# End-of-life decision making in circular economy using generalized colored stochastic Petri nets

Gautier Vanson, Pascale Marangé, Eric Levrat

<figure><img src="image_1.png" /></figure>

## To cite this version:

Gautier Vanson, Pascale Marangé, Eric Levrat. End-of-life decision making in circular economy using generalized colored stochastic Petri nets. Autonomous Intelligent Systems, 2022, 2 (1), pp.3. 10.1007/s43684-022-00022-6. hal-03652669

HAL Id: hal-03652669

https://hal.science/hal-03652669

Submitted on 13 Dec 2023

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

<figure><img src="image_2.png" /></figure>
<figure><img src="image_3.png" /></figure>

Check for
updates

# End-of-Life Decision making in circular economy using generalized colored stochastic Petri nets

Gautier Vanson<fnref n="1" />*, Pascale Marangé<fnref n="1" /> and Eric Levrat<fnref n="1" />

**Abstract**

Circular economy enables to restore product value at the end of life i.e. when no longer used or damaged. Thus, the product life cycle is extended and this economy permits to reduce waste increase and resources rarefaction. There are several revaluation options (reuse, remanufacturing, recycling, ...). So, decision makers need to assess these options to determine which is the best decision. Thus, we will present a study about an End-Of-Life (EoL) decision making which aims to facilitate the industrialization of circular economy. For this, it is essential to consider all variables and parameters impacting the decision of the product trajectory. A first part of the work proposes to identify the variables and parameters impacting the decision making. A second part proposes an assessment approach based on a modeling by Generalized Colored Stochastic Petri Net (GCSPN) and on a Monte-Carlo simulation. The approach developed is tested on an industrial example from the literature to analyze the efficiency and effectiveness of the model. This first application showed the feasibility of the approach, and also the limits of the GCSPN modelling.

**Keywords:** Circular economy, Sustainability, Decision making, Generalized colored stochastic Petri net, Monte-Carlo simulation

## 1 Introduction

According to the world population growth curve, we observe that the increase in the world's population is similar with the increase in energy, materials consumed and CO₂ in the atmosphere [1]. The transition of companies to Industry 4.0 brings new technological advances with a greater source of information to solve some of these issues. Nevertheless, product customization and planned obsolescence both generate today a significant end-of-life (EoL) product stream and for a long time [2].

The concept of circular economy based on the pillars of sustainable development is defined by the Ellen McArthur Foundation as follows: circular economy is based on the principles of designing out waste and pollution, keeping products and materials in use, and regenerating natural

system (https://www.ellenmacarthurfoundation.org/circular-economy/what-is-the-circular-economy). In circular economy, the goal is to reduce the waste produced by consumer society, in a circular cycle composed of a set of processes such as reuse/refurbishment/remanufacturing/recycling/... These loops enable a product to be renewed, reconstituted and to extend its life cycle. Nevertheless, one question seems essential: how does one choose the optimized process of revalorization? In fact, the process that allows a product to be regenerated for the longest time and at the lowest cost will be favored among all the possibilities offered to companies.

An analysis of the literature shows that there are many papers on these different processes, but very few that group these different processes into a unified process. The objective of our work over the last few years is to propose a global vision of product regeneration. In Diez 2015, we defined the regeneration paradigm as [3]: "set of actions, natural or technical, to restore a waste or its constituents

<footnote>*Correspondence: firstname.name@univ-lorraine.fr</footnote>

<footnote n="1">¹Université de Lorraine, CRAN, UMR 7039, Campus Sciences, BP 70239, Vandœuvre-lès-Nancy, 54506, France</footnote>
to an acceptable state (functional and operational) allow-
ing to extend its life cycle”. Nowadays this process is not
yet industrialized and regeneration activities are mainly
handmade [4]. When a manufacturer decides to regen-
erate a product, it should consider that revalorizing the
product enables to reduce implicit costs such as the end-
of-life product management costs, which are more and
more regulated, and to reduce the manufacturing costs
such as operating costs and raw material purchases. So, in-
dustrializing regeneration provides economic advantages
with reduced implicit costs and makes it possible to reach
a new clientele potentially demanding regenerated prod-
ucts, which are by definition less costly than new products.
From an environmental point of view, a life cycle analy-
sis of the product and in particular at the end of its life
would ensure the ecological advantages to regenerate a
product in such or such revalorization process. However,
there is no doubt that extending the life of a product by
reusing it, repairing it, remanufacturing it and recycling
it allows reducing the depletion of natural resources and
waste volume. So, the objective of this paper is to propose a
methodology and a decision making tool to determine the
best regeneration to perform, based on the product health
state, on information regeneration processes and market
needs.

In a sustainable point of view, we assume to assess each product in an individual way to find the best regeneration trajectory for each product. Indeed, if we assess a product set, the best regeneration trajectory mean could be not appropriate for all products. The second assumption is to work with products with high value added and multi-components to justify an individual assessment to each product. In addition, some regeneration processes are divergent because they require a partial or complete decomposition (reconditioning, remanufacturing, disassembly, ...). Thus, the decision making tool must also be sequential to be able to diagnose and assess each decomposed element. The third assumption is that there are several regeneration alternatives for a product. Furthermore, we consider that these alternatives are prioritized from an environmental and economic point of view. For example, reuse is the smallest recovery loop, therefore the least expensive and with the best environmental impact compared to remanufacturing and recycling. Thus, if reuse is not possible for a product then the next option such as remanufacturing can be considered. This assumption could be in opposition with current market but not from a sustainable development point of view.

Regarding these assumptions, the decision making must
be able to adapt the decision according to different vari-
ables and parameters impacting the decision. Thus, this
work considers complete loops in the regeneration cycle
(Collect product, Assess product and Regenerate prod-

uct) for decision-making and the assessment of differ-
ent regeneration alternatives depending on the product
health state, regeneration processes (reuse, remanufactur-
ing, recycling, ...) and information on the market. The
health state is built for a product, as a function of func-
tional, operational and dysfunctional variables. Indeed,
when considering an EoL product, the health state de-
pends on the information collected throughout the prod-
uct use phase. To take into account these uncertainties,
a modeling method for regeneration decision making is
proposed using generalized stochastic Petri net modeling
(GCSPN) combined with Monte-Carlo simulation. We ob-
tain a decision making tool to estimate the best regenera-
tion trajectory.

The paper is structured as follows: a review of decision
making in the regeneration process is proposed in Sect. 2.
Section 3 develops the proposed decision making method
for the EoL product. Section 4 is devoted to the applica-
tion of the method on an example from the literature and
its analysis. Section 5 concludes the paper and proposes
several perspectives for future research.

## 2 Literature review: decision making in regeneration process

The regeneration objective is to reuse products waste
.products, sub-assemblies and components) to remain in
use phase as long as possible (extend the product life cy-
cle) and delay the dismantling phase. This process is com-
posed of several activities as we can see in Fig. 1. The de-
cision activity is an indispensable building block to assess
the different regeneration alternatives and to choose the
best regeneration decision. Indeed, the assessment result
will depend on variables collected over the entire prod-
uct life cycle, but also on regeneration activities like refurb-
bishment, cleaning, disassembly, rework or re-assembly.
So, the assessment must take into account all the ele-
ments that may impact the regeneration trajectory, as well
as product, processes and market variables and parame-
ters. In addition, the decision making must consider that
some variables and parameters are uncertain. Thus, we
propose a literature review to answer the following ques-
tions:

* What does the end-of-life product assessment cover?

* What are the variables and parameters used in decision making?

* What are the tools used for decision making?

* What are the assessment criteria?

* Do the models take uncertainty into account in the assessment?

From the articles presented in Table 1, it is clear that
there are several assessment methods (graphical or algo-
rithmic). Each paper uses heterogeneous product, process,
market variables and parameters to perform assessment.
As the regeneration process has an input stream that is
<figure><img src="image_4.png" /><figcaption>**Figure 1** Example of regeneration activities</figcaption></figure>

<table><caption>**Table 1** End-of-life decision-making methods in literature</caption><thead><tr><th>Assessment type</th><th>Target</th><th>Variables parameters</th><th>Modeling tool</th><th>Solving tool</th><th>Criteria</th><th>Uncertainty</th></tr></thead><tbody><tr><td rowspan="3">EoL assessments</td><td>The best EoL alternative [5]</td><td>Market/Process</td><td>A multicriteria decision-aid</td><td></td><td>Economic/ Environnemental</td><td>No</td></tr><tr><td>The optimal EoL option [6]</td><td>Product/Process/ Market</td><td>PESTEL</td><td>Analytic hierarchy process and Cost/benefit analyses</td><td>Economic/ Environnemental</td><td>No</td></tr><tr><td>The best EoL alternative [7]</td><td>Process/Market</td><td>A Product Recovery Multi-Criteria Decision Tool</td><td></td><td>Economic/ Environnemental/ Social</td><td>No</td></tr><tr><td rowspan="3">Disassembly assessments</td><td>The best circularity strategies for a product and/or service [8]</td><td>Product/Process</td><td>A multi-criteria decision-making method</td><td></td><td>Economic/ Environnemental/ Social</td><td>No</td></tr><tr><td>The optimal disassembly sequence [9]</td><td>Product</td><td>Fuzzy Reasoning Petri Nets (FRPN)</td><td>Reasoning algorithm based on FRPN</td><td>Economic</td><td>Yes</td></tr><tr><td>The optimal disassembly sequence [10]</td><td>Product/Process</td><td>FRPN/matrix operation</td><td>A reasoning algorithm of formalisation</td><td>Economic</td><td>Yes</td></tr><tr><td rowspan="6">Other assessments</td><td>The best disassembly sequence [11]</td><td>Process</td><td>A mathematical model</td><td>Heuristic SS algorithm</td><td>Economic</td><td>No</td></tr><tr><td>The best disassembly sequence [12]</td><td>Product/Process</td><td>A mathematical model</td><td>A stochastic program</td><td>Economic</td><td>Yes</td></tr><tr><td>The optimal strategy between manufacturing or remanufacturing [13]</td><td>Product/Process</td><td>A mathematical model with prioritization</td><td>Tabu search/scatter search/neural networks</td><td>Economic</td><td>Yes</td></tr><tr><td>The best collection strategy [14]</td><td>Market</td><td>Fuzzy Coloured Petri Net models</td><td>Model simulation</td><td>Economic/ Environnemental</td><td>Yes</td></tr><tr><td>The optimal facility locations and flows between facilities [15]</td><td>Product/Process/ Market</td><td>A mathematical model</td><td>A stochastic multi-objective model/Pareto-optimal solution</td><td>Economic</td><td>Yes</td></tr><tr><td>The best PSS scenarios [16]</td><td>Process</td><td>A multi-criteria analysis/"ELECTRE I" method</td><td></td><td>Environmental/ Economic/Social</td><td>No</td></tr><tr><td></td><td>The best design for new products from post-used components [17]</td><td>Product</td><td>Multiagent systems</td><td></td><td>Economic</td><td>No</td></tr></tbody></table>
not controlled, it is essential to consider the uncertainty
on the product variables but also the effect of the regenera-
tion actions. As shown in Table 1, many papers do not take
uncertainty into account in their models. The applications
only deal with product or subassembly level equipment.
Another observation is that only 4 papers in Table 1 assess
more than three regeneration processes [5–8]. Else other
articles focus solely on one theme as disassembly or collect
or PSS scenario [9–17]. However, only papers working on
the disassembly activity such as [9, 10] and [11] propose
a customized decision making for each product to deter-
mine the best disassembly sequence with an industrializa-
tion perspective. On the contrary, papers [5, 6] and [7] as-
sess several regeneration options for a product individually
but at a strategic, static level, i.e., without considering the
product's health state but only according to the product's
characteristics such as its nomenclature and/or its com-
position in terms of material. Therefore, it is not possible
to use these decision making tools for the purpose of in-
dustrializing regeneration. So, to increase profit, the best
regeneration trajectory for each product should be deter-
mined in a personalized way rather than using the statis-
tical optimization of many products [12]. Indeed, even if
statistical optimization allows to reduce the importance of
the assessment time, this could lead to slowdowns in the
regeneration chain because the trajectory of a few prod-
ucts assessed globally could disrupt the process. The ma-
jority of the papers studied in Table 1 do not specify calcu-
lation times – which is an important omission – given that
the time criterion is crucial for the implementation of prof-
itable industrialization for manufacturers. So, [18] pro-
vides a review of decision making work in the solid waste
management area. In this review, several critical decision-
making points are highlighted: for example, most critical
issues are the assumptions and selection of evaluation cri-
teria and further the selection of the criteria weight val-
ues. In Table 1, the criteria column shows that all papers
use economic criteria, four papers use environmental cri-
teria, and three papers use criteria in all three sustainability
categories. The economic criteria are important because
they provide evidence for companies to demonstrate the
economic feasibility of product regeneration. In addition,
the articles using the most criteria are the articles assessing
several end-of-life alternatives. While the articles assessing
the disassembly activity focus only on the economic crite-
ria. In our work, we defined assumptions and the criteria
type (environmental, economic) in introduction. We pro-
posed consider all assessments with product, process and
market variables. Thus, the assessment trajectory of an
EoL product is more relevant since the variables used con-
cern the whole regeneration cycle. Thanks to this global

vision, it is obvious that the assessment of the regenera-
tion trajectory will be more accurate and more profitable.

3 Decision making method for EoL product

Following the state of the art, we will propose a decision
making approach that allows to determine the best regen-
eration strategy for a product depending on its health state
but also on the regeneration process and market informa-
tion. To do this, the proposed approach is divided into 3
steps:

* Identification of the variables that influence the health state and those that are related to the regeneration process and the market.

• Instantiation of the proposed models.

* Evaluation of the strategies by Monte-Carlo simulation.

As a reminder and to complete the assumptions men-
tioned in introduction: with an industrialization point
of view, we consider a single type of product, complete
(containing all its components), which arrives in the cy-
cle through a push flow (suppliers randomly provide EoL
products). In this way, products can arrive with informa-
tion about their usage, either provided by the supplier or
integrated into the product (smart product), or most fre-
quently any information requiring diagnosis and tests to
assess EoL product health state. Thus, we assume that it's
feasible to collect EoL product information with a certain
cost. To industrialize regeneration, the following activities
must be implemented (Fig. 1):

1. EoL products recovery through users or other collectors.

2. Diagnose the product health state:

(a) by recovering information on the product by
collecting all the information available on the
product and provided by the supplier. If
necessary, carry out additional tests to retrieve
missing information for decision making.

(b) by assigning a class to the product based on the
data collected, which allows the product to be
categorized to facilitate its assessment.

3. Determine the best regeneration route for the product to be regenerated according to its condition, criteria and constraints (cost, customers needs, environmental impact...). According to its assessment, the waste product will be oriented towards the most adapted process for its regeneration (reuse/remanufacturing/recycling/...).

4. Regenerate the EoL product and requalify it according to its new health state. The requalification objective is to position the product in a class in view of its reinsertion in remanufacturing or resale.

It is important to specify that the activities described
for regeneration are recursive for certain trajectories with
product decomposition. Indeed, for each decomposed el-
ement, activities 2 to 5 are repeated because the process
can deteriorate the product. Moreover, the diagnosis of the
physical and/or functional aspect at the lower level of the
nomenclature is not necessarily possible before decompo-
sition. Thus, the decision-making is important because it
is necessary to assess the regeneration trajectory for each
decomposed element. Nevertheless, before designing a de-
cision making system, it is required to identify and analyze
the variables and parameters impacting the decision mak-
ing.

The first sub-section proposes a formalization in order
to identify the variables influencing the decision making.
The second, a modeling into Petri Net following the regen-
eration activities. Finally, to obtain a decision making tool
that allows to determine the best regeneration trajectory
according to the product's health state, process and mar-
ket, a Monte-Carlo simulation is used. With this tool, we
obtain the best regeneration trajectory, success probability
and potential gain.

## 3.1 Variables identification impacting the decision-making

So, the need to identify variables and their influence on the
decision making is essential. Figure 2 presents a causal loop
diagram [19] which contains a non-exhaustive list of vari-
ables/parameters impacting the decision making. Thus, as
presented in the previous section and with a global vision,
three groups are identified: product variables, process vari-

ables and market variables. The interactions that exist be-
tween the identified variables are modeled through cause-
effect relationships. These interactions are represented by
polarized arrows. This polarity describes the type of in-
fluence that one variable has on another. The "S" (Same)
polarity means that the two variables evolve in the same
direction. Conversely, the "O" (Opposite) polarity means
that the two variables move in opposite directions. During
its regeneration, a product can be decomposed into sev-
eral elements, the causal loop diagram (Fig. 2) is instanti-
ated recursively for each sub-assembly and component of
a product.

The variable *DECISION MAKING* is therefore influ-
enced:

* by the variable PRODUCT HEALTH STATE with a "O" polarity because if the health state is acceptable and the product is still in use, then there is no need to take a decision.

* by the parameters of the regeneration processes grouped by the REGENERATION PROCESS variable to make the model more comprehensible with a "S" polarity.

* by the var MARKET DEMAND with a polarity "S" because the market demand prioritizes the regeneration processes. For example, if customer acceptability of regenerated products is low then the

<figure><img src="image_5.png" /></figure>
recycling process will be preferred over
remanufacturing.

### 3.1.1 Product variables

The PRODUCT HEALTH STATE stock variable repre-
sents the real state of a system in a vector indicators form
of various kinds. Theses variables can be deterministic,
uncertain and/or unknown. So, product variables are de-
fined according to the following families: functional, dys-
functional and operational for each level of the element
to be regenerated (equipment/sub-assembly/component).
The work of [20] proposes product prioritization to estab-
lish health assessment at each level, and a method to re-
construct the health state of the higher level or to project
for the lower levels. This principle is reused in disassem-
bly/reassembly activities.

In Fig. 2 the PRODUCT HEALTH STATE stock variable is seen as a stock that can be filled through the “Increasing” valve and emptied through “Decreasing”. The PRODUCT HEALTH STATE stock variable evolves according to the phase of the life cycle it is in. During the first manufactur- ing phase the health state is maximal, during the use phase the product’s health state is reduced while the regeneration actions increase its health state.

Thus, in the use phase we find the different functional,
dysfunctional and operational variables that are direct
indicators of the health state. Two types of variables
are used: Boolean variables for all-or-nothing indicators
(example: is the move function for product still avail-
able? YES/NO) and real variables for performance, rate
or percentage indicators (example: OperatingDuration,
RustPercentage...). Moreover, the polarity of the arrows
allows to know how the variables interact. For example,
if we consider the OperatingDuration variable its polarity
with FunctionalVariable is “O” because the more the op-
erating duration increases the more FunctionalVariable
will decrease. For the FunctionAvailable boolean vari-
able, it evolves in the same way as FunctionalVariable,
if the function in question is no more operational then
FunctionalVariable will decrease.

### 3.1.2 Process variables

Depending on the availability, capacity, performance, en-
vironmental impact, cost and other indicators of each re-
generation process, the decision on the trajectory will be
different. In addition, these variables evolve over time and
must be continuously updated so that the decision making
system can determine the best regeneration trajectory for
each product. For example, remanufacturing process vari-
ables such as the component inventory, the process avail-
ability, or the process environmental impact can influence
the decision to remanufacture the product following this

process and thus potentially improve the product condi-
tion. Thus, these variables have an indirect influence on
the product's health state, but are crucial in the decision
making process.

Process variables in Fig. 2 can be instantiated for each regeneration process (reuse, reconditioning, remanufacturing, recycling...) considered in the model. In addition, it is important to note that each activity in each regeneration process can have a negative influence on the health state. For example, a disassembly activity between two elements can physically damage one or both components. Indeed, the variability of the products' state to be disassembled makes the operations complex. The difficulty of adapting regeneration operations to the variability of the products justifies taking into account the potential failure of the activity in the decision making process. Thus, this possibility must be considered by inserting uncertainty for each regeneration process modeled in the decision making system. Thus, Fig. 2 shows that if the decision has been made to regenerate the product through the remanufacturing process based on the regeneration process's performance the product can either go well and thus improve the product's health state or not go well and further deteriorate the product. Thus, the polarity from "RemanufacturingPerformance" to "RemanufacturingSuccess" is "S" but is "O" to "RemanufacturingFailure".

### 3.1.3 Market variable

The MARKET DEMAND variable is also seen as a stock that can be filled through the “Inc” valve and emptied through “Dec”. This variable shows that customer demands for the new or regenerated product influence decision making through the target health state for regeneration. This is because the regenerated product health state depends on the market demand in terms of quantity and quality. If there is no demand, there will be no remanufacturing. In addition, as more customers purchase product, the demand will decrease.

## 3.2 Generalized colored stochastic Petri net formalization

The second step of our approach is to model the prod-
uct evolution throughout the regeneration process. This
model represents the impact of the regeneration actions
on the product variables, depending on the product health,
the company's strategy (cost, customer needs, environ-
mental impact, etc.) and the market demands.

The objective is therefore to propose the best regen-
eration trajectory of a product, considering uncertainties
(product and processes) and constraints (ranges, cost, cus-
tomer requirements, environmental impact...). We pro-
pose generalized colored stochastic Petri net (GCSPN) for
modelling all the previous aspects [21]. Colored aspect al-
lows information to be added to the tokens, which reduces the complexity and combinatorial explosion linked to the many variables considered in the model. The stochastic aspect allows the introduction of uncertainty when transitions occur (probability laws modelling uncertainty). A GCSPN is defined by: (P; T; C; Pre; Post; Inh; pri; M₀; W) where:

* P is a finite set of places;

* T is a finite set (disjoint from P) of timed transitions and immediate transitions;

* C: P ∪ T → ω is the color function that associates to each element s ∈ P ∪ T a color domain C(s) ∈ ω where ω is a set that contains finite and non-empty sets of colors. An element of C(s) is called the color of s;

* Pre, Post and Inh are the incidence and inhibition functions, which associate to each place p ∈ P and to each transition t ∈ T, an application of C(t) to Bag(C(p)) Notes Bag(X): the set of multi sets on X;

* pri : T → {0; 1} is the priority function which associates to each delayed transition value 0 and to each immediate transition value 1;

* M₀ the initial marking, is a function defined on P, such as ∀p ∈ P; M₀(p) ∈ Bag(C(p));

* W : ∀t ∈ T; W(t) : C(t) → ℝ⁺ is a function which associates to each delayed (or immediate) transition, for a given color c ∈ C(t) a crossing rate (or a weight).

The GCSPN model will allow to model:

* the dynamics of the regeneration process using the place/transition structure of the Petri net;

* The evolution of the product's health state, the different regeneration alternatives and the market needs using token colours;

Token color C(s) is a tuple (C<sup>Level</sup><sub>health</sub>, C<sup>Level</sup><sub>route</sub>, C<sup>Level</sup><sub>stock</sub>) allows the following information to be displayed:

* Health state based on the functional, dysfunctional and operational variables in Fig. 2: C<sup>Level</sup><sub>health</sub> and the resulted class from diagnosis. Color C<sup>Level</sup><sub>health</sub> is defined from a variable depending on the level in the nomenclature. Thus, this color is defined for each element of the product nomenclature using the product variables identified by the pattern of Fig. 2.

* Regeneration alternatives: C<sup>Level</sup><sub>route</sub> is an element of the set of regeneration processes identified in Fig. 2 = {reuse, remanufacturing, recycling, ...}.

* Stock based on sub-assemblies and components in reserve and updated by the regeneration processes:

$C_{\text{stock}}^{\text{Level}}$

The regeneration evolution is represented by GCSPN. Each transition of this GCSPN corresponds to a regeneration activity or a regeneration process. Figure 3 shows a GCSPN example that provides an overview of the regeneration activities: diagnosis (black), path selection activity (blue), regeneration processes (dark green) and inventory

(brown). The largest place at the bottom is the characteristics of a regenerated product possibly from reuse or remanufacturing. The following paragraph explains each type of activity (diagnostic, trajectory selection, regeneration process, stock) in more detail.

* The health diagnosis updates the product class variable based on the incoming product's health state. To do this, it is necessary to enter the thresholds of each variable upstream, which will allow the product to be categorized by assigning it a class (Fig. 4). Some variables can have a single threshold, for example for a functional variable: either the function is operational or it is not. Other variables can have several thresholds, especially for variables representing a percentage, for example the rust percentage of a product can be divided into three categories thanks to two thresholds: a normal level and an acceptable level. The pattern equation to determine the variable Classe<sub>Level</sub> in the diagnosis activity must verify equation (1). If the equation is verified, color Classe<sub>Level</sub> of the token takes the color of the class associated with the equation

$$ \left( \bigwedge_{j=1}^{K} \text{varBool}_j == a_j \right) \wedge \left( \bigwedge_{i=1}^{L} \left( \text{varThresholds}_i \bullet \text{Level}_i \right) \right) == \text{true} \quad (1) $$

with a<sub>j</sub> Boolean variables, level<sub>i</sub> real threshold, “•” represents a comparison operator, K number of boolean variables and L number of real variables.

* The activity that determines the regeneration path comes after the diagnosis activity. The activity determines the regeneration process for the product based on the product class, in-stock assemblies and components, market parameters, and strategic parameters (Fig. 5). For example, strategic parameters can be the prioritization of regeneration processes in terms of environmental impact and/or economic cost. Strategic parameters depend on the company's preferences and their sales strategy. Market parameters can for example indicate what the customer needs are in terms of product class preference. In our study, and according to the second assumption stated in the introduction, the strategy is to privilege the process with the lowest environmental impact, i.e., the smallest regeneration loops as a priority, such as reuse, then remanufacturing and finally recycling.
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>
To simplify the equation of this activity, we consider that there are as many test conditions as there are regeneration processes to select. In addition, we use a desired state (DS) parameter based on the company's decision-making strategy and market variables. The pattern equation to determine the model regeneration process is defined by the equation (2). If the equation is verified, color $C_{\text{route}}^{\text{Level}}$ of the token takes the color of the class associated with the equation:

$$C_{\text{route}}^{\text{Level}} = \begin{cases} \begin{array}{l} \text{if } (\bigvee_{i=1}^{O} \text{Class}_{\text{Level}} == \text{Class}_{\text{Level}}^{\text{Strat}_i} \\ \quad \wedge \text{DS} == \text{DS}^{\text{Strat}_i} \\ \quad \wedge \text{StockLevel} = \text{StockLevel}^{\text{Strat}_i} \\ \text{then (RegenerationProcess1)} \\ \text{else} \\ \quad \dots \\ \text{if } (\bigvee_{i=O+1}^{P} \text{Class}_{\text{Level}} = \text{Class}_{\text{Level}}^{\text{Strat}_i} \\ \quad \wedge \text{DS} = \text{DS}^{\text{Strat}_i} \\ \quad \wedge \text{StockLevel} = \text{StockLevel}^{\text{Strat}_i} \\ \text{then (RegenerationProcessN-1)} \\ \text{else (RegenerationProcessN)} \end{array} \end{cases} \quad (2)$$

* The inventory is updated if product components have been disassembled and requalified (Fig. 6).

The pattern equation for the inventory activity is therefore the following:

$$C_{\text{stock}}^{\text{Level}} = \begin{cases} \begin{array}{l} \text{if } (\text{Class}_{\text{Level}} = A) \\ \text{then } (\text{StockInit}_A + \mathbf{1}, \\ \quad \text{StockInit}_B, \\ \quad \text{StockInit}_C) \\ \text{else} \\ \text{if } (\text{Class}_{\text{Level}} = B) \\ \text{then } (\text{StockInit}_A, \text{StockInit}_B + \mathbf{1}, \\ \quad \text{StockInit}_C) \\ \text{else } (\text{StockInit}_A, \\ \quad \text{StockInit}_B, \\ \quad \text{StockInit}_C + \mathbf{1}) \end{array} \end{cases} \quad (3)$$

* Then, according to the determined trajectory, the product is sent to the chosen regeneration process to be revalored (reuse or remanufacturing or recycling or ...) (Fig. 7).

Reuse: only affects the highest level of the product nomenclature. Indeed, if the incoming product is similar to a new product or corresponds to a "class A" product, then a cleaning action is sufficient before requalifying the product.

Remanufacturing: is defined by [22] as a process which "consists of seven key activities to turn cores into remanufactured products/components including core acquisition, disassembly, cleaning, inspection, reworking, reassembly, and testing". In this work, we use only disassembly, reworking and reassembly activities (Fig. 8).

- Disassembly: determines the color of each sub-assembly or component obtained after decomposition. Each token has a color that depends on the health state of high level element, the disassembly route and the parameters of the disassembly process.

- Rework: used to increase the health of a sub-assembly or component. The new token has a color that depends on the incoming component's health state, the regeneration trajectory, and the parameters of the rework activities.

- Assembly: determines the color of the assembly from the health state of n sub-assemblies or components composed high level element, the assembly route and the parameters of assembly process.

- Recycling: is defined by [23] as "a product recovery option that involves techniques for creating new materials from wastes". The new token has a color that depends on the health state.

GCSPN transitions change the color of the tokens and thus the characteristics of the regenerated product, sub-assemblies and components. In addition, the parameters of the regeneration set of activities and thus the decision making depend on the business strategy of the company's

<figure><img src="image_9.png" /></figure>
<figure><img src="image_10.png" /></figure>

resources, the regeneration process performance and the product characteristics and its ability to be regenerated. For example, are all sub-assemblies disassemblable and re-workable? Thus, the structure of the GCSPN and the dif-ferent parameters depend on the strategy, constraints and resources of the company. So, the decomposition/recom-position equations of the product's health state must be de-termined.

In addition, the direction of the arrows for the different variables going to either the “Increasing” or “Decreasing” valves (Fig. 2) allows to know when (in which phase) and how EoL product color variables change through the GC-SPN transitions (improvement or deterioration). Once the model colors have been defined, regeneration activities are used to create recursive patterns to form the structure of the GCSPN. Thus, depending on the company’s resources and the regeneration processes compatible with the prod-uct, it is possible to integrate new process blocks (Fig. 7) into the model and/or to delete process blocks if a regen-eration path is now unusable. This hierarchical aspect of the PN modeling allows to simplify the modeling of the decision making system.

In addition, the variables used to define health state are not always known, some variables are formalized with

an uncertainty. In order to represent this uncertainty, the value of the variable is drawn randomly following a prob-ability law whose parameters depend on the color of the token:

* for unknown variables before disassembly, a value is drawn according to truncated probability laws (normal, uniform, ...) defined which represent the possible values variability. In this study, the probability laws of the stochastic variables are assumed to be known. They can be determined by statistical studies, by experts, from the monitoring of products in real time or by feedback.

* The success probability for a specific process operation, is modeled by discrete laws between 0 and 100%. Depending on the draw and the result of the operation, it is either a success (the variables follow the defined equations) or a failure (the variables are degraded).

The model equation for the regeneration processes takes place in three steps: first, a guard is used to activate or not the regeneration process depending on the regener-ation path selected in the upstream activity (Fig. 9). Sec-ondly, actions are associated with the transition to per-form the drawing of probability laws. For each activity of a regeneration process there is at least one probability
<figure><img src="image_11.png" /><figcaption>Figure 8 A remanufacturing activities GCSPN example</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 9 Input/Output of Regeneration process</figcaption></figure>

law (ProbabilityLaw<sub>i</sub>), the one corresponding to the uncertainty of the success or failure of the activity. Thirdly, according to the law0, the performance parameter of the resource N (PerfProcess<sub>N</sub>) and the process characteristics: a new color is created if the activity was a decomposition otherwise the input color is updated (improved or deteriorated state if the activity is a failure). In addition, after each pass in a regeneration process, the new color must pass again in the diagnostic activity to be assigned a new class (A, B or C). Indeed, the variables of the element leaving the process depend on the variables of the element entering and the regeneration process:

1. Guard: [ProcessN = C<sup>Level</sup><sub>route</sub>] "Transition activation".

2. Action: "Probability law drawing"

PerfDrawingProcess<sub>N</sub> = ProbabilityLaw<sub>0</sub>,

NewVar1 = ProbabilityLaw<sub>1</sub>,

…

NewVarN = ProbabilityLaw<sub>N</sub>.
3. Token color changes according to the following equation:

$$C_{\text{health}}^{\text{Level}} = \begin{cases} \text{if } (\text{PerfDrawingProcess}_N \le \text{PerfProcessN}) \\ \text{then } (\text{unknown}, \\ \quad \text{Var1Bool} = a_1, \dots, \\ \quad \text{VarNBool} = a_N, \\ \quad \text{NewVar1} + b_1, \dots, \\ \quad \text{NewVarN} + b_N, \\ \quad \text{SameVar1} + c_1, \dots, \\ \quad \text{SameVarN} + c_N) \\ \text{else } (\text{unknown}, \\ \quad \text{Var1Bool} = x_1, \dots, \\ \quad \text{VarNBool} = x_N, \\ \quad \text{NewVar1} + y_1, \dots, \\ \quad \text{NewVarN} + y_N, \\ \quad \text{SameVar1} + z_1, \dots, \\ \quad \text{SameVarN} + z_N). \end{cases} \quad (4)$$

Figure 10 shows the GCSPN of the disassembly activity for a product $P_A$ whose health state is defined by $C_{\text{health}}^{P_A} = (var_1, var_2, class_{P_A})$. Product $P_A$ is composed of a sub-assembly $S_{AB}$ and a component $C_C$. Sub-assembly $S_{AB}$ is composed of 2 components $C_D$ and $C_E$.

There are two possibilities to disassemble product $P_A$:

* Route DR1: if the health state of $P_A$ is classified B, it is decomposed into $S_{AB}$ and $C_C$.

* Route DR2: if product $P_A$ is classified C, it is first decomposed into $S_{AB}$ and $C_C$ and then $S_{AB}$ is decomposed into $C_D$ and $C_E$.

In the case of route 1, the following equation is used to determine the health state: $C_{\text{health}}^{S_{AB}} = (var_3, var_4, class_{S_{AB}})$. Component $S_{AB}$ has 2 variables and $var_3$ is not measurable. Thus, when product A is disassembled to obtain $S_{AB}$, the normal parameter law $(m, \sigma)$ models the variable uncertainty. In addition, the disassembly operation is also uncertain. In 70% of cases the operation is a success. In 30% of cases, variables are deteriorated. Function transition is therefore constructed as well as the place $(P_{S_{AB}})$ represented the sub-system $S_{AB}$ with color:

* Parameters for DR1 activity:
  - ProbabilityLaw$_0$ = Uniform(0, 100);
  - ProbabilityLaw$_1$ = Normal(50, 10).

* Parameters for disassembly $S_{AB}$:
  $PerfProcessN = P_{OP1} = 70, a1 = true, b1 = 0, x1 = false, y1 = -30,$

$$C_{\text{health}}^{S_{AB}} = \begin{cases} \text{if } (\text{Perf} = \text{Unif}(0, 100)) \\ \le 70 \\ \text{then } (\text{Unknown}, a1, \\ \quad \text{NewVar1} + b1) \\ \text{else } (\text{Unknown}, x1, \\ \quad \text{NewVar1} + y1). \end{cases} \quad (5)$$

* Parameters for disassembly $C_C$:

<figure><img src="image_13.png" /><figcaption>Figure 10 GCSPN for disassembly activity</figcaption></figure>
$$PerfProcessN = P_{OP2} = 80, a1 = true, c1 = 0, x1 = false, y1 = -15,$$

$$C_{\text{health}}^{C_C} = \begin{cases} \text{if } (\text{Perf} = \text{Unif}(0, 100)) \\ \le 80 \\ \text{then } (\text{Unknown}, a1, \\ \qquad \text{NewVar1} + c1) \\ \text{else } (\text{Unknown}, x1, \\ \qquad \text{NewVar1} + y1). \end{cases} \quad (6)$$

Section 3.2 showed how to use the causal diagram formalism in Fig. 2 to construct the colors:

* representing the product and component health state from the identified functional, dysfunctional, and operational variables of each level ($C_{\text{health}}^{\text{Level}}$).

* representing the choice of the regeneration path according to the number of regeneration processes available for the product to be regenerated ($C_{\text{route}}^{\text{Level}}$).

* representing the number of components, sub-assemblies in stock per class($C_{\text{stock}}^{\text{Level}}$).

Moreover, from the identified process parameters (the performance of the different operations, their capacity to increase the product's health state or to deteriorate it...) and the different regeneration activities (diagnosis, trajectory selection) the structure of the GCSPN is designed. Once the construction of the model is completed, it must be simulated to obtain results and to make a decision.

## 3.3 Simulation of the GCSPN

From this GCSPN model, it is possible to generate different regeneration trajectories as reuse, remanufacturing (disassembly and reassembly routes, reworking) and recycling. In order to assess them, a set of criteria such as the potential gain, the probability of reaching the desired state, the class of the remanufactured product, etc. are used. These criteria are added to the previous models in order to assess their value according to the health state of the incoming products and regeneration trajectory decisions.

By simulating the model, many times (Monte-Carlo process) with the same product variables and analyzing the

results, we obtain reliable estimates that allow us to decide on the regeneration trajectory. Indeed, the model repeats the simulation as long as the stopping criteria do not converge below a chosen step. One or more criteria can be chosen to stop the simulation.

To sum up, the stochastic aspect is found at several levels in the model: random drawing of the uncertain variables of the product according to probability laws, random drawing of the process operations by discrete laws and Monte-Carlo simulation.

To summarize, Sect. 3 outlined the methodology for building a decision making tool in GCSPN. The first step was to construct a causal diagram to identify and analyze the variables useful for determining the product's health state and the different variables useful for decision making. Then, from this formalism it is possible to define the colors and transitions of the GCSPN. Once the model is built, this section explains how to simulate the model to obtain results and to be able to make a decision in a personalized way to each product entering the decision making process. The next part, will highlight the proposed methodology through an application from the literature.

## 4 Numerical illustration to an industrial case

The model is tested with an example from the literature [12]: a Knorr-Brese EBS 1 Channel Module which allows us to verify the effectiveness and efficiency of the model (Fig. 11). We consider 3 sub-assemblies: the cover, the cap, the support which is composed of 2 sub-assemblies (electronic card and the body). The following variables are considered for the Knorr-Brese module level (Fig. 12). So, variables threshold are pre-defined. EBS module $F_1$ and $F_2$ functionalities are tested on test benches. The OperatingDuration is given by the supplier and the RustPercentage is determined by observation of the module. The color of the token corresponding to the EBS module health is formed of 5 variables:

$$C_{\text{health}}^{\text{EBS}} = (\text{class}_{\text{EBS}}, F_{1\text{EBS}}, F_{2\text{EBS}}, D_{\text{EBS}}, R_{\text{EBS}}). \quad (7)$$

In this scenario, we have 3 functional variables ($F_{1\text{EBS}}$, $F_{2\text{EBS}}$, $D_{\text{EBS}}$) and one dysfunctional variable ($R_{\text{EBS}}$).

<figure><img src="image_14.png" /><figcaption>Figure 11 EBS module and components of the returned module [12]</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>**Figure 12** Causal diagram of product, process and market variables impacting decision making for EBS Module</figcaption></figure>

<table><caption>**Table 2** Variables for the equipment level: EBS module</caption><thead><tr><th>Variables</th><th>Values</th></tr></thead><tbody><tr><td>Controlling brake pressures (F<sub>1EBS</sub>)</td><td>0: False<br>1: True</td></tr><tr><td>Containing the air (F<sub>2EBS</sub>)</td><td>0: False<br>1: True</td></tr><tr><td>OperatingDuration (D<sub>EBS</sub>)</td><td>0: D<sub>EBS</sub> &gt; AcceptableLevel_D<br>1: NormalLevel_D &lt; D<sub>EBS</sub> ≤ AcceptableLevel_D<br>2: D<sub>EBS</sub> ≤ NormalLevel_D</td></tr><tr><td>RustPercentage (R<sub>EBS</sub>)</td><td>0: R<sub>EBS</sub> &gt; AcceptableLevel_R<br>1: NormalLevel_R &lt; R<sub>EBS</sub> ≤ AcceptableLevel_R<br>2: R<sub>EBS</sub> ≤ NormalLevel_R</td></tr></tbody></table>

variable of the incoming product is first null and then updated after the diagnostic activity. Table 2 illustrates the variables categorized according to the thresholds determined by the company. The first two variables correspond to functions essential to the EBS module and are categorized with a binary state. Either the function is operational or it is not. For the *OperatingDuration* and *RustPercentage* variables, the threshold parameters must be filled in by the company.

The colors of each sub-assembly and component are defined in the same way. Some variables at module level allow, by deduction, to determine the elements variables (sub-assemblies, components) which represent elements that have just been disassembled (e.g., the usage support is the same for EBS duration). For variables that

cannot be inferred, uncertainty must be introduced (e.g., *TightnessLevel* of the valves).

The next step is to classify the classes (A, B and C) of each element (product, sub-assemblies and components) according to their variables (EBS module example in the Table 3). When crossing the transition and according to Table 3, the transition function determines the health state of the EBS module by the following color and parameters:

* EBS Parameters for *OperatingDuration*: NormalLevelD = 20,000 hours, AcceptableLevelD = 40,000 hours.

* EBS Parameters for *RustPercentage*: NormalLevelR = 20% rust, AcceptableLevelR = 40% rust.
<table><caption>**Table 3** Variables for the equipment level: EBS module</caption><thead><tr><th rowspan="2">Var</th><th colspan="2">F1</th><th colspan="2">F2</th><th colspan="2">D</th><th colspan="2">R1</th></tr><tr><th>1</th><th>0</th><th>1</th><th>0</th><th>2</th><th>1</th><th>0</th><th>2</th><th>1</th><th>0</th></tr></thead><tbody><tr><td>A</td><td>X</td><td></td><td>X</td><td></td><td>X</td><td></td><td>X</td><td></td><td></td><td></td></tr><tr><td>B</td><td>X</td><td></td><td>X</td><td></td><td>X</td><td></td><td></td><td>X</td><td></td><td></td></tr><tr><td>B</td><td>X</td><td></td><td>X</td><td></td><td></td><td>X</td><td></td><td>X</td><td></td><td></td></tr><tr><td>C</td><td colspan="10">If not A and not B</td></tr></tbody></table>

* Color evolution is defined by:

$$C_{\text{health}}^{\text{EBS}} = \begin{cases} \text{if } (F_{1\text{EBS}} = \text{true} \land F_{2\text{EBS}} = \text{true} \land (D_{\text{EBS}} \leq \text{NormalLevelD}) \land (R_{\text{EBS}} \leq \text{NormalLevelR})) \\ \text{then } (A, F_{1\text{EBS}}, F_{2\text{EBS}}, D_{\text{EBS}}) \\ \text{else} \\ \text{if } (F_{1\text{EBS}} = \text{true} \land F_{2\text{EBS}} = \text{true} \land (D_{\text{EBS}} \leq \text{AcceptableLevelD}) \land (R_{\text{EBS}} \leq \text{AcceptableLevelR})) \\ \text{then } (B, F_{1\text{EBS}}, F_{2\text{EBS}}, D_{\text{EBS}}) \\ \text{else } (C, F_{1\text{EBS}}, F_{2\text{EBS}}, D_{\text{EBS}}). \end{cases} \quad (8)$$

The activities: reuse, remanufacturing and recycling must be defined according to the equation (4). Disassembly/reassembly ranges, projection/reconstruction rules, costs, operations yield, stocks, customer demands and the laws of probability most representative of the uncertain variables must be defined. In this application, it's possible to:

* reuse EBS module with some minor cleaning activities.

* remanufacture EBS module. There are 2 ranges of disassembly ($DR_1, DR_2$) and 2 ranges of reassembly ($RR_1, RR_2$) possible. $DR_1$ aims at disassembling the EBS module into three sub-assemblies and $DR_2$ aims at obtaining the two sub-assemblies cap and cover as well as the two components electronic board and body. Rework is possible for the cap, the cover and the body. The electronic board is considered too complex to be reworked. For the reassembly, the company has the manufacturing resources to integrate the regenerated components in the manufacturing line: electronic board and body to obtain the support. The production line can also integrate the regenerated sub-assemblies to remanufacture an EBS module.

* recycle the EBS module's cap, cover and body. The electronic board is considered too complex to be recycled. So, in this scenario, if the electronic board is disassembled, another partner regenerates it.

Therefore, the color of the route is defined by: ($C_{\text{route}}^{\text{Level}}$) is an element of the set: {Reuse, $DR_1$, $DR_2$, Stock, Rework, Recycle, $RR_1$, $RR_2$, No$_R$}.

To simplify the company's decision-making strategy, we use a desired state (DS) parameter determined according to market variables. Thus, depending on the DS, the class of the incoming product and the stock, the decision making process will choose a regeneration process ($C_{\text{health}}^{\text{Level}}$).

The assumption in this application is that the remanufactured product market only requires class A or B products and in the event that a product or sub-assembly is diagnosed as C then it is disassembled and then recycled to recover raw material. In addition, for the reassembling process, we don't surpass the class element assembling. For example, for reach a DS product B, we only use sub-assemblies and components from class B. The three possible regeneration routes are defined as follows:

* If an incoming EBS module is class A, the DS is class A and the class A EBS module inventory is below the company's recommended threshold, then the module will enter the reuse process to be cleaned and requalified for resale.

* If an incoming EBS module is class A, the DS is class A but the inventory of class A EBS modules is below the company's recommended threshold, then the module will be disassembled following the DR1 range to store sub-assemblies for future reassembly. Another possibility for route DR1 to be chosen is that the incoming product is of class B with a DS equal to B and no matter how much stock of element B there is, or that the product is of class B and the DS is equal to A.

* In the case where an incoming product is class C, disassembly range number two DR2 is systematically selected with the objective of recycling class C sub-assemblies and components.

These three regeneration route are translated into the equation:

$$C_{\text{route}}^{\text{EBS}} = \begin{cases} \text{if } (\text{class}_{\text{EBS}} == A \land \text{DS} = A \land \text{stock}_{\text{EBS}}^{A} \leq \text{normalLevelEBS}_{A}) \\ \text{then } (\text{Reuse}) \\ \text{else} \\ \text{if } ((\text{DS} = A \land \text{class}_{\text{EBS}} == A \land \text{stock}_{\text{EBS}}^{A} \geq \text{normalLevelEBS}_{A}) \\ \quad \lor (\text{DS} == B \land \text{class}_{\text{EBS}} == B) \\ \quad \lor (\text{DS} == A \land \text{class}_{\text{EBS}} == B) \\ \text{then } (\text{DR}_1) \\ \text{else } (\text{DR}_2). \end{cases} \quad (9)$$

Decisions regarding the regeneration possibilities of the cap: If the DS is class A and cap is class B then the cap is rework to try to improve the cap's health state regardless of component stock. If the disassembled cap is class C then
recycling is chosen. Else the cap is stored.

$$C_{\text{route}}^{\text{Cap}} = \begin{cases} \text{if } (\text{DS} == A \land \text{class}_{\text{Cap}} == B) \\ \text{then } (\text{Rework}) \\ \text{else} \\ \text{if } (\text{class}_{\text{Cap}} == C) \\ \text{then } (\text{Recycle}) \\ \text{else } (\text{Stock}). \end{cases} \quad (10)$$

The decision to reassemble is only made once all the elements have been regenerated. In this scenario, the company prioritizes the regenerated components in the reassembly process. If this is not possible, then elements are taken from stock, with a preference for reassembling sub-assemblies (semi-finished product) rather than going through component assembly.

Regarding the body disassembly, 2 variables are uncertain because they cannot be diagnosed before the disassembly (*TightnessLevel* and *RustPercentage*). Thus, for the *TightnessLevel* valve, we use a truncated normal law between 0 and 100% with as parameter a fixed standard deviation assumed to be statistically determined and an average which is a function of the equipment level service life variable (EBS module). Once the parameters of the normal distribution have been determined, a random draw in the distribution assigns a value to the *TightnessLevel*.

Process uncertainty is modelled by discrete laws. Probabilities are defined according to the performance of each operation. Either the operation is a success or a failure. If it is successful, the values of the variables are affected normally according to the rules determined, otherwise the variables are deteriorated by adding or decreasing.

Before simulating, it is necessary to choose the variables of the input product supposedly determined in the diagnostic activity upstream of the decision-making model. We propose 6 possible scenarios:

1. Sc1: a class A product with a class A DS with a stock is supposed to be null for sub-assemblies but not for the components.

2. Sc2: a class A product and a class B DS with a stock is supposed to be null for sub-assemblies but not for the components.

3. Sc3: a class B product and a class A DS with a stock is supposed to be null for sub-assemblies but not for the components.

4. Sc4: a class B product and a class B DS with a stock is supposed to be null for sub-assemblies but not for the components.

5. Sc5: a class B product and a class A DS with null stock.

6. Sc6: a class B product and a class A DS with complete stock.

<table><caption>**Table 4** Alternatives costs</caption><thead><tr><th>Alternatives</th><th>Alternatives cost (€)</th></tr></thead><tbody><tr><td>Reuse</td><td>0.5</td></tr><tr><td>DR1</td><td>1.5</td></tr><tr><td>DR2</td><td>3.5</td></tr><tr><td>Rework Cap</td><td>4</td></tr><tr><td>Rework Cover</td><td>5</td></tr><tr><td>Rework Body</td><td>9</td></tr><tr><td>Recycle Cap or Cover or Body</td><td>0.5</td></tr><tr><td>RR1</td><td>1</td></tr><tr><td>RR2</td><td>2</td></tr><tr><td>No</td><td>0</td></tr></tbody></table>

<table><caption>**Table 5** Alternatives resale prices per class</caption><thead><tr><th>Alternatives</th><th>Alternatives cost (€)</th></tr></thead><tbody><tr><td>A</td><td>796,6</td></tr><tr><td>B</td><td>100</td></tr><tr><td>C</td><td>Not saleable</td></tr><tr><td>Material Recovery</td><td>1,49€</td></tr></tbody></table>

The alternatives that we are trying to assess, knowing the resale grid, are listed in the following tables. Table 4 and Table 5 are based on the work of [12]. The alternative costs are determined based on the disassembly operation cost which is “0.29 cents of € per second”. The alternatives resale prices per class are determined between 796.6€ which “represents an estimate of its actual average market price” and “the resale value of a remanufactured 1 channel EBS module [which] is about 100 €”. Regarding recycling, “The revenue which can be generated by recovering the material of this product is taken to be 1.49€.”

We observe the potential gain and the probability of reaching the desired state with a stopping step of 0.01 which corresponds to 1 euro-cent for the potential gain and 1% for the probability. As we observe two decision criteria, we have to wait for the two criteria to converge, before stopping the Monte-Carlo simulation. So, after many simulations to reach the stopping criteria and the processing of this data, it can be decided which will be the best regeneration trajectory. To illustrate, Fig. 13 shows for Sc4 scenario the number of Monte-Carlo simulations required for the potential gain and the success probability to converge. Thus, every 10 simulations the average of each criteria is recalculated and compared to the previous one until the difference is lower than the chosen stopping step. To perform the decision-making for the application presented and with an Intel(R) Core(TM)/i7-10850H CPU @ 2.70 GHz/2.71 GHz the time follow this equation: time = 0.053 × SimulationNbr. So, for a product that requires 1000 simulations to converge to the best decision, the decision making time would be: 53 s.

The results presented in Table 6 show that depending on the product's health state, the DS and the stocks, the regeneration trajectories change and do not necessarily lead to
<figure><img src="image_16.png" /><figcaption>**Table 6** Results of the six scenarios</figcaption></figure>

<table><thead><tr><th>Criteria</th><th>Sc1</th><th>Sc2</th><th>Sc3</th><th>Sc4</th><th>Sc5</th><th>Sc6</th></tr></thead><tbody><tr><td>Gain€</td><td>796.1</td><td>-2.39</td><td>150,66</td><td>67.3</td><td>-8.23</td><td>787.3</td></tr><tr><td>Success%</td><td><strong>1</strong></td><td><strong>0</strong></td><td><strong>0.23</strong></td><td><strong>0.7</strong></td><td><strong>0</strong></td><td><strong>1</strong></td></tr><tr><td>Reuse %</td><td><strong>1</strong></td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>DR1 %</td><td>0</td><td>0.48</td><td>0</td><td><strong>0.54</strong></td><td>0</td><td>0</td></tr><tr><td>DR2 %</td><td>0</td><td><strong>0.52</strong></td><td>1</td><td>0.46</td><td>1</td><td>1</td></tr><tr><td>Rework %</td><td>0</td><td>0</td><td><strong>0.8</strong></td><td>0</td><td><strong>0.8</strong></td><td><strong>0.85</strong></td></tr><tr><td>Recycle %</td><td>0</td><td>0.52</td><td><strong>0.78</strong></td><td>0.46</td><td><strong>0.8</strong></td><td><strong>0.81</strong></td></tr><tr><td>RR1 %</td><td>0</td><td>0</td><td>0</td><td><strong>0.54</strong></td><td>0</td><td>1</td></tr><tr><td>RR2 %</td><td>0</td><td>0</td><td>0.2</td><td>0.17</td><td>0</td><td>0</td></tr><tr><td>No %</td><td>0</td><td>1</td><td>0.8</td><td>0.3</td><td>1</td><td>0</td></tr><tr><td>nbr of simulations</td><td>20</td><td>160</td><td>860</td><td>440</td><td>160</td><td>180</td></tr></tbody></table>

the production of a regenerated product. Indeed, for Sc2 and Sc5 we obtain a negative potential gain, so, the regeneration decision could be to stock these products and wait for market variables (DS) to be favorable to regenerate them. Nevertheless, in cases where the probability of reaching the DS is less than 50%, for example, it is possible, thanks to the decision making system, to predict which regeneration trajectory has the best chance of leading to a result that is beneficial to the company. In addition, it is noted that the more a product has the possibility of trajectory as for example for Sc3 and Sc4, the more the number of simulation is high. The higher number of simulations can be explained by the fact that in these scenarios, depending on the drawing of the probability laws, several trajectories can be envisaged such as remanufacturing with disassembly, reworking and reassembly activities and recycling. The results show that the simulation number varies to converge and stop the data processing. However, with stopping criteria, no time is lost and the assessment can switch to another product to be regenerated.

One way to reduce trajectory fluctuations would be to reduce uncertainties in both the uncertain product variables and the regeneration processes that can deteriorate the product if the operation to restore the product fails.

In addition, the different scenarios show that the results vary depending on the DS to be achieved, the product condition, and the available inventory. Thus, decision making is needed because there are many possible alternatives. Moreover, the settings of the processes can also influence the regeneration trajectory, in particular the performance of the regeneration operations which, through a discrete draw, determines whether the operation is successful or not.

## 5 Conclusion

Regeneration processes plays a key role in the EoL's products valorization. They effectively enable used products to be reintroduced directly into the market. In this paper, we have proposed a formalization, in order to identify the variables influencing the decision making. This formalism is the starting point in the methodology to build a tool to help the assessment of the regeneration level by modeling generalized colored stochastic Petri nets. This proposed approach takes into account some uncertainties related to the product's health state and some uncertainties concerning the regeneration processes. The proposed model was applied to an industrial instance of the literature (EBS Module).

A first application showed the approach feasibility, but also the limits of the GCSPN modelling. If the proposed method was applied to large scale systems, the complexity of the model would explode because of a large number of sub-assemblies and components. Indeed, for each disassembled element a decision making activity is required to guide this element towards the most appropriate regeneration process for it. So, the structure of the model is based on the proposed regeneration processes (reuse/remanufacturing / recycling), and the product information is carried out by the token color. This choice of modelling implies redefining all the transition function equations if the product characteristics change. One perspective is to propose generic models regardless of the product nomenclature.
ture and the variables characterizing the health state. This generalization would allow the methodology to be applied to more complex products and processes.

In Sect. 4, the equation's time is determined for the application but this time depends on the capacity of the computer and the transition place number of the GCSPN model. Indeed, for a more complex product with more elements in the model, the simulation time would be longer. Thus, a relation between the number of place/transition in the model should be established to obtain an equation for the execution time. In addition, we should analyse the impact of the execution time in different regeneration processes when a trajectory implies several random drawings in probability laws. A perspective would be to compare this time with another decision making approach with the same product studied. Another perspective is to compare our approach with statistical optimisation approaches such as that of [12] in order to determine the characteristics of the products to be treated by our approach.

**Funding**

No funding was received for conducting this study.

**Availability of data and materials**

Not applicable.

**Code availability**

Not applicable.

**Declarations**

**Ethics approval**

All authors accept the ethical standards of the journal.

**Consent to participate**

Not applicable.

**Consent for publication**

Not applicable.

**Competing interests**

The authors declare that they have no competing interests.

**Authors' contributions**

All authors contributed to the study conception and design. All authors read and approved the final manuscript.

**Publisher's Note**

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 1 November 2021 Accepted: 15 February 2022

Published online: 12 March 2022

**References**

1. R.H. Arduin, G. Grimaud, J.M. Leal, S. Pompidou, C. Charbuillet, B. Laratte, T. Alix, N. Perry, Influence of scope definition in recycling rate calculation for European e-waste extended producer responsibility. Waste Manag. 84, 256–268 (2019)

2. B.J. Pine, B. Victor, A.C. Boynton, Making mass customization work. Harv. Bus. Rev. 71(5), 108–111 (1993)

3. L. Diez, Apport de la maintenance prévisionnelle au paradigme de régénération industrielle. PhD thesis, Université de Lorraine, Nancy, France, 2017

4. M.L. Bentaha, A. Voisin, P. Marangé, O. Battaïa, A. Dolgui, Prise en compte de l'état des produits pour la planification de leur désassemblage. J. Eur. Syst. Autom. 49(4–5), 579–605 (2016)

5. A. Bufardi, R. Gheorghe, D. Kiritsis, P. Xirouchakis, Multicriteria decision-aid approach for product end-of-life alternative selection. Int. J. Prod. Res. 42(16), 3139–3157 (2004)

6. A. Ziout, A. Azab, M. Atwan, A holistic approach for decision on selection of end-of-life products recovery options. J. Clean. Prod. 65, 497–516 (2014)

7. Y.A. Alamerew, D. Brissaud, Circular economy assessment tool for end of life product recovery strategies. J. Remanufacturing 9(3), 169–185 (2019)

8. Y.A. Alamerew, M.L. Kambanou, T. Sakao, D. Brissaud, A multi-criteria evaluation method of product-level circularity strategies. Sustain. 12(12), 5129 (2020)

9. M. Gao, M. Zhou, Y. Tang, Intelligent decision making in disassembly process based on fuzzy reasoning petri nets. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 34(5), 2029–2034 (2004)

10. S.-E. Zhao, Y.-L. Li, R. Fu, W. Yuan, Fuzzy reasoning petri nets and its application to disassembly sequence decision-making for the end-of-life product recycling and remanufacturing. Int. J. Comput. Integr. Manuf. 27(5), 415–421 (2014)

11. X. Guo, S. Liu, M. Zhou, G. Tian, Disassembly sequence optimization for large-scale products with multiresource constraints using scatter search and petri nets. IEEE Trans. Cybern. 46(11), 2435–2446 (2015)

12. M.-L. Bentaha, A. Voisin, P. Marangé, A decision tool for disassembly process planning under end-of-life product quality. Int. J. Prod. Econ. 219, 386–401 (2020)

13. N. Aras, V. Verter, T. Boyaci, Coordination and priority decisions in hybrid manufacturing/remanufacturing systems. Prod. Oper. Manag. 15(4), 528–543 (2006)

14. J. Hanafi, S. Kara, H. Kaebernick, Reverse logistics strategies for end-of-life products. Int. J. Logist. Manag. 19(3), 367–388 (2008)

15. M. Ramezani, M. Bashiri, R. Tavakkoli-Moghaddam, A new multi-objective stochastic model for a forward/reverse logistic network design with responsiveness and quality level. Appl. Math. Model. 37(1–2), 328–344 (2013)

16. B. Doualle, K. Medini, X. Boucher, D. Brissaud, V. Laforest, Selection method of sustainable product-service system scenarios to support decision-making during early design stages. Int. J. Sustain. Eng. 13(1), 1–16 (2020)

17. M. Bettinelli, M. Occello, D. Genthial, D. Brissaud, A decision support framework for remanufacturing of highly variable products using a collective intelligence approach. Proc. CIRP 90, 594–599 (2020)

18. A.C. Karmperis, K. Aravossis, I.P. Tatsiopoulos, A. Sotichros, Decision support models for solid waste management: review and game-theoretic approaches. Waste Manag. 33(5), 1290–1301 (2013)

19. J.W. Forrester, System dynamics, systems thinking, and soft or. Syst. Dyn. Rev. 10(2–3), 245–256 (1994)

20. B. Abichou, A. Voisin, B. Iung, Choquet integral capacity calculus for health index estimation of multi-level industrial systems. IMA J. Manag. Math. 26(2), 205–224 (2015)

21. N. Gharbi, M. Ioualalen, Evaluation des performances et de la fiabilité des systèmes multi-classes avec rappel à l'aide des réseaux de petri stochastiques colorés. PhD thesis, Université des sciences et de la technologie Houari Boumediène, Algérie, 2007

22. W. Ijomah, A model-based definition of the generic remanufacturing business process. J. Clean. Prod. (2002)

23. V. Ravi, Evaluating overall quality of recycling of e-waste from end-of-life computers. J. Clean. Prod. 20(1), 145–151 (2012)