D Louis

D Lacour

M Hehn

V Lomakin

T Hauet

F Montaigne

A tunable magnetic metamaterial based on the dipolar four-state Potts model

Metamaterials

, tunable artificial materials, are useful playgrounds to investigate magnetic systems. So far, artificial Ising spin systems have revealed unusual features like emergent magnetic monopoles[1,2] and charge fragmentation[3]. Here, we present a metasystem composed of a lattice of dipolarly-coupled nanomagnets. The magnetic spin of each nanomagnet is constrained to lie along a body diagonal, yielding four possible spin states. We show that the magnetic ordering of this metasystem (antiferromagnetic, ferromagnetic or spin-ice like) is determined by the spin states orientation relative to the underlying lattice. The dipolar four-state Potts model explains our experimental observations and sheds light on the role of symmetry, as well as short-and longrange dipolar magnetic interactions, in such non-Ising spin systems. Dipolar interactions in magnetic systems have been heavily studied because of their exotic physics and strongly affect essential technologies such as permanent magnets or hard disk drives [4-6]. In addition, statistical physics covers a host of applications ranging from fundamental physics to business analysis. In both cases, Ising model holds a central role. This two-spin state model has been widely used to grasp the essence of a variety of effects belonging to the field of condensed matter physics, such as phase transitions, spin glasses, magnetism, spin chains and geometric frustration. In 2006, Zhang and collaborators conducted the first experiments on artificial Ising spin systems at a

mesoscopic scale. Their findings were based on the exact determination of the spin configurations. This was made possible by the use of nano-magnets having a uniaxial anisotropy mimicking the two spin state of the Ising model. This approach proved to be particularly fruitful, especially to study original spin ices phenomena [START_REF] Ladak | Direct observation of magnetic monopole defects in an artificial spin-ice system[END_REF][START_REF] Mengotti | Real-space observation of emergent magnetic monopoles and associated Dirac strings in artificial kagome spin ice[END_REF][START_REF] Canals | Fragmentation of magnetism in artificial kagome dipolar spin ice[END_REF][START_REF] Anghinolfi | Thermodynamic phase transitions in a frustrated magnetic metamaterial[END_REF][START_REF] Gilbert | Emergent reduced dimensionality by vertex frustration in artificial spin ice[END_REF][START_REF] Perrin | Extensive degeneracy, Coulomb phase and magnetic monopoles in artificial square ice[END_REF]. So far, artificial spin systems have been restricted to the Ising variety, i.e., the two spin states being associated with the two possible magnetization states of a nanomagnet. However, beyond the Ising models, other spin lattice models, such as Potts, XY, Heisenberg or more generally n-vector models, have been theoretically studied and recognized to be of interest beyond the purely academic point of view. In particular, the Potts model [START_REF] Wu | The Potts model[END_REF] has found applications in many complex systems [START_REF] Schelling | Dynamic models of segregation[END_REF][START_REF] Sun | A discrete simulation of tumor growth concerning nutrient influence[END_REF][START_REF] Sanyal | Viscous instabilities in flowing foams: a Cellular Potts Model approach[END_REF].

Here, we introduce a variant of the Potts model: the dipolar four-state Potts model. This model system is experimentally produced as an array of artificial Potts spins, i.e. an array of dipolar coupled nanomagnets having four equivalent states of magnetization.

Artificial spins must meet several specifications. An important specification is the mapping between the micromagnetic configurations and the desired spin model. The magnetic anisotropy of the nanostructure is the key ingredient to ensure a mostly uniform magnetization state within the nanostructure. This is important to prevent non-homogenous equilibrium magnetization configurations, as they can lead to the appearance of internal degrees of freedom [START_REF] Rougemaille | Chiral nature of magnetic monopoles in artificial spin ice[END_REF][START_REF] Gliga | Broken vertex symmetry and finite zeropoint entropy in the artificial square ice ground state[END_REF]; this increases the complexity of the system under study, and thus deviates away from the ideal spin model. Up to now, artificial Ising spins have been implemented through uniaxial anisotropy induced either by the shape of the elements (in-plane magnetization [START_REF] Wang | Artificial 'spin ice' in a geometrically frustrated lattice of nanoscale ferromagnetic islands[END_REF][START_REF] Qi | Direct observation of the ice rule in an artificial kagome spin ice[END_REF][START_REF] Arnalds | A new look on the two-dimensional Ising model: thermal artificial spins[END_REF]) or by the material anisotropy (out of plane magnetization [START_REF] Zhang | Perpendicular magnetization and generic realization of the Ising model in artificial spin ice[END_REF][START_REF] Chioar | Nonuniversality of artificial frustrated spin systems[END_REF]).

Our four-states Potts artificial spins are made of a magnetic thin film having an in-plane magnetization and a well-defined cubic anisotropy [START_REF] Louis | Interfaces anisotropy in single crystal V/Fe/V trilayer[END_REF]. A 2 nm thin epitaxial iron film is shaped in 300 nm wide squares having their diagonals aligned with the [100] and [010] anisotropy directions, the in-plane easy axes of the Fe layer (fig. 1). At this thickness, the demagnetizing energy (scaling as the square of the thickness) is reduced enough to avoid the existence of a non-uniform ground state [START_REF] Vaz | Direct observation of remanent magnetic states in epitaxial fcc Co small disks[END_REF][START_REF] Mitsuzuka | Magnetic vortices in single crystalline Fe-V disks with four folds magnetic anisotropy[END_REF][START_REF] Li | Stabilizing a magnetic vortex/antivortex array in single crystalline Fe/Ag(001) microstructures[END_REF]. This geometry ensures a quasi-uniform ground state without any internal degrees of freedom and two equivalent directions with four possible stable states [START_REF] Louis | Four states magnetic dots: A design selection by micromagnetic modeling[END_REF] along the square diagonals. Moreover, in this thickness range, these artificial spins can be thermally excited (see SI1). This allows driving the system from a paramagnetic-like state toward the ground state by use of an external thermal bath as done in the case of artificial Ising spins systems [START_REF] Anghinolfi | Thermodynamic phase transitions in a frustrated magnetic metamaterial[END_REF][START_REF] Gilbert | Emergent reduced dimensionality by vertex frustration in artificial spin ice[END_REF][START_REF] Farhan | Direct observation of thermal relaxation in artificial spin ice[END_REF].

We focus here on artificial Potts spins on square lattices. While the element diagonals remain aligned with the Fe crystal to ensure the aforementioned magnetic properties, the lattice orientation can be varied. This provides a particularly interesting tuning knob as shown in the following parts. The state of each four-state Potts artificial spin can be determined directly by Magnetic Force Microscopy (MFM) as exemplified in figure 1.c. The magnetic configurations are stable at room temperature and not affected by the MFM measurement (SI1). However, 350 °C annealing leads to switch of spins towards one of the four accessible directions (SI1). (a,b,c) represent the magnetic configurations obtained after thermal treatment for the three lattice directions (𝛼 = 0°, 22.5° and 45°) having an identical period of 500 nm and a 30x30 size.

A usual approach to get insight into magnetic order consists in considering the magnetic structure factor as measured in neutron scattering experiments, which can be computed from the exact magnetization configurations [START_REF] Perrin | Extensive degeneracy, Coulomb phase and magnetic monopoles in artificial square ice[END_REF]. The magnetic structure factors associated with the three presented magnetization configurations bring to light the existence of different magnetic order (SI2). Going back to the real space, it is indeed possible to ascribe a large part (from 87 to 94%) of the magnetic configurations to four different local orders: (i) Ferromagnetic order in which all the spins have the same values (red color in figure 2). (ii) Collinear antiferromagnetic order with alternation of lines of opposite spins (cyan color in figure 2). (iii) Spin-ice order that is identical to the type I vertex introduced by Wang et al. [START_REF] Wang | Artificial 'spin ice' in a geometrically frustrated lattice of nanoscale ferromagnetic islands[END_REF] for the square Ising spin ice (green color in figure 2), which corresponds to a vertex obeying the "ice rules" with two spins pointing in and two spins pointing out.

Spin-ice domains can also be seen as crystals of spin loops with alternating chiralities. (iv) Zigzag order corresponds to lines of spins being perpendicular to one another in one direction and parallel in the other direction (magenta color in figure 2). It can be seen as a mixture of ferromagnetic and spin-ice orders.

 = 0°f ) e) d)  = 45° = 22.5°j ) i) h) g)
Fig. 2 | Experimental configuration and magnetic order analysis. a-c, Magnetic configurations deduced from Magnetic Force Microscopy (MFM) measurements for the three lattices orientations (resp. 0°, 22.°5 and 45°). The different domains corresponding to different magnetic order are colored (see g-j). White parts correspond to regions too small to be ascribed to particular magnetic order. d-f, Distribution of the different domain surfaces for the three lattices orientations (resp. 0°, 22.°5 and 45°). g-j, Schematics of the four magnetic orders: collinear antiferromagnetism (cyan), ferromagnetism (red), spin-ice (green) and zig-zag (magenta)

As evident from figure 2, whereas both the lattice and the spins are unchanged, the magnetization configurations depend drastically on their relative orientation 𝛼. For spin directions along the lattice directions (𝛼 = 0°), a large fraction of antiferromagnetic domain is observed (49% of the surface). The collinear antiferromagnetic order is easily understood by considering the dipolar interaction between the nearest neighbors and results from the anisotropic nature of the dipolar coupling. For spin oriented along the lattice diagonals (𝛼 = 45°), no antiferromagnetic domain is observed but spin ice domains are present (58% of the surface). Similar to the conventional square ice with Ising spins [START_REF] Wang | Artificial 'spin ice' in a geometrically frustrated lattice of nanoscale ferromagnetic islands[END_REF] this specific configuration is deduced from considerations of local interactions. For an intermediate angle, 𝛼 = 22.5°, large ferromagnetic domains are observed. This local order cannot be explained considering nearest-neighbor interactions (which would promote the two previous states). This is rather counter-intuitive as dipolar interactions in 2D and 3D systems generally destabilize ferromagnetism. A quantitative analysis of the dipolar four-state Potts model is thus necessary.

The dipolar four-state Potts model differs drastically from the usual Potts models (clock or standard [START_REF] Wu | The Potts model[END_REF]). Whereas the spin nature is the same within the dipolar four-state Potts model, the interactions between spins are anisotropic and cannot be described only by a function of spins values like cos(𝜃 𝑖 -𝜃 𝑗 ) or 𝛿 𝜃 𝑖 ,𝜃 𝑗 . The total Hamiltonian of the system can be expressed as:

𝐻 = 𝜇 0 . 𝑚 2 𝑁 ∑ 3(𝑺 𝒊 . 𝒆 𝒊𝒋 )(𝑺 𝒋 . 𝒆 𝒊𝒋 ) -𝑺 𝒊 . 𝑺 𝒋 𝑟 𝑖𝑗 3 𝑁 𝑖<𝑗
where 𝒆 𝒊𝒋 is a unit vector between sites 𝑖 and 𝑗, 𝑚 is the magnetic moment of the spins, and 𝑁 is the number of spins. To emphasize the specificity of the dipolar Potts model and its angular dependence, it is interesting to consider the interaction between two spins. For two Ising spins, this interaction results in two energy levels (figure 3.a) and, depending on the angle between the spin direction and their relative position (𝛼 = cos -1 𝑺 𝒊 . 𝒆 𝒊𝒋 ), the coupling is either ferromagnetic (𝛼 < 54.7°) or antiferomagnetic (𝛼 > 54.7°). For a given value of 𝛼, it is possible to describe the interaction by a single scalar 𝐽 𝑖𝑗 such that the energy writes -𝐽 𝑖𝑗 . 𝜎 𝑖 . 𝜎 𝑗 with 𝜎 𝑖,𝑗 = ±1. For the four-state dipolar Potts model, the 16 different possible configurations result in 6 different energy levels (figure 3.b) and the interaction cannot be described anymore by a single scalar. The lowest energy level is either parallel (𝛼 < 29.3°) or perpendicular (𝛼 > 29.3°) and can also be degenerate (𝛼 = 29.3°). For the case of the square lattice, the discussion is focused on the four typical configurations experimentally observed. Figure 3.c represents with dotted lines the energy of these configurations considering only the nearest-neighbor interactions (short range, 8 closest neighbors) for an infinite lattice (periodic boundary conditions). The collinear antiferromagnetic and the spin ice configurations appears to be the lowest energy configuration for 𝛼 inferior and superior to 22.5°, respectively. The ferromagnetic configuration (dotted red line of figure 3) observed predominantly at 22.5° is not the ground state of the short-range model. The same conclusions hold if only the 4 closest neighbors are considered, the difference between energy levels being larger. However, taking into account all longrange dipolar interactions (solid lines in figure 3.c,) leads to a reduction of the energy of the ferromagnetic state (solid red line of figure 3), which becomes the ground state for an angular range around 22.5°. On the other hand, taking into account all long-range dipolar interactions does not significantly affect the configurations with no net magnetization. Therefore, we find that the nature of the ground state depends strongly on the angle 𝛼 between spins and lattice. For spin directions close to the lattice directions (|α| < 13.2°), the lowest energy state is the collinear antiferromagnetic state. For spin directions close to 45° from the lattices directions (|α-45| < 13.2°), the lowest energy state is the ordered spin-ice state. For intermediate spin directions (13.2 < α < 31.8°), it is the ferromagnetic order that has the lowest energy state. Monte-Carlo simulations were used to confirm that these states are indeed the lowest energy ones for an infinite lattice. For finite lattice size, the same conclusions hold but there exist multi-domain configurations which further decreases the energy of ferromagnetic state. Note that the energy of the zigzag state is the average between the one of the ferromagnetic and spin-ice states, and the zig-zag state is the ordered state with energy just beyond the ground state for |α-45| < 22.6°. This theoretical analysis is based on the point dipole approximation. However, this approximation is questionable in our experimental case of nanomagnets with lateral extension not negligible compared to the lattice period (300 nm vs 500 nm). Therefore, we show in SI 3 with micromagnetic simulations that actually taking into account the magnetization distribution within the elements and their spatial extension, does not affect the previous conclusions.

From these main features of the dipolar four-state Potts model, it is possible to quantitatively analyze the experimental data. For the configurations represented in figure 2, the proportion of antiferromagnetic domains in the sampled area is 44% for 𝛼 = 0°, the proportion of ferromagnetic domains is 73% for 𝛼 = 22.5°, and the proportion of spin-ice domains is 64% for 𝛼 = 45°. These proportions are significant and well above the statistical noise level (which is below 8% for the sample size investigated here) and correspond to large ground-state domains. These experimental results are in agreement with our theoretical predictions.

The general behavior of the presented breed of artificial spin system is, therefore, well-explained within our theoretical framework. Beyond this analysis, as in many systems, the domain (or crystallite) structure reveals underlying mechanisms. It appears that the thermal procedure we used did not drive all the spins into a superparamagnetic state, as reminiscence of the initial state (saturated) are clearly observed for 𝛼 = 0 and 22.5°. This reminiscence is due to the low energy of this ferromagnetic configuration and not to properties of individual spins. For instance, in the case of 𝛼 = 22.5°, the proportion of ferromagnetic domains drops from 73% to 20% when the period of the lattice is increased from 500 to 700 nm, which corresponds to the reduction of the dipolar interactions by a factor of 2.7 (see SI4). The high temperature dynamics is also revealed by the orientation of the configurations. For 𝛼 = 0° (figure 2.a), the antiferromagnetic domains are mostly aligned perpendicularly to the initial saturation direction. For 𝛼 = 45°, the size of the spin-ice domains is limited by the number of spins opposite to the initial saturation direction, which is quite small at 12%. This indicates that the reversal of spins proceeds by a 90° rather than 180° rotation. The presented thermal procedure is therefore not fully efficient. The system does not explore the entire energy landscape and is trapped in a local energy minimum. Nevertheless, the system can be driven in domains of the expected low energy configurations. It is interesting to compare the energy scales of the four-state dipolar Potts model at 45° from a square lattice and the most studied square Ising ice. The ground state is identical for both systems (type I vertices). It can be reached almost perfectly in the square Ising ice via a superparamagnetic regime [START_REF] Farhan | Direct observation of thermal relaxation in artificial spin ice[END_REF]. But the energy difference between the two lowest energy states (type I and type II vertexes) is about 10 times the one between the two lowest energy states of the Potts system (considering a dipolar approximation). Achieving a perfect thermalization of the dipolar Potts system remains a challenge.

In conclusion, the dipolar four-state Potts square lattice is a versatile system; since a relative rotation between spin and lattice directions gives rise to very different properties and particularly different ground states. Interestingly, at α= 22.5°, a ferromagnetic phase originates only from dipolar interactions. This is a striking result since ferromagnetism is generally associated with short-range exchange interactions in a 2-or 3-dimensional systems, whereas long range dipolar interactions tend to decrease the net magnetic moment. The four state dipolar Potts model, therefore, adds up to the short list of 2D systems exhibiting dipolar ferromagnetism [START_REF] Puntes | Collective behaviour in twodimensional cobalt nanoparticle assemblies observed by magnetic force microscopy[END_REF][START_REF] Yamamoto | Direct visualization of dipolar ferromagnetic domain structures in Co nanoparticle monolayers by electron holography[END_REF]. The spin-ice phase is an alternative realization of artificial spin ice. Its ground state is identical to the square ice proposed by Wang et al. [START_REF] Wang | Artificial 'spin ice' in a geometrically frustrated lattice of nanoscale ferromagnetic islands[END_REF], following a two in / two out ice rule (type 1 vertex), but the number of different vertex type is much higher, being 15 instead of 4. More generally, the four-state dipolar Potts model offers an opportunity to precisely tune (via a geometrical parameter) the energy levels of the system.

At 𝛼 = 31.8° the ground state is degenerated with several ordered phases having the same energy.

The work presented here can be viewed as the first experimental realization of the dipolar four-state Potts model with artificial spins made of nanomagnets, which shows that main features of the theoretical model can be probed experimentally. We believe that this concept can be applied more generally to a host of other lattice types.

Beyond these fundamental aspects, we have shown that the system behaves like a magnetic metamaterial having an easily tunable magnetic order. Our experimental model system is based on thin epitaxial films but the concept can be extended with other nano-objects exhibiting a cubic anisotropy. Nano-objects with shape-driven anisotropy defined by lithography are interesting but bottom-up approaches relying on chemical synthesis are very promising [START_REF] Sun | Monodisperse FePt nanoparticles and ferromagnetic FePt nanocrystal superlattices[END_REF][START_REF] Yang | Controlled Synthesis and Assembly into Anisotropy Arrays of Magnetic Cobalt-Substituted Magnetite Nanocubes[END_REF]

  Figures 1. d, e, f represent the different geometries studied, defined by 𝛼, the angle between spins, and lattice direction; 𝛼 taking values equal to 0°, 22.5° and 45°.

Fig. 1 |

 1 Fig. 1 | Presentation of the system and geometry definition. a, schematic diagram of the artificial spin system. b, Atomic Force Microscopy (AFM) image of a portion of the lattice. c, Corresponding Magnetic Force Microscopy (MFM) image. The zoom indicates the correspondence between the MFM contrast and the spin value. d-f, AFM images describing the three

Figures 2 .

 2 Figures 2.(a,b,c) represent the magnetic configurations obtained after thermal treatment for the

Fig. 3 |

 3 Fig.3| Energy angular dependence. a,b, Energy levels for two Ising spins as a function of the angle made between the spins and the direction between them, 2 possible configurations result in 2 energy levels (a) and two Potts spins, 16 possible configurations result in 6 energy levels (b). c, Energy per spin for ferromagnetic state (red), collinear antiferromagnetic state (blue), spin-ice state (green) and zigzag state (magenta). Dotted lines correspond to first neighbor interactions and continuous lines to all interactions. All energies are normalized to 𝝁 𝟎 . 𝒎 𝟐 𝒂 𝟑 ⁄ , 𝝁 𝟎 being the vacuum permeability, 𝒎 the magnetic moment of an element and 𝒂 the closest distance between spins.

Acknowledgements

The authors want to thank N. Rougemaille and B. Canals for fruitful discussions. This work was supported by the Agence Nationale de la Recherche through project number ANR12-BS04-009 'Frustrated' and partially supported by the French PIA project "Lorraine Université d'Excellence" ANR15-IDEX-04-LUE.

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Author contributions

Competing interests

The authors declare no competing financial interest

Methods

Sample fabrication

The MgO//V(50 nm)/Fe (2 nm)/V(4 nm)/Au(4 nm) multilayer were grown by Molecular Beam Epitaxy as described in [START_REF] Louis | Interfaces anisotropy in single crystal V/Fe/V trilayer[END_REF]. The magnetic elements were patterned by ebeam lithography (P(MAA-MMA)/PMMA bilayer resist) and lift-off of an aluminum film (60 nm). The pattern was transferred by Ion Beam Etching down to the vanadium buffer layer. To reduce the interaction between the MFM tip and the magnetic elements, a 10 nm aluminum film was deposited over the entire surface by sputtering.

Thermal treatment and Magnetic Force Microscopy

The sample was heated at 350°C for 5 hours in vacuum. The MFM measurements were done with a lift height of about 25 nm on a Dimension 3100 nano-scope equipped with Quadrex module for phase detection. Asylum low-moment tips were used to minimize the sample-tip magnetic interactions.


A B Stract

Sarra Dahmani
email: dahmani@emse.fr

Xavier Boucher
email: boucher@emse.fr

Sophie Peillon
email: peillon@emse.fr

Didier Gourc
email: didier.gourc@mines-albi.fr

François Marmier
email: francois.marmier@mines-albi.fr

THE MANUFACTURER'S SERVITIZATION PROCESS: A PROPOSAL FOR A DECISION-MAKING MODELING FRAMEWORK AND DIAGNOSIS

Keywords: Servitization, decision process, reliability, risk occurrence

Servitization can provide significant economic opportunities, but its implementation in the industry remains surrounded by much uncertainty. This is due to the complexity of its decision making process. This research work focuses on decision making process followed by one industry during a servitization transition. The paper aims to introduce a framework of risk occurrence anticipation and analysis within servitization decision process. This is based on an innovative approach of decision reliability diagnosis. For this, we adopt a methodology typical of enterprise modeling and diagnosis domain, illustrated by a case study.

INTRODUCTION

Servitization and product service systems (PSS) concepts and applications have spread during the last decades in the academic and practitioner communities [START_REF] Hou | Barriers of Servitization: Results of a Systematic Literature Review[END_REF]. In many advanced economies, servitization is thought as a development approach able to provide opportunities to achieve sustainability, improve enterprise competitiveness, and better satisfy customer needs (Vandermer and Rada, 1998). Nevertheless, this paradigm shift requires questioning the business objectives and overall functioning. PSS are not based only on technical aspects, organizational aspects are also involved, which make the implementation of this concept in businesses more difficult [START_REF] Cook | The transfer and application of product service systems. From academia to UK manufacturing firms[END_REF]. Thereby, transition from product manufacturer into service provider constitutes a risky managerial challenge [START_REF] Oliva | Managing the Transition from Products to Services[END_REF][START_REF] Nudurupati | Eight Challenges of the Servitization[END_REF]. It involves the company in a dynamic and complex decision-making process. Therefore, it becomes necessary to provide decision makers with tools to manage the process, and to anticipate associated risks. This research work introduces a modeling framework of servitization decision making process, and decision reliability diagnosis. The purpose of the paper is to propose a method for risk occurrence anticipation in a servitization transition, to help decision-maker controlling the process. This paper consists of three main parts. First, we introduce the basics of the servitization process, risk analysis, and reliability notion. In the second part, we explain the decision modelling and the reliability assessment approach we propose. Finally, the aim of the third part is to illustrate the diagnosis approach and risk interpretation according to a case study of a French firm.

SERVITIZATION PROCESS, RISK ANALYSIS AND DECISION RELIABILITY

Servitization decision process

Servitization can be understood and formalized as a complex process to make enterprise transformation [START_REF] Oliva | Managing the Transition from Products to Services[END_REF]. Servitization leads to rethink the enterprise strategy, internal processes and competencies as well as its external networks. On the basis of a bibliographic analysis [START_REF] Oliva | Managing the Transition from Products to Services[END_REF][START_REF] Baines | Towards an operations strategy for product-centric servitization[END_REF]Gebauer and al,2012) and experience feedbacks from different leaders of industrial companies in a servitization transition, the global servitization process can be decomposed in 3 decisional issues which cover key dimensions of the business transformation [START_REF] Dahmani | Industrial transition through Product-Service Systems: proposal of a decision-process modeling framework[END_REF]: 1. The product service system (PSS) technical design; 2. The PSS business model transformation; and 3. The organizational changes, required to support the PSS implementation. Each of these decisional issues contributes to redesign the positioning of the firm in its ecosystem. Indeed, the first decision issue emphasizes services' intangible nature [START_REF] Baines | Towards an operations strategy for product-centric servitization[END_REF]. The second issue focuses on the importance of predicting the market behavior vis-à-vis this new offering (Vandermerwe and al, 1988). And, the third issue considers the importance of adapting organizational structures and processes to ensure congruence between the firm's resources and objectives [START_REF] Gebauer | Customer-focused and service-focused orientation in organizational structures[END_REF].

We define servitization as a transition decision-making process, which generates information and knowledge progressively through temporal sequences according to each specific context. This is the system adopted by an organization to move from a current economic model (product-oriented offer) towards a servitized economic model (integrated product and service offer). This decision-making process is decomposed in three decision Macroprocesses (MP) according to servitization decisional issues underlined: MP1: the product service system (PSS) technical design; MP2: the PSS business model transformation; MP3: the Organizational changes, required to support PSS implementation.

Coordinating these three decisional issues simultaneously can be problematic for the decision-maker; he has to take into account the complexity of the process and its dynamic evolution over time. This transition is then considered risky and full of uncertainties for the decision-maker. Then, preparing a servitization process may be compared to a project planning for the manufacturer. He will need a planning tool in order to control the process and to minimize uncertainties and attached risks.

Risk analysis in servitization process

The term risk is used in a wide range of meanings. According to literature, risk in business can mean either a feared event, or the probability of occurrence of the event, or its harmful consequences [START_REF] Courtot | La gestion des risques dans les projets[END_REF]. Project risk often refers to any departure from the planned objectives related to performance expectations [START_REF] Girard | Vers un modèle général du risque pour pilotage et la conduite des activités de biens et de services[END_REF]. According to a quantitative approach, risk refers to the exposure to loss/gain, or the probability of occurrence of loss/gain multiplied by its respective magnitude (Jaafari,200). Risk therefore can be defined as "an uncertain event which, occurrence would have an effect on achieving the objectives" [START_REF] Schmitt | Quality-Oriented Risk-Management-Approach for Service Innovations[END_REF]. Risks can be classified in several ways. It depends on the specificities of the project, the company and the ecosystem [START_REF] Schmitt | Quality-Oriented Risk-Management-Approach for Service Innovations[END_REF].

Generally in risk analysis modeling, it's important to differentiate risk factors from risk impacts. According to a reference model we adopt (Gourc, 2006), every risky situation is due to a combination of causes (origins) brought together in an occurrence area, and generates a set of impacts (consequences) brought together in an impact area. The risk event represents the intersection between both of the distinct areas. For a manufacturing company, servitization represents a transforming project which affects all strategic pillars of the business. Literature often emphasizes internal and external barriers faced by a firm in servitization process [START_REF] Mont | Product-Service Systems: Panacea or Myth?[END_REF][START_REF] Hou | Barriers of Servitization: Results of a Systematic Literature Review[END_REF]. Servitization barriers can be related to value creation issues [START_REF] Baines | Towards an operations strategy for product-centric servitization[END_REF]. Risk related to service is mainly due to coordinating issues between business ability and market expectations on one hand and organizational strategy, design and development on the other hand [START_REF] Sawhney | Creating growth with services[END_REF], service types can also affect considerably a servitization performance [START_REF] Benedettini | Service Types and their Differential Risk Effects for Manufacturing Firms: an Empirical Analysis[END_REF]. Studies oriented towards risk assessment for service innovations attempt to quantify risk occurrence probability and extent of losses in risk matrices, in order to propose risk management plans (schmitt and al, 2013). We consider risk in servitization decision process as the possibility for a decision-maker to fall in a risky decisionmaking position, which occurrence can lead to negative consequences for conducting the process as well as for the planned objectives.

In this paper we introduce a risk analysis approach for servitization decision process focused on risk occurrence domain. We propose a diagnosis method for the decision process of a pivot firm to deduce decision areas carrying potential occurrence of risk. For this, we need to evaluate the decision process accomplished by the firm. We proceed then to evaluate the decision reliability of servitization process.

Notion of decision reliability

To define the concept of decision reliability, we refer to Simon's researches on procedural rationality. In this approach, Simon has identified procedural rationality as an important information processing and decision-making approach. Procedural rationality is "problem solving by recognition, by heuristic search, and by pattern recognition and extrapolation [...]. They are not optimizing techniques, but methods for arriving at satisfactory solutions with modest amounts of computation" [START_REF] Simon | Invariants of human behavior[END_REF]. [START_REF] Dean | Does decision process matter? A study of strategic decision-making effectiveness[END_REF] redefined later procedural rationality as "the extent to which the decision process involves the collection of information relevant to the decision, and the reliance upon analysis of this information in making the choice" [START_REF] Riedl | Reducing uncertainty in supplier selection decisions: Antecedents and outcomes of procedural rationality[END_REF].

In the context of servitization decision process, we evaluate decision reliability according to the concept of procedural rationality. Our aim is to evaluate the procedural rationality of the decision maker through evaluating the reliability of the decisions made; our final purpose is to identify the least reliable areas of the decision process.

To evaluate the reliability of decisions we consider a theoretical reference model (i.e. a reference model providing a representation of the whole servitization decisional process) which reflects a "reference optimal procedural rationality", and which is complete and consistent. Thus, the decision reliability is considered here as an estimator of the proximity between (i) a reference decisionmaking process known and modeled a priori and (ii) an effective decision-making process, followed by decisionmakers according to a real case study.

SERVITIZATION DECISION PROCESS MODELING AND RELIABILITY ASSESSMENT

In this section we introduce the formalism used to represent servitization decision process. The diagnosis approach we propose is based on this formalism. This model has been fully explained in [START_REF] Dahmani | Industrial transition through Product-Service Systems: proposal of a decision-process modeling framework[END_REF] we just provide here a short synthesis.

Decision process modeling

We propose a model that considers the complexity of servitization decision process. This approach is inspired by the GRAI modeling formalism [START_REF] Doumeingts | Production management and enterprise modelling[END_REF] and is represented through servitization grid (in figure1).

Figure 1: Servitization reference decision model

To build this model, we represent decision making process through two axes. The vertical axis represents the decision's horizons: long, medium and short terms. This aspect of the decision clarifies crossing from high strategicdecision level to tactical and operational ones. The horizontal axis is related to three decision macro-processes described above: MP1, MP2, and MP3. The intersection between a decision macro process and a decision horizon represents a decision center (DC). The model results in a matrix containing 12 decision centers (figure1). Every DC consists of decision activities (DA) nets, and every DA is represented through critical characteristics, which are differentiated according to the type of the decision activity considered. Two generic types of decision activities are distinguished: decisional activities (noted D-DA, with outputs constituted by decision choices), or execution activity (noted E-DA with outputs constituted by simple informations). This model allows understanding the complexity of the overall transition process and identifying different interaction within the system. In addition, the servitization grid makes it possible to differentiate the granularity level of the decision process from strategic to operational level and from general to particular within each DC.

Definition, formalization and evaluation of decision reliability

We have defined reliability as the proximity between reference decision making process and effective one. To estimate this proximity, the proposed approach consists in:

• Characterizing in detail the actual servitization process followed by the firm according to decision-activities (DA) of the reference model;

• Assessing qualitatively whether the different features of the reference model are present or not in the actual decision making process;

• Aggregating the first level of assessment to measure the decision-making reliability at decision activities and decision centers of the grid.

As specified above, servitization decision activities are formalized using GRAI modeling formalism and include compounds of decision activities (D-DA and E-DA). In order to build the proximity measures mentioned, DA are modeled in detail through a set of "decision-making characteristics" and "decision attributes", each attribute is associated with a "reliability coefficient". Thus the conceptual modeling of the reference DA is as follows:

D-DA k , E-DA k = { C i } i=1 to n with C i = decision making reference characteristics C i = {name, description, {A j , Coef j } j=1 to m) with A j = decision attributes Coef j = Reliability coefficient associated to A j Referring to GRAI method, we describe decision activities (D-DA) by seven reference characteristics (Table 1) and execution activities (E-DA) only by five characteristics. The specific attributes of each characteristic have been identified in the reference model by detailed analysis of each specific activity of the servitization model. Table .1 provides example of characteristics and attributes for one D-DA Є PT2 (BM, MT) "Developing the internal value chain".

Table 1: D-DA 1 Є PT2 (BM, MT) "Developing the internal value chain" (reference model)

The construction of these descriptive detailed tables allows determining "reliability coefficients", which are fixed a priori for each attribute. These coefficients are determined in order to estimate an optimal reliability for each decision activity equal to 1 when effective decision activity satisfies correctly all attributes of the reference decision activity.

To determine these coefficients for each decision activity, the optimal reliability of 1 is equidistributed on the different characteristics Ci: this distribution provides a reliability coefficient of 1/7 for each C i of D-DA and 1/5 for each C i of E-DA. This coefficient is itself equidistributed among all the attributes (A j ) components of the "decision-making characteristic", which represents the "reliability coefficient".

In this paper, we choose an equidistribution for coefficients among different characteristics, and attributes within one characteristic in order to simplify the analysis process. We attribute similar importance to all components of the decision making process to illustrate the overall approach, before adding a specific part to explain the weightings. Thus, for D-DA 1 Є PT2 (BM, MT) "Developing the internal value chain", there is 17 attributes, each is characterized by its reliability coefficient:

D-DA 1 = {C 1 , C 2 , C 3 , C 4 , C 5 , C 6 , C 7 } C 1 =
{input, description of incoming information from the decision-making activity, {(A 1 = components of the internal value chain, Coef 1 = 1/14), (A 2 = characteristics of the internal value chain, Coef 2 = 1/14)}

We use then the theoretical modeling of decision activities and reliability coefficients to estimate the proximity between the reference decision process and the actual one. This assessment requires collecting information from the involved decision makers, to describe quite precisely how they decide: first identify whether each of the decision activities of the servitization reference model was performed or not, then in more details if each of the attributes characterizing these activities is present or not in the actual process followed by the decision maker. We represent this measure through a binary indicator of presence (1) or absence (0) for every "decision reference attributes" in the effective decision making process of the firm: the proximity between the effective decision process and the reference one is estimated according to the absence or presence of these attributes.

To formalize this notion, it is necessary to complete the conceptual model of the decision-making activity, first by adding an index of presence/absence for each decision attribute, and a reliability estimator F i for the decision activity.

For an effective decision-making process, the activity is described by: D-DA k , E-DA k = {(C i ) = 1 to n , F k } with F k = reliability estimator of the DA k ; C i = {name, description, (A j , Coef j , Ind j ) j = 1 to k }, with Ind j = presence / absence index for A j; with Ind j = 1 if A j is present; 0 otherwise.

The decision reliability F k for D-DA k activity is obtained by aggregating a sum of the reliabilities coefficients, taking into account the index of presence / absence for each of them:

F k = ∑ ( ) F k =0 if no
decision attribute is present; It is a lack of decision activity that should be detected in advance. F k =1 if all decision reference attributes are present in the actual decision-making process; It corresponds to an optimal situation when the decision maker proceeds by following fairly the reference decision process.

SERVITIZATION DECISION PROCESS DIAGNOSIS APPROACH: ILLUSTRATION ON A CASE STUDY

Industrial case study: General presentation

To illustrate our approach, we propose an application on a case study of a French SME named Ecobel. The main activity of the firm is manufacturing, sale and installation of shower heads based on an innovative technique that allows water savings and protection from legionella. Its current market considers establishments receiving general public like hospitals, campsites clubs…etc. Ecobel is planning to propose a service oriented offer, so it has initiated a debate on the implementation of servitization. This case study has aroused our interest for its positioning in the transition process. Ecobel currently offers two models simultaneously: the classic range selling only the showerhead product and the integrated PSS offer selling reliable showerheads over 5 years. The PSS offer includes service contracts for regular maintenance and periodic exchange of the showerhead product with a visual identification. Ecobel's leader highlights the difficulty of commercializing it. He remains cautious about the development of the servitization model over the entire range

The servitization decision process diagnosis approach according to a case study

In the previous section, we proposed an approach of formalization and evaluation of the decision reliability concept. Systemic vision shows that the reliability of any system depends on the reliability of its components, and their relationships among others. We suppose that an "unreliable" decision activity carries a potential occurrence of risk to the whole process which can cause negative losses for the firm.

We propose to carry out the reliability diagnosis for servitization decision process according to two complementary points:  Reliability assessment on decision activities with the interest of submitting a macroscopic view of the process;  Aggregated reliability assessment on decision centers with the interest of explaining weaknesses through a detailed analysis of DA.

Reliability assessment and diagnosis on decision activities

This analysis viewpoint aims to present a general state of servitization process accomplished by Ecobel at one moment. All decision activities are presented in the radar axes through a numbering from 1 to 48 (Figure 2). The advantage of this analysis viewpoint is to provide a macroscopic mapping of the reliability of the process according to the reliabilities of all DA. The contribution is to deduce a global mapping of potential occurrence of risk. Then, we proceed to classify the DA of the process at different reliability intervals as shown in the three first columns of Table 3. The intervals are predetermined. Since we focus on the planning phase of an innovative process, we can only evaluate a potential occurrence of a risk, instead of the classic probability of occurrence. Then we consider potential occurrence of risk inversely proportional to decision reliability. We interpret the reliability intervals according to a qualitative scale to evaluate the potential occurrence of risk which varies from "very high" to "low" as shown in the fourth column of Table 3. We can see that 31% of the process is caring a considerably high potential occurrence of risk which will impact the process advancement and the firm objectives negatively.

Reliability assessment and diagnosis on decision centers:

This analysis perspective is about to provide the decision maker with a reliability estimation at each decision center of the grid. For that purpose, we provide the decision-maker two useful indicators:  = Estimates the reliability of the decision center for the decision horizon i and Macro process j. This estimate is provided by the lowest reliability F k for DA k among all decision activities of a DC ij .

( )

CONCLUSION

The purpose of this study is to introduce a diagnosis approach for servitization process planning, which is based on risk occurrence anticipation. The diagnosis allows highlighting weaknesses of decision areas in the accomplished decision process. We consider these weaknesses as carrying a potential occurrence of risk. The main perspective of the study is to establish a global model that considers simultaneously potential risk occurrence and risk impacts, in order to prioritize the decision centers to deal with for the decision maker. This would facilitate controlling the servitization transition, and limiting time and effort losses. We recognize the limits of the study focusing only on the reliability of the decision process, the intuitive side of the decision making process is not taken into account in this work.

Table 3 : Distribution of reliabilities among intervals Figure 2: Reliability assessment on decision activities

 3

	Reliability	interpretations	Proportion of DA	Potential occurrence of
	intervals		in the servitization	risk interpretation
			process
	0	DA unreliable	6.25%
	] 0 ; 0.3]	DA with very	10.42%	DA carrying very high
		low reliability		potential occurrence of
				risk
	] 0.3 ; 0.6] DA with low	14.58%	DA carrying high
		reliability		potential occurrence of
				risk
	] 0.6 ; 0.9]	DA with	60.42%	DA carrying average
		average		potential occurrence of
		reliability		risk
	] 0.9 ; 1]	reliable DA	8.33%	DA carrying very low
				potential occurrence of
				risk

ACKNOWLEDGEMENTS

We would like to thank all the members of the Servinnov ANR project for their support and insightful comments, and Mr. Bosles (Ecobel's leader) for his time and for making his experience available to us in our research.

 N ij; F0 = Number of decision activities not taken into account by the decision maker within a DC ij .

The results interpretation in this analysis perspective is based on arbitration that takes into account simultaneously both of indicators ( and N ij; F0 ), and thus prioritizes DC on which decision-maker must act according to the importance of the potential occurrence of risk revealed by the indicators. We proceed to the classification of all DC according to the pre-established reliability intervals and potential risk occurrence interpretations, and also emphasizing the DC with N ij; F0 > 0. This classification shows that half of DC in the decision-making process accomplished by Ecobel are carrying a significantly high potential occurrence of risk. DC with N ij; F0 >0 are considered as unreliable. We consider these DC as the most prioritized DC for the remediation plan.

From this distribution, we proceed to prioritize the DC to deal with (in descending order of potential occurrence of risk), and then to detail the analysis of each DC in terms of DA. Then we can draw three kinds of general conclusions / remediation according to the predefined reliability intervals for DC and to the importance given to each DA:

-First, for unreliable DC, which include ignored DA, remediation would be turned to awareness raising and training effort for the decision maker for the ignored issues. This may have significant impact on the long-term performance of the firm.

-Then, for unreliable and very unreliable DC, they show that DA were actually treated, but decision maker referred often to an intuitive decision-making process strongly influenced by the initial organizational model. Then, in order to improve the reliability of these DA, remediation would demand efforts of analysis, anticipation and especially of changing dominant decision models in the business.

-Finally, for DC with average reliability, they show that their belonging DA have been processed and analyzed, but the problem lies in the lack of resources to carry out these decisions. The remediation effort will be directed more towards an allocation of resources in terms of competencies and investment to provide additional information necessary for these decisions. It's also important to consider the positioning of the prioritized DC on the grid. Actions plans should take into account the concerned decision horizon and MP in order to keep a global consistency of the process.