Gildas Morvan

Daniel Dupont
email: daniel.dupont@hei.fr

Philippe Kubiak Lgi2a -Ea
email: philippe.kubiak@ig2i.fr@fsa.univ-artois.fr

A SIMULATION-BASED MODEL OF ABDUCTION

Keywords: simulation-based reasoning, abduction, forensic entomology

Abduction, or Inference to the Best Explanation (IBE), is a reasoning process that generates possible explanations from a set of "surprising" observations. In this paper, a simulation-based model of abduction is introduced. This model is then implemented to develop a decision support system in the field of Forensic Entomology to help forensic scientists solve complex cases.

INTRODUCTION

Abduction, or Inference to the Best Explanation (IBE), is a reasoning process that generates possible explanations from a set of "surprising" observations. It has been widely studied in Philosophy and later on, in Artificial Intelligence (AI). The literature on this subject is so vast that a whole book would not be enough to undertake a complete survey. Therefore, only the key aspects of some of the abduction models developed in AI, set cover and logic based approaches, are briefly introduced. However readers may refer to [START_REF] Paul | Approaches to Abductive Reasoning[END_REF] for a more comprehensive presentation. Set cover based approaches define an abductive framework as a triplet < Φ, Ω, e > where Φ is a set of hypotheses, Ω is a set of manifestations, and e is a mapping from 2 Φ to 2 Ω . The abductive problem is then defined as follows: Let Ω ⋆ ⊆ Ω be a set of observed manifestations. The problem is to determine a minimal set Φ ⋆ ⊆ Φ such as e(Φ ⋆ ) = Ω ⋆ . Logic based approaches assume that domain knowledge is encapsulated in a theory T defined over a language L. Let A, a set of sentences of L, be a set of hypotheses. Let the sentence ω be a surprising observation (i.e., T ω). φ is an explanation of ω iff T ∪ φ ⊢ ω, T ∪ φ is consistent and φ ∈ A.

Existing abduction AI models rely on knowledge models structured a specific language, defining truth values for propositions or relations between facts. However, the knowledge about a given system can be encapsulated in many types of models, e.g., a computational model. The abduction model presented in this paper does not assume an a priori knowledge of causality relations nor a specific syntax for the knowledge base, but assumes that: (1) the set of possible explanations (hypotheses) can be defined as a metric spacei.e., distance measures can be computed between all the hypotheses; (2) it exists a deductive (predictive) model able, e.g., by simulating the causal history of the system, to compute a coherence measure between a hypothesis and a set of observations. Then, this model is applied to develop a decision support system in the field of Forensic Entomology, using agent-based simulations. Experimental results show that using this model to solve complex cases increases the efficiency and the precision of the results.

A SIMULATION-BASED MODEL OF AB-DUCTION

Definitions

Let Ω be a set of observations and Φ a set of hypotheses i.e., possible explanations -for Ω. We assume that a metric space (Φ, d Φ ) can be defined. A coherence measure between an hypothesis φ of Φ and Ω is computed by a deductive (predictive) model m : Φ × Ω → [0, 1]. This coherence measure is denoted c m,φ,Ω .

Definition 1 A hypothesis is called validated iff its coherence measure is known. Let Φ v ⊆ Φ be the set of validated hypotheses and R m,Φ v ,Ω : Φ × [0, 1] be the solution set computed by a model m:

R m,Φ v ,Ω = ∀φ∈Φ v (φ, c m,φ,Ω ).
(1)

Similarly, a metric space (R m,Φ v ,Ω , d R ) is defined. An estimated coherence measure c m,φ,Ω is associated to all φ in Φ-Φ v . This measure is estimated from R m,Φ v ,Ω
by interpolation. The resulting estimated solution set is defined as follows:

R m,Φ,Ω = ∀φ∈Φ (φ, c m,φ,Ω ) if φ ∈ Φ v (φ, c m,φ,Ω ) otherwise. ( 2
)
Goal g Definition pr.

Determine the most probable hypotheses el.

Eliminate unprobable hypotheses prs.

Determine c m,φ,Ω for probable hypotheses id.

Determine c m,φ,Ω for every hypotheses Table 1: Goals of an abductive task For convenience, in the rest of the paper, metric spaces will be denoted as their respective sets. During the abductive process, hypotheses to be validated are chosen iteratively by a heuristic

γ : R m,Φ,Ω → Φ -Φ v .
(3)

Abduction goals

It is generally accepted that abduction targets the best explanation of a given set of observations. However, in the context of a decision support system, this definition should be expanded. In the table 1, four different goals for an abductive task are defined. Thus, in its most general definition, abduction can be seen as an identification task (goal id.).

Quality measures can be defined to determine how goals have been achieved. A quality measure, q m,g , based on the root mean square deviation (RMSD) between R m,Φ,Ω and R m,Φ v ,Ω is defined for each goal g of the table 1. E.g., for the goal id. the quality measure is defined as follows:

q m,id. = P ∀φ∈R m,Φ,Ω ( c m,φ,Ω -c m,φ,Ω ) 2 |Φ| . (4)
For the other goals, subsets of R m,Φ,Ω and R m,Φ v ,Ω , meeting their defining conditione.g., for the goal prs., subsets are defined such as, for any φ in R m,Φ,Ω , c m,φ,Ω > 0 -are used.

Knowing the result set is mandatory to use these quality measures. Thus, they are useful to evaluate an implementation of the algorithm -and especially the heuristic γ -, not a particular solution.

Resolution algorithm

The algorithm 1 computes an estimated solution set R m,Φ v ,Ω using a deductive model m. One could argue that it is not an abductive algorithm as it does not return the best explanations for Ω. Nevertheless, as it explains possible causal histories of Ω, it is, in some ways, similar to the abductive framework developed in [START_REF] Lipton | Inference to the Best Explanation[END_REF]ch. 4). Moreover, the solution set can be used as an input for post-processing algorithms, e.g. to merge solution sets obtained with different deductive models, and then determine the best explanation. Halting of the algorithm 1 depends on a condition C. In the implementation presented in the next section, the following condition is used:

∀r∈R m,Φ v ,Ω cost(r) ≤ maxCost, (5)
where cost represents the cost necessary to select and validate an hypothesis and maxCost is the maximal cost allowed by the user to compute the solution. Of course the algorithm halts if all the φ in Φ have been validated. Using a condition of the type

c m,φ,Ω ≥ µ, (6)
where µ is a validity threshold, the algorithm would be a form of the hypothetico-deductive model [START_REF] Lipton | Inference to the Best Explanation[END_REF]ch. 4).

Algorithm 1 Inquisitive abduction algorithm

Input: A set of observations Ω Input: A halting condition C Input: A heuristic γ Output: A solution set R m,Φ v ,Ω 1: Determine Φ 2: Φ v ← ∅ 3: R m,Φ v ,Ω ← ∅ 4: compute R m,Φ,Ω 5: while not (C or (Φ = Φ v )) do 6: φ ← γ( R m,Φ,Ω ) 7: c m,φ,Ω ← m(φ, Ω) 8: Φ v ← Φ v ∪ φ 9: R m,Φ v ,Ω ← R m,Φ v ,Ω ∪ (φ, c m,φ,Ω ) 10: compute R m,Φ,Ω 11: end while 12: return R m,Φ v ,Ω
In the worst case, i.e., the halting condition C stays false during the entire execution, this algorithm needs |Φ| steps to halt and return R m,Φ,Ω , i.e., the exact solution set. However, the complexity of the algorithm depends mainly on the complexity of the heuristic and on the solution set estimation algorithm. This algorithm is very general; thus, it does not refer to the goal of the abductive task; this is handled by the heuristic γ. In the next section, elements for constructing such heuristics are presented. We focus on the goals el., prs. and id.: for the goal pr., classical optimisation heuristics can be used efficiently. Moreover, we target cases where the hypothesis set is not very big but the cost of validation of a hypothesis is important.

HEURISTIC DEFINITIONS Definition of the criteria

Many criteria that help an agent to choose the best hypothesis to validate have been identified in the literature. Among all of these, simplicity seems to have been the most used by computer scientists. Simplicity has been interpreted as logical simplicity. But it seems obvious that for a cognitive agent, simplicity is a little more complex [START_REF] Aliseda-Llera | Seeking Explanations: Abduction in Logic[END_REF][START_REF] Paul | Approaches to Abductive Reasoning[END_REF]). In the model presented in this paper, simplicity is defined as follows:

Definition 2 A hypothesis φ is simple if it exists a retrodictive model m -1 such as m -1 (Ω) = φ. More- over, if Φ is bounded, φ should be considered as simple if φ = inf (Φ) or φ = sup(Φ).
The simplicity criterion is used to preprocess the problem. Thus, all the hypotheses defined as simple will be primarily chosen and validated. Other criteria such as cost or utility have been quoted and should be used to handle the choice of a hypothesis in an abductive task [START_REF] Peirce | Collected papers of Charles Sanders Peirce[END_REF][START_REF] Mcgrew | Confirmation, Heuristics, and Explanatory Reasoning[END_REF]. The cost criterion can be defined informally as follows: the selected hypothesis should be, ceteris paribus, the one that minimises the cost of validation. This criterion should be used if it is possible to estimate empirically a cost function of the model m, cost m : Φ → R. Let u c : Φ → [0, 1] be a utility function of the cost criterion. For any φ in Φ

u c (φ) = 1 - cost m (φ) max(cost m ) . ( 7
)
The utility criterion can be interpreted in different ways.

In this approach, it could be defined as follows: the selected hypothesis should be, ceteris paribus, the one which maximises the knowledge of the agent. It means that the chosen hypothesis should be the one which best improves the quality of the solution -relatively to a quality measure. Of course in a real world problem, such an hypothesis as well as its effect on the quality measure cannot be determined with certainty. However, the metric of Φ or R m,Φ v ,Ω can be used to estimate a utility function for this criterion. First, consecutivity is defined:

Definition 3 Let (E,
The notion of maximal distance between two consecutive elements in a metric space is then defined as follows: Two different utility functions of the utility criterion can then be defined using the metrics of Φ -equation 9 -and R m,Φ,Ω -equation 10. Let u Φ u , u R u : Φ → [0, 1] be two utility functions of the utility criterion. For any φ of Φ

u Φ u (φ) = 0 if φ ∈ Φ v d Φ (φ,φ1) d Φ max (Φ v ) otherwise, (9)
where the hypothesis φ 1 = φ is the closest to φ in Φ v ∪ {φ}, i.e., there is no

φ 2 = φ in Φ v ∪ {φ} such as d Φ (φ, φ 2 ) < d Φ (φ, φ 1 ). For any r = (φ, c m,φ,Ω ) of R m,Φ,Ω -c m,φ,Ω may be unknown u R u (φ) = 0 if φ ∈ Φ v d R (r,r1) d R max (R m,Φ v ,Ω ) otherwise, ( 10
)
where the result r 1 = r is the closest to r in R m,Φ v ,Ω ∪ {r}, i.e., there is no

r 2 = r in R m,Φ v ,Ω ∪ {r} such as d R (r, r 2 ) < d R (r, r 1 ). For any φ of Φ, if φ belongs to Φ v , u Φ u (φ) = u R u (φ) = 0
. This is a desired property as it is useless to validate a hypothesis twice with the same deductive model.

Heuristic definitions

An aggregation operator is used to determine the global utility functions u Φ and u R -using respectively u Φ u and u R u -of a hypothesis. As an example, weighted sum is used here. However any aggregation operator can also be used. Let u Φ , u R : Φ → [0, 1] be two functions representing the utility of any hypothesis of Φ. For any φ of Φ

u Φ (φ) = α Φ • u c (φ) + β Φ • u Φ u (φ), (11)
with α Φ + β Φ = 1. For any r = (φ, c m,φ,Ω ) of R m,Φ,Ω u R (φ) = α R • u c (φ) + β R • u R u (φ), ( 12
)
with α R + β R = 1. Two heuristics γ φ and γ R can now be defined.

Definition 5 Let γ φ (resp. γ R ) : R m,Φ,Ω → Φ -Φ v be a function that chooses an hypothesis to be validated.

For any r = (φ, c m,φ,Ω ) of R m,Φ,Ω , γ φ ( R m,Φ,Ω ) (resp. γ R ( R m,Φ,Ω )) = r iff it does not exist a r ′ = (φ ′ , c ′ m,φ,Ω ) in R m,Φ,Ω such as u φ (φ ′ ) > u φ (φ) (resp. u R (φ ′ ) > u R (φ)) and φ does not belong to Φ v .
The efficiency of these heuristics is evaluated on a real world problem presented in the following section. The goals el. et id. are considered.

APPLICATION TO FORENSIC ENTOMOL-OGY Introduction to Forensic Entomology

Forensic entomology is widely used in criminal investigations to determine post-mortem intervals (PMI) from the insects found on a cadaver. A PMI is usually estimated by experts using retrodictive models. These models are very easy to use but they do not take into account the ecosystemic context; thus, estimations performed using these methods are often overestimated and not as precise as they could be. Modern PMI estimation methods are based on insect development models. These models consider that insect development speed is temperature-dependant [START_REF] Stinner | An Algorithm for Temperature-Dependant Growth Rate Simulation[END_REF]. It is given as a function f of the temperature T varying in the time t. When a cadaver is discovered, investigators take insect samples from the body. Entomologists determine the species and the accumulated rates of development, denoted ∆a, of the oldest individuals. Then, for each one of them, the laying time t 1 (generally close to the time of death) can be calculated from the following equation

∆a = t2 t1 f (T (t))dt, ( 13
)
where t 2 represents the time of the cadaver discovery. Data from the nearest meteorological station are usually used in order to estimate T (t). However, it is problematic for many reasons. First, one can notice that cadavers are rarely found at the foot of a meteorological station. Furthermore, corpse thermal inertia is important, especially in the first hours after death. Finally the heat generated by larva aggregates can raise the temperature locally up to 20 • C. Thus, in many cases, entomological expertises results are inaccurate and given with an important margin of error.

A decision support system for Forensic Entomology

To handle these issues, a predictive agent-based model of insect development and cadaver decomposition in a complex ecosystem has been developed. This model is used to determine if a hypothesis -a possible time of death -is coherent with the observations available on the ecosystem of the crime scene and the entomofauna found on the victim. More information about this model and the validation process can be found in [START_REF] Morvan | A Decision Support System for Forensic Entomology[END_REF].

As the model is stochastic, it is necessary to run a large number -about 100 in most of the cases -of simulations to compute a coherence measure statistically significant. For any

φ of Φ v c m,φ,Ω = s v m,φ s t m,φ , (14)
where s v m,φ represents the number of valid simulations and s t m,φ the total number of simulations. Possible times of death are obtained from police investigators. Let φ 0 be the last time the victime was seen alive and φ n the time she was found. The set of possible times of death is discretised hour by hour: Φ = {φ 0 , ..., φ n }; a finer discretisation would not be mandatory unless more precise climatic data can be obtained. The metric spaces (Φ, d) and (R m,Φ,Ω , d) are defined with the euclidean distance d. Simple hypotheses are φ 0 and φ n and any time of death determined with a "classical" method -cf. equation 13. Estimation of the result set is performed using linear interpolation. In the next section, some experimental results are presented.

Experimental results

To evaluate the quality of the heuristics, "realistic" solution setsi.e., with the same fuzzy Gaussian shape than real solution sets -are randomly generated with |Φ| = 200. The cost function of the agent-based model m has been approximated using the following function: for any

φ i of Φ cost m (φ i ) = λ • (1 - i |Φ| -1 ), ( 15
)
where λ represents the validation cost of the hypothesis φ 0 .

Here we assume that λ = 1. Thus, 50.25 would be necessary to validate all the hypotheses of Φ. The maximal cost allowed to compute the solution is arbitrary fixed to 20. 10000 result sets have been generated to determine the best affectation for α φ and α R . Results show that whatever the goal is α φ ≃ α R ≃ 0.001. An interesting feature of this result is that the quality of the solution is very bad if the cost criterion is not usedi.e., if α φ = α R = 0 -and best for very small values of α φ and α R . Results presented in the figures 1 and 2 show that γ φ and γ R perform well respectively for the goals id. and el.. Many different development models of Diptera can be found in the literature. It has been shown in [START_REF] Wagner | Modeling insect Development Rates: A Literature Review and Application of a Biophysical Model[END_REF] that it is impossible, in the general case, to establish that a model is better than an other. Thus, the multiagent model can be parametrised to use any development model. An experiment has been conducted on a real case using six different development models. Estimated result sets were merged using the Transferable Belief Model [START_REF] Smets | The Transferable Belief Model[END_REF]. The figure 3 shows the result of the merging after pignistic transformation. It allows to conclude that the victim died between φ 55 and φ 91 , and that the most probable time of death is φ 78 . This result is very interesting for at least two reasons. First, it shows that this new method allows to determine the most probable time of death more precisely than traditional methods, the agent-based model including more parameters than the classic analytic models. A discussion on the simulation results and their impact on the accuracy of PMI estimations can be found in [START_REF] Morvan | A Decision Support System for Forensic Entomology[END_REF]. Second, a confidence interval can be formally determined from simulations results whereas in traditional methods the confidence interval estimation relies on the expert's intuition and is often questionable. As the concept of proof is crucial in criminal investigations, such an improvement is particularly interesting.

CONCLUSION

In this paper, a model of abduction is introduced and heuristics are defined to implement efficiently this model. This model is then applied in the context of Forensic Entomology. However, hypothesis sets are unidimensional in this case and large experiments should be carried on multidimentional sets. But results are encouraging and the model is being applied to develop a decision support system for cervical cancer prevention. Moreover, such a model could be used to perform intelligent exploration of model predictions or solve complex inverse problems. More generally, this study attempts to show that simulation-based reasoning models can be very useful to produce inferences in the framework of complex systems that cannot be described in logic-based languages. It seems particularly true for abductive and inductive inferences that need a higher level description than deductive (predictive) inference.

  d) be a metric space. Two elements e i and e j of E are consecutive in (E, d) iff it does not exist an element e of E such as d(e, e i ) < d(e i , e j ) and d(e, e j ) < d(e i , e j ).

Definition 4

 4 Let (E, d) be a metric space and d max ((E, d)) the maximal distance between two consecutive elements in (E, d). For any consecutive elements e i and e j of E, d max ((E, d)) = d(e i , e j ) iff it does not exist two consecutive elements e k and e l in (E, d) such as d(e k , e l ) > d(e i , e j ).

Figure 1 :

 1 Figure1: Quality of the heuristics is evaluated for the goal id. and compared to a random choice of hypotheses.

Figure 2 :i

 2 Figure 2: Quality of the heuristics is evaluated for the goal el. and compared to a random choice of hypotheses.

Figure 3 :

 3 Figure 3: Pignistic probability for each hypothesis φ i of Φ


Martha Orellano
email: martha.orellano@emse.fr

Khaled Medini

Christine Lambey-Checchin

Gilles Neubert

Paul Stief
email: paul.stief@ensam.eu

Jean-Yves Dantan

Alain Etienne

Ali Siadat

A system modelling approach to collaborative PSS design

Keywords: Assembly, Design method, Family identification Product-Service Systems, Value co-creation, System modelling, Goal Modelling, MBSE Product-Service Systems, Value co-creation, System modelling, Goal Modelling, MBSE

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

Introduction

Due to the fast development in the domain of communication and an ongoing trend of digitization and digitalization, manufacturing enterprises are facing important challenges in today's market environments: a continuing tendency towards reduction of product development times and shortened product lifecycles. In addition, there is an increasing demand of customization, being at the same time in a global competition with competitors all over the world. This trend, which is inducing the development from macro to micro markets, results in diminished lot sizes due to augmenting product varieties (high-volume to low-volume production) [START_REF] Tran | Development of a strategic prototyping framework for product service systems using co-creation approach[END_REF]. To cope with this augmenting variety as well as to be able to identify possible optimization potentials in the existing production system, it is important to have a precise knowledge of the product range and characteristics manufactured and/or assembled in this system. In this context, the main challenge in modelling and analysis is now not only to cope with single products, a limited product range or existing product families, but also to be able to analyze and to compare products to define new product families. It can be observed that classical existing product families are regrouped in function of clients or features. However, assembly oriented product families are hardly to find.

On the product family level, products differ mainly in two main characteristics: (i) the number of components and (ii) the type of components (e.g. mechanical, electrical, electronical).

Classical methodologies considering mainly single products or solitary, already existing product families analyze the product structure on a physical level (components level) which causes difficulties regarding an efficient definition and comparison of different product families. Addressing this

Introduction

Manufacturing industry is shifting towards more servicized offerings as a way of responding to recent society needs. Customized solutions, sustainability, and collaboration between several actors (e.g. industrial partners, consumers) are some of the needs that an adequate business model (BM) should address nowadays. In this context, Product-Service Systems (PSS) represent an alternative to enable such BMs. Adopting PSS requires to identify the organizational elements to be taken into account for creating and capturing high value. For instance, collaboration between key actors is essential to achieve higher value, becoming a key driver for successful PSS BMs [START_REF] Tran | Development of a strategic prototyping framework for product service systems using co-creation approach[END_REF]. Designing collaborative BMs for PSS needs the alignment between the strategic objectives of the stakeholders and the operational development of the offer itself. However, the literature on PSS explains the content of a business model from an operational perspective, centered in the offer development. In contrast, the strategic perspective, which implies stakeholders' objectives in early phases of the value creation process, is undervalued. Solaimani et al. [START_REF] Solaimani | Networked enterprise business model alignment: A case study on smart living[END_REF] highlight the importance of the conceptualization of value in terms of "value expectations" from a strategic reasoning of the key actors, as the main principle to design feasible offers. The aim of the proposed modelling approach is to support the value co-creation process for PSS design, through the visualization of its various elements (e.g. goals, PSS offers, activities) from strategic and operational perspectives. The research questions guiding this research are:

• How to model the collaborative process of PSS design? • What are the elements of value creation concerning both strategic and operational perspectives?

To answer these questions, this study is based on the implementation of two modelling frameworks, Goal Modelling (GM) and Model-Based System Engineering (MBSE), developed in the field of system engineering and applied to requirement engineering in collaborative contexts [START_REF] Friedenthal | A Practical Guide to SysML: The Systems Modeling Language[END_REF][START_REF] Aljahdali | Goal oriented requirements engineering-a review[END_REF][START_REF] Yu | Modelling strategic relationships for process reengineering[END_REF]. The interest of using GM resides on the definition of high-level

Introduction

Manufacturing industry is shifting towards more servicized offerings as a way of responding to recent society needs. Customized solutions, sustainability, and collaboration between several actors (e.g. industrial partners, consumers) are some of the needs that an adequate business model (BM) should address nowadays. In this context, Product-Service Systems (PSS) represent an alternative to enable such BMs. Adopting PSS requires to identify the organizational elements to be taken into account for creating and capturing high value. For instance, collaboration between key actors is essential to achieve higher value, becoming a key driver for successful PSS BMs [START_REF] Tran | Development of a strategic prototyping framework for product service systems using co-creation approach[END_REF]. Designing collaborative BMs for PSS needs the alignment between the strategic objectives of the stakeholders and the operational development of the offer itself. However, the literature on PSS explains the content of a business model from an operational perspective, centered in the offer development. In contrast, the strategic perspective, which implies stakeholders' objectives in early phases of the value creation process, is undervalued. Solaimani et al. [START_REF] Solaimani | Networked enterprise business model alignment: A case study on smart living[END_REF] highlight the importance of the conceptualization of value in terms of "value expectations" from a strategic reasoning of the key actors, as the main principle to design feasible offers. The aim of the proposed modelling approach is to support the value co-creation process for PSS design, through the visualization of its various elements (e.g. goals, PSS offers, activities) from strategic and operational perspectives. The research questions guiding this research are:

• How to model the collaborative process of PSS design?

• What are the elements of value creation concerning both strategic and operational perspectives?

To answer these questions, this study is based on the implementation of two modelling frameworks, Goal Modelling (GM) and Model-Based System Engineering (MBSE), developed in the field of system engineering and applied to requirement engineering in collaborative contexts [START_REF] Friedenthal | A Practical Guide to SysML: The Systems Modeling Language[END_REF][START_REF] Aljahdali | Goal oriented requirements engineering-a review[END_REF][START_REF] Yu | Modelling strategic relationships for process reengineering[END_REF]. The interest of using GM resides on the definition of high-level objectives that stakeholders expect from the system in the early phases of the design process. Besides, MBSE complements the GM framework, since it consists in using formal modelling to support the engineering process of the system development.

In this paper, the application of those frameworks are complemented by a conceptualization effort associated to the literature of PSS and value co-creation, allowing to determine the specific elements related to the collaborative development of PSS. The structure of the paper is as follows. Section 2 presents briefly the theoretical background of value co-creation in the context of PSS, and provides an overview of existent methodologies implemented on PSS design. Section 3 introduces the model proposal for designing collaborative PSS. Section 4 discusses an illustration of the model proposal through a real example, modelling the provision of safety clothing of a French energy company. Finally, the conclusions and research perspectives are discussed in Section 5.

Conceptual background

Value co-creation in PSS

According to Xing and Ness [START_REF] Xing | Transition to product-service systems: Principles and business model[END_REF], the success of a PSS depends on the design of proper business models. In general, a business model is defined as the logic by which a company creates and captures value for its stakeholders [START_REF] Chesbrough | Business model innovation: Opportunities and barriers[END_REF][START_REF] Casadesus-Masanell | From strategy to business models and onto tactics[END_REF]. One of the most known works among industrial and academic is the Business Model Canvas (BMC) proposed by Osterwalder and Pigneur [START_REF] Osterwalder | Business Model Generation: A Handbook for Visionaries, Game Changers, and Challengers[END_REF], which is based on the definition of the value proposition. As in the BMC, the nature of value creation is commonly studied from the provider perspective, pointing out the maximization of profits while achieving customer satisfaction. However, some recent studies have started to investigate the design of collaborative business models, expanding the boundaries of a single company, and integrating customers and other actors (e.g., suppliers, competitors, community) to co-create higher value [START_REF] Nenonen | Business model design: Conceptualizing networked value co-creation[END_REF][START_REF] Romero | Collaborative networked organisations and customer communities: Value co-creation and co-innovation in the networking era[END_REF][START_REF] Hankammer | Degrowth and collaborative value creation: Reflections on concepts and technologies[END_REF]. In the context of PSS, the integration of the customer in the process of value creation is a crucial factor to have a successful business model [START_REF] Kindström | Development of industrial service offerings: A process framework[END_REF]. For instance, Tran and Park [START_REF] Tran | Development of a strategic prototyping framework for product service systems using co-creation approach[END_REF] highlight the concept of value co-creation as the fact of involving the customer from the early phases of the offer development. Furthermore, the process of value co-creation aims at aligning the objectives of the customer with the offer of the provider [START_REF] Nenonen | Business model design: Conceptualizing networked value co-creation[END_REF]. Indeed, the objective of the value co-creation process is to achieve the trade-off between the accomplishment of customer's value expectations and the development of feasible value propositions according to the capabilities of the providers' network. In this sense, Ballantyne et al. [START_REF] Ballantyne | Value propositions as communication practice: Taking a wider view[END_REF] define a value proposition as the "promise of value" between actors, materialized through products and services; while value expectations are the benefits that strategic actors expect to get through the business activity. This logic allows us to analyze the business model as two-fold model of value creation as shown in Fig. 1. A strategic perspective with a high-level abstraction related to global objectives and expectations [START_REF] Solaimani | Networked enterprise business model alignment: A case study on smart living[END_REF][START_REF] Casadesus-Masanell | From strategy to business models and onto tactics[END_REF]; and an operational perspective, with a low-level abstraction in terms of activities, resources, and actors for the development of value propositions [START_REF] Xing | Transition to product-service systems: Principles and business model[END_REF]. The collaborative development of PSS suggests that both the customer and the provider network are involved in the two perspectives of the model of value co-creation.

PSS design

There is an extensive literature about the techniques for the design of PSS offers. Regarding literature, we can group the existing works into three general categories: the engineering approaches, the organizational approaches, and the holistic approaches. The main part of the works refers to the engineering approaches [START_REF] Aurich | Life cycle oriented design of technical product-service systems[END_REF][START_REF] Durugbo | A review of information flow diagrammatic models for product-service systems[END_REF][START_REF] Geum | Designing the sustainable product-service integration: A product-service blueprint approach[END_REF][START_REF] Bertoni | Value visualization in product service systems preliminary design[END_REF], combining techniques from new product and new service development.

The contributions about the offer engineering cover several points. For instance, the integration of products and services along the life cycle, increasing offer variety, reducing costs, and reducing environmental impacts, among others. On the other hand, a few amount of works on organizational approaches have been proposed recently. Xing and Ness [START_REF] Xing | Transition to product-service systems: Principles and business model[END_REF] propose an improvement of the BMC by adopting the principles of the service-dominant logic (SDL), which is translated into collaborative practices between customers and providers. Most of the contributions in this category are based on the PSS typology proposed by Tukker [START_REF] Tukker | Eight types of product-service system: Eight ways to sustainability? Experiences from SusProNet[END_REF] (i.e., product-oriented, used-oriented and result-oriented). Some authors analyze each type of PSS through the lens of BMC as exposed by Barquet et al. [START_REF] Bezerra | Employing the business model concept to support the adoption of product-service systems (PSS)[END_REF], getting specific business model properties for PSS. Finally, holistic approaches group the works related to system modelling for supporting the offer development. For instance, Idrissi et al. [START_REF] Idrissi | Generic conceptual model to support PSS design processes[END_REF] elaborate a generic meta-model for the support of PSS design. The work in [START_REF] Hollauer | Towards a meta-model for the description of the sociotechnical perspective on product-service systems[END_REF] builds also a meta-model based on a socio-technical perspective of PSS offer development. Both contributions combine aspects of offer engineering with organizational modelling, providing integrated frameworks for the development of PSS. Nevertheless, the three categories of the PSS design adopt a provider perspective and an operational orientation. According to this overview, this work is placed in the holistic category, and it aims at putting the attention on the strategic perspective within the early phase of the value co-creation process. In the next section, the model proposal is explained.

Coupling Goal-MBSE modelling for collaborative PSS design

Background of Model-Based Systems Engineering (MBSE) and Goal Modelling (GM) approaches

Our model proposal is based on the conceptual model shown in Fig. 2. The model is operational through the application of two modelling approaches called GM and MBSE, applied to the context of PSS in this research work. GM is executed by using i * modelling language [START_REF] Yu | Modelling strategic relationships for process reengineering[END_REF], whereas MBSE by using SysML [START_REF] Friedenthal | A Practical Guide to SysML: The Systems Modeling Language[END_REF]. there are two main views of our model: strategic and operational. Firstly, for the strategic view, GM is applied. This approach is concerned with the intends and expectations of the stakeholders regarding the system to be designed. The main idea of GM is to get functional and non-functional requirements to be included in the offer design, aiming at satisfying strategic intends of the actors involved [START_REF] Aljahdali | Goal oriented requirements engineering-a review[END_REF]. In GM the central element is the actor, and each actor determines its own expectations according to its needs. Such value expectations contain goals to be achieved, criteria to qualify the goals, activities to be performed, and resources to be used for the offer development [START_REF] Yu | Modelling strategic relationships for process reengineering[END_REF]. Secondly, for the operational view of the model, MBSE is implemented. This approach is a formal application of modelling methodologies for developing integrated, consistent and coherent system models, aiming at representing any aspect of the reality [START_REF] Friedenthal | A Practical Guide to SysML: The Systems Modeling Language[END_REF]. MBSE is structured in four model views: requirement, structural, behaviour, and parametric views [START_REF] Friedenthal | A Practical Guide to SysML: The Systems Modeling Language[END_REF]. The requirement view is related to the concrete specifications of the stakeholders involved in the system. The structural view represents the primary elements constituting the entire system and it is the base of all the other views. The behaviour view explains the dynamics of the system in relationship with the actors, and represents the system states throughout its life time. Finally, the parametric view is supported by an evaluation model that allows assessing the performance of the system and facilitates decision-making.

The model proposal

Our proposal is a two-fold model which establishes the link between actors' strategic objectives expressed as value expectations, and operational capabilities of the provider network to develop suitable value propositions. Fig. 2 shows the detailed model, by using the notation of GM and MBSE. The left side of the model is designated as "goal model" and shows the strategic view of the value co-creation process. It is composed by three elements: value expectation, intentional element, and actor. The element value expectation represents the benefits that each actor aims to get from the PSS. The intentional element is the abstract representation of activities, goals, criteria and resources that stakeholders consider necessary to the development of the PSS. Finally, the element actor represents all the the strategic actors that motivate the offer development based on their needs. The output of the goal model are the value expectations. This view of the model is based on the i * meta-model proposed in [START_REF] Yu | Modelling strategic relationships for process reengineering[END_REF], which is a goal-oriented modelling framework developed to model socio-technical problems. i * allows modelling actors, their intentions and their relationships. In the following, we explain the objects contained in the model.

• Actor: strategic stakeholders involved in the goal model and their relationships. An actor could be a natural person or an organization, and it has a corresponding role. • Role: the behaviour of an actor according to its specialized domain of competencies. • Goal: high-level intentions of strategic actors. What they want to achieve through the design of the offer. • Criterion: evaluates the level of accomplishment of a goal.

• Activity: actions that an actor wants to be executed in order to achieve the set of goals.

• Resource: physical or informational elements required to perform an activity. • Value expectation: benefits that the stakeholders aim to get from the system. It is the output of the goal model.

The right side of the model is detailed on Fig. 2, which represents the operational view of the process of value co-creation. It is based on the generic PSS meta-model proposed in [START_REF] Idrissi | Generic conceptual model to support PSS design processes[END_REF]. The output of the operational view is the value created. Now, we present the concepts of the model in a simplified way, structured by the framework of MBSE.

Requirement model. It corresponds to the first step of the PSS modelling from the operational view. It allows to represent the functional and non-functional requirements of stakeholders, and draws out the set of functions needed to achieve them. Functions could be performed by products or by services.

Structural model. It refers to the set of primary elements on the system, products, services, actors, resources and activities involved in the required processes to develop the PSS offer, as well as the set of relationships between them.

• Product: physical or tangible part of the PSS offer. It integrates the technical specifications to accomplish the functions that fit the best actors' requirements. • Service: intangible part of the PSS offer. It needs a set of activities co-developed between the actors. • PSS offer: content of the value proposition. The PSS is a bundle of products and services that fit together in order to respond to complex customer needs. Several types of PSS can take place as defined by [START_REF] Tukker | Eight types of product-service system: Eight ways to sustainability? Experiences from SusProNet[END_REF], grouped into three main categories: product-oriented, use-oriented, and result-oriented. • Actor: set of actors involved in the process of value creation from an operational point of view. The key actors are partners, suppliers and customers. Each actor has a role, which depends on their competencies. • Activity: set of activities needed to perform the products and services contained in the PSS offer and depends on the role of each actor.

• Resource: set of assets, knowledge, people and technology that an actor deploys to support the process of value creation.

Behaviour model. It corresponds to the different configurations that the system could have throughout its life time. Each configuration is known as "scenario", and it is the result of combining the elements of the structural model: products, services, actors, resources, and activities. A scenario defines the responsibilities of the PSS actors to deliver the value proposition to the customer. Each scenario entails a different level of fitness between the value expectations and the value proposition, and then different level of value is created. Multiple scenarios could take place depending on the degree of knowledge and maturity of the actors and the quality of their relationships.

Parametric model. It allows the evaluation of the global performance of the system. In the context of business models, the parametric model provides the basis for assessing the value created in each different scenario. In PSS, the value is a multidimensional element, expressing economic, environmental, social, relational and functional values created through the business model from the point of view of each actor. The parametric model allows evaluating the global value created by the system and facilitates decision-making. This can be done by qualitative or quantitative indicators.

Illustrative example

The illustrative example is inspired by a real case company in France, which will be denoted as 'C' for confidentiality reasons. C main activity consists in the production and distribution of energy. One of the most important support activities of the company is the provision of safety clothing to the employees. Currently, the safety clothing belong to the employees (users), who are fully responsible for their usage, maintenance and recycling. This process limits the leverage of C regarding the management of the safety clothing through its life cycle. The company decided to launch an innovation project to transform the current offer, going from a product purchasing-based offer towards a PSS-based offer.

The development of the PSS offer needs the collaboration between several actors from inside and outside the boundaries of the company (i.e., actors from the entire value chain). In this context, C as a customer, needs to mobilize all the key actors in the value chain to meet its specific needs regarding the safety clothing offer: textile manufacturers, logistics providers, washing service providers, and service providers at the end of life. The application of the proposed model, described in Section 3.2, helps C to structure its problem and illustrates the collaborative process for the development of the PSS offer. In this section, the elements of the conceptual model presented in Fig. 2 will be instantiated using C context.

The first main step consists in defining the goal model. This requires the definition of the strategic actors participating in the definition of the value expectations, including their roles, goals, criteria of evaluation, and required activities and resources. Fig. 3 shows an extract of the goal model instantiated for C case. It is possible to observe the relationship between C (the customer) and the washing service provider. Each actor has a boundary space and a shared space of interaction. C main objective is the guarantee of employees safety, which is associated to secondary objectives as reducing environmental impacts, contributing to social development, controlling economic expenses, and improving relationships with providers. The accomplishment of these objectives should be evaluated through the definition of strategic criteria (e.g., level of knowledge sharing, job generation, environmental performance of providers, among others).

Besides, the main objective of the washing service provider is providing high-quality washing service to create profit. Two of the criteria to evaluate the main objective are the service rate and the efficiency of the traceability system. Here it is possible to identify the relationships of dependency between both actors, which is represented by an arrow. In this case guarantying C employees safety depends on the provision of the washing service. Some other dependencies could be modeled, for instance, the reduction of the environmental impacts is linked to the efficiency of the traceability system.

The second main step corresponds to the development of the operational view of the model, starting with the requirement model Fig. 4. Requirements are categorized into five dimensions of value (defined in the goal model in relation with the strategic objectives): functional, environmental, social, economic and relational values.

For instance, improvement of mechanic resistance, reduction of water consumption, variety of styles of the safety clothing, controlling economic expenses, and improving corporate image are some examples of requirements in each dimension of value. Once the requirements are clearly described, the elements of the structural model can be defined. Products, services, corresponding activities and resources are defined in the structural model (Fig. 5). The main product in this case is safety clothing, which can be made with ecological fibers, with integrated systems for traceability, recycling-oriented design, among others characteristics. Services include real-time data transfer (tracking), integral maintenance, training for efficient usability, re-manufacturing, among others. Different combinations of the structural elements lead to feasible scenarios, which are represented in the behaviour model. In this paper, we represent an extract of the behaviour model in Fig. 6, assigning actors to specific functions according to their skills. Finally, once the feasible scenarios are defined, they could be assessed through the parametric model, using quantitative or qualitative data. This model provides the basis for evaluating the performance of each scenario. A simplified overview of this model is provided in Fig. 7. A more detailed description of each of all the operational views should be developed during the next stages of the project development.

The modelling framework helped to visualize the information coming from the different actors in early steps of the PSS design. This visualization is of interest to decision-makers involved in the development process of PSS, or any system at large, as it supports more informed decisions and facilitates the collaboration between them. In this sense, the proposed model framework could be used as a starting point for collaboratively outlining requirement specifications by a given customer working with several suppliers.

Conclusions and future research

One of the major challenges for designing PSS business models lies in enabling the interaction between several stakeholders, and aligning their value expectations with the developing of feasible offers.

We proposed a two-fold model that helps to reconcile the strategic objectives of the stakeholders with the operational development of PSS offers. The illustration of the proposed model through a real case company witnesses the added-value of the model, particularly by supporting organizational design and offer development through a step-wise process coupling GM and MBSE. This systemic vision facilitates collaboration and decision-making in a context of multiple stakeholders. Moreover, the paper contributes to the decision support of disruptive innovation as the case of PSS, in relation with the early design phases. This decision support is particularly valuable, since it corresponds to early stages of offer development, where data is scarce. Certainly, deeper research is needed to develop the model in all its views. Although the effort made in this work for modelling the relation between strategic and operational perspectives, we consider that deeper research is needed for the model generalization. Furthermore, the application in other contexts is necessary for further validation of the model.

Fig. 1 .

 1 Fig. 1. Generic model for collaborative PSS design.

Fig. 2 .

 2 Fig. 2. System modelling for collaborative PSS design.

Fig. 3 .

 3 Fig. 3. Goal model extract based on the example of company 'C'.

Fig. 4 .

 4 Fig. 4. Requirement model based on the example of company 'C'.

Fig. 5 .

 5 Fig. 5. Structural model based on the example of company 'C'.

Fig. 6 .

 6 Fig. 6. Behaviour model based on the example of company 'C'.

Fig. 7 .

 7 Fig. 7. Parametric model based on the example of company 'C'.

Acknowledgements

Auvergne Rhone-Alpes Region (France) supports this research through a doctoral thesis allocation into the ARC 8. The sponsors of this research within the framework of the PEAK Cluster: THESAME, Funds F2I/UIMM, French Plan d'Investissement d'Avenir and its financial operator Caisse des Dépôts, the ACE project of PFA (French Automotive Industry).