D Guillon
email: d.guillon@estia.fr

E Villeneuve
email: e.villeneuve@estia.fr

C Merlo
email: c.merlo@estia.fr

E Vareilles
email: elise.vareilles@mines-albi.fr

M Aldanondo
email: michel.aldanondo@mines-albi.fr

Industrial Services Characterization for Bidding Process

Keywords: Knowledge management in production, Decision-support for human operators, Pricing and outsourcing, Industrial services, Offer modelling, Bidding process

When responding to call for tenders, many bidding companies offer services. This paper focuses on how to model industrial services during the bidding process to be able to easily develop them. A product offer configuration model is presented, then a reflection about the extension of this model to service offers is conducted. A study of the literature about service definition is dealt and services characteristics are identified. Their impact on the product offer model is analyzed and new characteristics are introduced. This work makes possible to propose a typology to adapt the product offer model to services.

INTRODUCTION

In the era of Industry 4.0, when responding to call for tenders, bidders have to cope with an increasingly complex environment, bringing together the well known problematic of customization and new challenges (digitalization, Big Data, etc.). To remain competitive, companies define offers that must be: in conformity with the customers requirements; attractive in terms of suggested solution, cost and time; and realizable in terms of delivery process, allocated resources and incurred risks. Thus, the OPERA 1 project aims at proposing a decision support system that will help bidders to elaborate offers by exploiting their knowledge and good practices. The OPERA project involves four companies with different types of activities and offers. According to the many interviews dealt with them, it has been identified that the most elements of an offer are built by reusing known elements, already defined and characterized in previous offers. The development of offers therefore corresponds mostly to a routine design situation [START_REF] Chandrasekaran | Design problem solving: A task analysis[END_REF], or even configuration [START_REF] Mittal | Towards a generic model of configuraton tasks[END_REF].

Two kinds of companies are involved in the OPERA project: two are manufacturing companies and two are consulting firms. Thus, we distinguish product offers, which require the development of a product, and service offers, which require the development of a service. Many papers discuss about product configuration, and joint configuration of a product and the associated delivery process [START_REF] Abeille | Formalization of an integrated system/project design framework: First models and processes[END_REF][START_REF] Robin | Management of a design system in a collaborative design environment using pegase[END_REF][START_REF] Sylla | Customer/Supplier Relationship: reducing Uncertainties in Commercial Offers thanks to Readiness, Risk and Confidence Considerations[END_REF]. In [START_REF] Guillon | Customer supplier relation: towards a constraint-based model for bids[END_REF], we have proposed a

1 Learn more about the OPERA project: https://research-gi.mines-albi.fr/display/OPERA generic model for product offer elaboration, which allows to configure a product and its delivery process, using information characterizing the context of the offer. However, the representation of service offers using this productcentered model is a key question. While [START_REF] Felfernig | Knowledge-Based Configuration: From Research to Business Cases[END_REF] emphasize the value of configuring services, there are few details describing how to apply it. Yet, the service market is growing [START_REF] Chae | An evolutionary framework for service innovation: Insights of complexity theory for service science[END_REF] and industrial needs are significant.

The work carried out through the OPERA project aims at extending the generic product offer configuration model to service offers. In this paper, we consider and analyze some key characteristics that will help to choose the relevant modeling approach for proposing a service offer model. Services and products being very different in nature [START_REF] Grönroos | Service management and marketing: customer management in service competition[END_REF], this model extension is not obvious. Among the notable differences, the presence or absence of a Bill Of Materials (BOM) seems particularly critical. For a product, the bidder must always configure a set of physical components. Considering a service, a reflection emerges about the existence of such a BOM and, if it exists, on its definition and representation. The diversity of industrial services could imply a service typology associated with a generic model typology. This article proposes to identify and discuss the characteristics allowing to set up a service typology. Such a typology may influence the way a service is modeled and represented.

Since this work is related to the OPERA project, this article will only focus on business-to-business (B2B) services in an industrial framework. These services cover, among others, the following ones: thermal calculation studies, organizational consulting, training, maintenance or rental of industrial equipment. The software edition is out of scope because software is commonly considered as a product. This paper is structured as follows. Section 2 presents the product offer generic model we intend to extend to service offers. Section 3 is dedicated to a literature review about service definition and description. Section 4 introduces a comparison between product and service offers issued from the OPERA feedbacks. Section 5 lays down service characteristics resulting both from literature and feedback, to define the bases of a service typology. Finally section 6 concludes this article and presents some perspectives.

GENERIC PRODUCT OFFER MODEL

This section introduces the generic product offer configuration model which will be extended to service offers. Many product models have been proposed in the last decades in order to help designers during all development phases, coupled or not with process models [START_REF] Eckert | On the integration of product and process models in engineering design[END_REF] in order to manage design during all development steps. Several models are based on Function-Behaviour-Structure approach [START_REF] Roucoules | Ippop: an opensource collaborative design platform to link product, design process and industrial organisation information[END_REF], where functions and structures are represented with a tree decomposition: functions into sub-functions and product into sets of components then components. System engineering as defined by INCOSE (2015) (International Council of Systems Engineering) proposes several views of the product, considered as a system, and defines several SYSML diagrams. At architectural design step, the product (or system) can be represented by applying the same mechanism of decomposition of block into sub-blocks. Other representations may participate to the definition of the context, requirements [START_REF] Faulconbridge | Managing complex technical projects: A systems engineering approach[END_REF], interactions between blocks, etc, according to the different steps of the chosen design process [START_REF] Mehnni | A sysml-based methodology for mechatronic systems architectural design[END_REF]. Elaborating an offer occurs before any design process starts, so existing product models propose too much details for the purpose of bidders: knowing that an existing solution can be reused is more important than the detailed characterization of this solution. The bidder must choose, among a set of known components, the key ones that fit the customer's needs [START_REF] Mittal | Towards a generic model of configuraton tasks[END_REF], in order to generate a BOM, which corresponds to configuration approach.

In [START_REF] Guillon | Customer supplier relation: towards a constraint-based model for bids[END_REF], a first open offer configuration model for products has been proposed (Fig. 1). To define a relevant offer and to be able to do case-based reasoning on passed cases, the bidder has to characterized (1) the offer context, (2) the product BOM, and (3) the product delivery process. Indicators (which are the evaluation parameters of the offer) characterize the product and the delivery process. These can be financial indicators (cost, price, margin), time-related indicators (delay, load, duration), but also confidence indicators [START_REF] Sylla | Customer/Supplier Relationship: reducing Uncertainties in Commercial Offers thanks to Readiness, Risk and Confidence Considerations[END_REF], depending on the bidder company. In this work, the product offer model is exploited with a constraint satisfaction problem (CSP) approach [START_REF] Montanari | Networks of constraints: Fundamental properties and applications to picture processing[END_REF].

(1) Context Four types of data characterize the context of the offer. They describe the customer profile, the offer characteristics, the bidder characteristics and the environmental factors. To describe the product, the bidder can choose between two approaches: a functional description of the needs, expectations and functions or a hierarchical description of the components. In both cases, we propose that the bidder configure the product BOM [START_REF] Aldanondo | Configuration for mass customization: How to extend product configuration towards requirements and process configuration[END_REF].

(3) Delivery process Whether the bidder has chosen a functional description or a hierarchical description of the components, it is important to define the delivery process, described as a set of operations to be performed to produce and deliver the product.

As part of the OPERA project, the configuration of the delivery process is based on a generic process consisting of the following operations: studies, manufacturing and delivery. The delivery process is not defined in detail because the study is done at pre-sale stage and therefore does not require to deepen this point. When defining the delivery process the objective is to identify the following key activities: evaluate process costs, provide a relevant delivery date, and identify key risks on the project. The results of these key activities allow to elaborate the offer. This section has introduced the generic product offer model. The possibility of extending this generic model to service offers will now be explored.

SERVICE : DEFINITIONS AND DESCRIPTIONS

In recent years, the tertiary sector has become increasingly important: the global economy has shifted from a productoriented vision to a service-oriented one and in most developed countries, the majority of revenues come from services [START_REF] Chae | An evolutionary framework for service innovation: Insights of complexity theory for service science[END_REF]. Today the development of services is such that tertiary sector companies are increasingly responding to calls for tenders. This section is dedicated to existing work on the definition and description of services.

Service definitions

Service as a process In [START_REF] Grönroos | Service management and marketing: customer management in service competition[END_REF] and [START_REF] Carlborg | Service process modularization and modular strategies[END_REF], a service is a process consisting of several activities more or less intangible. The activities often result from interactions between the customer and employees and/or physical resources or goods and/or systems of the service provider. Instead of trying to define what a service is, Grönroos proposes to define the characteristics of a service: main one is to characterize a service as processes composed of activities or series of activities rather than components. He defines a service as a process, as opposed to a product that is a thing.

Service as a performance Services are also sometimes defined as performances [START_REF] Lovelock | Whither Services Marketing[END_REF][START_REF] Parasuraman | A Conceptual Model of Service Quality and Its Implications for Future Research[END_REF][START_REF] Zeithaml | Problems and strategies in services marketing[END_REF]. In this case the description of the goal to be satisfied (the result) is more important than the resources or the process to be employed for (the means).

Service as a multi dimension concept

According to [START_REF] Bullinger | Service engineering -Methodical development of new service products[END_REF], a service is characterized by three different dimensions: (1) a structure dimension, which determines the capacity and willingness to provide the service in question, (2) a process dimension, and (3) a result dimension.

These definitions highlight interesting and varied aspects in the service definition which we will rely on later.

Service descriptions

The value of service configuration is the same as for product configuration: finding the balance between customization, to meet the varied needs of customers, and the benefits of standardization [START_REF] Felfernig | Knowledge-Based Configuration: From Research to Business Cases[END_REF]. The interest of the mass configuration applied to services has also been discussed [START_REF] Heiskala | Mass Customisation of Services : Benefits and Challenges of Configurable Services[END_REF]. After listing benefits and challenges of mass configuration for services, the authors conclude that the benefits are the same as for the products. Yet the results on mass customization for goods may not be directly applicable to services [START_REF] Felfernig | Knowledge-Based Configuration: From Research to Business Cases[END_REF]. This subsection aims to explore this problem from the literature review.

Service decomposition into elements Several authors discuss the service decomposition into elements. In [START_REF] Carlborg | Service process modularization and modular strategies[END_REF] and [START_REF] Goldstein | The service concept: The missing link in service design research[END_REF], a service is described as "a combination of physical and nonphysical elements integrated into various customer-specific configuration". They specify that the components are a combination of processes, human and material skills.

Service family In Ki [START_REF] Moon | Service representation for capturing and reusing design knowledge in product and service families using object-oriented concepts and an ontology[END_REF], the concept of service families is explored. The authors propose to extend the methods used on the product families to the service families. A service family is defined as a set of services based on common processes, activities, objects, and features. A service is composed of a set of service modules, themselves composed of a set of service components.

Service modularity In [START_REF] Løkkegaard | Towards a framework for modular service design synthesis[END_REF], the interest of modular services, which have the same advantages as modular products, is highlighted. In [START_REF] Böttcher | Providing a method for composing modular B2B services[END_REF], the authors propose a method for decomposing a B2B service into modules, and define a service module as follows: "A service module offers a welldefined functionality via interfaces described precisely. A service module can be used to decompose and can itself be part of a higher level service module". This definition is very close to the component definition for a product. They add: "This decomposition allows a specific configuration for a given client because the client can assemble a service offer from a set of given service modules". This definition is similar to product configuration, where a customer can assemble their product from a set of given components. However, this last model is not directly applicable to our case study because our aim is to model services whether they are modular or not. Process, element, and modular approaches suggest that a service might be represented by a tree structure.

PRODUCT OFFER VERSUS SERVICE OFFER

During the bidding process, the work focuses on the evaluation of deliverable cost, and in some cases, on the workload identification and the delivery time estimation. For a product offer, the bidder seeks mainly to put a cost on the product by using the BOM. In addition, the technical choices made on significant components may be of interest to the customer. For a service offer, the bidder mainly evaluates a cost for a process, and no longer on product components. The resources used in the delivery process are valued: mainly human resources, but also material occasionally. For example, equipment procurement and travel costs are related to the delivery process. Unlike a product where costs related to raw materials and components are important, for services, these costs are often negligible (except for maintenancelike services, where physical components are purchased for the customer). In addition, structuring a service offer often depends on the customer's request for costing, unlike product offers where the nomenclature is intrinsic to the product family and independent from the customer preferences. Then for services, structuring work packages can be very different from one customer to another, for a similar work. The application to a service of the product offer model presented in Fig. 1 raises several questions.

(1) Context First, the context of the offer, as described in section 2, remains unchanged. Whether for a product or a service, the bidder must characterize the context to elaborate the most relevant offer for the customer.

(2) Product or Service

Then, the description of the product proposed by the bidder is difficult to adjust for a service. Indeed, as mentioned above, the existence of a service nomenclature is questioned. As seen in 3.1, some authors define a service as a process. As a consequence, the bidder may develop the delivery process in order to describe the proposed service. In the product offer model, this would mean that the product modeling would become weak. Two possibilities exist: nothing would replace the product modeling part and the service is modeled only by the delivery process part; or we replace product modeling part by a WBS (Work Breakdown Structure), as in project management. We therefore have to determine whether product modeling part should be deleted or adapted to services. If the adaptation is chosen, it is necessary to identify a formalism allowing to describe a service and the information that characterized it, as for example WBS formalism for a process-centered description. This formalism must also consider representations such as service as a performance, and allow decomposition or modularity.

(3) Delivery process Then, as said before, the delivery process part is particularly important for a service. The choice of resources and the proper identification of the duration of each task are essential. The service offer model will therefore include a process and risk descriptions, surely more detailed than for product offers.

Section 5 lays down the characteristics of a service offer to identify criteria for a future service typology. The analysis of these characteristics will allow studying more differences between products and services. If the product configuration cannot be directly applied to the service configuration, listing the characteristics that distinguish a product from a service should help to understand how to extend the product offer model. The literature review presented here mainly comes from marketing, which is an area where the concept of service has been widely studied.

CHARACTERISTICS IMPACTING THE SERVICE OFFER MODEL

The service is often opposed to the product in the literature [START_REF] Edvardsson | Service portraits in service research: a critical review[END_REF]. If defining a service is difficult, it is possible to list its main characteristics, often opposed to those of the products. Historically, four characteristics have been identified: Intangibility, Heterogeneity, Inseparability (non-separability of production and consumption) and Perishability. These four characteristics are often grouped under the term IHIP. Although these features are often used in the literature, they have also been criticized in recent years [START_REF] Edvardsson | Service portraits in service research: a critical review[END_REF][START_REF] Lovelock | Whither Services Marketing[END_REF]). Indeed, it seems that IHIP characteristics can not be applied to all types of services. [START_REF] Lovelock | Whither Services Marketing[END_REF], therefore, propose either:

(1) to abandon the product / service distinction and to stop declaring them as two different domains, (2) to find a new paradigm for defining services, (3) to work on typologies of services.

Option 1 is not applicable in configuration. Indeed, as mentioned above, it is impossible to treat products and services identically because we can not define all types of services with a physical nomenclature. Option 2 is the option retained by the authors. They propose to characterize a service by the absence of ownership transfer. This characteristic and its possible impact on our model will be discussed in the following. Finally, option 3 seems best suited to configure services. Indeed, we think that we can define a typology of services, i.e. a generic model for developing offers associated to each type of service. We therefore propose to discuss about characteristics which are necessary to establish such a typology of services. IHIP characteristics and absence of ownership transfer will be discussed in 5.1 in order to identify if they are able to differentiate a product from a service, but also to differentiate different types of services. Then we will propose new characteristics in 5.2.

Characteristics from literature review

In this subsection, we present IHIP characteristics and ownership transfer. We discuss their limits, then we question their impacts on the offer model.

Intangibility Intangibility is the first characteristic of a service, the most cited, but above all the most critical, and the one from which all others derive [START_REF] Bateson | Why we need services marketing[END_REF]. This intangibility is defined as opposed to products tangibility, i.e. palpable and material [START_REF] Shostack | Breaking free from product marketing[END_REF]. Indeed, an intellectual service is intangible. This characteristic is essential because it impacts the key element of product offer configuration: the nomenclature. For a product, the bidder has to describe the characteristics of a tangible good and identify the components of a physical BOM. For a service, the service descriptors remain to be identified. This intangibility generates difficulty for the bidder to define a price for the service. Nevertheless, if a service is broken down into activities such as a WBS or elements following a batch decomposition, an intangible and not a physical nomenclature could be present.

Heterogeneity Heterogeneity concerns the potential for high variability in service performance. The service heterogeneity is due to human resources, as opposed to the machine repeatability. The quality and essence of a service can vary from one stakeholder to another and from one customer to another. The service performance for the same worker can also vary from day to day. This characteristic therefore leads to a standardization difficulty [START_REF] Zeithaml | Problems and strategies in services marketing[END_REF]. This characteristic has a strong impact on the delivered service quality, but has a very low impact on a generic offer model since human resources are always defined.

Inseparability

The characteristic of inseparability has several aspects, all related but with subtle differences. First, by definition, inseparability means that the production and the consumption of the service are simultaneous [START_REF] Parasuraman | A Conceptual Model of Service Quality and Its Implications for Future Research[END_REF][START_REF] Edvardsson | Service portraits in service research: a critical review[END_REF][START_REF] Lovelock | Whither Services Marketing[END_REF]. The involvement of the customer in the delivery process (e.g. in training) results from inseparability [START_REF] Heiskala | Mass Customisation of Services : Benefits and Challenges of Configurable Services[END_REF]. [START_REF] Grönroos | Service management and marketing: customer management in service competition[END_REF] adds that the added value of a service is the result of customer-supplier interactions, as opposed to the product, where value added is created within the factory. In the case of a service offer related to a calculation study, customers will provide specifications to the bidder at the beginning, and then they will be involved only at the end of the process for collecting the service deliverable provided by the bidder. This inseparability concept is not verified in this case. The training is, to some extent, produced and consumed at the same time since the training is part of both production by the worker and consumption by customers. Customers intervene directly in the service production because they are involved in the delivery process: they are present (and essential) during the training. The inseparability characteristic is verified in this situation. This characteristic does not inherently change the offer model. The part (3) Delivery process modeling will include activities involving the customer.

Perishability Perishability means that services cannot be retained. If the resource is not used at a time t, it cannot be retrieved or used later. A service has immediate consumption, unlike some products that may be used several years after delivery. Services have an ephemeral duration and cannot be stored. A rental service for example, is not lasting because the customer enjoys the service for a limited time, unlike the purchase of a product, where the customer enjoys the property durability. Service companies therefore often have a problem of synchronizing customer demand and their capacity [START_REF] Zeithaml | Problems and strategies in services marketing[END_REF]. This characteristic has no impact on the offer model.

Ownership transfer

Lovelock [START_REF] Lovelock | Whither Services Marketing[END_REF] propose to characterize the service by the absence of ownership transfer. For all the industrial services we have in mind, it seems this characteristics is applicable to all of them. It seems to be a very good one to distinguish services from products. But this characteristic has no impact on the offer model we propose.

Only intangibility has a direct impact on the generic offer model and will be selected as a criterion for establishing a typology of services. The characteristics of heterogeneity, inseparability, perishability and ownership transfer have no impact on the offer model. However, other features can differentiate services for bidding responses. In the following section, we propose some new characteristics, based on our observations during collaborations with industrial partners specialized in the production of services.

Identification of new characteristics

Obligation of means or results The customer's requirements may relate only to the result of the delivery process (the deliverable), or only to the progress of the delivery process. In the first case (obligation of result), we are getting closer to the customer's expectations in the case of products. No matter how it was manufactured and assembled, the customer's first expectation is about the product itself. This is the case, for example, for calculations studies. It does not matter to the customer that the study was done in one country or another, focus is on the fact that the calculations are done and that the report is clear and usable. In the second case (obligation of means), the customer is interested in the progress of the delivery process itself. For example, for training, duration, location and qualifications of the teacher are criteria that will interest the potential customer. If the customer is interested in the process progress, it will be better defined by the bidder. Indeed, while in the first case the process is defined only for the supplier, in the second case more attention must be paid to the process activities that interest the customer. The service offer model will have to be able to model an offer according to these two points of view.

Occurrence aspect

The temporal aspect of a service (project progress) is important. The service can take place once or iteratively. For example, the customer may be calling for tender for a single design study on a given part, or for getting a pool of engineers able to do as many calculations as necessary during a given duration. This characteristic can impact the (3) Delivery process modeling part, because this one can be more complex, with several sub-processes or the same process repeated several times.

Uncertainty

The bidder may commit himself to perform services with an identified level of uncertainty. This uncertainty can be on three types. It may be temporal uncertainty: in the case of maintenance for example, the bidder may commit himself for a given period to intervene in case of a predefined problem. He may have identified a range of activities, possible processes to perform, but does not know exactly how many times or when he will have to intervene. The second type of uncertainty can be related to resources: an estimated resource may be out of office when it is needed for acting. Finally, the uncertainty can be on the activities to be carried out. This uncertainty has a strong impact on the generic offer model. (2) Service modeling part and (3) Delivery process modeling part are impacted. In the case of temporal uncertainty, we can imagine that one or more processes are defined, and that the uncertainty relies on the number of repetitions and the duration of each process. In the case of resource uncertainty, the generic offer model is not modified but risk is impacted. Finally, in the case of a structural uncertainty, both (2) Service and (3) Delivery process modeling parts can be impacted.

Synthesis

To conclude, the three proposed characteristics have an impact on the elements that must be described for generating an offer, as well as the intangibility characteristic from literature review. Therefore, the modeling of a service by a structural representation correlated to the delivery process is not only possible but necessary. But their impact influences this modeling. We show that we need a structural representation to be able to describe: e.g. a service as a process composed of activities and sub-activities; or a set of service. In this case mechanism of decomposition is like product decomposition ones. We show also that even if the service modeling is "light", it is necessary to associate a description of the delivery process. This delivery process is strongly impacted by these highlighted characteristics: e.g. in some situations the level of granularity of the modeling must be very detailed; in some others, only activities dealing with the customer must be detailed. Moreover, some situations require to define specific parameters such as resources characteristics or duration.

CONCLUSION

This paper looks at industrial services in response to calls for tenders. A product offer configuration model is described, then the question of extending this model to service offers is studied. In the product offer model, (1) Context modeling and (3) Delivery process modeling are relevant for service offers. (2) Product modeling as a nomenclature is questioned. Four service characteristics from the literature (intangibility, heterogeneity, inseparability and perishability) are analyzed. It is demonstrated that only intangibility has a direct impact on the offer modeling. As it is not enough to build a new typology of services, new characteristics are proposed (relative to customer expectation, temporal aspect and uncertainty), and their identification allows to identify the first bases for extending the described product offer model to service offers. Subsequently, a global typology of industrial services based on these characteristics will be proposed in future work, as well as a global model for integrating both product offers and service offers. Finally, the work, presented in this article, considers only aspects related to B2B services. However, it seems appropriate to extend this work to other types of services in the near future to propose a global typology of services.

Fig. 1 .

 1 Fig. 1. Model of product offer (2) ProductTo describe the product, the bidder can choose between two approaches: a functional description of the needs, expectations and functions or a hierarchical description of the components. In both cases, we propose that the bidder configure the product BOM[START_REF] Aldanondo | Configuration for mass customization: How to extend product configuration towards requirements and process configuration[END_REF].

ACKNOWLEDGEMENTS

The work presented in the paper is funded by the French ANR (project n • CE10-0010). We thank all partners of the OPERA project for their involvement.


Alexandre Rio

Yoann Maurel

Olivier Barais

Yoran Bugni

Yoran Bugni Demand

Demand Side Management A model driven approach to promote energy self-consumption

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Case study: Industrial site self-consumption ► Self-consumption of renewable energies is defined as electricity that is produced from renewable energy sources ► The autonomy of sites with micro-generation capabilities is greatly increased by self-consumption of locally produced energy ► One of the keys is thus to align production and consumption either by planning processes differently or by relying on storage capabilities.

► Help for "What-if"/ "what-for" questions:

-How to size local renewable energy production units or storage to meet a site's energy consumption -Which region would be the most interesting for the expansion of a business?

-What organization of activities enables the best autonomy and selfconsumption?

Demand Side Management

A

  model driven approach to promote energy self-consumption Proposal: Simulator and Domain Language ► Model Driven Engineering (MDE) approach to address variability ► Energy Management System (EMS) in simulation or using real sensors ► Domain Specific Language (DSL) to represent an industrial site -Production, consumption, storage -Activities and constrains modeling ► Description files are used by a simulator. ► Simulator can be extended by experts, through plugins, to model complex devices behavior Experiences Evaluation: Activity shift recommendations ► Simulator can use third-party prediction to estimate future events: solar production, device usage ► Recommend actions based on context: battery state, user activity to optimize autonomy and self-consumption ► Take into account process constrains and flexibility and battery cycle usage► Improve autonomy from 30% to 50% or 70% activity to increase the autonomy fo the day from 9% to 20%


Abdourahim Sylla

Thierry Coudert

Élise Vareilles

Laurent Geneste

Michel Aldanondo

Rania Ayachi

E Vareilles

L Gesneste

Possibility theory and PROMETHEE II for decision aid in engineering design process

Keywords: Multi-Criteria Decision Making (MCDM), Possibilistic dominance degree, PROMETHEE II, Uncertainty and imprecision, Technical bid elaboration

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

In many engineering design problems, several design solutions are relevant to the requirements [START_REF] Renzi | A review on decision-making methods in engineering design for the automotive industry[END_REF]. In such a situation, one of the most critical phase in the design process is to select the most interesting design solution from a panel of several potential solutions El Amine et al. (2016). The selection, most of the time, involves several conflicting criteria such as: cost, duration and technical performances. Moreover, the criteria may have different importance to the decision maker. All of these aspects of the design problems must be taken into account in the decision process [START_REF] Renzi | A review on decision-making methods in engineering design for the automotive industry[END_REF].

In addition, in many situations, relevant information to evaluate the design solutions are not available. Due to this lack of information, the values of the decision criteria which characterize the design solutions are imprecise and uncertain (epistemic uncertainty and epistemic imprecision) [START_REF] Chapman | Incorporating uncertainty in competitive bidding[END_REF]. These imprecision and uncertainty are more important in the early phase of the design process where the design solutions are only partially designed El Amine et al. (2016). Therefore, in order to make the right decision when selecting the most interesting design solution, the uncertainty and imprecision related to the values of the decision criteria must be taken into account in the decision process [START_REF] Durbach | Modeling uncertainty in multi-criteria decision analysis[END_REF]. In this article, we consider the problem of the selection of a design solution in engineering design process where Sponsor and financial support acknowledgment goes here. Paper titles should be written in uppercase and lowercase letters, not all uppercase.

the evaluation of the design solutions is imprecise and uncertain.

In the scientific literature, many multi-criteria decision support approach which take into account uncertainty and imprecision, have been proposed (see the review in [START_REF] Durbach | Modeling uncertainty in multi-criteria decision analysis[END_REF]). In many of these research work, in order to take uncertainty and imprecision into account in the decision process, a standard multi-criteria decision analysis method (e.g. PROMETHEE, AHP, ELECTRE, etc.) is associated with an uncertainty theory (probability theory, fuzzy set theory, possibility theory, etc.) [START_REF] Durbach | Modeling uncertainty in multi-criteria decision analysis[END_REF]. As a multi-criteria decision analysis approach, PROMETHEE II method is well-known to be effective and easy to implement [START_REF] Behzadian | PROMETHEE : A comprehensive literature review on methodologies and applications[END_REF]. Moreover, possibility theory is relevant to simultaneously deal with uncertainty and imprecision [START_REF] Dubois | Ranking fuzzy numbers in the setting of possibility theory[END_REF]. Therefore, in this article, we propose an approach based on possibility theory and PROMETHEE II method for the complete ranking of alternative design solutions in engineering design process.

In our previous work [START_REF] Sylla | ETO Bid Solutions Definition and Selection Using Configuration Models and a Multi-Criteria Approach[END_REF], four possibilistic mono-criterion dominance relations have been proposed in order to compare two solutions with respect to a single decision criterion, and, to compute the possibilistic dominance relation between them. In the proposed approach, these four possibilistic dominance relations are used to compute a possibilistic dominance degree of a solution S i over another one S j . Then, based on this, PROMETHEE II ranking method is adapted in order : (i) to compare the potential solutions following all the decision criteria, and (ii) to compute a complete ranking of all the solu-tions while taking into account the degree of certainty or possibility of dominance.

The remainder of the article is structured as follows. The section 2, 3 and 4 present adequate background about PROMETHEE II method, the modeling of the imprecise and uncertain values of the decision criteria, and the four possibilistic mono-criterion dominance relations, respectively. In section 5, the proposed approach is detailed. In section 6, an illustrative application, dealing with the design of a technical bid solution (a couple of technical system and delivery process), is presented and discussed in order to show the validity and effectiveness of the proposed approach. Finally, section 7 presents conclusion and future research.

PROMETHEE II RANKING METHOD

The PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluations) methods belong to the family of outranking methods [START_REF] Behzadian | PROMETHEE : A comprehensive literature review on methodologies and applications[END_REF]. The first versions PROMETHEE I (for partial ranking) and PROMETHEE II (for complete ranking) have been developed by [START_REF] Brans | How to select and how to rank projects: The Promethee method[END_REF]. Since, several versions of PROMETHEE methods have been proposed, especially : PROMETHEE III (for ranking based on interval ) and PROMETHEE IV (for ranking when the set of solutions is continuous) [START_REF] Behzadian | PROMETHEE : A comprehensive literature review on methodologies and applications[END_REF]. They are wellknown to be simple and easy to implement compared to other MCDM methods [START_REF] Behzadian | PROMETHEE : A comprehensive literature review on methodologies and applications[END_REF]. In this article, PROMETHEE II ranking method is used in order to perform a complete ranking of design solutions in engineering design process. As the other outranking methods, PROMETHEE II is based on the pairwise comparison of potential solutions with respect to each decision criterion. Its main steps are presented in the following. Given: a finite set S = {S 1 , S 2 , ..., S m } (m ≥ 2) of m possible solutions; a finite set C = {C 1 , C 2 , ..., C n } (n ≥ 2) of n decision criteria. Each decision criterion C k is characterized with a weight w k which corresponds to its relative importance with regard to the other decision criteria. Each solution S i is evaluated following every decision criterion. The evaluation of a solution S i following a decision criterion C k is noted S k i .

Step 1: For each pair of potential solutions (S i ,S j ), the preference degree of the solution S i over the solution S j with respect to each decision criterion C k , noted P k (S i ,S j ), is computed. A preference degree P k (S i ,S j ) is a score between [0,1] which expresses how a solution S i is preferred to another one S j with respect to the criterion C k . Its computation is based on the evaluation S k i and S k j of the solutions S i and S j , and, it is done using a preference function F k (S i ,S j ).

P k (S i , S j ) = F k (S k i , S k j ) (1)
Step 2: For each pair of potential solutions (S i ,S j ), the overall preference degree of the solution S i over the solution S j with respect to all the decision criterion, noted π(S i , S j ), is computed. π(S i , S j ) is calculated as the weighted sum of the preference degree P k (S j ,S i ). The weight of each decision criterion, noted w k , is supposed known. It represents the relative importance of the decision criterion C k with regard to the other criteria.

π(S i , S j ) = n k=1 (P k (S i , S j ) * w k ) (2)
Step 3: Based on the overall dominance degree, the positive, negative and net outranking flows are computed for each potential solution S i . The positive outranking flow of S i , noted φ + (S i ), indicates how a solution S i is preferred over all the other solutions whereas the negative outranking flow of S i , noted φ -(S i ), indicates how the other solutions are preferred over the solution S i . The net outranking flow of S i , noted φ(S i ), takes into account both the positive and negative outranking flows. Is is calculated as the difference between the positive outranking flow and the negative flow. The net outranking flow is used to compute a complete ranking of the potential solutions. The solutions are ranked from the best to the worst. The more the net outranking flow of a solution S i is higher, the more it is preferred to the other solutions.

φ + (S i ) = 1 m -1 Sj ∈S/Si Sj (π(S i , S j )) (3) φ -(S i ) = 1 m -1 Sj ∈S/Sj Si (π(S j , S i )) (4) φ(S i ) = φ + (S i ) -φ -(S i ) (5)
In this article, the values of the decision criteria are imprecise and uncertain. They are modeled as possibility distributions (see section 3). Therefore, in section 5, the four possibilistic mono-criterion dominance relations, proposed in our previous work [START_REF] Sylla | ETO Bid Solutions Definition and Selection Using Configuration Models and a Multi-Criteria Approach[END_REF] and recalled in section 4, are used to compare two potential solutions S i and S j with respect to a single decision criterion and to compute the possibilistic dominance degree of the solution S i over the solution S j . As the preference degree in standard PROMETHEE II method, the possibilistic dominance degree expresses how the solution S i is preferred to the solution S j . Thus, the approach proposed in this article is based on possibilistic dominance degree instead of preference degree in a standard PROMETHEE II approach. In the next section 3, the way the values of the decision criteria are modeled is presented.

MODELING OF THE EVALUATION OF THE DECISION CRITERIA AS POSSIBILITY DISTRIBUTIONS

In scientific work, many framework have been used to model the imprecise and uncertain values of the decision criteria [START_REF] Durbach | Modeling uncertainty in multi-criteria decision analysis[END_REF]. In this article, possibility theory is used to model the evaluation of a solution S i with regard to a criterion C k as a possibility distribution µ S k i . The way the values are modeled is shown in Fig. 1 to be greater or smaller than another one µ S k j [START_REF] Dubois | Ranking fuzzy numbers in the setting of possibility theory[END_REF]. They are presented in the following. For the seek of simplicity and clarity, the computation method of these four indexes are not detailed here in this article. For more details, please consult [START_REF] Dubois | Ranking fuzzy numbers in the setting of possibility theory[END_REF]. to be smaller than the possibility distribution µ S k j . Based on these four indexes (POD, PSD, NOD and NSD), in our previous work [START_REF] Sylla | ETO Bid Solutions Definition and Selection Using Configuration Models and a Multi-Criteria Approach[END_REF], four possibilistic mono-criterion dominance relations have been proposed in order to compare two solutions S i and S j with respect to a single decision criterion, and to compute the possibilistic dominance relations of S i over S j . They are presented in the following.

Given: two potential solutions S i and S j ; their evaluation on a criteria C k , noted µ S k i and µ S k j , respectively; and their dominance indexes represented in the two vectors I Si Sj = [P OD P SD N OD N SD] for the solution S i and I Sj Si = [P OD P SD N OD N SD] for the solution S j . The notation S j S i means that the solution S i dominates (or is preferred to) the solution S j .

Certain Dominance (CD): In this situation, the two possibility distributions µ S k i and µ S k j are completely disjoints. One possibility distribution µ S k i is certainly smaller than the other one µ S k j . As a consequence, the solution S i dominates the other one S j with certainty. The value of the dominance index NSD of the vector I Si Sj is equal to 1 as shown in the equation 7.

I k Si Sj (4) = 1 (7)
Strong Possibility of Dominance (SPD): In this situation, the two possibility distributions µ S k i and µ S k j overlap. However, the four dominance indexes (POD, PSD, NOD and NSD) indicate that one possibility distribution µ S k i is generally smaller than the other one µ S k j . Therefore, it is not certain that the solution S i dominates the solution S j , however it is strongly possible. This relation is formalized in the equation 8.

[I k

Si Sj (4

) < 1] ∧ [∀ t ∈ {1, ..., 4}; D k Si Sj (t) > D k Sj Si (t)] (8)
Weak Possibility of Dominance (WPD): In this situation, the two possibility distributions µ S k i and µ S k j overlap and all the four indexes are not consistent for the ranking of the two possibility distributions. However, most of them indicate that one possibility distribution µ S k i is generally smaller than the other one µ S k i . As a consequence, it is not certain that the solution S i dominates the solution S j , however it is weakly possible. This relation is formalized in the equations 9, 10, 11 and 12.

[∃ t ∈ {1, ..., 4} :

I k Si Sj (t) ≤ I k Sj Si (t)] ∧[∀ l = t : I k Si Sj (l) > I k Sj Si (l)] (9) [∀ t ∈ {1, 4} : I k Si Sj (t) = I k Sj Si (t)] ∧[∀ l = t : I k Si Sj (l) > I k Sj Si (l)] (10) [∀ t ∈ {1, 4} : I k Si Sj (t) > DI k Sj Si (t)] ∧[∀ l = t : I k Si Sj (l) = I k Sj Si (l)] (11) [∃ t ∈ {1, ..., 4} : I k Si Sj (t) > I k Sj Si (t)] ∧[∀ l = t : I k Si Sj (l) = I k Sj Si (l)] (12)

INDifference (IND):

In this situation, the two possibility distributions µ S k i and µ S k j strongly overlap and all the four indexes are not consistent for the ranking of the two possibility distributions. In addition, in the contrary to the previous situations, none of the two possibility distributions have more dominance indexes which indicate that it is generally smaller than the other one. Therefore, the two solutions S i and S j are indifferent. This relation is formalized in the equations 13, 14 and 15.

[∀ t ∈ {1, ..., 4} :

I k Si Sj (t) = I k Sj Si (t)] (13) [∀ t ∈ {1, 4} : I k Si Sj (t) = I k Sj Si (t)] ∧ [I k Si Sj (2) > I k Sj Si (2)] ∧[I k Si Sj (3) < I k Sj Si (3)] (14) [∀ t ∈ {1, 4} : I k Si Sj (t) = I k Sj Si (t)] ∧ [I k Si Sj (2) < I k Sj Si (2)] ∧[I k Si Sj (3) > I k Sj Si (3)] (15)
In the following section, these four mono-criterion dominance relations are used to compare two solutions S i and S j with respect to a single decision criterion C k , and to compute the dominance degree of S i over S j .

PROMETHEE II RANKING METHOD BASED ON POSSIBILISTIC DOMINANCE DEGREE

The proposed approach follows the three main steps of PROMETHEE II ranking method presented in section 2. They are described in the following.

Computation of the possibilistic dominance degrees

The potential solutions are compared with respect to a single decision criterion. For each decision criterion C k and for each pair of potential solutions (S i ,S j ), the possibilistic dominance degree of the solution S i over the solution S j , noted δ(S i , S j ), is computed based on the equations presented in section 4. The computed dominance degree δ(S i , S j ) takes into account the possibility and necessity of dominance of the solution S i over the solution S j , but also, the possibility and necessity of dominance of the solution S j over the solution S i . Therefore, in this article, we consider that the possibilistic dominance degree (CD, SPD and WPD) are unidirectional. Meaning that once the dominance of a solution S i over the solution S j is equal to CD, SPD or WPD, the dominance degree of S j over S i is not relevant. It is noted NR. The function used to compute the possibilistic dominance degrees is presented in Algorithm 1.

Computation of the overall dominance degrees

For each pair of solutions (S i ,S j ), the overall dominance degree of the solution S i over the solution S j with respect to all the decision criteria is computed using the equation 16.

π (S i , S j ) = C k ∈C/Si Sj (δ k (S i , S j ) * w k ) (16)
δ k (S i , S j ) is the numerical score corresponding to the possibilistic dominance degree of S i over S j computed in section 5.1. The numerical values of the possibilistic dominance degrees are computed using the Table 1. For example, if the dominance degree of a solution S i over another one S j is SPD, then the numerical value δ k (S i , S j ) is equal to 0,66. In order to compute the net outranking flow, the positive and negative outranking flow are first calculated using the equation 17 and 18.

φ + (S i ) = 1 m -1 Sj ∈S/Si Sj (π (S i , S j )) (17) φ -(S i ) = 1 m -1 Sj ∈S/Sj Sji (π (S j , S i )) (18)
Then, the net outranking flow is computed as the difference of the positive outranking flow and the negative outranking flow (see equation 5 in section 2). The potential design solutions are ranked based upon their net outranking flow. The most interesting solution is the one that has the highest net outranking flow.

ILLUSTRATIVE APPLICATION

This application concerns the design of a technical bid solution (a couple of technical system/delivery process) during a bidding process. In the context of a bidding process, in order to propose an offer to a customer, a bidder must design several technical bid solutions that satisfy the customer's requirements. Then, based on several criteria, they must select the most interesting one to propose to the customer. In general, the time allowed by the customers to the suppliers to transmit their offers is short [START_REF] Kromker | BIDPREP-towards simultaneous bid preparation[END_REF]. Therefore, the bidders do not have sufficient time to perform a detailed design of their offers (couples systems/delivery process). The technical bid solutions are only partially designed. As a consequence, relevant knowledge to evaluate them are less available.

Due to this lack of knowledge the values of the decision criteria which characterize the technical bid solutions are imprecise and uncertain. Therefore, in order to make the right decision when selecting the most interesting technical bid solution, the uncertainty and imprecision related to the values of the decision criteria must be taken into account in the decision process. In this design problem, nine technical bid solutions have been designed using a configuration software. They are evaluated on two decision criteria : (i) the cost of the technical bid solutions, and (ii) the duration of the delivery processes. Based on these criteria, the bidder must select one technical bid solution from this panel of nine potential solutions. The detail about the decision problem is presented in Table 2. The possibilistic dominance degrees are computed using the function Algorithm 1 presented in section 5. They are presented in Fig. 3. The numerical values corresponding to the possibilistic dominance degrees, computed using the Table 1, are presented in Fig. 4.

Based on this, the overall dominance degree are computed using the equation 15. They are presented in Fig. 5.

Computation of the outranking flows and the complete ranking

Based on the overall dominance degrees, the positive, negative and net outranking flows of each technical bid solution are computed using the equation 17, 18 and 5, 3. The complete ranking of the solutions are computed based upon the net outranking flows φ. The more the net outranking flow of a solution is higher, the more it is preferred to the other solutions. The complete ranking of the nine solutions is presented in Fig. 6. The most interesting design solution is the solution S 6 .

Table 3. The outranking flows Solution φ + φ - φ S1 0,476 0,156 0,321 S2 0,518 0,031 0,488 S3 0,270 0,392 -0,122 S4 0,426 0,217 0,209 S5 0,021 0,812 -0,791 S6 0,570 0,072 0,498 S7 0,000 0,875 -0,875 S8 0,456 0,113 0,343 S9 0,291 0,362 -0,071 In this article, we have proposed a multi-criteria decision support approach based on possibility theory and PROMETHEE II ranking method. The application of the proposed approach has shown that it allows to effectively rank several potential design solutions while taking into account imprecision and uncertainty related to the evaluation of the design solutions. In the context of an engineering design process, this approach is very useful, especially in the early phases.

  Fig. 1. Modeling of the values of a decision criterion

  Given two possibility distributions µ S k i and µ S k j : • Possibility Of Dominance (POD) : provides the possibility of the possibility distribution µ S k i to be smaller than (or equal to) the possibility distribution µ S k j . • Possibility of Strict Dominance (PSD) : provides the possibility of the possibility distribution µ S k i to be smaller than the possibility distribution µ S k j . • Necessity Of Dominance (NOD) : provides the necessity of the possibility distribution µ S k i to be smaller than (or equal to) the possibility distribution µ S k j . • Necessity of Strict Dominance (NSD) : provides the necessity of the possibility distribution µ S k i

Fig. 2 .

 2 Fig. 2. Modeling of the values of a decision criterion

Fig. 4 .

 4 Fig. 4. Numeriacl values of possibilistic dominance degrees

Fig. 6 .

 6 Fig. 6. Complete ranking of the solutions

  Algorithm 1 δ k (S i , S j ) if (equation 7 is TRUE) then return CD # Sj dominates Si with certainty else if (equation 8 is TRUE) then return SPD # Sj dominates Si, not certainty, but with a strong possibility else if ((equation 9 is TRUE) ∨ (equation 10 is TRUE)

	∨ (equation 111 is TRUE) ∨ (equation 112 is TRUE))
	then
	return WPD # Sj dominates Si, not certainty, but
	with a weak possibility
	else if ((equation 13 is TRUE) ∨ (equation 14 is TRUE)
	∨ (equation 15 is TRUE)) then
	return IND	# S i and S j are indifferent to
	each other
	else
	return NR	# S i is dominated by S j
	end if

Table 1

 1

	. Numerical values of the possibilistic
	dominance degrees
	δ k (S i , S j ) δ k (S i , S j ) IND 0
	WPD	0.33
	SPD	0.66
	CD	1
	5.3 Computation of the outranking flows and the complete
	ranking of the potential solutions

ACKNOWLEDGEMENTS

The authors would like to thank all ANR OPERA partners for their implications in our research work.


Christian El
email: christian.s.hajj@gmail.com

Hajj Irstea

Ls2n

Saïd Moussaoui
email: saïd.moussaoui@ec-nantes.fr

Guylaine Collewet
email: guylaine.collewet@irstea.fr

Maja Musse
email: maja.musse@irstea.fr

Multi-Exponential Relaxation Times Maps Reconstruction and Unsupervised Classification in Magnitude Magnetic Resonance Imaging

Keywords: MRI, multi-exponential model, Maximum-Likelihood, Majorization-Minimization, K-means

In clinical and biological applications of T2 relaxometry, a multi-exponential decay model proved to be representative of the relaxation signal inside each voxel of the MRI images. However, estimating and exploiting the model parameters for magnitude data is a large-scale ill-posed inverse problem. This paper presents a parameter estimation method that combines a spatial regularization with a Maximum-Likelihood criterion based on the Rician distribution of the noise. In order to properly carry out the estimation on the image level, a Majorization-Minimization approach is implemented alongside an adapted non-linear least-squares algorithm. We propose a method for exploiting the reconstructed maps by clustering the parameters using a K-means classification algorithm applied to the extracted relaxation time and amplitude maps. The method is illustrated on real MRI data of food sample analysis.

I. INTRODUCTION

In MRI transverse relaxation (T 2 ) relaxometry, a series of T 2 -weighted images are acquired at a fixed sampling rate using a multiple spin echo (Multi-SE) technique. In most approaches, a mono-exponential decay curve is fitted to the measured signals, and one T 2 per voxel is estimated (time constants of the decay curve) alongside their corresponding amplitudes A 0 [START_REF] Adriaensen | MSE-MRI sequence optimisation for measurement of biand tri-exponential T2 relaxation in a phantom and fruit[END_REF]. However, in many cases a multiexponential decay curve can provide more relevant information on the micro-structure or composition of the tissue. Actually, since the size of an MRI voxel is of the order of 1 mm 3 , it may contain several different components such as lipids and water, or several kinds of water pools with different T 2 . The latter case is typically observed when analyzing MRI data of plants. Multi-exponential T 2 and A 0 maps can be used to access information about water status and distribution at the sub-cellular level. This is useful for example for the assessing of fruit characteristics in the monitoring of the post-harvest ripening of tomato fruit [START_REF] Musse | An investigation of the structural aspects of the tomato fruit by means of quantitative nuclear magnetic resonance imaging[END_REF]. The Multi-exponential behavior is attributed to different water pools corresponding to the compartments of the plant cell (vacuole, cytoplasm and wall). The same situation occurs for myelin water fraction in the brain [START_REF] Laule | Magnetic resonance imaging of myelin[END_REF], Thanks to Region Bretagne, École centrale Nantes, Irstea Rennes and GDR ISIS for funding.

where three major components are used to represent the water trapped between the myelin, the intra/extra-cellular water and the cerebro-spinal fluid. The multi-exponential aspect of the measured decay signal can be described by a continuous T 2 distribution, or by limiting the number of exponential decay curves inside each voxel to a small discrete number equal to the number of components. In this study, the expected number of components is considered as known and is of low value (triexponential model).

The noise in the MRI data is considered to follow an independent and identically distributed Gaussian noise in the complex domain. However, the phase of the complex data is both time and space dependent due to various uncontrollable factors related to the acquisition process. Thus, estimating the parameters directly in the complex domain proves to be a challenging problem. The common approach is to carry out the parameters estimation from the magnitude of the measured signals. This non-linear transformation, in the case of single coil acquisitions [START_REF] Henkelman | Measurement of signal intensities in the presence of noise in mr images[END_REF], results in a Rician distributed noise. Rician noise can be approximated by a Gaussian distribution for high signal to noise ratio (SNR). However, in relaxometry, the SNR of the acquired data is critically low, particularly at the end of the relaxation curve. If the Rician noise distribution is not properly accounted for, it leads to a bias on the estimated parameters [START_REF] Raya | T2 measurement in articular cartilage: impact of the fitting method on accuracy and precision at low snr[END_REF]. A Maximum Likelihood (ML) estimator based on the Rician probability density function is implemented in order to reduce the estimation bias caused by the Rician distribution [START_REF] Yokoo | Quantitative R2* mri of the liver with rician noise models for evaluation of hepatic iron overload: Simulation, phantom, and early clinical experience[END_REF]. Moreover, in order to stabilize the algorithm for low SNR values, a spatial regularization approach is implemented by taking advantage of low voxel to voxel variability at different image structures [START_REF] Hwang | Improved myelin water quantification using spatially regularized non-negative least squares algorithm[END_REF], [START_REF] Kumar | Bayesian algorithm using spatial priors for multiexponential T2 relaxometry from multiecho spin echo MRI[END_REF]. Due to the large amount of memory needed to directly implement the spatial regularization on the image level, it has been carried out either by dividing the image into overlapping patches [START_REF] Kumar | Bayesian algorithm using spatial priors for multiexponential T2 relaxometry from multiecho spin echo MRI[END_REF] or by carrying the estimation on regions of interest. The exploitation of the estimated parameters proves to be a complicated process due to the large amount of information in the reconstructed maps. A step of data classification in the space formed by the estimated parameters is suggested in order to extract a distribution of the estimated T 2 parameters at different spatial regions of the image. In this paper, an efficient penalized ML algorithm is presented in order to carry out the estimation on the whole image at once and obtain voxel level information [START_REF] El-Hajj | Spatially regularized multi-exponential transverse relaxation times estimation from magnitude magnetic resonance images under Rician noise[END_REF]. The first feature of the algorithm is to estimate all the image parameters simultaneously using a Majorization-Minimization (MM) approach leading to a quadratic majorant function to be minimized with an adapted Levenberg-Marquardt (LM) algorithm. The particularity is to compute a maximum step-size that guarantees the parameters positivity and to adopt a step search approach using a backtracking technique based on the Armijo line search to find a step-size that ensures the convergence of the algorithm. The main contribution of this paper is to propose an appraoch allowing to exploit the extracted parameters. It is based on clustering the estimated parameters using a non-supervised classification approach based on the K-means algorithm. Tests were conducted on simulated phantom and on experimental MRI images of a tomato. The performances are illustrated through numerical and visual results.

II. PARAMETERS ESTIMATION

Magnitude MRI data, measured with a Multi-SE sequence, are composed of N t images. The fixed sampling time between two successive images is given by ∆T E. For a multiexponential decay signal with N c components inside voxel j, the measured signal m j (τ ), can be fitted by a predicted signal model A j (τ, θ j ) expressed by :

A j (τ, θ j ) = Nc c=1 A 0 (c,j) e - τ T 2 (c,j) , for τ = τ 1 , ..., τ Nt (1)
where θ j = [A 0 (1,j) , T 2 (1,j) ... A 0 (Nc ,j) , T 2 (Nc ,j) ] is the vector of unknown parameters of length N p = 2N c .

A. Problem statement

In magnitude MRI data with a sufficiently high SNR, the noise distribution can be approximated by a Gaussian distribution. However, at low SNR the relaxation data statistics follows a Rician distribution model. Thus, in order to precisely estimate the parameters, the Rician probability density function P R should be taken into account. This leads to a Maximum Likelihood estimator that can be solved by maximizing the following likelihood function :

L (m, θ) = Nv j=1 τ N t τ =τ1 P R (m j (τ ) | A j (τ, θ j ) , σ) = Nv j=1 τ N t τ =τ1 m j (τ ) σ 2 e - m j (τ ) 2 +A j ( τ,θ j ) 2 2σ 2 I 0 (D jτ ) , (2)
where N v is the number of voxels, m is a vector containing the measurement at all the voxels of the MRI image, θ = [θ 1 ... θ Nv ], I 0 (•) is the zeroth order modified Bessel function of the first kind,

D jτ = mj (τ )Aj (τ,θj ) σ 2
, and σ 2 is the variance of the Gaussian noise. By taking the log of the likelihood function and reformulating, minimizing the likelihood function in (2) comes down to minimizing the following ML criterion :

J M L (m, θ) = Nv j=1 τ N t τ =τ1 A j (τ, θ j ) 2σ 2 2 -log (I 0 (D jτ )) , (3)
Estimating the high number of parameters under low SNR conditions is an ill posed problem. The spatial regularity at different image structure is accounted for by adding a penalizing function to the J M L criterion. For each voxel j, a penalty function ψ is computed such that high differences between parameters belonging to a predefined neighboring region V j are penalized :

R(θ) = Np p=1 β(p) Nv j=1 i∈Vj ψ (θ j (p) -θ i (p)) (4)
β is the vector that controls the weight attributed to the spatial regularization. An L1-L2 penalty function ψ was used in order to minimize the inter-tissue blurring [START_REF] Idier | Convex half-quadratic criteria and interacting auxiliary variables for image restoration[END_REF]. The penalized ML (PML) criterion to be minimized is thus given by :

F (m, θ) = J M L (m, θ) + R(θ) (5)

B. Proposed algorithm

The PML criterion does not have an analytical solution thus, an iterative approach must be adopted. In order to minimize the F (m, θ) criterion, two main challenges must be resolved; the first one is the non-convexity of the J M L (m, θ) criterion and the second one is the voxel-wise dependency of the solution due to the regularization penalty function. The optimization algorithm is based on a MM approach. It allows the minimization of a non quadratic, non convex and multidimensional criterion by transforming it into a separable quadratic criterion easier to minimize. Firstly, the strictly concave -log (I 0 (•)) function is majorized by its tangent :

-log I 0 D (k) jτ - m j (τ ) σ 2 R D (k) jτ A j τ, θ j -A j τ, θ (k) j (6)
where θ

(k) j

is the the vector of the estimated parameters at the kth iteration of the MM algorithm and R(•) = I1(•) I0(•) where I 1 (•) is the first order modified Bessel function of the first kind. The obtained quadratic ML criterion to be minimized at each step k of the MM algorithm is given by [START_REF] Varadarajan | A Majorize-Minimize framework for Rician and non-central Chi MR images[END_REF] :

QML(m, θ (k) ) = 1 2σ 2 Nv j=1 τ N t τ =τ 1 mj (τ ) R D (k) jτ -Aj (τ, θj) 2 .
Secondly, Erdogan and Fesslers's [START_REF] Hakan | Ordered subsets algorithms for transmission tomography[END_REF] method is used to construct a majorant function that establishes a voxel-wise variables separability. Accordingly, for each iteration k of the MM algorithm, the regularization function is majorized by :

ψ (z j -z i ) ≤ 1 2 ψ 2z j -z (k) j + z (k) i + 1 2 ψ 2z i -z (k) j + z (k) i . (7)
The obtained majorant criterion of the F (m, θ) is given by

Q P M L (m, θ (k) ) = Q M L (m, θ (k) ) + Q reg (θ (k) ) with : Qreg θ (k) = Np p=1 β(p) Nv j=1 i∈V j ψ 2θj (p) -θ (k) j (p) + θ (k) i (p) (8)
The minimization of Q P M L (m, θ (k) ) criterion is performed by a Levenberg-Marquardt (LM) algorithm witch is suited for solving non-linear least-squares problems. At iteration l of the LM algorithm, the parameters are updated in a direction d LM depending on the residual r and the Jacobian matrix J :

d (l) LM = J (l) T J (l) + λ (l) -1 J (l) T r (l) (9)
r is expressed such that

Q P M L (m, θ (k) ) = 1 2 r 2 2
. λ is a diagonal matrix that contains the LM parameters for each voxel. In order to ensure the physical coherence of the estimated parameters, a positivity constraint is imposed. In order to ensure a sufficient decrease of the criterion, the stepsize is computed using a backtracking technique based on the Armijo condition bounded by a maximum step-size that guarantees the positivity of the parameters. The proposed MM-PML algorithm is summarized in algorithm 1. initialize θ (0) ; for k ← 0 to k max or until convergence do For all j and i compute θ

LM = θ (l) LM + α (l) d (l) LM ; end update θ (k+1) = θ (lmax) LM end Output: θ (k+1)
Algorithm 1: The MM-PML algorithm

The algorithm is said to have converged if the value of the gradient is smaller than a tolerance value gtol or if the difference between the objective function at step k + 1 and k is smaller than a tolerance value f tol :

F (m, θ (k+1) ) -F (m, θ (k) ) F (m, θ (k+1) ) ≤ f tol. (10)

C. Classification of the results based on the estimated parameters

The reconstructed multi-exponential maps contain information that can be exploited in order to classify the image composition into coherent regions and extract the corresponding T 2 distribution. been treated in the context of MRI relaxometry. Actually, Each voxel j of the image have N p estimated parameter that can be considered as features. The K-means clustering algorithm was applied [START_REF] Hartigan | Algorithm as 136: A k-means clustering algorithm[END_REF] to the whole image as it is an unsupervised method of classification. The number of classes is chosen visually, such that, by adding another class, two classes with very close T 2 distributions will be obtained. Finally, the results of the classification with the corresponding T 2 distribution inside each class are shown.

III. MATERIALS

Both the MM-PML algorithm and the K-means clustering are tested on simulated and experimental MRI data.

A. Simulated data

A circular phantom was constructed on an image of 128 × 128 voxels, with an outside ring and 9 inner disks of different radii. At each part of the phantom, a different tri-exponential model was generated. T 2 and A 0 values that are close to parameters typically found in tomato fruits were used as seen in Figure 1. Then, a Rician noise was added with a variance of σ 2 = 49 in order to obtain a SNR value close to that found in experimental MRI data (212) (SN R = Nv j=1 mj (τ1) σNv

).

Reg. Ref.

Ref.

Non reg.

B. Experimental data

For the experimental MRI settings, a Multi-SE sequence was used on a 1.5T MRI scanner (Magnetom, Avanto, Siemens, Erlangen, Germany), with inter-echo spacing (∆TE) of 6.5 ms, bandwidth of 260 Hz/pixel, 512 echoes per echo train and a repetition time of 10s. The median planes of fruit (transverse section at middle height of fruit) were imaged with a total of 128 × 128 voxels and a slice thickness of 5 mm, resulting in voxel size of 1.19 × 1.19 × 5 mm 3 . The reference maps where acquiring with 32 scans in order to obtain higher SNR (687), and to verify the stability of the algorithm at the lowest SNR (145), the same images with only one scan were acquired.

C. Settings

In order to implement the regularization, the penalizing function ψ and the penalizing weight vector β must to be chosen. The choice of the penalizing function affects the intertissues smoothness, whilst the penalizing weight affects the degree of smoothness on the whole image. In this study, an L1-L2 penalizing function was used with a weight β chosen by trial and error so as to preserve a good separation between edges. The algorithm stopping criteria gtol and f tol were both set to 10 -6 . Numerical results were computed using the normalized root mean square error (NRMSE) that was computed as follows :

N RM SE = 100 1 N v N p Nv j=1 Np p=1 θ j (p) -θ * j (p) 2 θ * j (p) 2 (11)
with θ * j the reference parameters at voxel j.

IV. RESULTS In Figure 2 the value of the F (m, θ) criterion is shown at every step of the MM algorithm. By applying the LM algorithm to minimize surrogate criterion, a descent of the initial criterion is guaranteed, thus validating the correctness of the choice of the majorant functions.

A. Simulated data

In Figure 3 the results of applying the MM-PML on the phantom are shwon and are compared to a non-regularized version of the algorithm (β=0). Using the regularized algorithm, the maps are reconstructed with satisfying edge preserving which allows us to visually distinguish between the different structures even in low contrast situations. By adopting the spatial regularization, the noise of the reconstructed maps was reduced and the contrast between different image regions was more clearly pronounced. Numerically, a low NRMSE/voxel/parameter was obtained for the regularized version 4.47% whereas for the non regularized version we obtained a value of 22.85%. This further validates the importance of the regularization on reducing the estimation error. In Figure 4, the classification algorithm was applied to the results of the parameters obtained with the MM-PML algorithm. The different voxels were associated to their right class with a precision of a 100%. Furthermore, the distribution of T 2 values inside each class are shown and the centroids estimated using the k-means algorithm proved to be very close to the true values used for simulating the phantom.

B. Experimental data

Figure 5 shows that images reconstructed from low and high SNR data are quite similar for the different parameters, thus validating the robustness of the method to different noise levels. Furthermore, a low NRMSE equal to 5.56% was obtained. This shows that even with low SNR the algorithm converged to values close to those estimated from high SNR data which will allow us to reduce the acquisition time by scanning the fruit with only one repetition. Figure 6 shows the results of the k-means algorithm and the distribution of the T 2 values inside each identified class. Three different tissues (out of the anticipated five) of the tomato are well identified using the proposed classification technique and are attributed to a class with different T 2 distribution. The red class represents both the locular tissue and the extreme periphery, the green class represents the union of the outer pericarp and the core of the tomato and the blue class represents the placenta. The black class distribution is not shown because it is attributed to fraction volumes and it has no physical signification. The failure to identify the two other classes may be due to the non-homogeneity of the classes, a phenomena that can be related either to the physiology of the fruit tissues that is not 100% homogeneous or to the non-spatial homogeneity of the MRI antenna (observable on the A 0 maps). These points remain open for further studies by adapting the classification algorithm to those phenomena.

V. CONCLUSION

In this paper, algorithms for both the reconstruction and the exploitation of relaxtion times and their corresponding amplitude maps from noisy MRI magnitude data were presented. The estimation algorithm showed both stability and accuracy on real and simulated MRI data. The reconstruction was carried out, for the first time as far as our knowledge, on the whole image at once. Furthermore, we were able to plot the T 2 distribution inside different fruit tissues by using an automatic unsupervised classification technique and to identify characterizing fruit tissues. This work will enable the assessment of fruit characteristics at the subcellular level since the multiple T 2 are linked with the water status in the different cell structures.

  LM = θ (k) ; for l ← 0 to l max do Compute d (l) LM ; Compute the step-size α (l) using armijo line search.; Compute θ (l+1)

Fig. 1 .

 1 Fig. 1. T 2 and A 0 values used for the phantom.

FFig. 2 .

 2 Fig. 2. The evolution of the PML criterion (in blue) across the external MM steps (orange points) and the internal LM steps (blue points)

Fig. 3 .Fig. 4 .

 34 Fig.3. Reference (first columns) and estimated A 0 and T 2 maps on the phantom using a non-regularized (second columns) and the proposed regularized algorithm (third columns).

3 Fig. 5 .

 35 Fig. 5. A 0 and T 2 maps reconstructed from MRI images of a tomato acquired with 32 scans and 1 scan.

Fig. 6 .

 6 Fig. 6. Spatial partitioning of the 5 classes identified by the K-means algorithm and the joint distribution of parameters (T2;A0).The violet crosses represent the position of the estimated centroids.

  To our knowledge, this problem has not yet A 01 T 21 A 02 T 22 A 03 T 23

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Christian El Hajj

Guylaine Collewet

Maja Musse

Saïd Moussaoui

Spatially regularized multi-exponential transverse relaxation times estimation from magnitude MRI images under Rician noise

This work aims at improving the estimation of multi-exponential transverse relaxation times from noisy magnitude MRI images. A spatially regularized Maximum-Likelihood estimator accounting for the Rician distribution of the noise was introduced. This approach is compared to a Rician corrected least-square criterion with the introduction of spatial regularization. To deal with the large-scale optimization problem, a majoration-minimization approach was used, allowing the implementation of both the maximum-likelihood estimator and the spatial regularization. The importance of the regularization alongside the rician noise incorporation is shown both visually and numerically on magnitude MRI images acquired on fruit samples.

Purpose

Multi-exponential relaxation times and their associated amplitudes in an MRI image provide very useful information for assessing the constituents of the imaged sample. Typical examples are the detection of water compartments of plant tissues and the quanti cation of myelin water fraction for multiple sclerosis disease diagnosis . The estimation of the multi-exponential signal model from magnitude MRI images faces the problem of a relatively low signal to noise ratio (SNR), with a Rician distributed noise and a large-scale optimization problem when dealing with the entire image. Actually, maps are composed of coherent regions with smooth variations between neighboring voxels. This study proposes an e cient reconstruction method of values and amplitudes from magnitude images by incorporating this information in order to reduce the noise e ect. The main feature of the method is to use a regularized maximum likelihood estimator derived from a Rician likelihood and a Majorization-Minimization approach coupled with the Levenberg-Marquardt algorithm to solve the large-scale optimization problem. Tests were conducted on apples and the numerical results are given to illustrate the relevance of this method and to discuss its performances.

Methods

For each voxel of the MRI image, the measured signal at echo time is represented by a multi-exponential model:

with

The data are subject to an additive Gaussian noise in the complex domain and therefore magnitude MRI data follows a Rician distribution :

is the rst kind modi ed Bessel function of order 0 and is the standard deviation of the noise which is usually estimated from the image background.

For an MRI image with voxels, the model parameters are usually estimated by minimizing the least-squares (LS) criterion under the assumption of a Gaussian noise using nonlinear LS solvers such as Levenberg-Marquardt (LM). However, this approach does not yield satisfying results when applied to magnitude data . Several solutions to overcome this issue are proposed by adding a correction term to the LS criterion . In this study, the retained correction uses the expectation value of data model under the hypothesis of Rician distribution since it outperforms the other correction strategies:

stands for the sum of squares. We refer to this method as Rician corrected LS (RCLS).

A more direct way for solving this estimation problem is to use a maximum likelihood (ML) estimator which comes down to minimize:

To solve this optimization problem when dealing with the entire image, a majorization-minimization (MM) technique was adopted . The resulting MM-ML algorithm is summarized in gure 1, the LM algorithm used in this method minimizes a set of LS criteria derived from the quadratic majorization strategy.

A spatial regularization term based on a cost function was also added to both criteria ( and ) to ensure spatial smoothness of the estimated maps. In order to reduce the numerical complexity by maintaining variable separability between each voxel and it's neighboring voxels , the function is majorized by : where stands for the iteration number of the iterative optimization algorithm. The criterion to be minimized at each iteration of the MM algorithm is thus given by : contains the indices of voxel 's neighboors

T 2 I 0 1 2 T 2 T 2 j (τ ) y j τ (τ , ) = M j θ j ∑ c=1 Nc I 0 (c,j) e - τ T 2 (c,j) = [ , . . . , ] θ j I 0 (1,j) T 2 (1,j) I 0 ( ,j) Nc T 2 ( ,j) Nc 3 ( (τ ) , (τ , ) , σ) = ( ) P Rician y j M j θ j (τ ) y j σ 2 e - ⎡ ⎣ + y j (τ ) 2 M j (τ , ) θ j 2 2σ 2 ⎤ ⎦ J 0 (τ ) (τ , ) y j M j θ j σ 2 J 0 σ N v ∑ N v j=1 ∥ (τ ) - (τ , )∥ y j M j θ j 2 2 5 3,4 5 = , J RC LS ∑ j=1 Nv ∥ (τ ) - [ (τ , ) , σ]∥ y j E rice M j θ j 2 2 ∥ ⋅ ∥ 2 2 5 = [ -log( ( ))] . J M L ∑ j=1 Nv ∑ τ =τ 0 τn M j (τ , ) θ j 2 2σ 2 J 0 (τ ) (τ , ) y j M j θ j σ 2 6 ψ( (i) - (i)) θ j θ k J M L J RC LS j k ψ 7 ψ( (i) - (i)) ≤ ψ(2 (i) -( (i) + (i))) + ψ(2 (i) -( (i + (i )) f or i = 1, 2...2 * θ j θ k 1 2 θ j θ n j θ n k 1 2 θ k θ j ) n θ k ) n N c n 08/
The resulting methods are refereed to by RCPLS and MM-PML. The regularization function and the weight vector should be adapted to each problem.

Results

We rst acquired very high SNR images (128*128 voxels) on an apple with a 1.5T MRI (Avanto, Siemens) using a multiple-spin echo sequence allowing the acquisition of up to 512 echo times per relaxation decay. Then we estimated the reference tri-exponentiels and maps which were used afterwards for 100 Monte-Carlo signal simulations with a SNR of 108. Four methods were compared : RCLS, MM-ML, RCPLS and MM-PML. A quadratic penalty function was used with for all the variables. Figure 2 gives the relative bias, the relative standard deviation and the relative MSE per parameter. Figure 3 shows the reference maps and the estimated ones.

Results in gure 2 show that even though the bias is higher in the regularized version, the standard deviation was reduced signi cantly yielding to lower MSE. Figure 3 con rms the results by showing that the regularized version yields to visually better results. The fruit structures can thus be identi ed, and the maps are no longer drowned in the noise.

Concerning the numerical complexity, the MM-PML algorithm ran faster (41.5 s) than the RCPLS (107.96 s).

Figures

The algorithm of the MM-ML. The relative bias is given in percentage and is computed across the image and the 100 Monte-Carlo realizations. The standard deviation is also given in percentage and is computed across the 100 Monte-Carlo realizations and the average standard deviation of the di erent voxels is shown.

Table 1 shows the results of both algorithms without regularization; Table 2 shows the results of the regularized version.

First column contains the real maps used for this study, and then respectively : maps reconstructed using :RCLS, RCPLS and MM-PML algorithm. The MM-ML is not shown to avoid redundancy.

Acknowledgements

Christian EL HAJJ was supported in part by a scholarship from Brittany Region (France)

The work is funded by the GDR 720 ISIS as part of a call for exploratory projects 2016


François Fromard
email: francois.fromard@univ-tlse3.fr

Antoine Gardel

Christophe Proisy

Edward Anthony

Romain Walcker

Mangrove forest is a major ecosystem of tropical coasts that supports a wide range of ecosystem services, including protection against coastal erosion, a significant carbon storage capacity, nursery site for marine species, and retention of terrigenous pollutants.

2. However, mangroves have been largely threatened by direct human activities especially with the development of shrimp farming, urban expansion and agricultural practices over the last decades. Globally, around 40% of mangrove forests have been lost since the mid-twentieth century.

3 .

 3 Mangroves are particularly sensitive to climate change, with various impacts on ecosystem structuring, functioning and dynamics. Sea level rise, increase in air and water temperatures, changes in tropical storm frequency and intensity are the factors that are the most likely to affect the integrity of mangrove forests. Where they have not been directly disturbed by man, mangroves can be regarded as sentinels of climate change. This is true for the mangroves of French tropical overseas territories, that are specifically monitored in the framework of the CNRS/IRD's 2015 Year of the Mangrove initiative. The global distribution of mangroves : Western and Eastern areas and localisation of the French overseas territory mangroves respectively French Guiana and West Indies Islands (Atlantic area), Mayotte and Scattered Islands (West Indian Ocean), New Caledonia and Wallis Island (Pacific Ocean) 1. At the interface between land and sea and through adapted root system, mangrove forest acts (i) as a buffer protecting coastal areas from erosion and inland areas from high waves and flooding, and (ii) as a biofilter limiting coastal sewage pollution. Due to high photosynthetic efficiency and rapid growth, mangroves are characterized by high productivity and biomass and are considered as among the most carbon-rich forests in the tropics. Occupying less than 1% of the world coastal area, mangroves contribute up to 15% to coastal sediment carbon storage. (a) Mayotte Island (b) French Guiana. (a) (b)