Quantifying the benefit of SHM: what if the manager is not the owner?

Transcription

1 Bolognani, Denise and Verzobio, Andrea and Tonelli, Daniel and Cappello, Carlo and Glisic, Branko and Zonta, Daniele and Qigley, John (2018) Qantifying the benefit of strctral health monitoring : what if the manager is not the owner? Strctral Health Monitoring. ISSN , This version is available at Strathprints is designed to allow sers to access the research otpt of the University of Strathclyde. Unless otherwise explicitly stated on the manscript, Copyright and Moral Rights for the papers on this site are retained by the individal athors and/or other copyright owners. Please check the manscript for details of any other licences that may have been applied. Yo may not engage in frther distribtion of the material for any profitmaking activities or any commercial gain. Yo may freely distribte both the rl ( and the content of this paper for research or private stdy, edcational, or not-for-profit prposes withot prior permission or charge. Any correspondence concerning this service shold be sent to the Strathprints administrator: strathprints@strath.ac.k The Strathprints instittional repository ( is a digital archive of University of Strathclyde research otpts. It has been developed to disseminate open access research otpts, expose data abot those otpts, and enable the management and persistent access to Strathclyde's intellectal otpt.

2 Type of paper: Original manscript Denise Bolognani, University of Trento, Department of Civil, Environmental and Mechanical Engineering, via Mesiano 77, Trento, Italy Qantifying the benefit of SHM: what if the manager is not the owner? Denise Bolognani 1. Andrea Verzobio 2. Daniel Tonelli 1,3. Carlo Cappello 1. Branko Glisic 3. Daniele Zonta 1,2. John Qigley 4 1 University of Trento, Department of Civil, Environmental and Mechanical Engineering, via Mesiano 77, Trento, Italy. 2 University of Strathclyde, Department of Civil & Environmental Engineering, 75 Montrose Street, Glasgow, G1 1XJ, UK. 3 Princeton University, Department of Civil & Environmental Engineering, Princeton, NJ USA. 4 University of Strathclyde, Department of Management Science, 199 Cathedral Street, Glasgow, G4 OQU, UK. Abstract Only very recently or commnity has acknowledged that the benefit of Strctral Health Monitoring (SHM) can be properly qantified sing the concept of Vale of Information (VoI). The VoI is the difference between the tilities of operating the strctre with and withot the monitoring system. Typically, it is assmed that there is one decision maker for all decisions, i.e. deciding on both the investment on the monitoring system as well as the operation of the strctre. The aim of this work is to formalize a rational method for qantifying the Vale of Information when two different actors are involved in the decision chain: the manager, who makes decisions regarding the strctre, based on monitoring data; and the owner, who chooses whether to install the monitoring system or not, before having access to these data. The two decision makers, even if both rational and exposed to the same backgrond information, may still act differently becase of their different appetites for risk. To illstrate how this framework works, we evalate a hypothetical VoI for the Streicker Bridge, a pedestrian bridge in Princeton University camps eqipped with a fiber optic sensing system, assming that two fictional characters, Malcolm and Ophelia, are involved: Malcolm is the manager who decides whether to keep the bridge open or close it following to an incident; Ophelia is the owner who decides whether to invest on a monitoring system to help Malcolm making the right decision. We demonstrate that when manager and owner are two different individal, the benefit of monitoring cold be greater or smaller than when all the decisions are made by the same individal. Under appropriate conditions, the monitoring VoI cold even be negative, meaning that the owner is willing to pay to prevent the manager to se the monitoring system.

3 Keywords Vale of Information; Bayesian Inference; Expected Utility Theory; Decision-making; Bridge Management; Fiber Optic Sensors Introdction Althogh the tility of strctral health monitoring (SHM) has rarely been qestioned in or commnity, very recently a few pblished papers [1] [2] have clarified the way that the benefit of monitoring can be properly qantified. Indeed, seen from a mere strctral engineering perspective, the tility of monitoring may not be immediately evident. Wear for a minte the hat of the manager of a Department of Transportation (DoT), responsible for the safety of a bridge: wold yo invest yor limited bdget on a reinforcing work or on a monitoring system? A retrofit work will increase the bridge load-carrying capacity and therefore its safety. On the contrary, sensors don t change the bridge capacity, nor redce the external loads. So how can monitoring affect the safety of the bridge? The answer to this legitimate qestion goes roghly along these lines: monitoring does not provide strctral capacity, rather better information on the state of a strctre; based on this information, the manager can make better decisions on the management of the strctre, minimizing the chances of wrong choices, and eventally increasing the safety of the bridge over its lifespan. Therefore, to appreciate the benefit of SHM, we need to accont for how the strctre is expected to be operated and eventally recast the monitoring problem into a formal economic decision framework. The basis of the rational decision making is encoded in axiomatic Expected Utility Theory (EUT), first introdced by Von Nemann and Morgenstern [3] in 1944, and later developed in the form that we crrently know by Raiffa and Schlaifer [4] in EUT is largely covered by a nmber of modern textbooks (among the many, we recommend Parmigiani [5] to the Reader of SHM who is approaching the topic for the first time). Within the framework of EUT, the benefit of information, sch as that coming from a monitoring system, is formally qantified by the so-called Vale of Information (VoI). The concept of VoI is anything bt new: it was first introdced by Lindley [6] in 1956, as a measre of the information provided by an experiment, and later formalized by Raiffa and Schlaifer [4] and DeGroot [7]. Since its introdction, it has been continosly applied in manifold fields, inclding statistics, reliability and operational research [8] [9] [10] [11]. Its first appearance in the SHM commnity, however, is mch more recent and dates back, in or best knowledge, to a paper pblished in the proceedings of SPIE by Bernal et al. in 2009 [12], followed by Pozzi et al. [13], Pozzi and Der Kireghian [14], Thöns & Faber [1], Zonta et al. [2], - a recent state of the art can be find in Thöns [15]. In the last few years, qantifying the vale of SHM has known a renewed poplarity thanks to the activity of the EU-fnded COST action TU1402 [16]. Broadly speaking, the vale of a SHM system can be simply defined as the difference between the benefit, or expected tility *, of operating the strctre with the monitoring system and the benefit, or expect tility, of operating the strctre withot the system. Both * and are expected tilities calclated a priori, i.e. before actally receiving any information from the monitoring system. While in we assme the knowledge of the manager is his a priori knowledge, * is calclated assming the decision maker has access

4 to the monitoring information and is sometimes referred as to preposterior tility. The difference between these vales measres the vale of the information to the decision maker. Clearly, if the monitoring does not provide any sefl information, the preposterior * is eqal to the prior, and the vale of monitoring information is zero. Typically, it is assmed that there is one decision maker for all decisions, i.e. deciding on both investments as well as operations. This individal cold be for example an idealized manager of a DoT, as the fictitios character Tom who appears in [2]. We mst recognize that in the real world the process whereby a DoT makes decision over its stock is typically more complex, with more individals involved in the decision chain. Even oversimplifying, we always have at least two different decision stages. First a decision is made on whether or not to by and install the monitoring system on the strctre; this is a problem of long-term planning and investment of financial resorces. This decision is typically carried ot by a high-level manager, that in this paper we will conventionally refer to as owner, whose key performance measre is retrn on investment. The second stage concerns the day-to-day operation of the strctre which incldes for example maintenance, repair, retrofit or enforcing traffic limitations, once the monitoring system is installed; if installed these decisions may be informed by the monitoring system. Most of the time, the manager and the owner of the strctre are different individals. Both decision makers are motivated to maintain a high level of long term availability for the strctre, which is challenging as the state of the strctre is never known precisely while in operation. Operators balance two types of errors, either removing a strctre from operation prematrely for maintenance or operating too long reslting in a failre; both of which are based on imperfect information concerning the state of the strctre. Decision makers will differ in their choices nder ncertainty even when they have access to the same information if they have different appetites for risk. As sch, the owner needs to consider the operators appetite for risk when deciding whether to install a monitoring system, as this will indicate how the system will inference the operators decision making and as sch the vale of this information. The aim of this work is to formalize a rational method for qantifying the Vale of Information when two different actors are involved in the decision chain: the manager, who makes decisions regarding the strctre, based on monitoring data; and the owner, who chooses whether to install the monitoring system or not, before having access to these data. We start explaining why and how two different individals, both rational and provided with the same backgrond information, can end p with different decisions. Next, we review the basis of the VoI, which illstrates a method for evalating the VoI in SHMbased decision-problems, and revise the framework of Zonta et al. [2], to inclde the difference between the manager and the owner. To illstrate how this framework works we apply it to the same decision problem reported in [2]: the Streicker Bridge case stdy. This is a pedestrian bridge at Princeton University camps, which is eqipped with a continos monitoring system. Some conclding remarks are reported at the end of the paper. SHM-based decision In this section, we review the concepts of Bayesian dgment, expected tility and vale of information, as applied to SHM problems, following a similar path as in Cappello et al. [17] and Zonta et al. [2]. The Reader can find frther examples of SHM-based decision

5 problems in Flynn and Todd [18] [19], Flynn et al. [20] and Tonelli et al [21]. As observed in [17], SHM-based decision making (i.e., deciding based on the information from a SHM system) is properly a two-step process, which incldes a dgement and a decision, as depicted in Figre 1: first, based on the information from the sensors y, we infer the state S of the strctre; next, based on or knowledge of the state S we choose the optimal action aopt to take. Before proceeding with the mathematical formalization of this process let s confine the complexity of or problem throgh the following assmptions: - the monitoring system provides a dataset that can be represented by a vector y; - the strctre (e.g.: one bridge) can be in a one ot of N mtally exclsive and exhastive states S1, S2,, SN (e.g.: S1 = severely damaged, S2 = moderately damaged, S3 = not damaged, ); - the state of the strctre is generally not deterministically known, and can be only described in probabilistic terms; - the decision maker can choose between a set of M alternative actions a1, a2,, am (for example, a1 = do nothing, a2 = limit traffic, a3 = close the bridge to traffic, ); - taking an action prodces measrable conseqences (e.g.: a monetary gain or loss, a temporary downtime of the strctre, in some case casalities); the conseqences of an action can be mathematically described by several parameters (e.g.: the amont of money lost, the nmber of day of downtime, the nmber of casalties), encoded in an otcome vector z; - the otcome z of an action depends on the state of the strctre, ths it is a fnction of both action a and state S: z(a,s); when the state is certain the conseqence of an action is also deterministically known; therefore, the only ncertainty in the decision process is the state of the strctre S; - for simplicity and clarity, we refer here to the case of single shot interrogation, which is the case when the interrogation occrs only following an event which has a single chance to happen dring the lifespan; an extension to the case of mltiple interrogations is also fond in [2]. Figre 1. The process of SHM-based decision making.

6 Jdgment is abot nderstanding the state of the strctre based on the observation, which is exactly what SHM is abot from a logical standpoint. In the presence of ncertainty, the state of the strctre after observing the sensors data y is probabilistically described by the posterior information P(S y), and the logical inference process followed by a rational agent is mathematically encoded in Bayes rle [22], which reads: P( y) = p P p where P() indicates a probability and p() a probability density fnction. Eqation (1) basically says that the posterior knowledge of the ith strctral state P(Si y) depends on the prior knowledge P( ) (i.e., what I expect the state of the strctre to be before reading any monitoring data) [23] and the likelihood p(y Si) (i.e., the probability of observing the data given the state of the strctre). Distribtion p(y) is simply a normalization constant, referred to as evidence, calclated as: p = p P Decision is abot choosing the best action based on the knowledge of the state. When the state of the system Si is deterministically known, the rational decision-maker ranks an action based on the conseqences z throgh a tility fnction U(z). Mathematically, the tility fnction is a transformation that converts the vector z, which describes the otcome of an action in its entire complexity, into a scalar U, which indicates the agent s order of sbective preference for any possible otcome. When the state of the system is ncertain, and therefore the conseqences of an action are only probabilistically known, the axioms of expected tility theory (EUT) state that the decision maker ranks their preferences based on the expected tility, defined as: (a) = E where ES is the expected vale operator of random variable S, which we have assmed be the only ncertainty into the problem. To prevent confsion, note that in this paper capital U indicates the tility fnction, while lowercase denotes an expected tility. To better clarify the practical meaning of Eqation (3), let s start from the case of a strctre not eqipped with a monitoring system, where the manager decides withot accessing any SHM data. In this case the manager s prior expected tility (a) of a particlar action a, depends on their prior probabilistic knowledge P(Si) of each possible state Si: ( ) = P

7 and consistently with EUT, the rational manager will choose that actions aopt which carries the maximm expected tility payoff : = max a a opt = arg max a ab In contrast, if a monitoring system is installed, and data are accessible by the agent, the monitoring observation y affects the state knowledge, and therefore indirectly their decision. This time, the posterior expected tility (a,y) of actions a depends on the posterior probabilities P(Si y), which are now fnctions of the observation y: (, y) = P y 6 Becase the posterior probability depends on the particlar observation y, in the posterior sitation the expected tility is a fnction of y as well, and so are the maximm expected tility and the optimal choice: y= max, y a opt = arg max, y 7a,b Eqation (5a) and (7a) are the tilities calclates before and after a monitoring system is interrogated. Note that, in order to evalate the posterior tility of an action (a,y), we need to know the particlar realization of observation y, so we cannot evalate the posterior tility ntil the monitoring system is installed and its readings are available. How does the tility change if we have decided to install a monitoring system, bt we have still to observe the sensors readings? Technically, what we shold do is to evalate a priori (i.e., now that the system is not installed yet) the expected vale of the tility a posteriori (i.e., at the time when the system will be installed and operating). We denote this qantity preposterior tility, *, to separate it both from the prior and posterior tilities introdced above. The preposterior tility * is independent on the particlar realization and can be derived from the posterior expected tility (y) by marginalizing ot the variable y, [2] [17]: * = E max a,y max Dy a,y py dy 8 where distribtion py is the same evidence defined by Eqation (2). The preposterior expected tility encodes the total expected tility of a decision process, based on the information provided by the monitoring system, bt evalated before the monitoring system is actally installed. Finally, the Vale of Information of the monitoring system is simply the difference between the expected tility with the monitoring system (the preposterior tility *) and the corresponding tility withot the monitoring system (the prior tility ):

8 VoI = * = max Dy a,y py dy max a 9 In other words, the VoI is the difference between the expected maximm tility and the maximm expected tility. It is easily mathematically verified that * is always greater or eqal than, and therefore the VoI as formlated above can only be positive. This is to say that nder the assmption above SHM is always sefl, consistently with the principle that information can t hrt [24] as reported in Pozzi [25]. It is worth reminding that these assmptions are performed before acqiring the data. That means that the vale of those data is anticipated by the decision maker, even if the realized vale, once the decision is made, may be qite different. As well, it may be that the cost of data exceeds its vale, bt this wold be reflected in the calclation as we assess the tility associated with the cost of obtaining the data. The process of deciding on the monitoring system installation can be graphically represented as a two-stage decision tree, as shown in Figre 2. At the first stage the agent decides on whether to go or not with the SHM system, while at the second stage he decides on the action a1,, a to ndertake on the strctre. The realization of the state occrs at the following chance node and the otcome z depends on the action and the state. On the withot SHM branch of the tree, the state is determined by the prior information and the expected tility corresponds to in Eqation (9). On the with SHM branch of the tree, the second stage action is decided based on the information y from the monitoring system and the final otcome incldes the cost z SHM of the monitoring system. The best choice of stage one is the one that provides maximm tility, and this can be calclated by solving the two-stage tree by backward indction [5]. Figre 2. Graphical representation of the decision problem of whether or not to install a monitoring system (SHM). Two individals, two decisions In the classical formlation of the VoI stated above, we have implicitly assmed that the decision is taken at any stage by the same rational individal, characterized by a defined backgrond information and tility fnction. We address now the problem of qantifying

9 the VoI when two separate individals are involved in the decision chain. We conventionally denote manager (M) the one who makes decisions on the day-to-day operation of the strctre, and owner (O), the one who is in charge of the strategic investments on the asset and decide on whether to install the monitoring system or not. Referring to Figre 2, the manager is the one who takes decisions at stage two, while the owner decides at stage one. We will refer to the classical formlation of VoI, as stated in the previos section, as to nconditional - in contrast with the conditional VoI which we are abot to introdce. A common misnderstanding, not only in or commnity, is that two individals, if both rational and exposed to the same observation, shold always end p with the same decision. In the real world, there are a nmber of components in the SHM-based decision process that are inherently sbective, so different decisions by different individals shold not be necessarily be seen as an inconsistency. This concept needs a deeper explanation: with reference to Figre 1, the reasons whereby two individals, both rational, can take a different decision based on the same observation inclde: a) the two have a different prior knowledge of the problem i.e. they se different priors P(S); b) they interpret differently the observation i.e. they se different interpretation models, which are encoded in the likelihood fnction P(y S); c) they have a different expectation or knowledge of the possible otcome of an action i.e. they assme different otcome vectors z; d) they weight differently the importance of an otcome - i.e. they se different tility fnctions U(z). Differences in (a) (b) and (c) are merely abot backgrond knowledge and may actally occr in the real world; however, we expect that two individals with similar experience and edcation shold generally agree on any of that. For example, two strctral engineers with common backgrond will probably agree on the limited importance of a bending crack visible on an nprestressed reinforced concrete beam, while a non-expert cold be overconcerned. In this paper, we will assme that the two agents flly agree on (a), (b) and (c), while they only differ in the way how they weight otcomes (d), throgh their tility fnction. The tility fnction is not a matter of backgrond knowledge, rather it reflects the vale of the individal as to the conseqence of an action. Therefore, there is no logical argment to dge one tility fnction better than another one, as long as it does not violate the axioms of the expected tility theory. Even limiting or discssion to the case where the otcome z is st a monetary loss or gain, the tility fnction adopted by different people can be very different based on their particlar individal risk aversion [26] [27]. For instance, an agent is risk netral if his or her tility fnction U is linear with the loss or gain z, as shown in Figre 3. Since the expected tility is proportional to the probability of realization, as shown in Eqation (4), risk netrality implies indifference to a gamble with an expected vale of zero. So, for example, to a risk netral agent a 1% probability of losing $100 is eqivalent to a certain loss of $1. In practice it is commonly observed that individals tend to reect gambles with a netral expected payoff: in the example above individals often prefer to pay $1 off the pocket

10 rather than taking the risk of losing $100. This condition is referred to as risk aversion and can be graphically represented with a tility fnction with a concave (i.e., with negative second derivative) tility fnction, as shown in Figre 3. The condition of risk aversion is consistent with the observation that the marginal tility of most goods, inclding money, diminish with the amont of goods, or the wealth of the decision maker, as observed since Bernolli [26]. Figre 3. Utility fnction for risk seeking, risk netral and risk adverse agents. Dealing with losses, risk aversion respect to a loss depends on the amont of the loss with respect to the decision maker s own wealth or the extent of his or her own asset: when the loss is mch smaller than the whole vale of the asset, the agent tends to be risk netral, while they became risk averse when the loss is a significant fraction of their asset. In or sitation, the owner, who is in charge of the strategic development of the agency, typically manages a large stock of strctres, and the loss corresponding to an individal strctre is a mch smaller than the overall asset vale. In this case, it is likely that the owner is risk netral with respect to the loss compared to the vale of a single strctre. In contrast the manager is responsible for the safety of a single strctre: in this case the vale of the strctre corresponds to the vale of the asset, and their behavior is likely to be risk adverse respect to the loss of that particlar strctre. To proceed with the mathematical formlation, we have to acknowledge that the two agents involved in the decision chain, the owner and the manager, may have different tility fnctions. We re going to se indices (M) or (O) to indicate that a qantity is intended from one of the other perspective. The expected tility of the manager is calclated as: N M ( ) = M P

11 and we may calclate the optimal action and the maximm tility from the manager perspective as in the following: M = max M a M a opt = arg max M a ab If the owner was in charge of the entire decision chain, we wold end p with analogos O O expressions of optimal action a and maximm expected tility, this time from opt max the owner perspective. Observe that the optimal choice of the owner does not necessarily coincide with that of the manager, meaning that if the owner was in charge of the fll decision chain, they wold behave differently from the manager. Contining on this rationale, we can reformlate the expression of posterior tilities, preposterior tilities and VoI from the owner or the manager perspective. However, the sitation we are discssing is different: the owner is the one who decides on the monitoring system installation, bt the manager is the one who decides which is the optimal action at the second stage. Therefore, all tilities are from the owner perspective, bt shold be evalated acconting for the action that the manager, not the owner, is expected to choose. In other words, the tility of the owner is conditional to the action M chosen by the manager a. For example, the prior tility of the owner conditional to opt the decision expected by the manager reads: O M O = M a = opt O arg max M a O M where the index (O M) on the tility indicates that this tility is conditional to the O manager s choice, in opposition to the nconditional tility calclated assming the owner in charge of the fll decision chain. We can proceed accordingly to formlate the posterior conditional tility (the tility of the owner after the manager has observed the monitoring response): O M O = M a opt y = O arg max M a y and similarly the preposterior conditional tility (the tility of the owner in the expectation of what the manager wold decide if a monitoring system was installed): O M * M O = arg max a,y py dy 14 D y Eventally the conditional VoI is the difference between the preposterior and the prior conditional tilities:

12 O O M O M VoI = * M arg max a,y py dy D y = O arg max M a 15 The nconditional and conditional formlations are smmarized and compared in Table 1. At this point, it s interesting to compare the nconditional and the conditional tilities, and also the vale of information. The nconditional tility, prior or preposterior, is basically the owner s tility of their favorite choice, while the conditional tility is the owner s tility of the choice of someone else. If the two choices coincide, the conditional tility is eqal to the nconditional prior tility. If they do not coincide, the manager s choice can only be sboptimal from the owner s perspective, and therefore the conditional tility mst be eqal or lower than the nconditional. Therefore, the following relationships mst hold: In a sitation with one decision maker the VoI cannot be negative; if the decision maker anticipated misleading data it wold be optimal to discard it reslting in a VoI of 0. However, we consider a sitation of two decision makers and demonstrate that the VoI can be negative to one, i.e. the owner, as the new information is reslting in the other, i.e. the manager, making decisions that are less preferred by the owner than with no information. Table 1. Vale of Information of a monitoring system in the nconditional and conditional formlation. Unconditional formlation Conditional formlation Manager (M) = Owner (O) Manager (M) Owner (O) Prior tility withot monitoring O = max O O a = arg max opt O y = max a O a O O M O = M Posterior tility with monitoring a,y O O a y = arg max a opt,y O * O a = opt O O M O y = arg max M O M O y = arg max Preposterior tility with monitoring O M O = max a,y py dy * = arg max D y D y Vale of information of the monitoring system O O VoI = * O M M VoI = * a y opt M M a a,y a,y py dy O M

13 The Streicker Bridge case stdy To illstrate how the presence of two different decision makers in the decision chain affect the way how the VoI is evalated, we consider the case of Malcolm, the fictitios manager of an imaginary Office of Design and Constrction at Princeton University, protagonist in [2] and [17]. Malcolm is responsible for the Streicker Bridge, a pedestrian bridge located on Princeton University camps. The bridge and its monitoring system are illstrated in mch detail in a nmber of past pblications [28] [29] [30], we smmarise the main strctral featres, for clarity. The deck of the bridge is a continos thin concrete posttensioned deck featring a characteristic X-shape connecting for different sectors of Princeton Camps. From the strctral point of view, it consists of a thin post-tensioned spported by a high resistance steel lattice. The main span of the bridge overpasses Washington road, a bsy pblic road the camps (see Figre 4(a) and Figre 4(b)). The SHM-lab of Princeton University instrmented the bridge with two SHM systems: (i) global strctral monitoring sing discrete long-gage strain Fiber Optic Sensors (FOS), based on fiber Bragg-grating (FBG) [31], and (ii) integrity monitoring, sing trly distribted FOS based on Brilloin Optical Time Domain Analysis (BOTDA) [32]. These two approaches are complementary: discrete sensors monitor an average strain at discrete points, while the distribted sensors monitor one-dimensional strain field. Discrete FOS embedded in the bridge deck have gage length 60 cm and featre excellent measrement properties with error limits of ±4. Ths, they are excellent for assessment of global strctral behavior and for strctral identification. Instead, distribted FOS have accracy an order of magnitde lower than discrete sensors and so cannot be sed for accrate strctral identification; they are sed for damage detection and localization. Figre (4c) shows the sensors map in the main span, while Figre (4d) its cross section. Agents To make the case stdy easier to nderstand, we imagine the bridge managed by two agents with distinct roles: - Ophelia (O) is the owner responsible for Princeton s estate; she is Malcolm s spervisor and decides on whether to install the monitoring system or not. - Malcolm (M) is the manager responsible for the bridge operation and maintenance, gradated in civil engineering and registered as a professional engineer, who has to take decisions on the state of the bridge based on monitoring data, exactly as in Zonta et al. [2]. We assme that Ophelia and Malcolm are both rational individals and that have the same knowledge backgrond as for possible damage scenarios S of the bridge, prior information, and they have the same knowledge of the conseqence of a bridge failre. They only differ in the way how to weight the seriosness of the conseqences of a failre. It is probably nnecessary to remind that, while the Streicker Bridge is a real strctre, the two characters, Ophelia and Malcolm, are merely fictitios and do not reflect in any instance the way how asset maintenance and operation is performed at Princeton University.

14 a) b) c) d) Figre 4. The Streicker bridge: view of the bridge (a)(b), location of sensors in the main span (c), main cross section (d). States and likelihoods As part of this fictitios story, we sppose that both Ophelia and Malcolm are concerned by a single specific scenario: a trck, manevering or driving along Washington road, cold

15 collide with the steel arch spporting the concrete deck of the bridge. In this oversimplified example, we will assme that after an incident the bridge will be in one of the following two states: - No Damage (U): the strctre has either no damage or some minor damage, with negligible loss of strctral capacity. - Damage (D): the bridge is still standing bt has sffered maor damage; conseqently, Malcolm estimates that there is a chance of collapse of the entire bridge. Similar to the assmptions in [1], we assme Malcolm (and similarly Ophelia) focses on the sensor installed at the bottom of the middle cross-section between P6 and P7 (called Sensor P6-7d, see Figre 4(c)). We nderstand that for both Ophelia and Malcolm the two states represent a set of mtally exclsive and exhastive possibilities, which is to say that P(D) + P(U) = 1. On the basis of their experience, they both agree that scenario U is more likely than scenario D, with prior probabilities P(D) = 30% and P(U) = 70%, respectively. We can also assme that both se the same interpretation model, i.e. they interpret identically the data from the monitoring system. As Malcolm will pay attention only to the changes at the midspan sensor (labelled P6-7d in Figre 4(c)), we presme that he expects the bridge to be ndamaged if the change in strain will be close to zero. However, he is also aware of the natral flctation of the strain, de to thermal effects, and to a certain extent de to creep and shrinkage: he estimates this flctation to be in the order of ±300. We can represent this qantity with a probability density fnction pdf( U), with zero mean vale and standard deviation = 300, which describes Malcolm s expectation of the system response in the ndamaged (U) state, i.e. this is the likelihood of no damage. On the other hand, if the bridge is heavily damaged (D) bt still standing. Malcolm expects a significant change in strain; we can model the likelihood of damage pdf( D) as a distribtion with mean vale 1000 and standard deviation of = 600, which reflects Malcolm s ncertainty of expectation. Before the data are available, he can also predict the distribtion of, which is practically the so-called evidence in classical Bayesian theory, throgh the following formla: pdf = pdfd PD + pdfu PU. 17 When the measrement is available, both pdate their estimation of the probability of damage consistently with Bayes theorem: pdfd = pdfd PD pdf, 18 where pdfd is the posterior probability of damage. Figre 6(a) shows the two nnormalized posterior distribtions along with the evidence. Note that the posterior probability of damage starts exceeding the posterior of no-damage when the measrement exceeds the threshold p = 540.

16 Decision model After he assesses the state of the bridge, we assme that Malcolm can decide between the two following actions: - Do nothing (DN): no special restriction is applied to the pedestrian traffic over the bridge or to road traffic nder the bridge. - Close Bridge (CB): both Streicker Bridge and Washington Road are closed to pedestrians and road traffic, respectively; access to the nearby area is restricted for the time needed for a thorogh inspection, which both Ophelia and Malcolm estimates to be 1 month. Ophelia and Malcolm agree that the costs related to each action, for each scenario, are the same as estimated in Glisic and Adriaenssens [28], and reported in Table 2. Table 2. Costs per action and state. Action DN (do nothing) Action CB (close bridge) Scenario U (no damage) nothing happens yo pay nothing 1-month downtime cost =$139,800 Scenario D (bridge fails) failre cost = $881,600 1-month downtime cost = $139,800 However, Ophelia and Malcolm differ in their tility fnctions, which is the weight they apply to the possible economic losses. Ophelia is risk netral, meaning that according to her a negative tility is linear with the incrred loss, as illstrated in Figre 5. Strictly speaking, a tility fnction is defined except for a mltiplicative factor, therefore it shold be expressed in an arbitrary nit sometime referred to as til [33]. Since Ophelia s tility is linear with loss, for the sake of clarity we will deliberately confse negative tility with loss, and therefore we will measre Ophelia s tility in k$. Unlike Ophelia, Malcolm is likely to behave risk adversely, i.e. his negative tility increases more than proportionally with the loss. We can describe mathematically the risk aversion classically defined in Arrow-Pratt theory [34] [35], where the level of risk aversion of an agent is encoded in the coefficient of Absolte Risk Aversion (ARA), defined as the rate of the second derivative (crvatre) to the fist derivative (slope): Az = U"z U'z 19 To state Malcolm s tility fnction, we can make the following assmptions: - Malcolm s and Ophelia s reaction are virtally identical for a small amont of loss, while their way of weighting the losses departs for bigger losses. - For small losses, therefore, the two-tility fnction may be confsed, and we will adopt for Malcolm s the same conventional nit (call it eqivalent k$) for measring tility. Malcolm s tility fnction derivative for zero loss is eqal to 1.

17 - We assme that Malcolm s tility has constant ARA; it is easily demonstrated that a fnction with constant ARA and nitary derivative at zero [36] takes the form of an exponential: 20 where is the constant ARA coefficient: A(z) = - To calibrate, we assme that for a loss eqal to the failre cost, Malcolm s negative tility is twice that of Ophelia s. This reslts in a constant ARA coefficient = M$ -1. Using these assmptions, the reslting Malcolm s tility fnction is plotted in Figre 5. We wish now to verify how the different tility fnctions affect the decision of the two a priori and a posteriori. Figre 5. Representation of Ophelia s and Malcolm s tility fnctions. Prior tility Consider the case where Malcolm has no monitoring information. Based on his tility, Malcolm estimates the tilities involved in each action. Action CB depends only on the downtime cost z DT, while action DN depends also on his estimate of the state of the bridge:

18 (M) = (M) z DN F PD = k (M) = (M) z CB DT = k 21 Since the tility of action CB is clearly less negative than the tility of action DN, Malcolm wold always choose to close the bridge after an incident if he has no better information from the monitoring system. Therefore, Malcolm s maximm expected tility withot the (M) (M) monitoring system is = = k$. CB Now imagine Ophelia in charge of the decision: her prior tilities are different from Malcolm s and their vales are somewhat closer: (O) = (O) z DN F PD = k (O) = (O) z CB DT = k 22 bt in the end, in this particlar case, her optimal action wold be again close the bridge. Posterior tility Now imagine that the monitoring system is installed and let s go back to Malcolm. Since now Malcolm can rely on the monitoring reading, in this case the expected tility of an action is calclated sing the posterior probability of damage pdfd rather than the prior: = z DT, DN = z F pdfd. 23ab Note that since the cost of closing the bridge is independent on the bridge state, the monitoring observation does not affect the posterior tility of closing the bridge (CB), which is always eqal to k as in the prior case. On the contrary, the expected tility of doing nothing (DN) does depend on the probability of having the bridge damaged, and this probability, in trn, depends on the monitoring observation throgh Eqation (23b). Malcolm s posterior expected tilities (i.e. after observing data from the monitoring system) for actions DN and CB are plotted in the graph of Figre 6(b) as fnctions of the observation. As a rational agent, Malcolm will always take the decision that maximizes his tility. For very small vales of, sggesting a small probability of collapse, Malcolm s tility of DN is bigger than the tility of CB, and therefore Malcolm will keep the bridge open. Malcolm s tility of closing the bridge starts exceeding the tility of doing nothing (M) above a threshold of strain of = 170, and therefore Malcolm will always close the bridge above this threshold. Note that this threshold is mch smaller than the threshold p whereby Malcolm wold dge the damage more likely, so there is a range of vales whereby Malcolm, in consideration of the possible conseqences, will still prefer to close the bridge even if it is more likely the bridge is not damaged. Malcolm s maximm expected tility is plotted in bold in the graph of Figre 6(b). Assme now that Ophelia is in charge of the decision. Since she weights the losses differently, her tility crves as fnctions of are different from Malcolm s, and are plotted in the graph of Figre 6(c). For the same reason, the threshold above which she wold close (O) the bridge, = 310, is different and mch higher than Malcolm s, reflecting Ophelia s risk netrality in contrast to Malcolm s risk aversion. Therefore, there is a range of vales of measrements, from 170 to 310, where the two decision makers, both

19 rational, behave differently nder the same information, simply becase of their different level of risk aversion. Figre 6. Representation of Malcolm s estimation of the state of the bridge a priori (a), Malcolm s decision model with monitoring data (b), Ophelia s decision model with monitoring data (c), Ophelia s decision model based on Malcolm s own (d).

20 Preposterior tility and Vale of Information In this scenario Ophelia and Malcolm are both involved in the decision chain. Malcolm is the operational manager who decides whether or not to close the bridge in the occrrence of an incident. Ophelia is the owner who decides on the prchase of the monitoring system. This is illstrated as a decision tree in Figre 7. We seek the VOI as anticipated by Ophelia (she has to decide), which explicitly acconts from Malcolm reacting to the signals from the monitoring system. Figre 7. Decision tree for the Streicker Bridge case stdy. Before attacking this problem, let s first see what happens if the decision chain was in the hands of a single individal. Let s start, for example, with Malcolm. His preposterior tility (i.e., the prior tility of operating the bridge with the monitoring system) can be calclated with the eqation: M M * M = argmax (a,) p d = k 24 D where the index (M) indicates that all the tilities are calclated from Malcolm s perspective. Malcolm s VoI is simply the difference between the preposterior tility (i.e. the prior tility of operating the bridge with the monitoring system) and the prior tility (i.e. the tility of operating the bridge withot the monitoring system): VoI = M M - = k k =.435 k 25 Note that the VoI is a tility, not an actal amont of money, and is measred in Malcolm s tility nit, which in or case is Malcolm s dollar-eqivalent as defined above. Now we can calclate the VoI from Ophelia s perspective, assming that she takes decisions at any stage of the decision chain. In this case being Ophelia less risk adverse

21 O than Malcolm, her tilities will be O = k and = k, so eventally Ophelia s VoI wold be: O VoI = O - = -84,600 k + 139,800 k = k$ 26 This practically means that, if Ophelia was in charge of all the decisions, she wold be willing to spend p to k$ for the information from the monitoring system. In reality, Ophelia is only in charge of the prchase of the monitoring system, while the one who is going to se it is her colleage Malcolm. So, in taking her decision, Ophelia has to figre ot how Malcolm is going to behave both with and withot the monitoring system. In other words, we have to calclate the prior and preposterior tility from Ophelia s perspective, bt conditional to the action that Malcolm will ndertake. For example, to calclate the prior (i.e. the tility of Ophelia of operating the bridge withot the monitoring system, conditioned to Malcolm s actions) conditional tility, Ophelia thinks: what will Malcolm do after an accident if no monitoring system is installed? I know Malcolm, and I know he will close the bridge right away (I wold do the same, bt that s irrelevant). My tility, if he closes the bridge, is: O M (O) = argmax M (O) (a ) = k 27 CB which in this case is the same as the nconditional. And what Ophelia contines to think wold Malcolm do if a monitoring system was installed. I know that he wold look at the strain and he wold close the bridge if >170 and keep the bridge open otherwise. I personally wold NOT do the same, bt that s it, I have to live with Malcolm s decision! The way Ophelia evalates the tility on Malcolm s decisions is explained in Figre 6(d): her tilities for each possible Malcolm s choice are calclated sing her tility fnction, hence all individal crves are identical to those of Figre 6(c), However, the threshold whereby she expects the bridge is closed is Malcolm s threshold, i.e. the same as in Figre 6(d). Ophelia s tility of Malcolm s choice is, for any vale of : = argmax (M) (a,) - k$ 28 and therefore the preposterior tility conditional to Malcolm is: O O M * M = argmax (a,) p d = k 29 D Eventally, Ophelia s VoI, conditional on Malcolm s decision, is: O M VoI O M = * O M - = k k = k. 30 Again, this qantity is the money Ophelia believe is worth spending on a monitoring system, having accepted that Malcolm, not her, is going to se it. The conditional O M O VoI k is slightly lower than the nconditional VoI = k$. Generally, it is clear from Ophelia perspective, that when Malcolm s decision is different

22 from hers it is always sboptimal. Therefore, the conditional prior and pre-posteriors are O M O O M O always smaller than the corresponding nconditional:,. In O M O the present example, Ophelia and Malcolm agree on what to do a priori, the O M O conditional VoI is necessarily smaller than the conditional VoI. In simple words, Ophelia s rationale goes along these lines: I can exploit the monitoring system better than Malcolm, therefore the benefit of the monitoring system wold be greater if I was sing the monitoring system rather than Malcolm. However, this is not the most general case. Assme for example the prior probability of (O) damage P(D) is 10%: Ophelia s prior tility of action DN = k$, small enogh DN (M) for Ophelia to keep the bridge open; on the contrary Malcolm s prior tility = - DN k$, is still big enogh for Malcolm to close it. In this case the nconditional prior is mch bigger than the conditional one, since Ophelia doesn t agree with Malcolm s O M choice, and the conditional VoI = k$ is mch bigger than the nconditional O VoI = k$, meaning that monitoring is mch more sefl in this case. We can almost hear Ophelia commenting: This Malcolm can t make the right decision alone, hopeflly some monitoring will help him! For sre a monitoring system is more sefl to him rather than me! Negative Vale of Information? We noted above that in the nconditional case (i.e. when Ophelia is both owner and manager), the preposterior tility * is always greater or eqal than the prior, hence the VoI cannot be negative. In simpler words, if a monitoring system if offered to Ophelia at no cost, she has no reason not to accept it. Of corse, if at any time Ophelia realizes that the monitoring system yields nk data, she can always decide to disregard this information, bt she has no economic reason to refse a priori to see the data ( Take each man s censre, bt reserve thy dgment ). We also noted that in the nconditional case (i.e. when Ophelia is the owner bt someone else, Malcolm, is the manager who decide based on the SHM data) there is no logical necessity whereby Ophelia s preposterior tility mst be greater than her prior. So in principle we can always find a combination of prior probabilities and tility fnctions which ltimately yield a negative conditional VoI. We illstrate this concept with an example. Imagine that Malcolm, instead of being risk adverse, is risk seeking. This is to say that his tility fnction is convex (i.e., with positive second derivative), as shown in Figre 8: for this exercise we can again assme an Arrow-Pratt s tility model, as in Eqation (20), bt this time with a positive ARA coefficient M$ -1. Also, assme, both for Ophelia and Malcolm, a high prior probability of damage, say P(D) = 55%. Using these assmptions, Ophelia s prior tilities for doing nothing (DN) and closing the (O) (O) bridge (CB) are = k$ and = k$ respectively, while Malcolm s (M) DN (M) CB are = k$ and = k$. For both, closing the bridge (CB) is DN CB the action that yields the maximm expected tility a priori: so they both agree that, withot a monitoring system, the best thing to do is to close the bridge. Their decisions start departing after receiving data from the monitoring system. Figre 9 shows how Ophelia s and Malcolm s decision models change based on the new assmptions.

23 We note that: - becase of the high prior risk of collapse, risk-netral Ophelia is very conservative and thinks it is a good idea to close the bridge as soon as the elongation recorded is (O) greater than = 70 ; - risk-seeking Malcolm doesn t take a collapse so seriosly and he wold rather keep (M) the bridge open nless the sensor reads an elongation greater than = 423. So there is a very wide range of vales, from 70 to 423, whereby Malcolm wold keep the bridge open in disagreement with Ophelia, who believes this is a dangeros practice which can potentially reslt in a big loss. Based on these premises, Ophelia s conditional preposterior (i.e. Ophelia expected tility conditional to Malcolm s decision) O M is calclated, sing eqation (29), in * = k, and eventally her conditional vale of information is: O M VoI O M O M k k k Contrary to the example above, now the conditional vale of information is negative, meaning that Ophelia s perceives the monitoring information as damaging. Ophelia thinks that, in observing the monitoring data, Malcolm may wrongly decide to keep the bridge open even when, in her opinion, it shold be closed. She concldes that, after all, it is better not to install the monitoring system at all. In Ophelia s own words: Malcolm is an irresponsible and shold not se the monitoring system! I wold rather pay money than letting him se the system! Indeed, the negative vale of information is exactly the amont of money Ophelia is willing to pay to prevent Malcolm sing the monitoring system. Figre 8. Representation of Ophelia s and risk-seeking Malcolm s tility fnctions.