Franck Marle

Min Xie

An integrated

Keywords: risk response planning, project management, complexity, design structure matrix, resource constraints, genetic algorithm

HAL is

Introduction

Engineering projects require the timely accomplishment of a number of tasks, which are exposed to risks of delay, erroneous or low quality completion, incompletion, etc. The Project Management Institute (PMI) defines a project as "a temporary endeavor undertaken to create a unique product, service or result", and a risk as "an uncertain event or condition whose occurrence affects at least one of the project objectives, e.g., scope, schedule, cost, and quality" [START_REF] Xie | A Guide to the Project Management Body of Knowledge (PMBOK)[END_REF]. The classical Project Risk Management (PRM) process includes risk identification, risk analysis, risk response planning, risk monitoring & control and lessons learned. In particular, project risk response planning aims at identifying actions that can reduce the threats to the realization of the project objectives at minimum cost. It includes the identification and assignment of one or more persons (the "risk response owner") to take responsibility for each agreed-to and funded risk response action. Risks are addressed by their priorities in terms of their impact on the project. Resources are then assigned to the budget and risk response actions are scheduled in the project plan.

Risks are generally identified using more or less structured methods involving a combination of experience, expertise and information search [START_REF] Maytorena | The influence of experience and information search styles on project risk identification performance[END_REF], with classical methods, for example, based on analogy [START_REF] Riek | From experience: Capturing hard-won NPD lessons in checklists[END_REF], heuristics [START_REF] Chapman | The controlling influences on effective risk identification and assessment for construction design management[END_REF] or analysis [START_REF] Shimizu | Reliability problem prevention method for automotive components-development of GD'3' activity and DRBFM method for stimulating creativity and visualizing problems[END_REF]. They are generally assessed with respect to their probability and impact [START_REF] Xie | A Guide to the Project Management Body of Knowledge (PMBOK)[END_REF][START_REF] Raz | Use and benefits of tools for project risk management[END_REF][START_REF] Williams | A classified bibliography of recent research relating to project risk management[END_REF]. For risk prioritization, a very common tool in risk management practice for projects and other contexts is the "risk matrix" or "probability-impact grid" (PIG) or "probability-impact graph" [START_REF] Chapman | How to manage project opportunity and risk: why uncertainty management can be a much better approach than risk management[END_REF][START_REF] Cox | What's wrong with risk matrices?[END_REF][START_REF] Ahmed | A review of techniques for risk management in projects[END_REF]. Top-ranked critical risks are then subject to budget allocation and action planning for prevention or mitigation. The other risks identified are not treated, because the risk is regarded acceptable (in terms of both probability and impact) or the action is too expensive and there is no sufficient budget remaining.

However, engineering projects are growing in complexity, of both structure and context due to the involvement of numerous, diverse and strongly interrelated elements [START_REF] Baccarini | The concept of project complexity -a review[END_REF][START_REF] Chu | Theories of complexity -Common denominators of complex systems[END_REF][START_REF] Tiwana | Functionality risk in information systems development: an empirical investigation[END_REF]. This situation exposes projects to a number of diverse and interdependent risks, which implies that identifying and analyzing their causes and effects is an important aspect. For instance, Failure Modes and Effects Analysis (FMEA) consists in a qualitative analysis of dysfunction modes and their effects [START_REF] Bradley | An alternative FMEA method for simple and accurate ranking of failure modes[END_REF]. Initially developed for product-related risks, it has been expanded to process-related and project-related risks, where the focus changes, but the principle is the same, consisting in identifying direct causes and effects of a potential failure. Fault Tree and Cause Tree Analyses determine the conditions which lead to an event, and link them through logical connectors in a tree-structure which clearly displays causes and effects of the particular risk analyzed [START_REF] Geymayr | Fault-tree analysis: a knowledge-engineering approach[END_REF][START_REF] Ferdous | Fault and event tree analyses for process systems risk analysis: uncertainty handling formulations[END_REF]. Some methods have been considered for analyzing the interrelationships among risks, such as Bayesian Belief Networks [START_REF] Lee | Large engineering project risk management using a Bayesian Belief Network[END_REF][START_REF] Fan | BBN-based software project risk management[END_REF], System Dynamics [START_REF] Lyneis | System dynamics applied to project management: a survey, assessment, and directions for future research[END_REF][START_REF] Xu | Optimizing circular economy planning and risk analysis using system dynamics[END_REF][START_REF] Williams | Modelling complex projects[END_REF][START_REF] Williams | Project risk: systemicity, cause mapping and a scenario approach[END_REF], and Influence Diagrams [START_REF] Dikmen | Using fuzzy risk assessment to rate cost overrun risk in international construction projects[END_REF].

In addition, risk analysis methods and risk response planning methods do not share the same objectives. Risk analysis methods can help to identify actions (for instance, preventive actions by inferring the causes of a risk from a bow-tie diagram), but they do not indicate how to decide on which actions to undertake or not. Within the risk decision-making process, these methods perform the step of searching alternatives, not the step of sorting / ranking the alternatives. In the end, risk responses must be appropriate, cost effective, and realistic within the project context. Selecting the best risk response from several options is often required. To measure the effectiveness of an action or of a portfolio of actions is not easy, since it affects an uncertain event with the additional uncertainty inherent to the planning and execution of the action itself.

In our work here presented, the complexity underlying the web of interconnections among project risks is modeled and represented in terms of a risk network [START_REF] Fang | A simulation-based risk network model for decision support in project risk management[END_REF]. Such network representation captures the individual risks and the interactions which may trigger global phenomena, like chain reactions or loops. For instance, a single source risk such as project schedule delay, may impact on the risk of cost overrun, which influences a technical risk, and propagates looping back to amplify the original delay. Then, the effects of response actions designed for mitigating the exposure to one or several risks may impact other parts of the network, so that the overall effects of risk response actions may be very different from the expectation of project managers. The challenge of risk response planning is rendered more difficult by the limitation of resource. As constraints become tighter, balancing risks is more critical and less intuitive. For these situations, reliable analytical methods can help project managers plan risk response actions that optimize resource allocation [START_REF] Dillon | Programmatic risk analysis for critical engineering systems under tight resource constraints[END_REF][START_REF] Dillon | Optimal use of budget reserves to minimize technical and management failure risks during complex project development[END_REF][START_REF] Borgonovo | A study of interactions in the risk assessment of complex engineering systems: an application to space PSA[END_REF].

In this paper, a novel integrated five-step framework is introduced to guide the risk response planning process, which is to determine and implement preventive and corrective actions to avoid, reduce or transfer project risks. A matrix-based method is used to facilitate identifying and assessing risk interactions, and build the representative project risk network. This enables the risk propagation behavior in the network to be analyzed. It is then possible to anticipate the global effects of response actions identified by the project management team.

Thus, the framework supports project managers" decision-making process in coping with the complexity of project risks and resource constraints. An example of application to a real industrial engineering project, which consists in implementing a tramway system in a medium-sized city in Europe, is considered.

For this case study, a genetic algorithm is developed to optimize the plan of response actions under given budget constraints. Genetic algorithm (GA) is a probabilistic search method introduced by Holland in 1970s [START_REF] Holland | Adaptation in natural and artificial systems: An introductory analysis with applications to biology, control, and artificial intelligence[END_REF]. It is based on Darwin"s principle of "survival of the fittest", and has rapidly become a popular evolutionary technique for solving complex combinatorial optimization problems, in a wide range of applications [START_REF] Mitchell | An introduction to genetic algorithms[END_REF]. For example, they have been extensively used for the optimization of system reliability and maintenance [START_REF] Marseguerra | Basics of genetic algorithms optimization for RAMS applications[END_REF][START_REF] Wattanapongskorn | Fault-tolerant embedded system design and optimization considering reliability estimation uncertainty[END_REF][START_REF] Cadini | Optimal expansion of an existing electrical power transmission network by multiobjective genetic algorithms[END_REF][START_REF] Ye | Some improvements on adaptive genetic algorithms for reliability-related applications[END_REF], index fund portfolio management [START_REF] Oh | Using genetic algorithm to support portfolio optimization for index fund management[END_REF][START_REF] Gilli | Heuristic optimisation in financial modelling[END_REF], project scheduling [START_REF] Hartmann | A self-adapting genetic algorithm for project scheduling under resource constraints[END_REF][START_REF] Peteghem | A genetic algorithm for the preemptive and non-preemptive multi-mode resource-constrained project scheduling problem[END_REF][START_REF] Huang | Project-scheduling problem with random time-dependent activity duration times[END_REF] and machine scheduling problems [START_REF] Chang | Generating artificial chromosomes with probability control in genetic algorithm for machine scheduling problems[END_REF][START_REF] Vallada | A genetic algorithm for the unrelated parallel machine scheduling problem with sequence dependent setup times[END_REF]. The GA results are compared with those obtained by using a greedy algorithm, which is based on Sequential Forward Selection (SFS) [START_REF] Whitney | A direct method of nonparametric measurement selection[END_REF], where the search for the optimal solution proceeds by making the locally optimal choices at each step, with the hope of finding the global optimum.

The remainder of the paper is organized as follows. Section 2 introduces the integrated framework for risk response planning under resource constraints. Section 3 describes the process of building the project risk network and a risk propagation model. In Section 4, the remaining steps of the framework and the developed algorithms for optimizing the risk response plan are presented in details. Section 5 illustrates the application of the proposed approach to a real industrial project. Finally, we conclude the paper in Section 6.

An integrated framework for risk response planning

In this Section, a five-step framework for project risk response planning is presented (Fig. 1):

1) Building project risk network;

2) Defining objective function;

3) Identifying budget constraints; 4) Identifying potential response actions; 5) Optimizing risk response plan.

Building the project risk network allows us to follow risk propagation in the project.

Potential risk response actions can then be proposed, given the risk management objectives and budget constraints. The effects of these response actions can be traced and anticipated in the risk network model. Embedding these analyses within an optimization algorithm (like the SFS greedy algorithm or the genetic algorithm used in this paper) allows searching for an optimal project risk response plan.

The details of each step of the framework are discussed in the following Sections 3 (step 1, which consists of a few sub-steps) and 4 (steps 2 to 5). In practice, the implementation of the proposed framework requires the involvement of the project management team in each step, to provide the necessary project knowledge and expertise and to take decisions.

-7 -Fig. 1. Framework for risk response planning

Building project risk network (step 1)

The project risk list containing previously identified potential risks is provided by the project management team (step 1.1). It serves as an input for studying risk interactions in order to build the project risk network.

Identification of risk interactions (step 1.2)

The Design Structure Matrix (DSM) method introduced in [START_REF] Steward | The Design Structure Matrix: a method for managing the design of complex systems[END_REF] has proven to be a practical tool for representing and analyzing relations and dependencies among system components [START_REF] Browning | Applying the design structure matrix to system decomposition and integration problems: a review and new directions[END_REF][START_REF] Danilovic | Managing complex product development projects with design structure matrices and domain mapping matrices[END_REF]. For example, it has been extensively used in process modeling and project scheduling problem for design and product development projects, such as in [START_REF] Browning | Modeling impacts of process architecture on cost and schedule risk in product development[END_REF][START_REF] Lévárdy | An adaptive process model to support product development project management[END_REF][START_REF] Cho | A simulation-based process model for managing complex design projects[END_REF][START_REF] Qian | A novel approach to DSM-based activity sequencing problem[END_REF][START_REF] Smith | Identifying controlling features of engineering design iteration[END_REF].

In this work, we use the DSM method to identify risk interactions, for determining the causeeffect relationships among project risks. It provides a simple and concise way to represent the inter-relationships in a complex system. This helps the project manager and the experts focusing on one risk and its dependency with other risks (causes in row and effects in column) during the identification and also the subsequent assessment process, while not getting confused in the complex interrelationships among risks. In addition, the possible existing DSMs representing the interrelations among project objects, such as tasks, actors and product components, can be used to guide the identification of the interactions among the risks associated to these objects. For example, an object-object relationship (whether functional, structural or physical) means that risks, which may be related to product function, quality, delay or cost, can be linked, since a problem on one object may have an influence on another.

For instance, the project schedule gives information about task-task sequence relationships; this enables identifying relationships among risks of delay on these tasks.

Moreover, a number of DSM tools and algorithms have been developed to facilitate systemic information acquisition and matrix-based analysis, e.g., in [START_REF] Lindemann | Structural complexity management: an approach for the field of product design[END_REF][START_REF] Eppinger | Design structure matrix methods and applications[END_REF]. Although applying these DSM tools/algorithms is not in the scope of this paper, using the DSM methods may provide possible solutions (e.g., in risk grouping and risk owner assignment) for other managerial purposes.

Risk interaction consists of a precedence relationship between two linked risks. We can represent this by the Risk Structure Matrix (RSM), which is a square matrix whose generic element:

1 if there is an interaction between risks and 0 otherwise

ij i j ij RSM R R RSM        (1)
Fig. 2 shows an example of a risk structure matrix capturing the relationships in the risk network.

Fig. 2. Risk network and Risk Structure Matrix (adapted from [START_REF] Fang | A simulation-based risk network model for decision support in project risk management[END_REF])

In the process of building the risk network structure, a sanity check is performed.

Suppose we know that R j has R i as a cause: if R i does not have R j as a consequence, then there is a mismatch. All identified mismatches are studied and solved, like in [START_REF] Sosa | The misalignment of product architecture and organizational structure in complex product development[END_REF]. Multiple experts are engaged for this task, after being made aware of the possible confusion between direct and indirect interactions among risks, and being asked to concentrate on direct dependencies.

For solving mismatches, the two actors involved at each end of the edge are asked to confirm or to deny their initial proposal by discussing together. Generally, people are more easily aware of potential causes that may affect them, rather than potential effects of their own failures. This is why these discussions are mandatory and useful, both for creating a reliable input matrix and for creating links among people.

Assessment of risk interactions (step 1.3)

In the assessment task, we not only evaluate risk characteristics such as impact and probability, but also assess the strength of risk interactions (interpreted as transition probability between risks). Risk impact may be assessed on a qualitative scale (ordinal or cardinal scale with 5 or 10 levels for instance) or on a quantitative scale (financial loss for instance). Risk impact is assessed by classical methods, based upon a mix of previous experience and expert judgment [START_REF] Xie | A Guide to the Project Management Body of Knowledge (PMBOK)[END_REF][START_REF] Chapman | Project Risk Management -Processes, Techniques and Insights[END_REF].

For the probability assessment, we make a distinction between the probability of a risk to be triggered by another risk inside the network, and its probability caused by external events or risks which are outside the system. Spontaneous probability can be interpreted as the likelihood of a risk which is not the effect from other activated risks inside the system. On the other hand, transition probability measures the likelihood of direct cause-effect relation between two risks. For the example in Fig. 2, Risk 5 occurs only by spontaneous probability;

and Risk 6 may arise from both its spontaneous probability and the transition probability between Risk 5 and Risk 6.

Qualitative scales are often used to express risk probability with 5 to 10 levels (e.g., very rare, rare, unlikely, likely, etc.), which typically correspond to non-linear probability values (e.g., 10 -4 , 10 -3 , 10 -2 , 10 -1 , etc.) [START_REF] Cox | What's wrong with risk matrices?[END_REF][START_REF] Vose | Risk Analysis: a Quantitative Guide[END_REF].

A risk propagation model (step 1.4)

Some DSM-based work has been done to model the propagation or transmission behavior in the design process. For example, Clarkson and Hamilton proposed a "signposting" model to identify the next design tasks based on the confidence in key design parameters [START_REF] Clarkson | Signposting", a parameter-driven task-based model of the design process[END_REF]; Smith and Eppinger introduced a work transformation matrix based on the DSM method to model the engineering design iteration process [START_REF] Smith | Identifying controlling features of engineering design iteration[END_REF]. In the domain of project risk management, a matrix-based risk propagation model has been presented in [START_REF] Fang | Modelling risk interactions to re-evaluate risks in project management[END_REF]. This risk network model can be used to predict the global effects of response actions on the entire risk network.

Suppose there are N identified project risks in the network. Let vector s represent their spontaneous probabilities, i.e. the initial vector of risk probabilities before propagation in the network. Let the N-order square matrix T denote the matrix of transition probabilities. We make the assumption that a risk may occur more than one time during the project (as witnessed in practical situations). Risk probability is thus cumulative if arising during propagation from different causes or several times from the same cause. After m steps of propagation, the probability vector of risks is thus equal to m Ts  and the cumulative risk probability vector P is given by the following equation:

1 1 0 ( ) ( ) m m m i i i i i i P s T s I T s T s               ( 2
)
where I is the N-order identity matrix. In the limit of infinite propagation steps in the project development, 0 lim( )

m i m i P T s     (3)
Multiplying both sides of Eq. ( 3) by (I -T),

1 0 ( ) ( ) ( ) ( ) m i m i I T P I T T s I T s             (4)
-12 -It is not guaranteed that the infinite product of the transition matrix T would converge to 0, as shown in the following equation:

lim 0 m m T   (5)
Sufficient conditions for the convergence of an infinite product of matrices have been given, e.g., in [START_REF] Holtz | On convergence of infinite matrix products[END_REF][START_REF] Daubechies | Sets of matrices all infinite products of which converge[END_REF][START_REF] Bru | Convergence of infinite products of matrices and inner-outer iteration schemes[END_REF]. Since in our case T is the risk transition matrix, which is usually sparse and composed of transition probability values less than 1, convergence is usually satisfied. Thus, the cumulative risk probability vector can be re-evaluated as:

1

() P I T s     (6)
Response actions performed on the risk network translate in changes in the values of the parameters of the model, e.g., the spontaneous probabilities in vector s, the transition probabilities in matrix T. The global effects of these actions in terms of the new values of the risk probabilities in the vector P after actions implementation can then be obtained by running the propagation model.

Formulating and solving the optimization problem

Defining objective function (step 2)

Generally, risk response actions with allocated budget are conducted to achieve two different goals: the local mitigation of particular risks and the global risk exposure mitigation.

In this paper, we only consider minimizing the overall risk exposure or expected financial loss in global sense. In this regard, the objective function OF can be defined as:

1

N ii i OF P G    (7)
-13 -where P i and G i indicate the probability and impact (G for gravity scale or financial value) of Risk i.

Identifying budget constraints (step 3)

Given the project scope, a budget for project risk management, called B RM , is initially established by the project manager. This budget is dependent on the total budget of the project, the evaluated overall level of risk exposure, and also the risk attitudes of the stakeholders.

The budget B RM is normally comprised of three parts. Besides the expense for performing risk analysis B RA (not significant compared with the other parts) and the reserve for risk contingency B RC , the remaining amount B RR is for the execution of the risk response plan:

RR RM RA RC B B B B    (8)
It should be noted that based on the results of the project risk analysis and of the evaluation of the costs of actions in Step 4 (Fig. 1), the budget for performing the risk response plan B RR can be updated according to the new knowledge acquired with regard to the risk management tasks.

Identifying potential response actions (step 4)

The identified project risks can be analyzed and prioritized using classical methods or a simulation model based on the risk network [START_REF] Fang | A simulation-based risk network model for decision support in project risk management[END_REF] (step 4.1). However, it is not the main concern of this paper. Aiming at achieving the objectives defined for risk management, for example, mitigating the global risk exposure as mathematically captured by the OF in Eq. ( 7), potential response actions can be identified based on the project risk analysis results (step 4.2). The response action list may include different types of risk response actions on risks and their interactions, in terms of risk sharing, risk avoidance, risk mitigation and risk acceptance, etc. These actions are, for instance, adopting less complex processes, conducting more tests, enhancing internal communication, choosing a more stable supplier, etc. From the point of view of the framework of modeling and analysis, conducting the response actions has the effects of changing the values of some of the parameters of the risk network model. For example, a classical response action on a particular risk reduces its spontaneous probability or impact; a complementary preventive action is to cut off the input links or reduce their transition probabilities; blocking the output links can be regarded as the action of confining the further propagation of such risk to subsequent risks.

Risk response actions always consume time, money and other resources. In order to perform the optimization, the cost of each identified action is evaluated by the project management team (step 4.3). Actions should be worthwhile, i.e., more valuable than the expected value of the risk impact. Before the next step of optimization, the response action list shall be examined by the project manager to exclude the unfeasible actions.

Optimizing risk response plan (step 5)

For each risk response action identified in Step 4, the project manager can decide whether to implement it or not. Given a list of n candidate actions, there are 2 n -1 combinations for the risk response plan aiming at mitigating the overall risk exposure (the global objective function). An exhaustive test of all the combinations is impractical.

Considering the resource constraints, heuristic algorithms can be exploited to optimize the portfolio of response actions: here, we provide two examples of such algorithms which are then applied on a real case study in Section 5.

A SFS greedy algorithm

A greedy algorithm based on Sequential Forward Selection is developed for the optimization of a risk response plan under constraints. At each step the action with the best test performance is chosen until the budget is completely allocated. The risk propagation model presented in Section 3.3 can be used to evaluate the mitigation performance of actions in terms of the OF in Eq. ( 7).

The SFS greedy algorithm is sketched as follows:

Usually such greedy algorithm for optimization under constraints can achieve only a locally optimal solution because it makes commitments to certain choices too early, which

( * i A A A   ); Remove A i * from L : ( * \ i L L A  );
Allocate the corresponding amount of budget ( * ()

RR RR i B B C A  );
END RETURN A as the optimal portfolio of actions.

prevent it from finding the best overall solution later. For example, choosing at an early stage an action with positive effects but expensive reduces the budget remaining for future actions, with the risk of sacrificing opportunities.

A genetic algorithm

In our work, a genetic algorithm is devised for the optimization of a project risk response plan. The aim is to find an optimal portfolio of actions, whose performance is measured by an objective function (fitness) which integrates the budget constraint. The synergic effects (positive or negative) of the actions in the portfolio are taken into account, because the entire portfolio is tested on the risk network model, while not just the single actions separately.

The basic genetic algorithm-based optimization process is described as follows:

1) Basic Scheme indicates whether the corresponding action A i is chosen in the portfolio or not.

1) Representation

2) Fitness

We integrate the budget constraint into the objective function (fitness) of the optimization problem, aiming at minimizing the value:

1 ( )+(1 )( / ) N i i RR i Fitness f P G C B           ( 9
)
Here C is the total cost of the action plan; i P  and i G  are the probability and impact of Risk i after the implementation of the response plan. The penalty value The details of the GA process are introduced in the Appendix.

Application to a real industrial project

The framework proposed has been implemented to a real engineering project aimed at building a tramway infrastructure and associated systems. The project includes the construction and implementation of tramway, equipment, and civil work.

Build the project risk network (step 1)

An original project risk list has been provided by the project manager. It contains 56 identified risks at the main level, with their names, domains and qualitatively evaluated characteristics, as shown in Using the DSM-based method introduced in Section 3.1, the interactions among the 56 risks have been identified with the help of the project manager and the team of experts, composed of the 11 risk owners. For each risk, experts were asked to provide information about the potential causes and effects (to explore the row and the column corresponding to the considered risk in the risk structure matrix). The aggregation of local cause-effect relationship identifications enables to build the global risk network.

As anticipated in Section 3.2, the assessment of the identified risk interactions was then performed on a 10-level Likert scale, due to the high expertise of interviewees. This requires the participation of several experts involved in the project, since it necessitates a wide overview of the project elements and stakes. In this case study, four risk owners, including the project manager, were mostly contributing to the data gathering. The other owners and an external risk manager were only solicited to give some specific and local information, and to validate existing data. In the end, the binary risk structure matrix can be transformed into the matrix of the transition probabilities between risks.

Table 1. Tramway project risk list and related characteristics

Define the mitigation objective and budget constraint (steps 2 and 3)

In this prototype application, the aim is to mitigate the global risk exposure, and for this, the objective function in Eq. ( 7) is used as the function for which minimization is sought. The impact of risks is assessed in terms of qualitative severity scale (from 1 to 10) for this case study, as shown in Table 1.

We suppose in this case study that the budget reserve for implementing the risk response plan is B RR = 300 k€.

Build the action list (step 4)

With the help of the project management team, a list of potential risk response actions is proposed, as reported in Table 2. The 21 proposed actions are based on a refined analysis, taking into account interactions between risks and eliminating some unfeasible ones. The actions are intended to mitigate the risk nodes (reduce risk spontaneous probability or risk impact) or the risk interaction edges (reduce transition probability between risks). The local effects of the response actions are estimated (Table 2). The global effects of the actions can be predicted using the risk propagation model described in Section 3.3. The cost for executing these actions is also estimated by the project management team.

Optimize the portfolio of actions (step 5)

Optimization results obtained using both the SFS greedy algorithm and the genetic algorithm are illustrated and compared in this Section.

Greedy algorithm results

The SFS greedy algorithm devised in Section 4.4.1 has been used to obtain a portfolio of actions, given the budget constraint B RR = 300 k€. The results are reported in Table 3, following the successive iterations of optimal action addition to the portfolio.

Genetic algorithm results

In the genetic algorithm, the population size is set to M = 100 individuals. The Roulette Wheel Method is used for selecting the parents for the next generation. The crossover fraction is set to 0.8, and the mutation rate is set to 0.01 by testing. The termination condition is set as either 1) the maximum number of generations GEN* = 100; or 2) there is no improvement in the best fitness value for 20 successive generations.

For the parameters of the fitness function f of Eq. ( 9), we set =0.9  , =0.95  and =20  by experience and testing. We have run the GA for twenty times with different random seeds and selected the best solution among them. In that run performed, the algorithm terminates at the 48 th generation and the best individual is the chromosome x * = [1, 1, 0,

, corresponding to the decoded optimal portfolio A * = [A1, A2, A4, A5, A6, A7, A8, A9, A11, A12, A13, A16, A21]. The best fitness value is equal to 39.052. The total cost of implementing action plan A * is 295 k€. The objective function of global risk exposure in Eq. ( 7)

is reduced to the value of 43.169.

Comparison with the results of the SFS greedy algorithm (Table 4) shows that in the optimal solution obtained by the genetic algorithm, the action A3 has been replaced by the combination of A7, A11 and A21. In this case, the required budget for the portfolio is the same, but the optimal risk response plan has better effects on the objective of mitigating the global risk exposure. The parameters of the genetic algorithm can be modified to reflect the adjustment of strategy by the risk management. For example, if we set 0.8   to strengthen the control over the budget, the optimal portfolio becomes A * = [A1, A2, A4, A5, A6, A7, A8, A9, A12, A13, A16, A21]. We can see that A11 has been removed from the action plan so that the required budget has decreased to 275 k€, with an objective function value of 43.443.

On the other hand, if we increase the balance factor  to 0.95 for emphasizing the mitigation effects, the optimal portfolio becomes A * = [A1, A2, A3, A4, A5, A6, A8, A9, A12, A13, A16, A21]. In this case, A3 has replaced the actions A7 and A11. As a result, the objective function has improved to 42.963. However, extra budget is required to achieve such result, for a total cost of the risk response plan equal to 320 k€.

Discussion

This study has been motivated by questions and requests by practitioners, who are ready to apply more sophisticated techniques to make decisions about their risk response plans. They were confident in the results of the case study on the tramway construction project, since both algorithms confirmed their priorities.

Apparently, the comparison of the results on the case study indicates that the genetic algorithm provides a superior search for the optimum than the greedy algorithm. The deficiency of the SFS greedy algorithm is that only the effect rather than the cost of actions is considered as the basis for local searches, which may prevent it from finding the global optimal solution. On the contrary, through testing the entire response plans while not individual actions in the risk network model, the genetic algorithm takes into account the synergy or co-effects of different actions for mitigating. Moreover, by adjusting the parameters of the fitness function, the project manager is able to achieve a trade-off between improving risk management results and lowering the budget.

However, the practitioners were attached to the sequence of inclusion of actions in the portfolio by the SFS greedy algorithm, even if in a global optimization algorithm, like GA, this could not have any importance. Specifically, they were confident on inclusion of actions A16, A5 and A9, rather than A13, A12 and A3. On this last action A3, they were ready to include it in their action plan, and both greedy algorithm (since this was the last action included in the portfolio) and genetic algorithm (since it was not included in the optimal portfolio, but embraced after relaxing the budget constraint) proved helpful in convincing them to change their plan in such direction. In general, it is to be expected that the optimization should change only some elements of an action plan, and not make a complete revolution, since decision-makers are capable of identifying the most important and efficient response actions. The optimization work can help in the decisions for actions which are close from inclusion or exclusion, and in the identification of possible big surprises, although less frequent.

One may wonder when to perform this process of data gathering and related analysis. In most cases, the earlier, the better. Indeed, it changes the risk response plan, with its associated budget, resources and actions, so it is recommended to change decisions before they are applied.

However, information may be neither available nor reliable at the very beginning of the project, which may result in irrelevant action plans. The decision about the schedule (one or several times during the project) should thus be a balance between the necessities to do it early enough and to have enough reliable information. The best moment depends on the degree of uncertainty on data.

If projects are recurrent and some historical data are available, both on risks, risk interactions and risk response alternatives, then this process may be run at the earliest phases. But generally, if the context is new (country, subcontractors), or if the objectives are significantly different, then it is better to wait to have enough and more reliable data. In the case study presented here, the project had already been launched before the beginning of the study. Eight risk review meetings had been conducted before our intervention.

It should be admitted that there exist limitations of applying the proposed approach in practice. For example, the difficulties and uncertainties are unavoidable in identifying and quantifying the risk interactions using the DSM methods. First, the issue of a correct risk identification and particularly risk formulation is relevant. In this regard, efforts should be made by the project management team to determine the proper level of details and the way to formulate risks in less ambiguous ways. Second, it is sometimes difficult to differentiate direct and indirect interactions between risks, although the interviewees have been reminded to concentrate on direct dependencies. Third, dealing with project risks, especially the probabilities being used, includes subjective assessment and thus uncertainties. Subsequently, we have to be very careful when manipulating uncertain / unreliable data using optimization algorithms, since the output depends on the reliability of the inputs. One should not apply blindly the optimization results, but should analyze carefully the gap between the proposed solution and its neighbors.

Also, we have to be careful when using quantitative data, since we cannot have all the data which are quantitative, so the danger is to mix qualitative and quantitative data.

Conclusion and perspective

In this paper, we have presented an original framework for decision support in project risk response planning, and showed how it is applied to a real case study of a tramway construction project. Through modeling risk interactions, the framework makes it possible to analyze risk propagation behavior and thus to anticipate the overall effects of response actions on the global risk network. It can guide the project manager design some non-conventional actions on risk interactions which mitigates risk propagation instead of risk occurrence. For optimally allocating tight resources for risk mitigation, i.e., selecting the best risk response plan from an action list with many options, a Sequential Forward Selection greedy algorithm and a genetic algorithm have been investigated, taking into account budget constraints. The comparison of the results obtained by these two optimization algorithms shows that the genetic algorithm has superior performance. The proposed framework and quantitative methods are expected applicable to a wide set of engineering projects for risk management.

In addition to the limitations of the approach discussed above, for potential improvements, the stakeholders" or the project manager"s preferences would be included into the risk response planning process. For example, the mitigation of several particular risks is sometimes mandatory.

In addition, the portfolio of actions may be more complex. In practice, for instance, if more funds are allocated on the reinforcement of a component or task, the probability of its failure risk will decrease. In this regard, an action for mitigating risks, for example, A2 can be subdivided into several alternatives (e.g., A2.1, A2.2, and A2.3) with different levels of cost, which will undoubtedly generate different levels of mitigation effects. In this case, we need not only to decide whether to choose an action or not, but also to optimize the level of investment on each action and related risks. Furthermore, the developed DSM-based tools/techniques can be considered for managerial purposes concerning risk management, e.g., in risk grouping and risk owner organization. This work will also be considered for program management of multiple related projects with regard to risk management.

Appendix. The process of Genetic Algorithm for this study

1) Initial population

An initial population of M individual solutions is created randomly. Each individual is a risk response plan, namely a portfolio of actions. Population diversity (i.e., differences in the individuals) is encouraged to investigate more broadly the search space [START_REF] Nsakanda | Ensuring population diversity in genetic algorithms: A technical note with application to the cell formation problem[END_REF].

2) Selection of the parents

During each successive generation, a proportion of the existing population is selected to breed a new generation. Individual solutions are selected through a fitness-based process, where fitter solutions (with lower values of the fitness function) are more likely to be selected. We employ the straightforward Roulette Wheel Selection method [START_REF] Fogel | An introduction to simulated evolutionary optimization[END_REF][START_REF] Rajkumar | An improved genetic algorithm for the flowshop scheduling problem[END_REF]: the chromosome k

x is selected if:

1 1 1 1 1 ( ) ( ) ( ) ( ) k k j j j j M M j j j j f x f x r f x f x         ( 10
)
where r is the generated random number with   0, 1 r  .

3) Crossover and mutation

Crossover allows combining two parents to form a child. We employ a conventional scattered crossover as sketched in Fig. 3 [START_REF] Popov | Genetic algorithms for optimization[END_REF]. A random binary vector is created as bit mask. It selects the genes from parent 1 where the mask bit is "1", and the genes from parent 2 where the mask bit is "0", and combines the genes to form the child. It should be noted that in Fig. 3 the symbols a~h and 1~8 are replaced by binary bits in the work here presented. A crossover fraction of value in [0, 1] specifies the portion of the individuals in the next generation that are produced by crossover, other than the elite individuals (the number of individuals that are guaranteed to survive to the next generation). Elitism is the process of selecting individuals with a bias towards the better ones, which is based on fitness ranking in the developed GA. Indeed, elitism is important for allowing the solutions to get better over generations. Fig. 3. Illustration of the crossover operation Mutation inserts small random changes in the individuals of the population, which further favors genetic diversity. It thus enables the GA to extend the search to a broader space. A mutation rate is introduced as the probability that a bit in a chromosome will be reversed (0->1, 1->0). The mutation rate for a single bit is usually taken very low for binary encoded genes [START_REF] Senouci | Genetic algorithm-based multi-objective model for scheduling of linear construction projects[END_REF].

4) Reduction of population for the next generation

We use the conventional GA with fixed population size in this work. In this regard, fitness ranking is used to guide the reduction of the population for the next generation [START_REF] Marseguerra | Basics of genetic algorithms optimization for RAMS applications[END_REF]: the individuals with lowest fitness are removed from the enlarged population of parents and children, where the original size M is reestablished.

5) Termination condition

In this work, the search iterations of the GA are terminated simply when an a-priori fixed number of generations GEN * is reached, or when the top ranked solution's fitness has stabilized, i.e., a fixed number of successive iterations no longer improve it.

  budget constraint. Thus, breaking constraints is penalized by the decrease of the fitness. The parameter 1   reflects the project manager"s degree of aversion to budget overruns. The project manager can adjust the parameter [0,1] to balance the trade-off between budget constraints and mitigation effects.

  The optimal portfolio A * contains 11 actions:A * = [A1,A2, A3, A4, A5, A6, A8, A9, A12, A13, A16]. The total cost is 295 k€ and the value of the objective function, namely the overall risk exposure, has been reduced from 63.128 to 43.599 thanks to the identified actions.

Table 1 .

 1 The project risks identified with negative effects belong

	to	different	categories	such	as	Technical,	Contractual,	Financial,
	Client/Partner/Subcontractor, and Project management on construction site.

Table 2 .

 2 List of risk response actions[START_REF] Williams | Project risk: systemicity, cause mapping and a scenario approach[END_REF]

Table 3 .

 3 Optimization results using the SFS greedy algorithm

	Iteration	Selected Action ID	Cost (k€)	Objective Function Value	Added Effects	Current Portfolio	Allocated Budget (k€)
	Initial Status	-	0	63.128	0.000	[Null]	0
	1	A16	40	59.572	-3.556	[A16]	40
	2	A5	20	56.521	-3.051	[A16,A5]	60
	3	A9	5	54.367	-2.154	[A16,A5,A9]	65
	4	A2	10	52.558	-1.808	[A16,A5,A9,A2]	75
	5	A4	20	50.777	-1.781	[A16,A5,A9,A2,A4]	95
	6	A6	10	49.222	-1.555	[A16,A5,A9,A2,A4,A6]	105
	7	A1	35	47.910	-1.313	[A16,A5,A9,A2,A4,A6, A1]	140
	8	A8	20	46.641	-1.269	[A16,A5,A9,A2,A4,A6, A1,A8]	160
	9	A13	60	45.434	-1.208	[A16,A5,A9,A2,A4,A6, A1,A8,A13]	220
	10	A12	20	44.424	-1.009	[A16,A5,A9,A2,A4,A6, A1,A8,A13,A12]	240
	11	A3	55	43.599	-0.825	[A16,

A5,A9,A2,A4,A6, A1,A8,A13,A12,A3] 295

Table 4 .

 4 Comparison of the results obtained by the greedy and genetic algorithms

			Number of	Required	Objective
	Method	Optimal Portfolio
			Actions	Budget (k€)	Function Value
	SFS Greedy Algorithm	[A1, A2, A3, A4, A5, A6, A8, A9, A12, A13, A16]	11	295	43.599
	Genetic Algorithm	[A1, A2, A4, A5, A6, A7, A8, A9, A11, A12, A13, A16, A21]	13	295	43.169


Enrico Zio
email: enrico.zio@polimi.it

Francesco Di Maio

FAILURE PROGNOSTICS BY A DATA-DRIVEN SIMILARITY-BASED APPROACH

Keywords: Prognostics, Residual Useful Life, Pointwise Fuzzy Similarity Analysis, Particle Filtering, Crack Growth

This paper presents a data-driven, similarity-based approach for prognostics of industrial and structural components. The potentiality of the approach is demonstrated on a problem of crack propagation, taken from literature. The crack growth process is described by a non linear model affected by non-additive noises. A comparison is provided with a Monte Carlo-based estimation method, known as particle filtering.

Introduction

When the conditions of a component or structure can be monitored, maintenance can be planned dynamically on the basis of the observed conditions [START_REF] Williams | Condition-based maintenance and machine diagnostics[END_REF]Marseguerra et al., 2002]. This requires prognostic capability for predicting the evolution of the degradation state of the component or structure in the future. The aim of the prediction is to verify whether the component or structure can continue performing its function throughout its lifetime and, in case it cannot, to estimate the Remaining Useful Life (RUL), i.e. the time remaining before it can no longer perform its function [START_REF] Jardine | A review on machinery diagnostics and prognostics implementing condition based maintenance[END_REF]. In practice, the estimate of the RUL of a component or structure may be difficult to obtain, since its degradation state may not be directly observable and/or the measurements may be affected by noise and disturbances.

Approaches to failure prognostics can be categorized broadly into model-based and datadriven [START_REF] Chiang | Fault detection and diagnosis in industrial systems[END_REF]. Model-based prognostics attempts to set up physical models of the component or structure for the estimation of the RUL. However, uncertainty due to the assumptions and simplifications of the adopted models may pose limitations on this approach.

Many researchers have focused on the problem of building exhaustive models of deteriorating components and structures to implement model-based prognostic tools. Markov and semi-Markov models have been widely exploited for achieving analytical results [START_REF] Samanta | Degradation modeling with application to ageing and maintenance effectiveness evaluations[END_REF][START_REF] Lam | Optimal maintenance policies for deteriorating systems under various maintenance strategies[END_REF][START_REF] Hontelez | Optimum condition-based maintenance policies for deteriorating systems with partial information[END_REF]Kopnov, 1999;Yeh, 1997;[START_REF] Grall | Optimal dynamic inspection/replacement planning in condition-based maintenance[END_REF][START_REF] Bérenguer | Simulation and evaluation of conditionbased maintenance policies for multi-component continuous-state deteriorating systems[END_REF]. On the basis of these models, several approaches have been proposed to analyze reliability-based and condition-based maintenance policies [START_REF] Pulkkinen | Optimal operational strategies for an inspected component[END_REF][START_REF] Vlok | [END_REF][START_REF] Castanier | Stochastic maintenance planning for a repairable system which is inoperative during maintenance operations[END_REF].

The most promising approaches rely on Bayesian methods to combine a prior distribution of the unknown degradation states with the likelihood of the observations collected, to build a posterior distribution [CUED-F-ENGTR310, 1998;[START_REF] Doucet | An introduction to sequential Monte Carlo methods[END_REF]. In this setting, the estimation method most frequently used in practice is the Kalman filter, which is optimal for linear state space models and independent, additive Gaussian noises [START_REF] Anderson | Optimal filtering[END_REF]. In this case, the posterior distributions are also Gaussian and computed exactly, without approximations. However, in most realistic cases the dynamics of degradation is non linear and/or the associated noises are non-Gaussian. Various approximate methods can be proposed to tackle these cases, e.g. the analytical approximations of the Extended Kalman (EKF) and the Gaussian-sum filters and the numerical approximations of the grid-based filters [START_REF] Anderson | Optimal filtering[END_REF]. Recently, Monte Carlo sampling methods are gaining popularity for their flexibility and ease of design [Kitagawa, 1987].

These methods go under the name of particle filtering because the continuous distributions of interest are approximated by a discrete set of weighed particles, where each particle represents a random trajectory of evolution in the state space and the weight is the probability of the trajectory [START_REF] Djuric | Particle filtering[END_REF][START_REF] Doucet | On sequential Monte Carlo sampling methods for Bayesian filtering[END_REF][START_REF] Cadini | Monte Carlo-based filtering for fatigue crack growth estimation[END_REF].

Data-driven techniques utilize monitored operational data related to system health. They can be beneficial when understanding of first principles of system operation is not straightforward or when the system is so complex that developing an accurate model is prohibitively expensive. Furthermore, recent advances in sensor technology and refined simulation capabilities enable us to continuously monitor the health of operating components and living structures, and to manage the related large amount of reference data.

Data-driven techniques can be divided into two categories: statistical techniques (regression methods, ARMA models, etc.) and Artificial Intelligence (AI) techniques (neural networks, fuzzy systems, etc.). The most direct data-driven techniques for RUL estimation attempts at fitting available data of component or structure degradation by regression models and then extrapolating the evolution up to failure. However, in practice, the component or structure degradation history available may be short and incomplete, and extrapolation may lead to large errors [START_REF] Yan | A Prognostic Algorithm for Machine performance Assessment and its Application[END_REF].

With respect to AI techniques, the most commonly used prediction methods are based on Neural Networks [Peel et al., 2008;[START_REF] Barlett | Nuclear Power Plant Status Diagnostics Using an Artificial Neural Network[END_REF][START_REF] Santosh | Diagnostic System for Identification of Accident Scenarios in Nuclear Power Plants using Artificial Neural Networks[END_REF]. For prognostic tasks, promising methods are Recurrent Neural Networks (RNNs) [START_REF] Campolucci | On-Line Learning Algorithms of Locally Recurrent Neural Networks[END_REF], Neuro-Fuzzy (NF) systems [START_REF] Wang | Prognosis of Machine Health Condition using Neuro-Fuzzy Systems[END_REF] and Support Vector Machines (SVMs) [START_REF] Sotiris | [END_REF]. In spite of the recognized potential of empirical, data-driven techniques, limitations still exist for their use in safety critical applications, e.g. in nuclear technology, because of the lack of a systematic approach for selecting the structure and parameters of the models and their black-box character which limits intuition with respect to the understanding of their performance [START_REF] Wang | A similarity based prognostic approach for Remaining Useful life estimation of Engineered Systems[END_REF]].

An opportunity for increased transparency and openness of empirical, data-driven models is offered by fuzzy logic-based methods, which are sustained by a formal mathematical framework for dealing with the vagueness of everyday reasoning [Zadeh, 1965] and accommodating measurement uncertainty and estimation imprecision [START_REF] Yuan | Data driven identification of key variables[END_REF][START_REF] Zio | Identification of nuclear transients via optimized fuzzy clustering[END_REF].

The subject of the present paper is the estimation of the RUL of a component or structure on the basis of measurements of its degradation state taken at predefined inspection times, which are possibly only few due to the fact that the lower the number of measurements, the lower the computational time and the cost associated to the inspection procedures. The computational framework proposed manipulates a set of degradation-to-failure patterns (hereafter called reference patterns) within a fuzzy-based, data-driven similarity analysis procedure [Angstenberger, 2001] for predicting the remaining life of a newly developing degradation pattern (hereafter called test pattern). The adoption of a fuzzy definition of pattern similarity allows capturing and integrating the possibly imprecise information carried by the measured degradation signal. An application is presented with reference to a non linear fatigue crack growth process, typical for a certain class of industrial and structural components [START_REF] Oswald | Realiability of deteriorating structures[END_REF][START_REF] Sobezyk | Random fatigue: From data to theory[END_REF][START_REF] Bolotin | Probabilistic model of early fatigue crack growth[END_REF][START_REF] Myotyri | Application of stochastic filtering for lifetime prediction[END_REF].

The paper contents are structured as follows. Section 2 contains the description of the algorithm for fuzzy similarity analysis, at the basis of the prognostic approach to RUL estimation.

Section 3 presents the dynamic model of fatigue crack growth. In Section 4, the results of the application of the approach to the case study are presented and compared with those obtained by particle filtering. Finally, some conclusions on the advantages and limitations of the approach here propounded are given in Section 5.

Methodology

It is assumed that J measurements taken at predefined inspection times are available for N degradation-to-failure trajectories (reference patterns) of components or structures of the type of interest; these trajectories last all the way to component or structure failure, i.e., to the instance when the degradation state reaches the threshold value beyond which the component loses its functionality.

A degradation trajectory (test pattern) is developing in the actual component or structure under analysis, which is monitored at the predefined inspection times. The RUL estimation for the degrading component or structure is performed by analyzing the similarity between the test pattern and the N reference patterns, using their RULs weighted by how similar they are to the test pattern [Angstenberger, 2001]. The estimation procedure is adapted from a data-driven framework of recovery time prediction proposed by the authors for accident management [Zio et al., 2009].

            || 11 ˆ| | | | i f j f j ii ii j j f j i j i t T i t T f j f j RUL T MTTF T t T RUL T i t T i t T        (1)
          1 , 1 ,..., 1 , j j j j j j f T f T n T f T n f T f T          is built
        1 ,, nj k n j i j f k r i k        , 1, 2,..., iN  , 1, 2,..., jJ  (2)
The matrix   NJ   contains the difference measures  

, ij  between all n-long segments of the reference patterns and the test pattern. between "similar" and "non-similar" [START_REF] Binaghi | Fuzzy Reasoning Approach to Similarity Evaluation in Image Analysis[END_REF][START_REF] Joentgen | Dynamic Fuzzy Data Analysis based on Similarity between Functions[END_REF]. This can be achieved by resorting to a fuzzy logic modeling paradigm in which the pointwise difference of two patterns is judged for similarity with respect to an "approximately zero" fuzzy set , d i j between two pattern segments is then computed as:

    , 1 , d i j i j   , 1, 2,..., iN  , 1, 2,..., jJ  (3)
Common membership functions can be used for the definition of the FS, e.g. triangular, trapezoidal, and bell-shaped [START_REF] Dubois | Weighted Fuzzy Pattern Matching[END_REF]. In an exploration of the effects of these membership functions, all defined on the same support of the fuzzy set "approximately zero" and centered in the value zero, no particular sensitivity of the RUL estimation results

has been recorded with respect to the particular shape. In the application illustrated in this work, the following bell-shaped function has turned out to give more robust results due to its gradual smoothness:

      2 2 ln , , ij i j e           (4)
The arbitrary parameters  and  can be set by the analyst to shape the desired interpretation of similarity into the fuzzy set: the larger the value of the ratio

  2 ln    , the
narrower the fuzzy set and the stronger the definition of similarity. The choice of the values of  and  depends on the application; one may proceed to determining the value  of the difference value  which must have a degree of membership  equal to  [Angstenberger, 2001;Zio et al., 2009].

-Step 4: weight definition. To assign the weight i w given to the i-th reference pattern accounting for how similar it is to the test pattern, the minimum distance * i d along the i-th row of the matrix of Eq. ( 3) is first identified:

  * 1,..., min , i j J d d i j   , 1, 2,..., iN  (5)
The weight i w is then computed, resorting to the arbitrarily chosen decreasing monotone function, which guarantees that the smaller the minimum distance the larger the weight given to the i-th reference pattern:

  * 1 * 1 i d ii w d e         , 1, 2,..., iN  (6)
The value of β in ( 7) is arbitrarily taken equal to that used in Eq. (4).

Then, the weight w i is normalized: ). Thus,   i RUL t is the remaining time before failure on the reference pattern starting from the end time of the latest-in-life segment of minimum distance from the developing test pattern (Figure 2). This allows a conservative RUL estimation, biased towards "pessimistic" predictions of the RUL because in the case that more than one segment along the i-th reference pattern is closest to the developing test pattern, the latest one is taken, i.e. the one closest to failure.

1 N i i
  1 j fT    j fT
The ideas behind the weighting of the individual   i RUL t is that: i) all reference patterns bring useful information for determining the RUL of the degradation pattern currently developing; ii) those segments of the reference patterns which are most similar to the most recent segment of length n of the currently developing degradation pattern should be more informative in the extrapolation of the occurring pattern to failure.

The model of the fatigue crack growth process

Let us consider the process of crack growth in a component subject to fatigue. The common Paris-Erdogan model is adopted for describing the evolution of the crack depth x as a function of the load cycles t [Pulkkinen, 1991]:

() m dx CS dt  ( 10
)
where C and m are constants related to the material properties [Provan, 1987;[START_REF] Kozin | Probabilistic Models of Fatigue Crack Growth: Results and Speculations[END_REF]],

which can be estimated from experimental data [START_REF] Bigerelle | Bootstrap Analysis of FCGR, Application to the Paris Relationship and to Lifetime Prediction[END_REF] and ΔS is the stress intensity amplitude, roughly proportional to the square root of x [Provan, 1987]:

Sx   ( 11
)
where  is again a constant which may be determined from experimental data.

The intrinsic stochasticity of the process may be inserted in the model by modifying equation ( 10) as follows [Provan, 1987]:

  m dx e C x dt    ( 12
)
where 2 ~(0, ) N   is a white Gaussian noise. For t  sufficiently small, the state-space model ( 12) can be discretized to give:

    1 () j m kk x t x t e C S t       (13)
which represents a non-linear Markov process with independent, non-stationary degradation increments of the degradation state x .

At the generic inspection time T j , the degradation state   j xT is generally not directly measurable. In the case of non-destructive ultrasonic inspections a logit model for the observation   j fT can be introduced [Simola et al., 1998]:

        01 ln ln jj k jj f T x T d f T d x T        ( 14
)
where d is the component material thickness, 0 ( , )     and 1 0   are parameters to be estimated from experimental data and  is a white Gaussian noise such that 2 ~(0, )

N   .
The following standard transformations are introduced:

      ln j j j fT yT d f T   (15)       01 ln j j j xT T d x T      (16)
In the case study here considered, taken from [START_REF] Myotyri | Application of stochastic filtering for lifetime prediction[END_REF], the parameters of the state equation ( 13) are C = 0.005, 1.3 m  and γ = 1, whereas those in the measurement Eq. ( 14) are

Results

Application of the RUL estimation procedure on the fatigue crack propagation case study

The application of the procedure for computing at the inspection time T j the estimate   ˆj RUL T (Section 2) on the fatigue crack propagation case study (Section 3) is hereafter illustrated. otherwise, is the observation   ft of Eq. ( 14). This means that the actual safety parameter driving the RUL, i.e., the state variable x of Eq. ( 13), is followed only indirectly through the observable variable, thus challenging further the robustness of the approach with respect to non-observable safety parameters. At each inspection time j T , 1, 2,...,30 j  , its value is appended and stored in the vector containing the 1 j  values of the signal collected at the previous times. At the current time Step 2: pattern pointwise difference computation.

          1 , 1 ,..., 1 , j j j j j j f T f T n T f T n f T f T          is
The pointwise difference   , ij  between the test pattern           1 , 1 ,..., 1 , j j j j j j f T f T n T f T n f T f T        
 and   , r i k is evaluated (Eq. ( 2)).

Step 3: pattern pointwise similarity and distance score computation.

The pointwise differences   3), 1, 2,...,50 i  , 1, 2,...,30 j  .

Step 4: weight definition.

The minimum distances * i d are evaluated by Eq. ( 5), and the relative normalized weights i w calculated through Eqs. ( 6) and ( 7), 1, 2,...,50 i  .

Step 5: RUL i (t) and RUL(T j ) estimation.

For each reference pattern, an estimate   i RUL t is computed by Eq. ( 8), 1, 2,...,50 i 

; then, the   i RUL t are aggregated in the weighted sum of Eq. ( 9) with the weights i w previously calculated.

For the crack propagation pattern plotted in Figure 3

Comparison with a model-based particle filtering method

The Monte Carlo-based particle filtering method has been applied to the same case study of Section 3 in [START_REF] Cadini | Monte Carlo-based filtering for fatigue crack growth estimation[END_REF]. The application of particle filtering for RUL estimation entails the evaluation of the conditional cumulative distribution function (cdf) of the stochastic observable variable related to the degradation state. For more details on the procedure, the interested reader may refer to [START_REF] Cadini | Monte Carlo-based filtering for fatigue crack growth estimation[END_REF].

From Eq. ( 15) it follows that the transformed observation 

     2 ~, jj Y T N T   is a
Gaussian random variable with cdf:

                    j jj j j j j j YT y T T cdf y T x T P Y T y T x T             (17)
where   u  is the cdf of the standard normal distribution   0,1 N .

The conditional cdf of the stochastic measurement variable   j FT related to the stochastic degradation state   j XT is then:

                      1 ln ln jj jj j j j j F T Y T jj f T f T cdf f T x T cdf x T T d f T d f T                             (18)
with corresponding probability density function (pdf): The prognostic procedure has been applied to the crack propagation dynamics of a component subject to fatigue cycles. Comparison with particle filtering demonstrates that the method is capable of successfully handling nonlinear dynamics and of dealing with non-Gaussian noises.

                   

Figure 1

 1 Figure1shows a schematic sketch of the computational framework, with reference to

  component or structure failure time of the i-th pattern (i.e., the time when the signal value exceeds the threshold beyond which the component or structure loses its functionalitythe remaining useful life on the reference pattern i starting from j T . At the following inspection times, the algorithm continues to update the estimate of is detected upon inspection because the degradation signal has deviated from its nominal value beyond its range of allowed variability, which is a priori gauged with respect to the range of possible fluctuations in the signal; at this time, the RUL estimation algorithm is put to work to match the similarity of the developing degradation signal to the reference patterns and combine their failure times for estimating the component or structure RUL.-Step 2: pattern pointwise difference computation. At the current time j T , the latest n -

  by linear interpolation of the values   1 j fT  and   j fT measured at the successive inspection times 1 j T  and j T . The test pattern f is also normalized in [0.2,0.8]. The pointwise difference     between the n normalized values of pattern f and of the reference pattern segment   , r i k is computed:

Figure 1

 1 Figure 1 The flowchart of the fuzzy-based, data-driven approach

(

  FS) specified by a function which maps the elements   the condition of "approximately zero". The distance score  

  Figure 2 The remaining time before failure

Figure 3

 3 Figure 3 Crack growth pattern used as test pattern

  The interval between two successive inspections is equal to 1100 n [min] if the estimated RUL>200 [min], otherwise 2 10 n [min], reflecting a more frequent inspection of the component or structure integrity as the component or structure is approaching the end of life. A database of N = 50 reference crack propagation patterns of differing initial conditions is organized in the are then compared for similarity with a test pattern containing the values of the measured signal of the developing degradation pattern. For each of the test patterns the procedural steps are performed as follows:Step 1: fault detection.The signal monitored starting from time 1 t  [s] to the time horizon of observation T = 1000 [min], with time step of n 1 =100 [min] until the estimated RUL>200 [min], or 2 10 n [min]

  successive inspections. The Mean Time to Failure   j MTTF T is calculated resorting to Eq. (1) and   ˆj RUL T is set equal to   j MTTF T for each time step, until fault detection when the on-line data-driven, similarity-based RUL estimation is activated.

  " FS. The bell-shaped function of Eq. (4) is taken with parameters values 0are then computed by Eq. (

  4 in thin continuous lines with the bars of one standard deviation of the samples time at which the crack depth x exceeds the material thickness d and thus failure occurs. The   ˆj RUL T estimates obtained based on pattern segments of 1 [min], respectively, are plotted in bold circles; at the beginning of the test pattern, the predictions match the (dashed thick line). In the Figure, the bold vertical line indicates the time of crack depth exceedance of the limit on the material thickness.

Figure 4

 4 Figure4RUL estimation for the crack propagation pattern of Figure3

  the RUL obtained resorting to particle filtering are plotted in Figure5(together with those obtained by the data-drive, similarity-based approach of Section 2) in thin continuous lines with the bars of one standard deviation of the samples   at the inspection times of Figure4, are indicated in bold squares. The predictions provided by the two methods are comparable: after fault detection, the particle filtering towards the real RUL (dashed thick line). In the Figure, the bold vertical line indicates the time of crack depth exceedance of the limit on the material thickness.

Figure 5

 5 Figure 5 Comparison of the RUL estimations for the crack propagation pattern of Figure 3 provided by the similarity based, data-driven approach and particle filtering

: RUL i (t) and RUL(T j ) estimation. With

	whose element is equal to * i d
						w w 	e  	e w	(7)
	-Step 5i f tt  , the value	i RUL t is first determined as:  
					i RUL t t t f    , j	1, 2,..., iN 	(8)
										iM
	where	t	  ji   M * max arg , n i j d    			is the final time of the latest-in-life segment of the i-
			j
	th pattern among those which have minimum distance * d from the developing test pattern (n
										i
	is the test pattern length,	 max arg			  , i j d	* i			,   i
					j

respect to the generic i-th reference pattern for which  gives the largest column index j of  