
	3.3. Apply SVD to the	following equation (13),
	3.4. Calculate the separating matrix	following equation (12),
	3.5. Apply SVD to the	following equation (14),

1.1 Make a first test of the healthy structure, 1.2 Make other tests of the healthy structure with enough time lag between tests, 1.3. Build the measurement matrix of the healthy structure, 1.4. Build the measurement matrices ̃ of the healthy structure. .1. Make one test of the structure at unknown state, 2.2. Build the measurement matrix of the structure at unknown state. 3.6. Calculate the damage index following the proposal 1 (see equation (19)). Step 4: Calculate the bound associated to the damage index 4.1. Take back the result of step 3.2, and repeat for i=1 to the following steps:

Table 1 :

 1 Mechanical properties of the composite material

	Property
	Value

Table 2 :

 2 Mechanical and electrical properties of the PZ29[START_REF] Ferroperm | Material Data Based on Typical Values for Piezoceramic[END_REF]

	Property	(	)	(	Curie temperature C°
	Value				235

Table 3 :

 3 Simulation states for the composite plate model

	Simulation n°	State
	1-50	Healthy state: used to build the analytic bound
	51	Damaged 1: Reduced stiffness of 5% for the element 17
	52	Damaged 2: Reduced stiffness of 5% for the element 56:
	53	Damaged 3: Reduced stiffness of 5% for the element 140
	54	Damaged 4: Reduced stiffness of 5% for the element 37
	55	Damaged 5: Reduced stiffness of 5% for the element 184

Table 5 :

 5 Value of the simulated temperature under the perturbation of the moduli

	Simulation n°	Simulated temperature [ ]
		26
		27
		28
		29
		30

Table 6 :

 6 Effect of the temperature in the damage detection

	Simulation n°

Table 7 :

 7 Simulation sate of the composite plate model: temperature effect

	Simulation n°	State of the composite model by taken into account of the
		temperature effect Healthy state: α = 0.0105% and 0.75% variance noise Healthy state: α = 0.0115% and 0.25% variance noise Healthy state: α = 0.0125% and 0.25% variance noise Healthy state: α = 0.0135% and 0.50% variance noise Healthy state: α = 0.014% and 0.50% variance noise Damaged 6: Reduced stiffness of 5% for the element 17, α =
		0.014% and 0.25% variance noise Damaged 7: Reduced stiffness of 5% for the element 56, α =
		0.013% and 0.5% variance noise Damaged 8: Reduced stiffness of 5% for the element 140, α =
		0.0125% and 0.75% variance noise Damaged 9: Reduced stiffness of 5% for the element 37, α =
		0.0115% and 0.5% variance noise Damaged 10: Reduced stiffness of 5% for the element 184, α =
		0.014% and 0.75% variance noise

Table 8 :

 8 Results of the damage detection by taken into account the temperature effect

	Simulation n°

Table 9 :

 9 Result of the damage detection of the CALS using PCA: damage 1

	Damage1 of the CLAS: hole	0.3159	0.0347

Acknowledgement

This work was supported by the project: Smart Materials and Structures for Electromagnetics funded by the consortium ASTech Ile de France. The authors wish to thanks all the persons involved in this project.

Damage detection result: damage 2

Once the damage 1 detected, we have provoked a delamination of an antenna network part from the CLAS (figure 11), we have done tests of this structure state and we have applied the damage detection procedure (the result is depicted in table 10). In this case, the damage index , and it is upper than the bound . Moreover the damage rate . Then damage 2 is also well detected. In order to consolidate our results, we have performed other tests of the healthy and the damaged (damage 2) state of the CLAS using PZT 3 as an actuator. To test the damage detection methodology regarding the alarm false, we have done other tests of the healthy state (tests1-4); with enough time lags between them. Figure 12 depicts the damage rate versus the test number. We can observe that the damage rate is greater than one for the damaged states.

One can also see that no false-positive alarms (NFPA) were detected.


H Shidpour

A Bernard
email: alain.bernard@irccyn.ec-nantes.fr.

M Shahrokhi

Forty

Keywords: New product development, Three-Dimensional Concurrent Engineering, Multi-objective programming model, Intuitionistic fuzzy set

This paper proposes a group decision-making process by using multi-objective programming to address three-dimensional concurrent engineering (3D-CE) problems involving product, process and supply chain design. This paper uses opinion of decision makers to evaluate of the candidate suppliers and to determine importance of criteria by considering lack knowledge/information in the early design stages. For identifying impact of the lack knowledge/information on selecting the best configuration of product design, assembly/manufacturing process and suppliers of components, a numerical example is represented for two states of intuitionistic fuzzy and fuzzy. The evaluation showed that the configuration selected for two states are completely different.

Introduction

NPD process is a series of necessary activities for developing a new product that it is including of the growth of its idea, production and introduction into the market. The product design is one of the most important activities in developing new product. A good design process should guarantee both the fulfilment of customer needs and business goals. So, evaluation of product design plays a critical role in the early phase of product development. It can save both cost and time in product development by decreasing the risk of re-design of new product. It is widely accepted that more than 70% of the total product development costs are committed by decisions taken at the early stages of design [START_REF] Dieter | Engineering Design-A Material and Processing Approach[END_REF]. The subject is more important when to know that much information about criteria in the early stages of design is not available or is uncertain.

Today, considering of simultaneous different criteria from diverse areas in the early stages of design is inevitable. Fine [START_REF] Fine | [END_REF] introduced the term "Three-Dimensional Concurrent Engineering (3D-CE)" that consider, simultaneously, different aspects of the design, process and supply chain in the early stages of the product development.

Selecting of design alternatives is a multi-criteria decision-making process which involves many factors of both customer needs and business constraints. In the early design stages, evaluation of design alternatives is difficult to precisely express by crisp data because the information available is usually imprecise, incomplete or subjective. So, an effective method to evaluate design alternatives in the early stage of design process is inevitable.

In real life, however, the information of an object corresponding to a fuzzy concept may be incomplete; a decision maker (DM) may have a hesitation or uncertainty about the membership degree. Atanassov [3] introduced the concept of intuitionistic fuzzy set (IFS) to deal with this challenge. Expression of hesitation is particularly helpful for decision makers (DMs) when they need to select suppliers in a highly uncertain supply network such as a design product [START_REF] Wu | Supplier selection under uncertainty: a switching options perspective[END_REF]. An IFS-based method can be applied to derive the most appropriate suppliers that its output can be further utilized by a multi-objective optimization model to determine the most appropriate design alternatives.

The paper is organized as follows: section 2 review literature related to 3D-CE. Section 3 describes the basic concepts used in the paper. Section 4 represents the proposed method to choice the best configuration of product design, assembly/manufacturing process and suppliers of components. Section 5 expresses a numerical example to show efficiency of the method. Section 4 represents conclusion and future research.

Literature

In the recent years, some papers have argued impact of product design, process and supply chain design on NPD process. Ellram et al. [START_REF] Ellram | Product-process-supply chain: an integrative approach to three-dimensional concurrent engineering[END_REF] surveyed literature related to mutual fields of 3D-CE (product/process, product/supply chain, and process/supply chain) and done a good review of the 3D-CE. Huang et al. [START_REF] Huang | Towards integrated optimal configuration of platform products, manufacturing processes, and supply chains[END_REF] integrated platform product decisions, manufacturing process decisions, and supply sourcing decisions by developing a mathematical model to quantify the relationships among various design decisions. Petersen et al. [START_REF] Petersen | Supplier integration into new product development: coordinating product, process, and supply chain design[END_REF] explained how to integrate suppliers into the new product development process and showed their impacts on process design and supply chain decisions. Fine et al. [START_REF] Fine | Modeling trade-offs in threedimensional concurrent Engineering: a goal programming approach[END_REF] proposed a goal-programming model to address three-dimensional concurrent engineering (3D-CE) and surveyed relationship between product and supply chain structure. Wang et al. [START_REF] Wang | Product-driven supply chain selection using integrated multi-criteria decision-making methodology[END_REF] described relation of product characteristics to supply chain strategy and developed an integrated analytic hierarchy process (AHP) and preemptive goal programming (PGP) methodology. Fixon [START_REF] Fixon | Product architecture assessment: a tool to link product, process, and supply chain design decisions[END_REF] developed a multi-dimensional framework as a coordination mechanism that builds on existing product characteristic such as component commonality, product platforms, and product modularity. Nepal et al. [START_REF] Nepal | Integrated fuzzy logic based model for product modularization during concept development phase[END_REF] proposed a method based on fuzzy logic to model both product structure and supply chain based on experts` information. Gehin et al. [START_REF] Gehin | Integrated design of product lifecycles -The fridge case study[END_REF] presented a method to support designers in the definition of the product lifecycle scenario, including component lifecycle scenarios, when designing the elements of the structure of the product. Blackhurst et al. [START_REF] Blackhurst | PCDM: a decision support modeling methodology for supply chain, product and process design decisions[END_REF] used a short network approach to develop the Product Chain Decision Model (PCDM) for considering decisions related to product design and manufacturing process and the impact of such decisions on the supply chain. Ellram and Stanley [START_REF] Ellram | Integrating strategic cost management with a 3DCE environment: Strategies, practices, and benefits[END_REF] used the 3D-CE to integrate environmentally responsible manufacturing (ERM) and NPD and surveyed its benefit. Shahrokhi et al. [START_REF] Shahrokhi | A hybrid method to select best process and suppliers, in the concurrent engineering environment[END_REF] proposed hybrid method by integrating multiobjective programming and fuzzy AHP to address CE problem to select processes and suppliers in an uncertain environment. Shidpour et al. [START_REF] Shidpour | A multi-objective programming approach, integrated into the TOPSIS method, in order to optimize product design; in three-dimensional concurrent engineering[END_REF] developed a multiobjective linear programming (MOLP) model integrated to TOPSIS method to determine the best configuration product design, assembly process and suppliers by considering qualitative and quantitative criteria in early stages of new product development (NPD) process.

This paper proposes a group decision-making process by using multi-objective programming model to select the best configuration of product design, assembly/manufacturing (A/M) process and suppliers of components. Since in the early design stages much information about criteria is uncertain, we develop a new method based on group decision-making process and intuitionistic fuzzy set. We use opinion of decision makers to construct intuitionistic fuzzy decision matrices by considering lack knowledge/information. These intuitionistic-based decision matrices are used to early evaluate of the candidate supplier and to provide the final set of suppliers as well as to determine importance of criteria. For identifying impact of the lack knowledge / information on selecting the best configuration, a numerical example is represented for two states of intuitionistic fuzzy set and fuzzy set. Results of the evaluation showed that the best configuration for two states is completely different.

Background

In this section, we present the basic concepts used in our method.

Intuitionistic fuzzy sets

Intuitionistic fuzzy sets (IFSs) that are recognized as a generalization of fuzzy set theory is characterized by a membership function and a non-membership function. Let X be a universe of discourse, then a intuitionistic fuzzy set B is defined as:

{ , ( ) , ( ) } B B B x x x x X    (1) 0 ( ) ( ) 1 B B x x x X       (2)
where

: [ 0 ,1] B X   and : [ 0 ,1] B X   are membership
and non membership functions, respectively. The hesitation degree is calculated as:

1 ( ) ( ) B B B x x       (3)
If U and V be two intuitionistic fuzzy numbers (IFN), then [3]:

{ ( ) ( ) ( ) . ( ) , ( ) . ( ) | } u v u v u v U V x x x x x x x X            (4) { (1 (1 ( ) ) , ( ) } , 0 u u U x x          
(5)

Multi-objective linear programming

Zimmermann [START_REF] Zimmermann | Fuzzy programming and linear programming with several objective functions[END_REF] proposed one method based on fuzzy set theory to solve multi-objective linear programming (MOLP). This method develops membership functions for each objective with considering its upper ( m a x m a x , r r

LU

) and lower bound

( m in m in , rr LU
) [START_REF] Blackhurst | PCDM: a decision support modeling methodology for supply chain, product and process design decisions[END_REF]. They are determined by solving the multi-objective problem, when only one objective is used at each time. Membership function for minimizing objective is obtained as:

m in m in m in m in m in m in m in 1 () () 0 , 1, , r rr r r z r r r r r rr zL U z x x L z U UL zU r n                  (6)
After constructing membership functions, the multiobjective problem converts into a weighted single objective problem as follows:

1 m in m in m in 1 m a x ( ) ( ) , 1, , () 1 0
, [ 0 , 1] m rr r r r r r m r r r v U L U z x r n g x b v x                (7)
where r v is weight of objectives.

The proposed method

In this section, we present the method to evaluate design alternatives in the 3D-CE.

Let A= {A 1 , …, A S } be set of supplier alternatives, C= {C 1 , C 2 , … C n } be set of criteria, P={p 1 ,p 2 ,…,p y }be set of components, DM={d 1 ,…,d k } be set of decision makers (DMs) and w = (w 1 , w 2 ,…, w n ) be weight vector of criteria.

To determine weight of criteria

The weight values of criteria can be determined by asking all k decision makers to express their opinion on each intuitionistic preference for each pair of criteria as following:

Step1. Calculate degree of membership function ( ij  ), the degree of non-membership function ( ij  ) and hesitation degree ( ij  ).

Suppose that k 1 DMs express that C i would be preferred to C j , k 2 DMs answer that C i wouldn't be preferred to C j and k 3 DMs don't answer due to their lack of knowledge/ indeterminacy about the criteria. So:

1 k i j k k w    , 2 k i j k k w    , 3 k i j k k w    (8) and 1 2 3 k k k n   
Step2. Construct an intuitionistic preference relation of criteria

The DMs provides his/her intuitionistic preference for each pair of criteria and constructs an intuitionistic preference relation

( ) ( , , ) i j n n i j i j i j X x       , 0 1 ij ij     , ji ij   , ji ij   , 0 .5 ii ii    for all , 1, 2 , ..., i j n  that ij  , ij  and ij  .
Step3. Use the intuitionistic fuzzy arithmetic averaging operator:

1 1 , , 1, 2 , ..., n i ij j x x i j n n     (9) where ( , ,
)

i i i i x    
Step4. Calculate weight of criteria by using following equation:

) i i i i i i n i i i i i i w              (10) 1 ( ( ) ) ( ( )

Pre-evaluate of all candidate suppliers for different components

Assume that appropriateness of s suppliers have to been evaluated on criteria for different components. There, we use a group decision-making method based on IFS to compute weight of objectives, as follows:

Step1. Develop intuitionistic fuzzy decision matrix for each component.

1 1 1 2 1 2 1 2 2 2 1 2 n n s s s n r r r r r r R r r r                     (11)
where

( , , ) ij ij ij ij r     and i=1,2, …, s; j=1,2, …, n.
For constructing intuitionistic fuzzy decision matrix, from all k decision makers are asked to express their opinion whether each supplier is appropriate or not with respect to each criterion. . Number of answers "Yes", number of answers "No" and number of answers "I do not know, I am not sure, etc." determine values of ij  , ij  and ij  , respectively in the way was described in the stage 3.1.

Step2. Determine the intuitionistic fuzzy positiveideal (IFP) and negative-ideal solutions (IFN). Let L1 and L2 be benefit criteria (Larger the better) and cost criteria (smaller the better), respectively. If A  be intuitionistic fuzzy positive-ideal solution and A  be intuitionistic fuzzy negative-ideal solutions, then values of them are obtained as follows: If ( , )

j j A     
 and ( , )

j j A      
, then:

1 2 { ( m a x | ) , ( m in | ) } j ij ij i i j L j L          (12) 1 2 { ( m in | ), ( m a x | )} j ij ij i i j L j L          (13) 1 2 { ( m in | ) , ( m a x | ) } j ij ij i i j L j L          (14) 1 2 { ( m a x | ) , ( m i n | ) } j i j i j i i j L j L          (15)
Step3. Calculate differences between alternatives with A  and A  . In order to measure difference between alternatives on intuitionistic fuzzy set, distance measure proposed by Szmidt and Kacprzyk [START_REF] Szmidt | Distances between intuitionistic fuzzy sets[END_REF], is used as follows:

2 2 2 1 1 ( , ) [( ) ( ) ( ) ] 2 n j ij j ij j ij j j d A A w                              ( 16
) 2 2 2 1 1 ( , ) [( ) ( ) ( ) ] 2 n j ij j ij j ij j j d A A w                              ( 17
)
where j w is weight of criterion j. Step4. Calculating the relative distance of each supplier alternative using the following equation:

1, 2 , , i i i i d d i s d d        (18)
Step5. To select the proper set of suppliers for final evaluation. The suppliers are selected that their values of i d be equal or more than 0.50.

These stages are repeated for all components that have to provide from suppliers.

Developing a multi-objective linear programming (MOLP) model

Our proposed approach employs a MOLP model to calculate the best design alternative (version), the appropriate assembly/manufacturing process and the optimum order allocated to the selected suppliers. Notations of the model are defined as follows: k=1,2,...,K Index of configurations i=1,2,...,y Index of components Defect rate of the j th supplier for component i

j o
Ordering cost to supplier j M A large number Decision variables:

ij x Amount of component i Purchased from supplier j s u 1 if supplier s is selected, 0 otherwise k z 1 if configuration k is selected, 0 otherwise
The objectives and constraints of the model are represented as follow:

1 2 1 1 1 1 1 1 1 y y K s K k i j i j k k i k i j k i s j j j M i n y z x p z p o u                (19) 1 2 1 1 1 2 1 1 1 y K s k i j i j k i j y K k k i k i M i n y z x t m D z t           (20) 3 1 1 0 1 y K s k i j i j k i j M i n y z x q m D       (21) 1 ( ) 0 , S k i j j z x D k i      (22) , , k i j i j z x c ap i j k   (23) 1 1 K k k z    (24) 1 , y k i j j i z x M u j k     ( 25
) , { 0 , 1} , j k u z j k   (26) 0 , ij x i j   (27)
First objective (19) shows the cost of components purchased from supplier, assembly cost and order cost. The next objective (20) identifies the lead time and assembly time. The third objective (21) represents the average of defect rate. Constraints ( 22) and (23) show demand and supplier capacity for each component, respectively. Constraint (24) shows that just one configuration is selected. Constraint (25) combines the order cost of several components into one single order for each supplier. Constraints (26) and ( 27) show binary and continue variables. By solving this MILP model, the order quantity allocated to each supplier is determined.

Numerical example

A manufacturer wants to produce a new product to compete with other firms. Experts of production department propose three version of new product. Table 1 shows the alternatives with their possible A/M process and candidate suppliers. Because components 1, 2, 3, 4 and 5 are same in all design alternatives and three components 6, 7, 8 make differences between alternatives, so we evaluate alternatives based on different components. Three criteria cost (C1), lead time (C2) and defect rate (C3) are selected to evaluate design alternatives. The proposed method is applied to determine the best configuration of design product, the A/M process and suppliers as following:

1. To determine weight of criteria described in stage 3.1. In this stage, from 5 DMs is wanted to express their opinion about preference of criteria with asking following question: Do you prefer C i to C j ?

Dependence to answer of DMs, membership function, non-membership function and hesitation degree are determined. Number of answers "Yes", number of answers "No" and number of answers "I do not know, I am not sure, etc." determine values of ij  , ij  and ij  , respectively in the way was described in the stage 3.1. The preference matrix of criteria is shown in Table 2.

2. Pre-evaluate of all candidate suppliers for different components to construct final set of suppliers. This step selects the final set of suppliers among of the six candidate suppliers.

There, we use a group decision-making method based on IFS to compute weight of objectives, as follows:

Step1. Develop intuitionistic fuzzy decision matrix for each component

Step2.

Determine the IFP and IFN by using equations ( 12) to [START_REF] Shahrokhi | A hybrid method to select best process and suppliers, in the concurrent engineering environment[END_REF]. In this example all criteria are cost type.

Step3. Calculate differences between alternatives with A  and A  by using equations ( 16) and [START_REF] Zimmermann | Fuzzy programming and linear programming with several objective functions[END_REF].

Step4. Calculate the relative distance of each supplier alternative using the equation [START_REF] Szmidt | Distances between intuitionistic fuzzy sets[END_REF].

Step5. Select the proper set of suppliers for final evaluation. The suppliers are selected that their values of i d be equal or more than 0.50. 4).

For identifying impact of the hesitation -that is obtained from lack of knowledge or indeterminacy about the alternatives-on the results of the proposed method, we evaluate this example again. We assume that DMs have sufficient knowledge or information about accept or reject a decision. So, from all k decision makers are asked to express their opinion about preference of criteria to each other and appropriateness suppliers on criteria. So, Number of answers "Yes" and "No" determine values of ij constructing preference matrix to evaluate criteria, weight of criteria is obtained as: w 1 =0.44, w 2 =0.27, w 3 =0.29. By solving MOLP model for this state, the configuration 7 including design alternative 3, assembly/manufacturing process h 7 and suppliers is determined (Table 5).

The Figure 1 compares the orders allocated to suppliers for two states IFS and FS.

Results of two evaluations based on fuzzy and intuitionistic fuzzy shows that lack of knowledge or indeterminacy has direct effects on selecting the best configuration thus whole NPD activities. Because data or information in the early stages of product design is not always available, So, IFS can be used as a tool to consider impact of lack of knowledge.

Conclusion

One of the most important activities in developing new product is evaluation of design alternatives in the early design stages. Because of existing imprecise and incomplete information in the early design stages, IFS can be used as a more adaptive tool for expressing uncertainty by defining membership and nonmembership functions and hesitation degree.

This paper proposes a new group multi-criteria decision-making method by using IFS and multiobjective linear programming for: a) To determine weight of objectives; b) To determine final set of suppliers from among all candidate suppliers by using a new method based on IFS; c) To determine the best configuration of design alternative, A/M manufacturing process and suppliers by applying MOLP model.

Since in the early stages of product design, we are faced with lack information/ knowledge about design, process and supply chain, so we use from approach intuitionistic fuzzy because of considering hesitation degree. This paper uses opinion a group of DMs to form decision matrix and pairwise comparison matrix based on IFS to determine final set of suppliers and weight of criteria. By applying the method for two states of fuzzy and IF, it is shown that to consider lack of knowledge or indeterminacy by IFS can be impact on evaluation of design alternatives, A/M process and supplier selection and thus all NPD process.

There are a number of opportunities to expand the proposed research. The results obtained from the proposed method can be verified by applying a real case and by considering other criteria-especially qualitative criteria-of product design, supply chain design and assembly/ manufacturing process.

  have not hesitation about decision. So, the IFS convert to fuzzy set (FS). By

Fig. 1 .

 1 Fig.1. The comparison of orders allocated to suppliers for IFS and FS

Table 1 .

 1 The information about design alternatives and A&M process

	Design	The	The A&M	A&M	A&M
	alternative	components	proposed	cost	time(Min.)
	A1	1,2,3,4,5,6,7	h1,h2	U[30,50] U[2,8]
	A2	1,2,3,4,5,6,8	h3,h4,h5	U[36,60] U[1,6]
	A3	1,2,3,4,5,7,8	h6,h7	U[27,45] U[2,7]

Table 2

 2

	. The preference matrix of criteria
	C1	C2	C3	IF	weight
	C1 (0.5,0.5)	(0.65,0.2) (0.8,0.2) (0.67,0.27) 0.46
	C2 (0.2,0.65) (0.5,0.5)	(0.4,0.3) (0.37,0.46) 0.29
	C3 (0.2,0.8)	(0.3,0.4)	(0.5,0.5) (0.34,0.55) 0.25

Table 3 .

 3 The results of steps of 1, 2, 3, 4 and 5 for component 6

	P6		C1	C2	C3	d	i 	d	i 	d	i
	A1		(0.45,0.4)	(0.7,0.1)	(0.5,0.2)	0.3	0.37 0.55
	A2		(0.6,0.3)	(0.55,0.25) (0.35,0.45) 0.2	0.43 0.68
	A3		(0.55,0.35) (0.5,0.3)	(0.75,0.1)	0.33 0.31 0.48
	A4		(0.75,0.15) (0.6,0.25)	(0.7,0.2)	0.47 0.19 0.29
	A5		(0.65,0.2)	(0.35,0.45) (0.8,0.2)	0.33 0.36 0.52
	A6		(0.85,0.15) (0.45,0.25) (0.6,0.25)	0.44 0.21 0.32
	A		(0.45,0.4)	(0.35,0.45) (0.35,0.45)
	A		(0.85,0.15) (0.7,0.1)	(0.8,0.1)

  Table 3 shows the final set of suppliers for component 6 [A 1 , A 2 , A 5 ]. The final set of suppliers for components 7 and 8 are obtained as follows: Component 7: [A 1 , A 3 , A 5 ] and Component 8:

	[A 1 , A 2 , A 5 ].
	3.3. Using the multi-objective linear programming
	(MOLP) model of section 4.3 for evaluation of design
	alternatives. By solving MOLP model, the configuration
	1 included design alternative 1, A/M process h 1 and
	suppliers is determined (Table

Table 4 .

 4 The final result for IF evaluation

	Component	Supplier	order
	6	1,3,5	450, 0,550
	7	1,2,5	0,650, 350

Table 5 .

 5 The final result for fuzzy evaluation

	Component	Supplier	order
	7	1,3,5	0,650,350
	8	1,2,5	700,300, 0